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Transcript of Wiley Capital Asset Investment Strategy, Tactics
Capital Asset Investment
Strategy, Tactics & Tools
Anthony F. Herbst
JOHN WILEY & SONS, LTD
Copyright C© 2002 John Wiley & Sons Ltd, Baffins Lane, Chichester,West Sussex PO19 1UD, England
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Library of Congress Cataloging-in-Publication Data
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN 0-470-84511-2
Typeset in 10/12 pt Times by TechBooks, New Delhi, IndiaPrinted and bound in Great Britain by Biddles Ltd, Guildford and King’s LynnThis book is printed on acid-free paper responsibly manufactured from sustainable forestryin which at least two trees are planted for each one used for paper production.
Contents
Preface xiii
Acknowledgments xv
1 Introduction: the big picture, environment, terminology, and preview 1Magnitude of capital investment 1General perspective on capital investment 3Capital budgeting 5Cash flows 6Cost of capital 6Risk and uncertainty 7
Risk 7Uncertainty 8
2 The Objective of capital budgeting 9A normative model for capital budgeting 10Basic valuation model 11
Operational adaptation 11The cash flows 12
Cash flows and the public sector 12
3 Estimating basic project characteristics 13Project types 13Project characteristics 14Initial cost 15
Sunk cost 15Components of initial cost 16
Useful life 17Physical deterioration 18Technological obsolescence 19
Cash flows 19Cash inflows (cash receipts) 21Cash outflows 22
vi Contents
Taxes and depreciation 23Depreciation 23Straight-line depreciation 24Double-declining-balance depreciation 24Sum-of-the-years’ digits depreciation 24Comparison of the basic depreciation methods 25Example 3.1 25Example 3.2 26Example 3.3 26ACRS Depreciation 27Canadian depreciation 28Investment tax credit 29Inflation 30
4 Cost of capital 31Introduction 31Cost of capital components 31
Debt 31Preferred stock 32Common stock and retained earnings 33Example 4.1 34
Overall cost of capital 34Optimal capital structure 36Interaction of financing and investment 38Cautionary note 38
5 Traditional methods that ignore time-value of money 39Payback and naive rate of return 39
Payback 39The naive rate of return 41Strong points of payback 41Weak points of payback 41Unrecovered investment 43
Accounting method: alias average return on average investment 43Strong points of accounting method 43Weak points of accounting method 44
Comprehensive example 44
6 Traditional methods that recognize time-value of money: the net present value 47Unequal project size 51
The profitability index 51Unequal project lives 51
Level annuities 53Summary and conclusion 53
Strong points of NPV 53Weak points of NPV 54
Contents vii
7 Traditional methods that recognize time-value of money: the internal rateof return 55Definition of the IRR 55A caution and a rule for IRR 58Payback and IRR relationship 58Mathematical logic for finding IRR 59
Interval bisection 60Newton–Raphson method 61Strong points of IRR 62Weak points of IRR 62
A digression on nominal and effective rates 62Investment–financing relationship 62Nominal rate and effective rate 63Clarification of nominal and effective rates 63IRR with quarterly cash flows 64
8 Reinvestment rate assumptions for NPV and IRR and conflictingrankings 65Reinvestment rate assumptions for NPV and IRR 65Conflicting rankings and fisher’s intersection 68Relationship of IRR and NPV 72Adjusted, or modified, IRR 72Summary and conclusion 73
9 The MAPI method 75The concept of duality 75The MAPI framework 77
Challenger and defender 77Capital cost 77Operating inferiority 78Physical deterioration 79Technological obsolescence 79Two basic assumptions 79Adverse minimum 80First standard assumption 80Second standard assumption 80
Application of the MAPI method 80The problem of capacity disparities 82
Conclusion 83
10 The problem of mixed cash flows: I 85Internal rate of return deficiencies 85
Example 10.1 85Descartes’ rule 85
Example 10.2 86The Teichroew, Robichek, AND Montalbano (TRM)analysis 87
viii Contents
The TRM algorithm 89Example 10.3 89Example 10.4 91Example 10.5 92
The unique, real internal rate of return: caveat emptor! 94A new theorem 96
Theorem 96Proof 96Corollary I 96
11 The problem of mixed cash flows: II 97The Wiar method 97
Example 11.1. An application of the Wiar method 98Sinking fund methods 99
The initial investment method 100The traditional sinking fund method 100Initial investment and traditional sinking fund methods 100Example 11.2 101Example 11.3 102Example 11.4 103Example 11.5 104
The multiple investment sinking fund method 107Strong points of the methods 108Weak points of the methods 109
11A Appendix: the problem of mixed cash flows III — a two-stage methodof analysis 110Relationship to other methods 110The two-stage method 111
Example 11A.1 111Example 11A.2 114
Formal definition and relationship to NPV 114Payback stage 115“Borrowing” rate 116Example 11A.3 116
Conclusion 117A brief digression on uncertainty 117
12 Leasing 119Alleged advantages to leasing 119Analysis of leases 120
Traditional analysis 120Example 12.1 121Alternative analysis 122Analysis 124Implications 125
Contents ix
Practical perspective 126Summary and conclusion 127
Appendix 128
13 Leveraged leases 131Definition and characteristics 131Methods: leveraged lease analysis 132Application of the methods 132
Example 13.1 132Example 13.2 133
Analysis of a typical leveraged lease (Childs and Gridley) 137Example 13.3 137
Conclusion 139
14 Alternative investment measures 141Additional rate of return measures 141
Geometric mean rate of return 141Average discounted rate of return 142Example 14.1 142
Time-related measures in investment analysis 144Boulding’s time spread 145Macaulay’s duration 146Unrecovered investment 147
Summary and conclusion 148
15 Project abandonment analysis 149The Robichek–Van Horne analysis 149An alternate method: a parable 151
Comparison to R–VH 155A dynamic programming approach 156Summary and conclusion 158
16 Multiple project capital budgeting 159Budget and other constraints 159General linear programming approach 159
Mutual exclusivity 161Contingent projects 161
Zero–one integer programming 162Example 16.1 164
Goal programming 165Summary and conclusion 169
16A Appendix to multiple project capital budgeting 170
17 Utility and risk aversion 177The concept of utility 177
x Contents
Attitudes toward risk 178Calculating personal utility 180
Whose utility? 184Measures of risk 185
J. C. T. Mao’s survey results 187Risk of ruin 188Summary 189
18 Single project analysis under risk 191The payback method 191Certainty equivalents: method I 192
Example 18.1 192Certainty equivalents: method II 193Risk-adjusted discount rate 194
Example 18.2 195Computer simulation 195
Example 18.3 196Lewellen–Long criticism 200
19 Multiple project selection under risk: computer simulation and otherapproaches 203Decision trees 203
Example 19.1 205Other risk considerations 207
Example 19.2 A comprehensive simulation example 207Conclusion 214
20 Multiple project selection under risk: portfolio approaches 215Introduction 215
Example 20.1 215Generalizations 216Project independence 217Project indivisibility 217
Multiple project selection 218Finding the efficient set 220The Sharpe modification 222
Relating to investor utility 223Epilogue 223
21 The capital asset pricing model 225Assumptions of the CAPM 225The efficient set of portfolios 225
Portfolio choices 226Enter a risk-free investment 226
The security market line and beta 228The CAPM and valuation 230The CAPM and cost of capital 231
Contents xi
The CAPM and capital budgeting 232Comparison with portfolio approaches 232Some criticisms of the CAPM 233The arbitrage pricing theory (APT) 234
Factors — what are they? 235APT and CAPM 236Development of the APT 237
22 Multiple project selection under risk 239Example 22.1: Al’s Appliance Shop, revisited 240Generalizations 241
Project independence — does it really exist? 242Project indivisibility: a capital investment is not a security 243
Example 22.2: Noah Zark 244Generalization on multiple project selection 246
Securities 247Capital investments 247
Summary and conclusion 248
23 Real options 251Acquiring and disposing: the call and put of it 251More than one way to get there 252Where are real options found? 252Comparison to financial options 253Flashback to PI ratio — a rose by another name . . . 253Types of real options 254Real option solution steps 255Option phase diagrams 256Complex projects 257Estimating the underlying value 258What to expect from real option analysis 258Identifying real options — some examples 259Contingent claim analysis 260The binomial option pricing model 260
Example 23.1 Value of strategic flexibility in a ranch/farm 262A game farm, recreational project 264
Option to switch operation 265Case 1: 100 percent switch 265Case 2: Mix operation 266Option to abandon for salvage value 267The option to expand operation (growth option) 268Interaction among strategic options 269Example 23.2 270Contingent claim valuation: 270The optimal exercise of the growth option and firm value 271The abandon option and firm value 272Current value of the firm 273
xii Contents
The strategic dimension of the real-option analysis 273Conclusion 273
Appendix: Financial mathematics tables and formulas 275A.1 Single payment compound amount. To find F for a given P 276A.2 Single payment present worth factor. To find P for a given F 283A.3 Ordinary annuity compound amount factor. To find F for a given R
received at the end of each period 291A.4 Ordinary annuity present worth factor. To find P for a given R received at
the end of each period 299
Bibliography 307
Index 315
Preface
The aim of this book is to tie together the theory, quantitative methods, and applications ofcapital budgeting. Consequently, its coverage omits few, if any, topics important to capitalinvestment. My intention is to effect a harmonious blend of the old, such as the MAPI methodof capital investment appraisal, with the new, such as the capital asset pricing model (CAPM).I have tried to provide a balanced treatment of the different approaches to capital projectevaluation and have explored both the strengths and weaknesses of various project selectionmethods.
A work on this subject necessarily uses mathematics, but the level of mathematical sophis-tication required here is generally not above basic algebra. Although I have favored clarity andreadability over mathematical pyrotechnics, the book’s level of mathematical rigor should besufficiently high to satisfy most users.
The book’s treatment of risk is deliberately deferred to later chapters. The decision to dothis, rather than treating risk earlier, was based on my belief that readers new to the subjectare less overwhelmed by the added complexities of risk considerations — and better able tocomprehend them — after they have become thoroughly familiar with capital budgeting in anenvironment assumed to be risk-free.
In Chapter 21 I try to present a balanced treatment of the CAPM, including some of theimportant criticisms of its use in capital budgeting. I know that some readers might prefer anearlier introduction of the CAPM, as well as its subsequent use as a unifying theme. I chosenot to employ that structure for three reasons.
First, although the CAPM may be adaptable to capital budgeting decisions involving majorprojects (e.g. the acquisition of a new company division), serious questions exist concerning itsapplicability to more typical projects for which estimation of expected returns alone is difficult,to say nothing of also estimating the project’s beta.
Second, company managers increasingly appear to be placing primary emphasis on thesurvival of the firm rather than on consideration of systematic risk in their capital investmentdecisions, thereby diluting the implications of the CAPM. In other words, top managementdoes indeed care about unsystematic (or company) risk, to which portfolio diversification may,in some cases, give little importance. To the management of a company, such risk may notalways be reduced easily, and, if neglected, may imperil the company.
Third, the CAPM is concerned with risk. For the reasons stated earlier, I felt that the bookwould better serve its audience if it examined capital investment under assumed uncertaintyfirst, without the added complexities that a simultaneous treatment of risk would entail.
xiv Preface
For those who may wish to obtain them I have developed a computer program and spreadsheettemplates for several applications illustrated in the book. Also, for those who adopt the book forteaching, I have a separate book of end-of-chapter questions and problems, and an instructor’ssolutions manual. Please contact the following website http://www.utep.edu/futures
A. F. H.Department of Economics and Finance/CBAThe University of Texas at El PasoEl Paso, Texas 79968-0543
E-mail: [email protected]
Acknowledgments
At this stage of my life, many of those who contributed to the development of the person I am,and the accomplishments that I have achieved, are no longer here to receive my expression ofappreciation. Yet I would be remiss if I did not mention them anyway. First are my parents,who sacrificed that I obtain a university education, and who I think would be pleased withwhat their efforts achieved. Second is Joseph Sadony Jr. who, upon my father’s illness, tookup the task of keeping the teenage me on track to success later in life. And I should mentionHarvey Nussbaum, a professor at Wayne State University in Detroit who nudged me into theacademic life from my career in industry and banking. Then there is my wife Betty, who nolonger types my manuscripts, but is supportive in many other ways that facilitate my work. Itis to these persons, and my children Mya and Geoff that I dedicate this work.
Many persons contributed to the technical, academic content of this work. Professor JamesC. T. Mao merits special mention for inspiring me early in my teaching a research careerwith his work in quantitative analysis of financial decisions to undertake my own work in thatrealm. Numerous of my students and colleagues contributed to this work in various ways, fromencouraging me to do it, to helping with it. Special recognition belongs to Jang-Shee (Barry)Lin for co-authoring Chapter 23, and to Marco Antonio G. Dias of Brazil, for reviewing thatchapter.
1Introduction: the Big Picture,
Environment, Terminology, and Preview
Once one consciously thinks about the nature of capital investment decisions, it becomesapparent that such decisions have been made for millennia, since humans first awakened to theidea that capital1 could improve life. The earliest investment decisions involved matters thattoday would be considered very primitive. But to the early nomadic hunter-forager the firstcapital investment decisions were quite significant. To the extent that time and energy had to bediverted from the immediate quest for food, short-term shelter, and the production of tools forthe hunt, into defensive installations, food storage facilities, and so on, capital investments weremade. Such fundamental capital creation required significant time and effort. And the benefitsthat could have been expected to result were uncertain; it took foresight and determination tobuild capital.
As society evolved, the benefits of capital accumulation gradually became more indirect andcomplex, involving specialization and cooperation not previously envisioned, and the associ-ated commitment of resources more permanent. Additionally, social norms and institutionshad to be developed to facilitate the evolution. For example, the changeover from nomadicto agrarian life required a great increase in capital in the form of land clearing, constructionof granaries, mills, irrigation canals, tools, and fortifications. Fortifications were necessary todeter those outsiders who would use force to seize the benefits achieved by investment. Thechangeover required a commitment that tended to be irreversible, at least in the short run. Andit became more and more irreversible as the changes caused social and economic institutions toadapt or be developed to support it and coordinate the various requisite activities. (C. NorthcoteParkinson writes of such differences between agricultural and nomadic societies in his book[122].)
MAGNITUDE OF CAPITAL INVESTMENT
In the United States in 2001, business capital expenditure on new plant and equipment amountedto more than 20 percent of gross domestic product (GDP) according to data from the FederalReserve Bank of St Louis [47]. In the United Kingdom in 2001 the percentage of GDP ac-counted for by investment was slightly less, between 15 and 20 percent. Comparable statisticsfor Germany and France are 20–22 percent; Canada and Italy about 20 percent, and Japanin a category by itself at more than 25 percent. Figure 1.1 displays saving and investment inthe G7 industrial nations. It is apparent that capital investment is a very important compo-nent of GDP in every one of these countries. The larger share of GDP allocated to savingand investment by Japan may arise from that nation’s relatively lesser spending on militaryhardware and weapons development, and also the relative emphasis on electronics manufacture
1The word “capital” is used in several senses. It may refer to physical plant and equipment (economic capital) or to the ownershipclaims on the tangible capital (financial capital). In this book, unless otherwise indicated in a specific instance, the word shall refer tophysical or economic capital.
2 Capital Asset Investment: Strategy, Tactics & Tools
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Introduction: the Big Picture, Environment, Terminology, and Preview 3
Table 1.1 Saving and investment as a percent of gross domestic product, 1973–97
United West UnitedStates Canada Japan France Germanya Germanyb Kingdom
SavingNet savingc 6.6 7.7 18.5 9.3 10.4 7.9 4.7Personal savingd 6.2 7.6 11.6 7.8 8.1 7.8 4.0Gross saving
(net savingplus cons. offixed capital)e 17.5 19.7 32.5 21.6 22.6 20.9 15.9
InvestmentGross
nonresidentialfixed capitalformation 13.8 15.1 24.0 14.8 14.5 14.9 14.0
Gross fixedcapitalformation 18.3 21.1 30.2 21.0 20.5 21.6 17.6
a The statistics for West Germany refer to western Germany (Federal Republic of Germany before unification).The data cover the years 1973–95.b The statistics for Germany refer to Germany after unification. The data cover the years 1991–97.c The main components of the OECD definition of net saving are: personal saving, business saving (undis-tributed corporate profits), and government saving (or dissaving). The OECD definition of net saving differsfrom that used in the National Income and Product Accounts published by the Department of Commerce,primarily because of the treatment of government capital formation.d Personal saving is comprised of household saving and private unincorporated enterprise.e The main components of the OECD definition of consumption of fixed capital are the capital consumptionallowances (depreciation charges) for both the private and the government sector.Source: Derived from National Accounts, Organization for Economic Cooperation and Development (OECD)Statistical Compendium 2000. Prepared by the American Council for Capital Formation Center for PolicyResearch, Washington, DC, June 2001.
that requires hefty capital equipment investment to produce items with short product lifecycles.
Table 1.1 reinforces the impression of the graphs in Figure 1.1. It contains comparativesaving and investment statistics for six of the major industrialized nations over the span 1973–97. It is apparent that in every case gross nonresidential fixed investment is greater than 1/8of GDP, ranging from a low of 13.8 percent for the United States to a high of 24.0 percent forJapan. This category takes into account only the plant and equipment and other fixed assets2
(i.e. depreciable assets, lasting more than a year). Important investments that are not easilymeasured are excluded, but are nevertheless important, such as “human capital” and technologyresearch and development.
GENERAL PERSPECTIVE ON CAPITAL INVESTMENT
Because resources are scarce (as everyone learns in a first course in economics) and becausecapital investment figures so prominently in the economy, decisions on capital budgets ought
2Housing, a major component of investment, is excluded from this measure but included in the bottom row item, gross fixed capitalinvestment.
4 Capital Asset Investment: Strategy, Tactics & Tools
to be made on a sound, rational basis. The general irreversibility of capital investments, andtheir legacy for future costs as well as benefits, make such decisions of great importance bothfrom the standpoint of the individual firm and at the level of national policy.
For any given level of technology, considerations arising from unemployment, and fromdemands for increased standards of living, are important. These require that for given levelsof technology the stock of capital goods increases apace with growth in the labor supply or, iftechnological growth tends toward the more capital-intensive side, to exceed it. As long as thegoals are the same, this will hold for every form of government because political philosophycannot change the feasible mix of land, labor, and capital for a given technology.
Few in business and industry would dispute the view that the press and media generally tendto emphasize labor’s employment problems while generally ignoring the problems of under-employment of capital, reduced or negative capital accumulation, and return on investment.Such imbalanced editorial policy, however, is understandable to the extent it exists becausemuch of the public identifies their interests with labor rather than with capital. However, it rein-forces popular notions that, although not clearly incorrect, are at least suspect and thus tend tocontribute further to problems faced by labor and capital alike. The notion that the interests oflabor and capital are mutually exclusive may be useful to rhetoricians at the political extremes,but it fails to address the problems of underemployment and sagging productivity in societiesin which labor and capital are undichotomized. In the industrial nations of North America,Europe, and Asia, for example, it is common for workers to own shares in their own employeror other corporations either directly or through their pension funds. Worker representation oncorporate boards has been accepted in some European nations for many years and the practiceseems to be growing. Thus the interests of labor and capital are not easily separable, if at all,because so many are neither entirely laborer nor entirely capitalist.
A proper blend of the factors of production is necessary to minimize unemployment of laborwithout resorting to government “make work” projects. But it depends on wages that reflectproductivity and product value. Adoption of government policies that ignore capital in orderto aim at direct treatment of labor unemployment, or low minimum wages, is bound to failin a broader sense. Still, such policies are often advocated, whereas the capital investmentnecessary to create a real expansion in jobs is largely ignored. Return on capital investment inthe industrial nations today tends to be generally less than can be earned on the same fundsif put into government bonds, after taking capital consumption into account. This has theeffect of dampening enthusiasm for capital investment and consequently moving funds to lessproductive uses. Since capital must complement labor, fewer productive jobs are created, whichleads to higher labor unemployment, more public policies to treat the symptoms, and still moredisincentive to capital investment. In fact, studies at the University of Chicago concluded thatin the United States, capital was being consumed at a faster rate than it was being created;in other words, industry was paying liquidating dividends. This has serious implications forgovernment economic policies.
Concern that capital will replace labor may be legitimate in the short run, if not longer term.If the relative costs of labor and capital should favor a capital-intensive production mix offactors, this should present no problem provided the distribution of benefits is equitable. Inother words, replacement of some labor by capital may increase the quality of life for workersprovided that the benefits are shared by labor. The dismissal of some workers while othersare put on extended overtime or continue to work a normal number of hours does not furtherequitable distribution. But a reduction in the standard work week for everyone may. If it werenot for capital accumulation and improved technology yielding increased productivity, workers
Introduction: the Big Picture, Environment, Terminology, and Preview 5
would not have a 40-hour or less standard week but one of 60 hours or so, as was once thestandard. (Determination of what is equitable and what is not is outside the scope of this book.)
Most of the developing nations are capital-poor, but have abundant labor that is not highlyskilled or educated. For those countries it makes sense to adopt labor-intensive methods of pro-duction, gradually shifting to capital-intensive technologies as their capital stock is increasedand the quality of labor increases. In the developed, relatively capital-rich nations, it is notsensible to adopt labor-intensive methods of production because that would waste productivecapacity. It is ironical that while developing nations strive to accumulate capital, developednations often follow policies that discourage capital growth and may even cause the capitalstock to shrink, relatively if not absolutely. One may well wonder if it would not be morehonest and efficient simply to donate capital to the developing nations to raise their capitalintensity while lowering that of the donor nation.
Capital investment decisions have repercussions that may extend far beyond the immediatetime frame, because they involve long-term commitments that are not readily undone. Thedecision not to invest is a capital investment decision also. And future repercussions may becompounded by the often very long planning periods for capital goods and the fact that it takescapital and labor together to produce more capital goods. The “production process” to create“human capital” — skilled labor — is a lengthy one, and the more sophisticated the capitalgoods to be produced the higher the quality of human capital generally required.
CAPITAL BUDGETING
In the private sector, capital investment and the analysis it requires are generally referred to ascapital budgeting. Capital budgeting focuses on alternative measures of project acceptability.Tangible factors are emphasized. However, to the extent that their effects can be factoredinto the process, intangibles must be considered. Capital investment in the private sector hasperhaps tended to pay less attention to intangibles than its public sector counterpart, cost–benefit analysis. This may well be due to the inherently more qualitative nature of social andpolitical goals and constraints. Some would contend that intangibles in the public sector areoften exaggerated with the result that projects are undertaken that cost more than they yieldin benefits. But to the extent this is a problem the remedy falls under the rubric of politicalprocess rather than investment evaluation.
In private enterprise, profitability provides the principal criterion for the acceptability ofparticular prospective investment projects through the effect they are expected to have onthe market value of the enterprise. Capital budgeting centers on an objective function thatmanagement seeks to maximize, subject to various constraints. Often the primary constraint isimposed by the funds available for investment. When this constraint is binding, it is referred to ascapital rationing. In the public sector, a reasonable alternative is provided by minimizing a costfunction, constrained by specifying minimum levels of services to be provided. The aggregateeffect of private and public support for capital investment is to raise a nation’s standard ofliving, both in tangible benefits and in terms of intangible benefits such as greater security andless social stress. To the extent that the benefits accrue to a few rather than to the many, it isthe social and political institutions that are responsible, not the economic framework. (For alucid presentation of a rationale for aggressive national policy encouraging capital formation,see the classical work by E. A. G. Robinson [132].)
Emphasis throughout this book is on private sector capital budgeting. Yet, some of themethods covered are applicable without modification to public sector investment, and others
6 Capital Asset Investment: Strategy, Tactics & Tools
are applicable with modification. The following sections set forth some principles that will beused throughout the remainder of the book.
CASH FLOWS
Emphasis is on net, after-tax cash flows. Pre-tax net cash flow is defined as the total cashinflow associated with the capital investment less its net cash outflow. Then, after removingthe portion that will be paid out for income taxes we have the net, after-tax cash flows for thetime period over which we have measured or estimated the flows. Cash flows are estimated foreach period of a capital investment’s life. Some may well be zero. Normally the time perioddivisions will be annual, but they can be quarterly, semiannual, or for some other time periodif desired and sensible to do so. The major focus in this book is on what is to be done with thecash flow estimates once they are obtained, rather than on how to make cash flow estimates,although Chapter 3 does address that subject.
In practice, cash flow estimates will normally represent an amalgamation of experiencedjudgment by persons in such diverse functional areas as production engineering, marketing,and accounting more than the result of objective measurements and data analysis. We canestimate future results a priori, but of course we can only measure them ex post, after theyoccur. Managers who are responsible for capital budgeting must ensure that adequately preciseestimates are obtained and, where possible, objective, unbiased forecasts are prepared. It maybe necessary to emphasize to those participating in the process by which estimates are obtainedthat it is cash flows that are sought, not accrued profits or cost savings. It is cash that may beused to pay dividends, employee wages, and vendor bills and it is cash that may be investedin new plant and equipment. Likewise, it is cash flow on which methods of capital investmentanalysis are based.
An early survey in the US revealed that many firms, even some fairly large ones, failed toinclude all associated cash flows in their analysis. Any such omissions of relevant cash flowscan seriously bias the measures of project acceptability and destroy the usefulness of whatevertechniques are employed. It is vitally important that all relevant cash flows attributable to acapital project be included.
COST OF CAPITAL
The organization’s cost of capital, expressed as a decimal or percentage, is used in two basicways in capital budgeting: as a minimum profitability rate (i.e. hurdle rate) that prospectiveproject returns are required to exceed and as a discount rate applied to cash flows.
The literature dealing with cost of capital is extensive. However, the concept and the mea-surement of cost of capital are still somewhat unsettled. For our purposes, cost of capital willbe assumed to be independently derived, along the lines suggested in Chapters 4 and 21, ex-cept where it is explicitly stated otherwise. The importance of cost of capital should not beignored. An adequate estimate of cost of capital is crucial to properly apply capital-budgetingtechniques, because all but the crudest techniques incorporate it in one way or another.
In the author’s experience the importance of obtaining good estimates of the firm’s costof capital is often overlooked. The result is that capital budgeting may, in practice, becomesomewhat of a burlesque: sophisticated techniques yielding accept/reject decisions based oncrude and incorrect data. One of the largest industrial firms in the United States in the 1970swas using 8 percent as its overall, marginal cost of capital. At the time, by generally accepted
Introduction: the Big Picture, Environment, Terminology, and Preview 7
measure, the firm had a cost of capital above 10 percent. In capital investment methods thatemploy discounted cash flows, and these methods are all conceptually better, this was a seriouserror. Paradoxically, the same firm went to considerable effort to obtain finely detailed projectcash flow data from its marketing, engineering, production, and accounting staffs at the sametime. As a result, projects were undoubtedly undertaken that, had an adequate cost of capitalbeen used, would have been rejected. The stock market performance of this firm during the1970s, vis-a-vis comparable firms in its industry, tends to support this view.
RISK AND UNCERTAINTY
For better or worse we live in a world of probabilities, with little more certain than the proverbialdeath and taxes. In general, capital budgeting projects are no exception, although some specificclasses of capital investment, such as we find in leasing, may approach certainty sufficientlyto be treated as if they were risk free.
Risk
Risk is usually defined to prevail in situations in which, although exact outcomes cannot beknown in advance, the probability distributions governing the outcomes are either known ormay be satisfactorily estimated. In a risky environment, probabilities may be associated withthe various results that can occur. Life insurance companies, for example, can predict withina close range of error how many policyholders in any age group will survive to age 65, eventhough they cannot predict accurately which specific individuals within the group will reachthat age. Those poker players who are successful in the sense of being net winners over the longrun understand risk. Winning card combinations have associated probabilities, and successfulplayers must take these into account, at least intuitively, to win on balance. In gambling gamesprobabilities can generally be determined with precision, whether the players know the oddsexplicitly or through a sense gained by experience in playing.
In games of chance the thrill of risk-bearing itself may be more important than the prospectof gain. And in capital investment the taking on of risk for the sake of doing so may hold appealto some individuals in management. However, if management is to serve the interests of theorganization’s owners and creditors, it must manage risk, not be managed by it. Managersmay undertake risky investments (later we show that this can actually be beneficial to theenterprise), but they must strive to commit funds to investments that in the aggregate promise agreater probability of gain than loss. This is not to say that individual projects offering a smallprobability of huge gain in return for a great probability of small loss must never be undertaken.Often such probabilities can be altered by managerial action. Moreover, such projects normallywill constitute a relatively small proportion of the firm’s total capital budget. Managers coulddo worse than be guided by the Machiavellian-like principle that a single loss on a clearlyhigh-risk project may do more harm to their reputation than a string of gains on other projects.When portfolio effects are taken into account, this is an unhappy state of affairs, but one thatthose making capital investment decisions must be aware of.
Later we treat in detail the topic of portfolio risk — risk of the total of the enterprise’s in-vestments. Although it is both informative and a necessary beginning to examine individualinvestment projects by themselves, it is important to recognize such interrelationships of in-vestments that may exist and to analyze the effect on the total enterprise that acceptance ofany particular project is likely to cause.
8 Capital Asset Investment: Strategy, Tactics & Tools
Uncertainty
The generally accepted distinction between risk and uncertainty is that in the case of uncertaintywe know that the possible outcomes are random variables, but we do not know the probabilitydistribution that governs the outcomes, or its parameters, and cannot estimate them a priori.Because capital-budgeting decisions are usually one of a kind, there is insufficient prior expe-rience with similar situations to grasp the probabilities associated with the possible outcomes.The uniqueness of such capital investments means they are not amenable to Bayesian revisionbecause they will not be repeated. In the more extreme cases even the entire range of outcomesthat could reasonably be expected to occur may be unknown.
In the following chapters we shall first consider risk and uncertainty as if a certainty envi-ronment existed for all project parameters. This has the advantage of allowing us to concentrateon basic principles and thus to gain a solid understanding of them before factoring in the com-plicating matters of risk or uncertainty. Later this assumption of certainty is relaxed in order todeal realistically with risk in individual projects and with the risk relationships between capitalinvestment projects and the enterprise. Prior to that, risk and uncertainty considerations willbe mentioned where appropriate as a complicating factor requiring attention and planning forunforeseen contingencies.
2
The Objective of Capital Budgeting
Unless one restricts attention to the very general social goal of accumulating capital in orderto increase national welfare, it is difficult to define only one objective for capital budgetingto achieve. The classical economics assumption of profit-maximizing entrepreneurs cannotbe considered appropriate for government or not-for-profit, private institutions. Furthermore,that assumption is not operationally feasible, and doubts have been expressed as to whether itrepresents the true motivation of managers in either private enterprise or in government.
Alternatives to the classical assumption of profit-maximizing behavior have been proposed.Among these, the more prominent include the concepts of “satisficing” and organizationaldecision-making. Satisficing owes much of its development to Herbert Simon [145], who ob-served that “Administrative theory is peculiarly the theory of intended and bounded rationalityof the behavior of human beings who satisfice because they have not the wits to maximize.”This view is supported by the observation that managers make decisions without the completeinformation classical economic theory assumes they possess. Managers may intuitively takeinto account the classical concepts of rationality, including marginal analysis and game theory.However, there is no evidence to support the notion that managers attempt to perform the com-plex calculations demanded by classical economic theory in other than rare, specific instances.Even if they wished to do so, managers usually do not have the detailed and exhaustive datathat classical theory would require, nor can they obtain it.
Other behaviorally oriented theories of management decision-making have been developed,including those of Cyert and March [26], which expand and supplement the satisficing concept,and Galbraith [51], who views managements of large, widely held corporations as serving theirown interests above those of the owners, and in some cases above those of the nation. Becauseof the diversity of human behavior and the evolution of attitudes and institutions, it is likely thatadditional behavioral theories will be developed that attempt to elaborate or offer alternativesto those proposed to date.
The behavioral theories, although realistically portraying management in human ratherthan mechanistic terms by incorporating a wider spectrum of behavioral assumptions thanprofit maximization, have contributed to better understanding of organizations. Like the clas-sical model of profit maximization, however, they are not operational. Although profit max-imization provides a normative model of management behavior, the behavioral models aredescriptive. They describe what management does rather than what management should door, more specifically, they fail to specify what objective criteria that management should usein reaching decisions that best serve those who employ them.1 Also, although profit max-imization provides a nomothetic model, the behavioral models to date are ideographic, ornonuniversal.
Our intent is to develop capital budgeting as much as possible within a normative framework.Therefore, we shall leave the behavioral models at this point and proceed to define a normative
1An enlightening critical examination of popular theories of managerial behavior is found in the controversial, now out of print bookby James Lee [84].
10 Capital Asset Investment: Strategy, Tactics & Tools
model that can be operationalized. The model that is adopted can be applied to both privateand public enterprise.
A NORMATIVE MODEL FOR CAPITAL BUDGETING
A serious deficiency of the classical economics principle of profit maximization, which preventsits adoption as operating policy, is that it does not indicate whether long-run or short-run profitsare to be maximized. A firm’s management, for example, might maximize short-run profits byformulating a policy that would simultaneously alienate customers, employees, and creditorsover a period of a few months. It might attempt charging the highest prices the market will bearwhile lowering quality, paying minimum wages, letting the firm’s financial structure, plant,and equipment deteriorate, and so on. Such behavior would surely injure the firm’s chances tosurvive beyond the short run. Once owners of a firm have decided to liquidate it, they mightadopt such an irresponsible mode of operation, but they could not thus operate and expect thefirm to survive for long.
A less extreme manifestation of short-run profit syndrome is not uncommon. It has takenroot in firms that neglect proper maintenance of plant and equipment and steadfastly refuseto abandon worn or technologically obsolete equipment until a new replacement can “payfor itself” in one or two years. This thinking blossoms into the specious yet appealing notionthat plant and equipment, even though seriously worn and obsolete, should not be abandonedbecause they have been “paid for” years ago. In other words, such assets are considered “free”resources to the firm because they were fully depreciated in the past. In truth, the firm paysincreasingly more in high scrap rates, in higher than necessary labor content, and machinerepair and tooling costs as time goes on. The assets were paid for when acquired, not whendepreciated. Depreciation merely enabled the firm to recover part of the cost of the investmentsthrough tax remission over the useful, economic asset lives.
The opposite of this obsession with short-run profits is preoccupation with the prospect ofprofits in the long-run future. In the extreme this is a much less common pathological conditionthan short-run profit obsession. This is, at least in part, due to the fact that firms that seriouslyneglect short-run profits cannot survive to the long run. A firm may be able to neglect long-runprofits and still survive as an economic cripple; a firm that neglects short-run profits is likelyto be an early fatality.
A tendency toward overly great emphasis on long-term profits may be observed in firms that,although showing poor current performance, spend lavishly on public relations, landscapingand lawn care around factory and office facilities, excessive employee benefits, research anddevelopment, and so on. The key word here is “lavishly.” All firms are expected to spendreasonable amounts on such indirectly beneficial things that make the world more pleasantand that may not be strictly defensible in terms of expected tangible benefits. It is ultimatelyfor the owners and creditors of the enterprise to determine what is reasonable, although inreality management may to a considerable extent be protected by the shield of ignorancesurrounding the enterprise’s operations. It is uncommonly difficult for ordinary stockholdersto obtain the detailed information necessary to successfully challenge mismanagement andlargess by management.
It would appear that management is caught between the horns of a dilemma. Short- andlong-run profit maximization seem to be contradictory. Is there any way to resolve the conflict,the ambiguity? Yes, there is, if we recognize that money, our economic numeraire, has time-value.
The Objective of Capital Budgeting 11
BASIC VALUATION MODEL
The model of managerial behavior we adopt is that of modern financial management, whichis not part of classical economics, although it might be considered a direct descendant:
Maximize wealth, or value, V =∞∑
t=0
Rt
(1 + k)t(2.1)
subject to governmental, economic, and managerial constraints. The terms of this basic val-uation model are as follows: t is the time index, Rt the net cash flow in period t, and k theenterprise’s cost of capital.
This model resolves the ambiguity over whether it is long- or short-run profits that shouldbe maximized. It is the total discounted value of all cash flows that is to be maximized. Thetheoretical value of the enterprise is defined in terms of its profitability over time. One may,of course, adopt the continuous analog of this discrete-time model, although traditionally thishas not been common or especially useful.
The nature of this model is such that its maximization may be powerfully facilitated by aneffective financial management that can raise adequate funds at minimum cost. This becomesclear if one considers that denominator terms containing k are raised to progressively higherpowers. And it underscores the need for good estimates of the enterprise’s cost of capital if itis to yield correct decisions. If the firm will accept only capital investments that have a positivenet present value then the value of the firm will be increased, irrespective of the timing of thenet cash flows.
Contemporary literature recognizes that there is interaction between the cost of capital kand the risk characteristics of the cash flow stream R, over time. In other words, if the firmundergoes changes that alter the variability of its overall cash flows over time, this will have aneffect on its cost of capital. The theory behind this notion has not yet become operational, and,although we shall deal with it in a later chapter, for now it will suffice to adopt the principlethat the cost of capital will not be increased if the enterprise invests its funds in capital projectsof similar risk to the existing assets of the firm. If projects are accepted that are less risky thanthe existing asset base of the firm, a tendency will be for the cost of capital to decrease. Thisprinciple will be useful for now; however, it ignores portfolio effects that imply that certaininvestments that are more risky than the existing asset base can sometimes actually serve toreduce the enterprise’s risk and thus its cost of capital. Portfolio effects are considered in detailin later chapters.
Operational Adaptation
We adopt the simple convention of accepting prospective investments that add to the valueof the enterprise. In an environment in which there is no capital rationing — that is, wherefunds are sufficient to accept all projects contributing to the value of the enterprise — weaccept all profitable projects. In the more common capital-rationing situation in which fundsare scarce compared to the investment cost of the array of acceptable investments, we shouldtry to accept those projects that contribute the maximum amount to the value of the firm.Nothing in this contradicts the behavioral theories of management. Other motivations may deterachievement of maximum value increase. But even satisficing management, or managementmerely interested in remaining in control, should still try to attempt to increase the value of theenterprise, even if not to the fullest possible extent. Management that habitually does otherwise
12 Capital Asset Investment: Strategy, Tactics & Tools
does so at its peril because the owners may replace the existing management. And increases invalue of the firm may indicate that management is alert and responsive to competitive forcesin the market in other ways as well. Thus, a firm that fails to increase in value may be signalingthat there are problems in the current management of which shareholders should be aware.
The Cash Flows
For individual projects that constitute only one of many items of capital equipment, it isdifficult, if not impossible, to associate net cash flows or even accounting revenues directly.In these cases it may sometimes be possible to treat the package of such individual items ofplant and equipment as a single, large capital-budgeting proposal. This is likely to be the mostuseful when the package alone is to support a new product line, or when it is to replace anentire production facility.
Alternatively, the cash flows associated with the individual capital equipment componentsmay result from cost reduction. Cost reduction may be considered equivalent to positive cashflow because it represents the elimination of an opportunity cost. Such costs are defined asthose attributable to inaction or the result of adopting some alternative to the best availableaction.
CASH FLOWS AND THE PUBLIC SECTOR
Because the equivalent of positive cash flows may be obtained from a reduction in costs, thebasic valuation model can be useful in public sector cost–benefit analysis. Cost reduction freesresources that would otherwise be wasted, so that they may satisfy other public demands.Also, maximization of the value of public enterprise may be considered beneficial, becausesuch institutions belong to all citizens of the community. Maximization of the enterprise valuetherefore serves to maximize the public wealth of society. Value in such cases is drawn fromthe tangible and intangible benefits provided to the citizens and not in the funds accumulatedby the institution, or the wages and salaries of its workers and managers, which should becompetitive with those in corporate industry. Any surplus of money accumulated by publicenterprise should be distributed in the form of social dividends, paid through reduction inrequired external funding (by taxes or government borrowing), or by an increase in the servicesprovided. Unfortunately, the performance of some government institutions may lead one tobelieve that this, in fact, is not done, perhaps because the bureaucrats managing them are notanswerable to the citizens nor held accountable by elected officials. The US Postal Serviceprovides an excellent example.
3
Estimating Basic Project Characteristics
In order to apply any objective, systematic method of capital-budgeting evaluation and projectselection, it is first necessary to obtain estimates of the relevant project characteristics orparameters. The conceptually superior methods of evaluation also require an estimate of thefirm’s cost of capital, which is taken up in the next chapter. The present chapter focuses onthose characteristics of a particular project or aggregation of projects that become manifest incash flows attributable to the project.
PROJECT TYPES
Let us define those capital investment projects as major projects, which in themselves generatenet cash inflows, and as component projects those that do not in themselves directly generate netcash flows. Examples will serve to clarify the distinction. Investment in plant and equipment toproduce a new product line that will generate sales revenues as well as production, marketing,and other costs would be classified as a major project. Investment in a new tool room lathewould be classified as a component project. The lathe will be used to service other capitalequipment in the plant or produce prototype parts. The lathe thus will not produce directlyattributable cash inflows, but will necessitate directly attributable cash outflows for operatorwages and fringe benefits, electric power, and so on. A new spline rolling machine that willreplace several milling machines in a plant producing power transmission shafts and gearsis also a component project. The spline rolling machine in such a production facility wouldnot produce a product that is sold without many other machining, heat treatment, inspection,and assembly operations being performed on it. Thus, product revenue cannot be directlyassociated with this machine except through rather tenuous cost accounting procedures notlikely to properly reflect its revenue contributions.
Major projects have both cash inflows and cash outflows directly associated with themselves.Component projects ordinarily do not. Therefore, although attention to value maximization maybe appropriate for major projects, cost minimization will generally be a more suitable approachto component project analysis. For instance, cost reduction brought about by replacing an older(component project) machine by a newer and more efficient machine contributes to net cashinflow as much as an increase in revenue with costs held constant does. In this case the net cashflow is attributable to eliminating opportunity cost that is associated with inefficient productionequipment for the particular operation.
To further clarify the distinction between project types, assume that, for a given costof capital and risk level, we want to maximize net cash flows by holding costs constantwhile increasing cash revenues. For major projects this may be appropriate. On the otherhand, for component projects we may more easily achieve the same result by minimizingcosts for a given level of cash revenue. An analogy may be drawn using the terminologyof mathematical programming, in which the major project is considered the dual prob-lem of cost minimization. We may expect equivalence between maximizing some objective
14 Capital Asset Investment: Strategy, Tactics & Tools
function, subject to cost constraints, and minimizing cost subject to some performance con-straints.
Most capital-budgeting techniques have been oriented to selecting projects that will con-tribute toward maximization of some measure of project returns. However, the MAPI methodproposed by George Terborgh departs from these by selecting projects that contribute to min-imization of costs. The classical MAPI method is treated in detail in Chapter 9.
PROJECT CHARACTERISTICS
The quantitative parameters of an investment that are relevant to the decision to accept or rejectthe project are:
� Initial cost project� Useful life� Net cash flows in each time period� Salvage value at end of each time period
These parameters, together with the enterprise’s cost of capital, provide information upon whicha rational decision may be based, using objective criteria. In addition to the above quantitativeparameters, sometimes qualitative considerations will also affect the investment decision. Forexample, one production facility may use a collection of custom-designed machinery, whereasanother employs more or less standard production machines. If the specialized machinerycannot readily be converted to producing other machined products, and if the probability thatthe particular products for which this specialized machinery is acquired will be abandonedprematurely is high, then the decision to adopt standard production machinery may be supe-rior even though it may promise somewhat lower benefits if things go well for the productline.
Qualitative considerations may be crucial, especially in contingency planning. In answeringthe “what if” question of what alternative use may be made of capital investment projects ifthings do not go as they are expected to, management may well prefer the array of projectsthat offers flexibility over the somewhat more efficient, but highly specialized, alternative.For example, a firm that produced automobile fabric convertible tops and decided in 1970to acquire new automated equipment to stitch the seams, predicated on a 15-year useful ma-chine life, would have found in 1976 that original equipment market sales were vanishing.In 1976, Cadillac, the last of the United States automakers to produce fabric-covered con-vertibles, announced it was phasing out such models. Unless the replacement market wouldcontinue to provide sufficient sales into the mid-1980s, the firm would need to find alterna-tive products suitable for production on the specialized equipment. Perhaps fabric camper-trailer tops or similar items could be produced as profitably on the specialized machinery. Ifnot, the firm would find that its decision in 1970 was, with benefit of hindsight, the wrongdecision.
No general rules or procedures have been developed for contingency planning in capitalbudgeting. Each case has its own unique attributes that prevent uniform application of rigorousprinciples. It is in such aspects of decision-making that there is no reasonable alternative tothe judgment of management. For instance, how should a firm that produces barrel tubes forshotguns incorporate in its decisions the possibility that a Congress will be elected that isdisposed to outlaw private firearms ownership or restrict ownership drastically? Given thatsuch a Congress is elected, what is the contingent probability that it will find the motivation
Estimating Basic Project Characteristics 15
and time to act? Will a later change in Congressional composition reverse such legislation?If so, would consumer demand return to its former level? Could alternative uses be found forthe machinery, such as making hydraulic cylinders? Such a firm must incorporate factors suchas these into its capital-budgeting decisions in order to ensure flexibility in using its physicalcapital in case the environment in which it operates suddenly changes. And is there any firmnot subject to environmental changes?
INITIAL COST
Capital-budgeting projects will generally require some initial cash outlay for acquiring theproject and putting it into operation. Such cost may arise from construction outlays, purchasecost or initial lease payment, legal fees, transportation, and installation, and possibly from taxliability on a project that is being replaced by the new one or from penalty costs associated withbreaking a lease on the replaced project. Cash costs attributable to the decision to accept a capitalproject should be included in the initial project cost. Costs that would be incurred regardlessof whether or not the project were accepted are “sunk costs” that should not be included inthe project cost. With some projects, especially large ones involving lengthy design work andconstruction that extends over several years, initial costs should be considered as the negativenet cash flows incurred in each period prior to the one in which net cash flow becomes positive.In such cases there is a series of initial costs instead of a single outlay.
Sunk Cost
Unrecoverable costs associated with previous decisions should not be allocated to a new projectunder consideration.1 For instance, assume that a firm purchased a $400,000 machine 5 yearsago, which had at the time an estimated useful life of 20 years, and that $228,571 has still tobe claimed as depreciation against taxes on income at a rate which yields a tax reduction of$118,857 spread over the next 15 years. Assume now that a technologically improved machineis available to replace it. Should the unrecovered tax reduction be added to the other costs ofthe new machine? The answer is no, in this case. However, the unrecovered tax reductions maybe included as costs in the years in which they would have been realized. Now, what if the$400,000 machine suddenly breaks down and cannot be repaired? In this event the unrecoveredtax reductions from depreciation should not be charged to a replacement machine, either in theinitial cost or in the cash flows over the remaining depreciable life of the broken machine. Fortax purposes the broken machine’s remaining value will be charged to the firm’s operationsas a loss which will have no effect on the replacement decision. Similarly, equipment that isdiscarded because of a change in the firm’s operations should not have any of its cost chargedto new equipment that is subsequently acquired.
The problem of what costs to include and what costs to exclude from a particular capitalinvestment project will be resolved by focusing on cash flows and ignoring accounting costs.Does acceptance of the candidate project increase or decrease cash flow in years zero throughthe end of its anticipated economic life? If acceptance of the project precipitates changes incash flows, then these must be taken into account in evaluating the project. Noncash items,and items such as sunk costs, must not be included in project evaluation, even though specificdollar figures are associated with them.
1Of course, for accounting reasons related to minimizing tax liability, they may be included in the bookkeeping for the project.
16 Capital Asset Investment: Strategy, Tactics & Tools
What if costly preliminary engineering studies have been completed and research and de-velopment costs incurred? Should these be included in the cost of the project for capital-budgeting evaluation? No, they should not, for they represent sunk costs — water over thedam and subsequent acceptance or rejection of the project will not affect them. The handlingof such sunk costs in the accounting framework may well be a different matter; for tax orcontrol reasons they may be associated with the project. Since in this book our concern iscapital budgeting and not accounting, such matters will be ignored. They are irrelevant tothe decision of whether or not to accept a project, except to the extent that they affect cashflows.
Components of Initial Cost
In addition to the obvious component of initial cost, namely, the basic price of the capitalproject in question, some less obvious costs must be included. Among them are:
� Transportation and insurance charges� Installation costs, including special machine foundations, movement of other equipment to
get the new project to its location in the plant, installation of service facilities such as electric,hydraulic and pneumatic lines, and so on
� License or royalty cost� Required additional working capital investment� Operator training costs
Transportation and insurance costs may be included in the vendor’s price. In many in-stances, however, delivery will be FOB the vendor’s plant. In these cases failure to include thetransportation and insurance costs will understate the project’s initial cost, perhaps seriouslyso for large, heavy equipment that is difficult and expensive to ship.
Installation costs include the full expense of project installation. With industrial machinerythat will be placed in an existing plant, it may be necessary to move intervening equipmentto allow room for the new equipment. In some instances the plant structure itself may haveto be temporarily or permanently modified or the disassembled equipment moved in pieces toits site within the plant. Worker safety may require installation of noise-dampening materials,special ventilation equipment, dust collectors, fire extinguisher systems, and so on. Heavymachines often require a concrete “anchor” foundation to be poured prior to installation,with the foundation sometimes being as much as a meter deep and nearly as large as themachine attached to it. Machinery often requires these foundations to be built so that vibrationsemanating from them are not communicated through the plant floor to other machines, wherethey would affect the quality of the operations they perform. Adequate vibration dampeningcan increase installation cost considerably, but it is required in environments in which precisionmay be affected by this type of unwanted disturbance.
License or royalty cost for use of patented equipment or processes may require an initialpayment as well as the customary periodic payments as production gets underway. Theseshould be included in the project cost.
Required additional working capital investment is an item that is easy to overlook. Usually,this will not be a significant factor with component projects, but will be important for mostmajor projects as defined earlier in this chapter. Additional net working capital required tosupport accounts receivable, inventory, or other current asset increases (net of current liabilityincreases) are included in project cost. Recovery of such net working capital requirements at
Estimating Basic Project Characteristics 17
project termination may be incorporated in salvage value or the last period’s cash flow, whichare equivalent means of handling this factor.
Operator training costs, if not included by the vendor in the basic equipment cost, mustbe added to the project cost by the purchaser. Such training is to be expected with capitalequipment employing a new technology or capital equipment whose operation by nature iscomplex. For instance, it is generally required when large-scale computer or industrial controlsystems are acquired, that operators and support personnel be trained properly in correct useof the equipment. This is true even though the new machine replaced an older model of thesame vendor, or a smaller machine of the same series with a different operating system orfewer options than the new one.
In addition to these initial cost items, others may be found in particular cases. The rule tofollow in determining whether a cost item should be included in the initial cost of a capitalinvestment project is to answer the two following questions. If the answer to both is in theaffirmative, the cost should be included in the initial cost of the project.
1. Is the cost incurred only if the capital project is undertaken, that is, accepted?2. Is the cost represented by a cash outflow?
If the answer to both is “yes,” then only the cash outflow associated with the cost should beincluded in the initial cost of the project. Noncash costs should be excluded. For this purpose weassume net working capital increases to be cash flows, although “internal” to the firm. Increasedworking capital requirements must be funded, and the money committed will generally not bereleased until the end of the project’s useful life.
USEFUL LIFE
The investment merit of a project will depend on its useful economic life. Useful economic lifeof capital equipment may end long before it becomes physically deteriorated to the point ofinoperability. Economic life, and decline in the value of capital equipment over the economiclife, may mean that project abandonment prior to the end of the originally anticipated projectlife will be of greater benefit to the firm than holding the project to the end. This topic is takenup in detail in Chapter 15.
Terborgh, in his classical method, defined the cumulative effects on decline in capital ser-viceability over the period the equipment is held as operating inferiority [151]. This is a usefulconcept, and it is adopted here. Operating inferiority is determined by two components: phys-ical deterioration and technological obsolescence. Physical deterioration is what is normallyconsidered to be the determinant of project life. However, technological obsolescence will befar more important in determining the economic life of some projects.
Consider an accounting firm that purchased a large number of mechanical calculating ma-chines in 1970, assuming an economic life of 10 years. If the machines were to be kept inservice over the full 10 years, they could be expected to undergo steady physical deteriora-tion. As time went on, more frequent and more serious breakdowns would be expected, andexpenditures for repairs would increase accordingly.
Some rotary, mechanical calculators cost $1000 or more in 1970. At about the same pointin time, due to advances in technology, a variety of electronic calculators came on the market.Not only was the cost substantially less, as little as one-fourth the cost of their mechanicalpredecessors for some of them, but they were superior in several respects. The electronic modelswere substantially faster, immensely quieter, less subject to mechanical problems because
18 Capital Asset Investment: Strategy, Tactics & Tools
of the dearth of moving parts, and provided number displays that were larger (sometimes),illuminated, and generally easy to read. Thus it was that the mechanical calculator fell victimto technological obsolescence. They simply were inferior in most respects to their moderndescendants, even if still quite serviceable.
Or imagine the firm that bought a large quantity of personal computers in the mid-1990sunder the false assumption that they would be quite adequate for at least a decade. Perhaps theywould remain functional, if maintenance were not required, but clearly a PC even four yearsold presents problems in terms of both physical function and obsolescence. After a few years,hardware and software vendors no longer find it worthwhile to support older machines, andthus keeping them running presents an ever-increasing headache for system administrators.And as newer and better software becomes available it will not run on old machines at all, orat best suboptimally.
In firms whose employees devote a large portion of their working time to calculation, by1975 few mechanical calculators were in use. And by the year 2002 few personal computersof vintage more than three years are in use in companies. Why? Because replacement withthe best technology allows less wasted time while waiting for results, a quieter atmospheremore conducive to productive work, and reduced maintenance and repair costs. If an employeeworks half a day with a calculator or computer, and the new technology is 25 percent faster,the same employee can do the equivalent of one hour’s additional work in the same four hours.Such efficiency gains are easily translated into money terms. An office employing four suchpeople, by equipping each with the new technology, could avoid hiring the fifth person whenthe workload expanded by as much as 25 percent.
Physical Deterioration
For many types of capital, experience with similar facilities in the past may provide usefulguidelines. For instance, we would expect that the physical life of a punch press purchasedtoday would, on the average, be similar to that of a new punch press of 20 years ago.
With capital that embodies new technology, or a new application of existing technology,managers and engineers experienced with production equipment may provide useful estimates.However, capital goods produced by a firm not likely to stay in the business and maintain asupply of replacement parts may negate an otherwise good estimate of useful life. Unavail-ability of a crucial part from a supplier will mean either producing the part in the adoptingfirm’s tool room, contracting to have it custom made, modifying the machine to take a similarstandard part, or abandoning the machine. This last alternative obviously ends the useful life.
Physical deterioration of major projects, as defined earlier, may well be affected by thatof plant and individual equipment components. It is not meaningful to speak of the physicaldeterioration of a major project unless this is taken to be synonymous with deterioration of thebuildings housing the operation, or unless the major project is indeed one major item of capitalequipment that dominates all others. If our major project is a division of the firm composed ofone or more buildings, each housing 100 or more items of capital equipment, what meaningcan we attach to physical deterioration of the project? The answer is none; at least this is true ifcomponent projects are added and replaced as time goes on. On the other hand, if our divisionis based on operation of one dominating item of capital, such as a toll bridge or a carwashor parking garage, it may well be meaningful to refer to physical deterioration of the majorproject.
Estimating Basic Project Characteristics 19
Technological Obsolescence
Although we may often be able to obtain workable estimates of physical life for capitalequipment and the corresponding physical deterioration over time, it is a very different matterfor technological obsolescence. Technological innovations that contribute to the obsolescenceof existing capital tend to occur randomly and unevenly over time. Sometimes technologicalchanges are implemented rapidly during relatively short intervals of time: The technologicaladvances in computer equipment since the 1950s have been profound, and now may continueat a rapid pace for some time yet. Theoretical developments, such as holographic, laser-directed computer memories, once the engineering obstacles have been overcome, and artificialintelligence software promise yet further waves of innovation in the industry.
Terborgh’s approach to incorporating technological obsolescence into the operating infe-riority of capital equipment is difficult to improve upon. His recommendation, basically, isto assume, in the absence of information to the contrary, that technological obsolescence ofexisting capital will accumulate at a constant rate as time goes on [151, p. 65]. In the absenceof information to the contrary, such extrapolation from the past into the future is reasonable.However, should information be available that implies more rapid or less rapid technologicalchange, this information should be employed, even if it is only qualitatively.
Should the firm delay an investment when technological changes are expected in the nearfuture? The answer to this question may be found in evaluating the merit of investing today withreplacement when the technologically improved capital becomes available, and evaluating themerit of the alternative — that of postponing investment until the improved capital is available.Comparison of the merits of the alternatives will serve to determine the better course of action.Methods for performing such analysis, once the net cash flows are determined, are covered inChapters 9 and 23.
The subject of technological obsolescence in capital budgeting is not limited to the firm’sphysical capital; it may apply also to the human capital of the enterprise, and certainly doesapply to the product lines on which the cash flows of investment projects are predicated. Forexample, the replacement of vacuum tubes with transistors and integrated circuits, and laterwith miniaturized circuits made possible by large-scale integration, caused radios and televi-sions incorporating vacuum tubes to become obsolete. A firm that based its capital investmentdecisions on continued demand for sets with vacuum tubes found that later it had still ser-viceable equipment for a product no longer in demand. Similar examples are to be found inmechanical versus electronic calculators, automatic versus manual automobile transmissions,and piston versus jet propulsion aircraft engines, to name just a few of the more obvious.
CASH FLOWS
In capital budgeting we must base our analysis on the net cash flows of the project underconsideration, not on accounting profits. Only cash can be reinvested. Only cash can be used topay dividends and interest and to repay debt. Only cash can be used to pay suppliers, workersand management, and tax authorities.
Successful application of any method of capital project evaluation requires forecasts of es-timated cash flows. Like all forecasting, this is a “damned if you do, damned if you don’t”proposition — one must forecast, but in doing so is destined to be in error. A successful fore-cast is one wherein the forecast error is minimized. It is far beyond the scope of this bookto delve into the arcane art and science of forecasting. That is left to books that specialize
20 Capital Asset Investment: Strategy, Tactics & Tools
in the subject. However, those who need to use forecasts, or to make them, should consultcredible references on the subject and become familiar with the merits and pitfalls of thecurrently available methods. (For example, one can find good insights and guidance in bookssuch as that by Spyros Makridakis et al. [96].) Whether one makes forecasts or commis-sions others to do so, the manager should be familiar with the strengths and weaknessesof the methods used to develop forecasts. This book assumes that the capital investmentdecision-maker has the best forecasts possible when analyzing projects. This is somewhatof a fiction, but it will keep us on track here instead of sending us on a detour leading farfrom this book’s main theme, that of capital investment management strategy, tactics, andtools.
Over the long run, both total firm cash flow and total accounting profits provide measuresof management performance. However, in the short run the two will generally not be highlycorrelated. For example, the firm may have a very profitable year as measured by “generallyaccepted accounting principles” and yet have no cash to meet its obligations because the“profits” are not yet realized but are tied up in accounts receivable that are not yet collected,and in inventories.
Determination of net cash flows involves consideration of two basic factors: (1) those thatcontribute to cash inflows or cash receipts and (2) those that contribute to cash outflows or cashcosts. Major projects, as defined earlier, will have both these factors affecting them throughouttheir economic lives. Component projects, after the initial outlay, will be directly involved withcash outflows, but only indirectly with cash inflows.
Estimation of cash inflows for major projects generally requires the joint efforts of specialistsin marketing research, sales management, and design engineering, and perhaps staff economistsand others as well. If the firm does not have the required expertise itself, it will have to hirethe services of appropriate consultants.
For reliable cash outflow estimation, the joint contributions of design, production, industrialengineering, production management, cost accounting, and perhaps others, are required formajor projects. Additionally, staff economists and labor relations personnel can contributeinformation relevant to probable cost increases as time goes on.
Financial management is responsible to top management for analyzing the effects on thefirm’s financial strength if the project is undertaken and for obtaining the funds necessary tofinance the undertaking. Finance personnel will undoubtedly be involved in recommendingwhich major projects should be accepted and which should be rejected based on their analyses,project interactions with the existing assets of the firm, and consequently their profitability.Accounting staff will be concerned with the project’s effect on reported profits and tax liabilitiesand all the attendant details.
Cash inflows for a major project will be determined by (1) the price at which each unitof output is sold; (2) the number of units of output sold; and (3) the collection of accountsreceivable from credit sales. Estimation of these items is not easy, especially for a productdissimilar to product lines with which the firm may already have had experience. Properestimation, as mentioned earlier, involves persons from different functional areas within thefirm, and possibly external to it.
For component projects we will be concerned with cash outflows. If a given task must beperformed by capital equipment, then we shall seek to obtain capital that will do so at low-est cost. Thus, on initial selection of mutually exclusive candidate projects, we will selectthe project that minimizes cost or, alternatively and equivalently, the project for which thesaving in cash opportunity cost is maximized. Of course, we must assume that all compo-nent projects admitted to candidate status are capable of performing the tasks that must be
Estimating Basic Project Characteristics 21
done within the environment of a major project. Projects that cannot should not be treated ascandidates.
Cash Inflows (Cash Receipts)
Assume that we are examining a combination of several machines that will produce a singleproduct our new firm plans to sell. The principals of our firm are experienced in sales andengineering of such a product; both worked for years for a larger company that producedsimilar products. Some preliminary orders have already been obtained from firms that willpurchase the product we will produce. It is estimated that at a price of $17.38 per unit, our firmcan reasonably expect sales of 100,000 units per year.
If we assume that sales and cash receipts on sales will be uniform throughout each year,and that unit price and sales volume will be constant from year to year, the task of cash inflowestimation is trivial. If we have no uncollectable accounts receivable, our cash inflows eachyear will be unit price times number of units, that is, $1,738,000.
Life is usually not so simple as this, however. Sales probably will not be uniform throughoutthe year, but will have seasonal variations. Sales from year to year will seldom be even nearlyconstant. Possibly sales will grow from year to year along a trend of several years’ duration,or decline for several years. Such variations are very difficult to predict in advance, and are ag-gravated by unforeseen developments in competition, the national economy, and other factors.
In the final analysis, estimation of cash inflows will depend on managerial judgment, con-ditioned by the economic environment and knowledge of the firm’s competition and trends inproduct design and improvement. In many industries revenues may be influenced by advertis-ing expenditures so that firms may in fact influence the demand for their products. Seldom willa firm be in the position of having a mathematical model that provides truly reliable demandand revenue forecasts, especially for periods beyond one year. There are too many qualitative,vague, and intangible factors at work that cannot be quantified given the current state of theart in mathematical modeling. Proper incorporation of these factors requires human judgment,and perhaps not a little luck.
It will often be useful to prepare several forecasts, including a “worst case” forecast inwhich it is assumed that whatever can go wrong for the firm’s sales will go wrong. Of course,unanticipated factors and events may make actual events still worse, but we would assume thatfor a proper “worst case” forecast, actual experience would be no worse, say, 95 times out of100. In other words, a good “worst case” forecast should have a small probability associatedwith it that actual events will turn out worse. “Best case” and “most likely” forecasts maysimilarly be made. The population projections of the United States Bureau of the Census, infact, have been prepared on the basis of high, low, and most likely. Because we can neverexpect an exact forecast, it is extremely useful to be able to bracket the actual outcome, sothat we may say that the probability is p percent that the actual outcome will fall betweenthe best case and worst case estimates, where p is a number close to 1.0. For this approachto be useful, the best case and worst case forecasts cannot be so far apart as to make themmeaningless.
Let us now go back to our example. If $17.38 per unit and 100,000 units are taken to beour most likely forecast for the coming year, we may determine that $14.67 and 40,000 unitsis the worst that is likely and $23.00 and 200,000 the best that is likely to happen. If we canattach (albeit subjective) probability of 0.05 or 5 percent that the actual revenue will fall shortof the worst case, we have useful information. The chances of sales revenues being less thanthe worst case are then only 5 in 100.
22 Capital Asset Investment: Strategy, Tactics & Tools
Cash Outflows
To some extent estimates of cash outflows may be made with more confidence than thosefor cash inflows. For example, the initial investment outlay is made at the beginning of theproject’s life, and may often be estimated precisely. In fact, the supplier may provide a firmprice for the cost of the investment. However, other cost components may prove to be almostas difficult to forecast as factors affecting cash receipts.
What specific items do we consider as cash outflows? Anything requiring cash to be paidout of the firm, or to be made unavailable for other uses. Suppose that the machines we areconsidering for purchase are to be treated as one project. Assume that the project costs $3 millionfor purchase and installation. For simplicity, assume also that we plan to run production on aconstant level, so that labor, electricity, and materials will be constant for at least two years.In addition, cash will be tied up in raw materials and finished goods inventory and in accountsreceivable. Setting aside the notion of worst and best case forecasts for now, let us deal withthe most likely production and cost forecast, and the most likely annual cash revenue forecast.We obtain the following:
Year 0 Cash outflow Cash inflow
Initial investment $3,000,000Wages 100,000Fringe benefits 50,000FICA, etc. 20,000Raw materials inventory 200,000Finished goods inventory 500,000Accounts receivable investment 400,000Electric and other direct variable costs 10,000
$4,280,000Cash receipts on sales, net of discounts,
and bad debt losses $1,738,000Net cash flow ($2,542,000)
Note that no overhead or sunk cost items have been included in the cash flows. In the secondyear of operations, if no new investment in inventories or accounts receivable is required,and the same amounts of labor, electricity, and so on, at the same rates are employed, weobtain:
Year 1 Cash outflow Cash inflow
Wages $100,000Fringe benefits 50,000FICA, etc. 20,000Electric and other direct variable costs 10,000
$180,000Cash receipts on sales, net of discounts,
and bad debt losses $1,738,000Net cash flow $1,558,000
Wages and fringe benefits include those of direct labor plus the portion of indirect labor thatservices the machinery: machine setup, materials handling, and so on.
Estimating Basic Project Characteristics 23
Ordinarily, in expositions of the various capital-budgeting techniques, we assume for sim-plicity of explanation that cash flows occur only at the end of a period (usually a year) and arenot distributed throughout the period. Period zero in such treatment includes only the installedcost of the capital equipment. Subsequent periods include net cash flows arising from cashreceipts minus cash disbursements, the latter including whatever additional investment thatmay be required in capital equipment, inventories, and so on. Interest expenses are specificallyexcluded from cash costs because methods of project evaluation that employ discounting al-ready incorporate the interest costs implicitly — they are imbedded in the discount rate andnot in the cash flows.
TAXES AND DEPRECIATION
Since firms generally pay income taxes on earnings and depreciation is deductible as anexpense, it has an effect on cash flow. Cash that does not have to be paid as tax to the governmentserves to increase net cash flow because it is a reduction in cash outflow.
The rationale behind allowing depreciation to be tax deductible is that it represents recoveryof investment rather than profit. And since the benefits of capital investment occur over theeconomic life of the project, it is deemed appropriate to spread recognition of the investmentoutlays, as expenses, over the same period. Various accounting conventions and tax authorityrulings on how the depreciation charges may be calculated, and what depreciation lifetimesmay be used for various asset types, have complicated a basically simple concept. Profitabledisposal of capital equipment may subject the firm to additional taxes on residual salvagevalue. Such details are covered by texts on accounting and on financial management. Herewe are concerned only with basic concepts and the effect of depreciation on cash flow, not onthe details of tax rules. Tax laws change from time to time, not only for various theoreticalreasons, but as part of the occasional economic “fine-tuning” by the federal government aimedat encouraging or discouraging new investment, as the situation of the national economy mayindicate.
Several methods of calculating depreciation for each year of a capital investment’s lifehave been devised. Because cash may be invested, it is generally better to charge as muchdepreciation as possible in the early years of a project’s life, thus deferring taxes to later yearsand simultaneously retaining more cash in the early years. This is especially true during timesof rapid inflation. Unless the firm has tax losses larger than it can use to offset taxable income,it will charge the maximum allowable depreciation in the early years of a capital investment’slife. Given that money has time value, to do otherwise would not be in the best interests ofthe owners of the enterprise. The more rapid the rate of price inflation, the more incumbentit is to charge the maximum depreciation in the early years for tax purposes. However, formanagement control purposes the firm may use the depreciation schedule that is considered tomatch most closely the actual economic deterioration in the capital project from year to year.Yet, with high rates of price inflation in capital goods, the depreciation charged against theoriginal cost if unadjusted may be of little usefulness. Such adjustment, however, is beyondthe scope of this book.
Depreciation
Until 1981 there were two tax depreciation frameworks in the United States: (1) the GeneralGuidelines (GG) and (2) the class life asset depreciation range (ADR) system. The latter could
24 Capital Asset Investment: Strategy, Tactics & Tools
be considered a precursor to the current modified accelerated cost recovery system (MACRS),which in 1986 replaced the accelerated cost recovery system (ACRS) adopted in 1981.
The GG could be used for any depreciable asset. The ADR system was authorized by theRevenue Act of 1971. Examination of these and the fundamental methods of depreciation willhelp in understanding MACRS depreciation. And, given the propensity of Congress to changethe tax laws, we have not seen the end of changes in allowable methods of depreciation. Byunderstanding the basic methods one can easily grasp what is involved in new procedures.
A firm could select the ADR system in preference to the GG for any depreciable assetacquired after 1970 until ACRS came into being in 1981. A firm elected each year eitherto use or to not use the ADR system for those depreciable assets acquired during the fiscalyear. Under the ADR system, assets corresponded to various classes. For example, one classwas 00.241 — light general purpose trucks. The range for that class was three years at thelower limit, four years “guideline,” and five years upper limit. The firm could choose the lowerlimit to achieve the most rapid depreciation. What the change to ACRS depreciation did wastantamount to mandating that all firms would use ADR depreciation with the lower limit.
It will be helpful to understanding if we now review the fundamental methods of depreciation.Then we shall examine the GG and ADR system before considering MACRS, which derivesfrom ADR.
Straight-line Depreciation
The notion behind straight-line depreciation is that the investment’s residual value declines bya constant dollar amount from year to year uniformly over the useful life. Therefore, the initialinvestment is divided by the number of years of useful life, and the result used as the annualdepreciation charge. This is the least useful method for deferring taxes to the later years of theproject’s life.
Double-declining-balance Depreciation
With declining-balance depreciation it is assumed that the remaining depreciable value of theinvestment at the end of any year is a fixed percentage of the remaining depreciable valueat the end of the previous year. Alternatively, we take as declining-balance depreciation aconstant percentage of the remaining depreciable value at the end of the previous period. InUnited States practice, the method used is that of double-declining balance (DDB), in whichthe percentage value of decline is multiplied by a factor of two. This results in acceleratingthe depreciation charges and deferring larger amounts of taxes to later years. Salvage value isexcluded from the calculations in this method.
Sum-of-the-years’ Digits Depreciation
The method of sum-of-the-years’ digits (SYD) depreciation is implemented by writing theyears in the asset’s lifetime in reverse order, and then dividing each by the sum of the yearsin the useful life. Depreciation for each year is then determined by multiplying the originalasset cost by the factor corresponding to each year. A project lasting five years will thereforehave 5/15 (or 1/3) of the original value charged to depreciation in the first year and 1/15charged in the fifth and last year. This method, like that of double-declining balance, providesfor accelerated asset depreciation.
Estimating Basic Project Characteristics 25
Table 3.1 Comparison of depreciation methods under generalguidelines, with zero salvage
Double-declining Sum-of-years’Year Straight line balance digits
1 $10,000 $20,000 $18,1822 10,000 16,000 16,3643 10,000 12,800 14,5454 10,000 10,240 12,7275 10,000 8,192 10,9096 10,000 6,554 9,0907 10,000 5,243 7,2738 10,000 4,194 5,4559 10,000 3,355 3,636
10 10,000 13,422a 1,819
a The $13,422 in year 10 is $2684 plus the remaining $10,738 undepreciatedbalance, to arrive at zero salvage value.
Comparison of the Basic Depreciation Methods
Assume we have an asset that cost the firm $C, an estimated salvage value of $S and has adepreciable lifetime2 of N years. Then the annual depreciation charges with the three methodsfor 0 < t ≤ N are:
Straight line:
(C − S)/N
Double-declining balance:
C(1 − 2P)t−12P where P = 1/N
Sum-of-years’ digits:
(C − S)(N − t + 1)/[N (N + 1)/2] sinceN∑
t=1
t = N (N + 1)
2
Example 3.1 For illustration, let us take a project with C = $100,000, N = 10, and calcu-late depreciation at the end of each year with each method. Table 3.1 contains a comparisonof the methods when salvage is assumed to be zero. We assume zero salvage value.
The total depreciation charged with each method is equal to the cost of the project. Totaldepreciation cannot, of course, exceed the investment acquisition cost less estimated salvagevalue. Note that with DDB depreciation, the final year’s depreciation charge is the sum of theDDB amount plus the undepreciated balance.
In practice, firms will usually switch from one method of depreciation to another when itis advantageous to do so, and when the tax authorities will allow the change. For instance,under GG a firm may switch from double-declining-balance depreciation to straight-line de-preciation. Of course, the straight-line depreciation charge will not be based on the original
2The depreciable lifetime of an asset will often be different from the useful economic life. This happens because tax authority rulingsconcerning the lifetime that may be used for depreciation may not properly reflect the useful economic life of such asset in any particularfirm.
26 Capital Asset Investment: Strategy, Tactics & Tools
Table 3.2 Comparison of depreciation methods under generalguidelines, with salvage
Double-declining Sum-of-years’Year Straight line balance digits
1 $12,000 $26,000 $21,8182 12,000 20,800 19,6363 12,000 16,640 17,4554 12,000 13,312 15,2735 12,000 10,650 13,0916 12,000 8,520 10,9097 12,000 6,816 8,7278 12,000 5,754a 6,5459 12,000 5,754 4,634
10 12,000 5,754 2,181
$120,000 $120,000 $120,000
a Note the straight-line depreciation in years 8, 9, and 10 on the remainingdeclining balance. No more than 100 percent of the asset may be depre-ciated. Salvage is expected to be $10,000 and $102,738 has been chargedby the end of year 7. The straight-line amount is obtained by dividing$(120,000 − 102,738) by 3. This yields $5,754, which is larger than theDDB charge of $5,453 would be, so the switch is advantageous.
cost and lifetime, but on the undepreciated balance and remaining depreciable life at the timeof the switch.
In this example the firm could, under the GG, switch to straight-line depreciation in year 6. Inthis year straight-line depreciation of the remaining balance yields the same dollar depreciationas the double-declining-balance method. However, in years 7, 8, and 9 the amount chargedto depreciation is larger with straight line. In no case may the firm depreciate below salvagevalue. This means that if an asset costs $C, and is expected to have a salvage value of $S, nomore than $(C – S) may be depreciated. Let us now consider an example in which salvagevalue must be taken into account. Note that DDB depreciation ignores salvage, although nomore than $(C – S) may be charged.
Example 3.2 Let us take a project with C = $130,000, N = 10, and S = $10,000. We shallassume the firm will switch to straight-line from DDB depreciation, as allowed by the GG assoon as this is advantageous. Table 3.2 contains a comparison of the methods.
Example 3.3 With the ADR system, salvage is treated the same way as with DDB underthe GG; that is, it is ignored. As always, however, no more than 100 percent of the asset valuemay be depreciated. The following example shows the results of using ADR with the sameasset just considered. In practice, because ADR allows a choice of depreciable asset life, weshould not expect that the asset life will remain 10 years. If a shorter life is allowed, the firmwill take it to maximize accelerated write-off. Tables 3.3 and 3.4 contain a comparison of themethods, including optimal depreciation.
GG allows the firm to switch from DDB to straight line only; ADR allows the firm toswitch to SYD from DDB. This allows for the greatest amount of depreciation in the earlyyears. Switching from DDB to SYD will always be advantageous in the second year of the
Estimating Basic Project Characteristics 27
Table 3.3 Comparison of depreciation methods underADR system, with salvage
Year Straight line Double declining Sum-of-years
1 $13,000 $26,000 $23,6362 13,000 20,800 21,2733 13,000 16,640 18,9094 13,000 13,312 16,5455 13,000 10,650 14,1826 13,000 8,520 11,8187 13,000 6,816 9,4558 13,000 5,453 4,1829 13,000 4,362 0
10 3,000 3,490 0
$120,000 $116,043 $120,000
Table 3.4 Comparison of optimum depreciation to DDB and SYD,with salvage of $10,000 and cost of $130,000
Optimum Method Double-declining Sum-of-years’Year depreciation used balance digits
1 $26,000 DDB $26,000 $23,6362 20,800 SYD 20,800 21,2733 18,489 SYD 16,640 18,9094 16,178 SYD 13,312 16,5455 13,867 SYD 10,650 14,1826 11,556 SYD 8,520 11,8187 9,245 SYD 6,816 9,4558 3,865 SYD 5,453 4,1829 0 SYD 4,362 0
10 0 SYD 3,490 0
$120,000 $116,043 $120,000
project’s life. Applying this to Example 3.2 yields the following results: actual depreciationin the second year will be the same; SYD will yield greater depreciation in the subsequentyears.
ACRS Depreciation
As a part of the Economic Recovery Tax Act of 1981, new mandatory depreciation rules werepromulgated. The new rules were termed the accelerated cost recovery system (ACRS). Thepurpose in the new depreciation rules was to stimulate investment. Under ACRS depreciation,assets belong to one of several asset life classes. ACRS depreciation is based on the assumptionthat all assets are placed into service at the midpoint of their first year (the half-year assumption)regardless of when during the year they are acquired. For example, after the change in the lawthree-year class assets are depreciated over four years. The reason for this change is that thehalf-year convention was made to apply to the last year of service as well as the first by the1986 Act.
28 Capital Asset Investment: Strategy, Tactics & Tools
Table 3.5 Depreciation rates for ACRS property other than real propertya
Recovery 3-year 5-year 7-year 10-year 15-year 20-yearyear (200% DDB) (200% DDB) (200% DDB) (200% DB) (150% DDB) (150% DDB)
1 33.33 20.00 14.29 10.00 5.00 3.752 44.45 32.00 24.49 18.00 9.50 7.223 14.81 19.20 17.49 14.40 8.55 6.684 7.41 11.52b 12.49 11.52 7.70 6.185 11.52 8.93b 9.22 6.93 5.716 5.76 8.92 7.37 6.23 5.297 8.93 6.55b 5.90b 4.898 4.46 6.55 5.90 4.529 6.56 5.91 4.46b
10 6.55 5.90 4.4611 3.28 5.91 4.4612 5.90 4.4613 5.91 4.4614 5.90 4.4615 5.90 4.4616 2.95 4.4617 4.4618 4.4619 4.4620 4.4621 2.24
a Assumes the half-year convention applies. Accuracy to two decimal places only. Rates are percentages.b Switchover to straight-line depreciation at optimal time.
After 1985 the ACRS schedule was to have been based on DDB depreciation in the first yearof service, with a switch to SYD depreciation in the second year. However, since the law wasfirst enacted there have been several changes, and doubtless there will be more modificationsto the law. Despite the tendency to tinker with the law every year or two, an understanding ofthe fundamental depreciation methods will enable one to adapt that knowledge to subsequentchanges in the law.
The Tax Reform Act of 1986 changed the system to the Modified ACRS and established sixasset classes in place of the four that had existed. Except for real estate, all depreciable assetsfall within one of the six classes. The law also changed by requiring a switch from DDB tostraight-line in the year for which the straight line amount exceeds the DDB amount.
Table 3.5 contains the MACRS schedule for the 1998 tax year (Form 4562). If history isany guide, by the time you read this the schedule may have changed again, and possibly morethan once. Cost — salvage is always ignored — is multiplied by the percentages. For example,a three-year life asset costing $100,000 would have year 1 depreciation of $33,330.
Canadian Depreciation
Depreciation is generally called “capital consumption allowance” in Canada, or “capitalcost allowance.” Because the Income Tax Act was changed in 1949, only the declining-balance method has been generally allowed. This is not a DDB as discussed earlier, but adeclining balance based on assigned, fixed rates. The rate is applied to the undepreciated
Estimating Basic Project Characteristics 29
book balance of the asset, and the firm need not charge depreciation in years when it haslosses.
All assets acquired within a tax year qualify for a full year’s depreciation. There are 25asset classes, each assigned a fixed capital cost allowance rate. All assets of a class are pooledtogether. Total capital consumption is calculated by multiplying the book balance of each poolby its corresponding capital cost allowance rate and adding the products together. Capital gainsand losses result only when a given asset pool, not an individual asset, is sold. Capital costallowances due to a specific asset can remain in effect indefinitely if the given asset expireswithout salvage value, if there are other assets in the same asset pool, and if the firm generatesincome from which the capital cost allowance can be deducted.
Summary on Depreciation
1. Under the general guidelines, the only switch that can be made is from DDB to straight-linedepreciation.
2. Under the general guidelines, only the DDB method ignores salvage value.3. With the ADR system the depreciable life is chosen from an IRS guideline range for the
type of asset to be depreciated.4. In the ADR system, salvage value is ignored (as it is with DDB under the general guidelines)
in calculating depreciation with SYD and straight line as well as DDB.5. Under the ADR system, it is permitted to switch from DDB to SYD when this becomes
advantageous to the firm. The switch will be made in the second year in order to maximizethe early write-off of an asset.
6. Although salvage is ignored under ADR (and DDB with the general guidelines), the maxi-mum that may be depreciated is the difference between the asset cost and its salvage value.
It can be shown that under the ADR system, maximization of the tax deferral in the project’searly years will always be achieved by using DDB depreciation in the first year, and SYDdepreciation in the second and subsequent years. Because depreciation rules change from timeto time, it is wise to consult the current tax code.
Investment Tax Credit
From time to time the federal government has provided special tax credit on new asset purchasesin order to encourage aggregate investment in the economy. In recent years the credit has beenincreased from 7 to 10 percent (more in certain special cases) on new investment. The effecton cash flow is much the same as that of depreciation: it facilitates cash recovery during thefirst year of the asset’s life. Because the rules governing application of the investment taxcredit contain some complications, and may change from year to year, the current tax codeshould be checked when estimating the cash flows for an investment project that may qualifyfor the investment tax credit. At the time of this writing the investment tax credit has beenrepealed. However, history shows it is likely to be reinstated, especially when the economygoes into recession, creating a desire to stimulate investment. Therefore, it may be instructiveto examine previous investment tax credit rates. If the asset life is less than three years, nocredit may be claimed. The credit applies to one-third the asset cost if the asset has an economiclife of three years but less than five; and two-thirds if the asset cost is five years but less thanseven. The credit applies to assets described as “qualified investment” under Section 38 of the
30 Capital Asset Investment: Strategy, Tactics & Tools
Tax Code. It is equal to the amount allowed on new assets under Section 38 plus as much as$100,000 of the cost of newly acquired used assets qualified under Section 38. In no case maythe investment tax credit exceed the firm’s total tax liability for the year.
Unused portions of the investment tax credit are treated in the manner of capital losses:they may be carried back three years and forward five. The firm is restricted in applying thetax credit. If its tax liability is above $25,000, the credit claimed for the year may not exceed$25,000 plus 50 percent of the tax liability exceeding $25,000. If an asset is abandoned priorto the end of its estimated life, a portion of the tax credit claimed may have to be added tothe firm’s tax bill in the year. The amount will equal the difference between the credit actuallyclaimed and the amount that would have been used had the actual asset life been used originallyto calculate the credit.
Inflation
Over a span of time when price levels are fairly constant, depreciation rules and practicesmay be reasonably equitable in allowing the cost of the project to be recovered. A capitalinvestment costing, for example, $150,000 will provide recovery of $150,000, which may beused to purchase a successor project at the end of its useful life. However, if capital equipmentprices were to increase at 12 percent per annum, at the end of only 10 years it would cost$465,870 just to replace the worn machine with an identical new one. Existing depreciationrules do not take this into account, and therefore capital recovery is often inadequate to providefor replacement investment when required.
4
Cost of Capital
The cost of capital is a complex and still unsettled subject. It is discussed in finance texts indetail far beyond what we can devote to it in this text. In this chapter, intended primarily asa review, some of the more important considerations from the theory on cost of capital willbe discussed, and some operational principles illustrated. Cost of capital is treated further inChapter 21, within the context of the capital asset pricing model.
INTRODUCTION
Stated succinctly, the traditional view is that the firm’s cost of capital is the combined costof the debt and equity funds required for acquisition of fixed (that is, permanent) assets usedby the firm. Under this definition even such things as permanent, nonseasonal working cap-ital requirements are acquired with capital funds. Short-term financing with trade credit andbank lines of credit is generally excluded from cost of capital considerations. “Short term” isgenerally understood to be one year or less, in which such balance sheet items as accountspayable and line-of-credit financing are expected to be turned over at least once, or eliminated.Alternatively, the firm’s cost of capital is the rate of return it must earn on an investment sothat the value of the firm is neither reduced nor increased.
In terms of the firm’s balance sheet, cost of capital relates to the long-term liabilities, andcapital section to the firm’s capital structure. Although the specific account titles to be foundfor the various components of capital structure may differ, depending on the nature of thefirm’s business, the preferences of its accountants, and tradition within the industry, certaincommonalities exist. There will usually be long-term debt items in the form of bond issues out-standing or long-term loans from banks or insurance companies. There will always be equitysince a firm cannot be solely debt financed and there must be an ownership account. Equityfor corporations means common stock, retained earnings, and perhaps “surplus”; for propri-etorships and partnerships it may be just an undifferentiated “equity” account. For a varietyof reasons the corporate form of business organization is dominant. However, the principlesfor dealing with corporate organization can be applied straightforwardly to proprietorshipsand partnerships, and thus we will concentrate on the corporate form of organization. Eachcomponent item in the firm’s capital structure has its own specific cost associated with it.
COST OF CAPITAL COMPONENTS
Debt
An important characteristic of debt is that interest payments are tax deductible,1 whereasdividend payments are not; the latter are a distribution of after-tax profits. Thus the effective
1Although interest payments are tax deductible, principal repayments are not. This is a point often overlooked, although not aconceptually difficult one to understand.
32 Capital Asset Investment: Strategy, Tactics & Tools
after-tax cost of debt is (1 − τ ) times the pre-tax cost, where τ (tau) denotes the firm’s marginaltax rate.
If a firm borrows $1 million for 20 years at annual interest of 9 percent, its before-taxcost is $90,000 annually and, if the firm’s marginal income tax rate is 48 percent, the after-tax cost is $46,800, or 4.68 percent. The cost of debt is defined as the rate of return thatmust be earned on investments financed solely with debt,2 in order that returns available tothe owners be kept unchanged. In this example, investment of the $1 million would need togenerate 9 percent return pre-tax, or equivalently 4.68 percent after-tax, to leave the commonstockholders’ earnings unaffected.
For purposes of calculating the component cost of an item of debt, it is not important whetherthe particular debt component is a long-term loan from a bank or insurance company, whetherit is a mortgage bond or debenture, or whether it was sold to the investing public or privatelyplaced. There is one important exception, however — that of convertible debentures. Suchbonds are convertible at the option of the purchaser into shares of common stock in the firm.Because of this feature, they are hybrid securities, not strictly classifiable as either debt orequity. It is beyond the scope of this book to treat such issues, and the reader is referred tostandard managerial finance textbooks as a starting point in the analysis of those securities.
Preferred Stock
Preferred stock fills an intermediate position between debt and common stock. Ordinary pre-ferred stock has little to distinguish it from debt, except that preferred dividends, in contrast tointerest payments on debt, are not tax deductible. And the firm is under no more legally bindingobligation to pay preferred dividends than it is to pay dividends on common stock. However,preferred stock dividends must be paid before any dividends to common shareholders may bepaid, and unpaid preferred dividends are usually cumulative. This means that if they are notpaid in any period, they are carried forward (without interest) until paid.
The cost of preferred stock may be defined similarly to that of debt. It is the rate of returnthat investments financed solely with preferred stock must yield in order that returns availableto the owners (common stockholders) are kept unchanged. Since preferred issues generallyhave no stated maturity, they may be treated as perpetuities,3 as may securities issues withexceptionally long maturities.4 Therefore, the component cost of a preferred stock that pays adividend Dp and can be sold for a net price to the firm of Pp is given by
kp = Dp
Pp(4.1)
There are many variations of preferred stock, including callable issues, participating, voting,and convertible stocks. Convertible preferreds, like convertible bonds, present problems of clas-sification that are beyond the scope of this book. Principles established for treating convertiblebonds are also applicable to convertible preferreds.
2Note that this view ignores risk and the interactions between cost of capital components. In practice, the firm should evaluate potentialinvestments in terms of its overall cost of capital whether or not the actual financing will be carried out by debt, equity, or somecombination.3A perpetuity is a security that has a perpetual life, such as the British consols issued to finance the Napoleonic Wars and, in thetwentieth century, issues of the Canadian Pacific Railroad and the Canadian central government.4The noncallable 4 percent bonds issued by the West Shore Railroad in 1886 and not redeemable before the year 2361 (475-yearmaturity) could be considered perpetuities. Unfortunately for the bond owners, many firms do not have perpetual life, and thus thereis risk that the firm will fail and the bonds will be rendered worthless.
Cost of Capital 33
Common Stock and Retained Earnings
We will consider equity to exclude preferred stock and to include only common stock andretained earnings. In other words, we take equity to mean only the financial interest of theresidual owners of the firm’s assets: Those that have a claim (proportionate to the shares held)of assets remaining after claims of creditors and preferred shareholders are satisfied in theevent of liquidation.
The cost of equity capital has two basic components: (1) the cost of retained earnings and(2) the cost of new shares issued. In general terms, the cost of equity can be defined as the mini-mum rate of return that an entirely equity-financed investment must yield to keep unchanged thereturns available to the common stockholders, and thus the value of existing common shares.
There are two different but theoretically equivalent approaches to measuring the firm’s costof equity capital. The first is a model premised on the notion that the value of a share ofcommon stock is the present value of all expected cash dividends it will yield out to an infinitetime horizon. We shall refer to this as the dividend capitalization model, or Gordon model.(For detailed development of this and related models, see James C. T. Mao [98, Ch. 10].) It isderived under the assumption that the cash dividend is expected to grow at a constant rate gfrom period to period. The model, for retained earnings, is
kre = D
P+ g (4.2)
where D is the expected annual dividend for the forthcoming year, P the current price pershare, and g the annual growth rate in earnings per share.5
Unfortunately, estimation of the cost of equity capital is not simple and objective asequation (4.2) suggests. Dividends may be quite constant for some firms, but price per shareis subject to substantial volatility, even from day to day. And some firms do not pay dividends;Microsoft has never paid a dividend, at least up to the first quarter of 2002.6 This may requirethat one obtain an average, or normalized price. For corporations whose shares are not activelytraded there may be no recent market price quotation. Growth is affected not only by the in-dividual firm’s performance, but also by the condition of the economy. Therefore, estimationof the firm’s cost of equity capital is not simply employing a formula, but also a substantialamount of human judgment. More complex formulations than equation (4.2) have been de-vised, but still no means for bypassing the need for exercising judgment have been seriouslyproposed. The more complex formulas suffer from the same problems as equation (4.2).
The second approach to estimating the firm’s cost of equity capital is with what has come to beknown as the capital asset pricing model, or CAPM. Although the dividend capitalization modelcould be characterized as inductive, the CAPM might be better characterized as deductive. TheCAPM yields the following equation:
ke = RF + β(RM − RF) (4.3)
where RF is the rate of return on a risk-free security, usually meaning a short-term UnitedStates government security7 such as treasury bills, and RM is the rate of return on the marketportfolio — an efficient portfolio in the sense that a higher return cannot be obtained without
5The cost of new shares is found similarly, except that the share price must be adjusted for flotation costs. Thus, if we denote flotationcosts, as a proportion of the share price, by f , the cost of new equity is given by ke = (D/P(1− f )) + g.6Valuation and cost of capital for nondividend-paying firms presents problems not fully resolved. See for instance Mark Kamstra [78].7Even federal government bonds are not entirely risk free because there is risk of change in their value caused by change in the marketrate of interest. With short-term securities, however, such risk is slight, at least for magnitudes of market changes usually seen.
34 Capital Asset Investment: Strategy, Tactics & Tools
also accepting higher risk. The beta coefficient relates the returns on the firm’s stock to thereturns on the market portfolio. It is obtained by fitting a least-squares regression of thehistorical returns on the firm’s stock to the historical returns on the market portfolio; it isthe slope coefficient of the regression. β (beta) measures the risk of a company’s shares thatcannot be diversified away, and provides an index that indicates the responsiveness of returnson a particular firm’s shares to returns on the market portfolio. (In calculating betas, capitalappreciation is taken into account explicitly along with dividends in the returns calculations.)
The CAPM thus provides us with a means for estimating the firm’s cost of capital with marketdata and the beta coefficient, which relates the firm to the market. The dividend capitalizationmodel, in contrast, requires only the current market price of our firm’s common shares and itscash dividends. Because of this, the CAPM data requirements are greater. Furthermore, thestability of betas over time for individual firms is not assured.
Example 4.1 If we assume that we have obtained the beta for our firm, which is 1.80, thatthe risk-free interest rate is 6 percent and the return on market portfolio is 9 percent, we obtainas our estimate:
ke = 6% + 1.80(9% − 6%) = 11.4%
The CAPM deals with risk explicitly through the firm’s beta coefficient. The dividend cap-italization model, on the other hand, implicitly assumes risk is fully reflected in the marketprice of the firm’s shares.
OVERALL COST OF CAPITAL
The overall cost of capital is obtained by calculating an average of the individual components,weighted according to the proportion of each in the total. Suppose a firm has the followingcapital structure:
Debt (debentures maturing 2018) $30,000,000Preferred stock 20,000,000Common stock 15,000,000Retained earnings 35,000,000
—————–Total $100,000,000
Assume further that the after-tax costs of the components have been estimated as shown inTable 4.l. Calculation of the weighted average cost of capital is performed as shown.
Note that this average cost of capital is based on historical, balance sheet proportions, andon debt and preferred stock costs that were determined at time of issue of these securities.8
In capital investment project analysis, we are not concerned with average cost of capital. Weare interested in the marginal cost, for that is the cost of funds that will be raised to undertakeprospective capital investments. We cannot raise money at average historical cost, but at today’smarginal rate.
Now, what if the firm must raise an additional $10 million? If the capital structure is judgedto be optimal (more about optimal structure later), funds should be raised in proportion to the
8Once a bond or preferred stock issue is sold, the firm is committed to paying a fixed, periodic return per security. Even though capitalmarket conditions may subsequently dictate higher or lower yields for similar securities of comparable risk if they are to be sold now,the interest and dividend payments on such securities sold in the past do not change.
Cost of Capital 35
Table 4.1 Calculation of weighted average cost of capital (WACC)
(1) (2) (3)$ Amount Proportion % (4)(millions) of total Cost (2) × (3)
Debt 30 0.30 4.16 1.25%Preferred 20 0.20 9.00 1.80Common 15 0.15 15.00 2.25Retained earnings 35 0.35 14.00 4.90
WACC 10.20%
Table 4.2 Calculation of marginal cost of capital (MCC)
(1) (2) (3)$ Amount Proportion % (4)(millions) of total Cost (2) × (3)
Debt 3 0.30 4.68 1.40%Preferred 2 0.20 12.00 2.40Common 1.5 0.15 17.00 2.55Retained earnings 3.5 0.35 15.00 5.25
MCC 11.60%
existing capital structure. If its profits are adequate, the firm may utilize retained earnings ratherthan float new common shares, providing that dividend policy will not be seriously affected.Assume that the $10 million will be raised in amounts and at costs illustrated in Table 4.2.
This 11.60 percent marginal cost of capital suggests that market conditions have changed,and that investors require higher yields than formerly. The result of raising additional fundsat a marginal cost higher than the average historical cost will be to raise the new average costfigure somewhat. The new average cost is given by
10.33% = (100 × 10.20% + 10 × 11.60%)/110
One may be tempted to ask why the entire $10 million should not be raised with debt, thus atthe lowest attainable marginal cost. This question arises naturally, but ignores the interrelatednature of financing decisions. Investors and creditors have notions about the proper mix of debtand equity for firms. Therefore, although today the firm might raise the entire $10 million withdebt, at a later date it could find it has no reserve borrowing capacity, and also cannot borrowon favorable terms or at acceptable cost. In such a situation the firm may find that to raise fundsit must float a new issue of common stock at a time when required yields are much higherthan normal, in a depressed stock market. If that were to occur, it would be a disservice to thecurrent stockholders. The cost of any single capital component by itself cannot be consideredthe true cost of capital for yet another reason. The cost of capital that is associated with anyparticular component applies to that component as a part of the whole firm, within the contextof the firm. Bond purchasers are not merely buying bonds, they are buying bonds of a firmwith a balanced financial structure. Because the overall results of the firm are what mattersto the suppliers of funds, and because funds are not segregated by their origin, it would beinappropriate to use the cost of a component as a substitute for the overall cost.
36 Capital Asset Investment: Strategy, Tactics & Tools
Analysts have observed that stock and bond prices often move in opposite directions. There-fore, the financial manager has some flexibility in establishing the appropriate financing mixover the short run. Indeed, it is expected that financial management will use its best judgmentin such matters. Sometimes it is better to raise funds with bonds, at other times with commonstock. However, future price and yield trends must be anticipated; investors and creditors willnot willingly tolerate marked deviation in capital structure from established norms over longperiods. And the indenture agreements of prior bond issues or loans may well restrict the firm’slatitude in using more leverage.
OPTIMAL CAPITAL STRUCTURE
In the early 1960s considerable controversy erupted over the theory proposed by Modiglianiand Miller. They contended that the firm’s cost of capital is invariant with respect to its capitalstructure [108], depending only on the risk class to which the firm belongs. The original M andM theory did not take taxes into account, particularly the tax deductibility of interest paymentson debt.9 Subsequent modification of the M and M theory to include tax effects weakenedtheir original conclusions. Since business income taxes are reality in most nations, and interestpayments for businesses a tax-deductible expense, most authorities today agree that there isan optimal capital structure or range of optimal structures for any particular firm. The theoryof M and M is elegantly developed and its repercussions are still affecting financial economicstheory. We shall examine the implications of optimal capital structure now, rather than ventureany further into the M and M arguments that hold only in a world true to their restrictiveassumptions.
Existence of an optimal capital structure for any given firm suggests that financial manage-ment should aim to obtain the optimal, or at least to approach it. Examination of the basicvaluation model introduced early in this book reveals why this is desirable. Attainment of thelowest overall cost of capital will do proportionately more to increase the value of the firmthan will an increase in the net cash flows, because of the compounding of terms containing k,the cost of capital.
We must recognize, however, that world conditions do not remain stationary. The quest foroptimal capital structure requires that one follow a moving target, adjusting and readjustingsights as capital market conditions and investor and creditor attitudes change. Optimal capitalstructure, in practice, is not a once-and-for-all-time achievement. Rather, it requires periodicreview and adjustment. Within a range of leverage, the firm may at times choose to financemore heavily with debt than with equity. Such a process, however, cannot continue indefinitelyor the firm’s leverage will become excessive, and with this the risk of insolvency, so that the costof debt increases and onerous indenture conditions are imposed, not to mention the financialleverage effect on variability in common stockholder returns.
At other times the firm may finance more heavily with equity, or employ preferred shares.An analogy may be drawn to a driver who must use accelerator and brake to adjust to a speedthat is optimal for road conditions — a speed that is safe and yet gets the driver to a destinationin minimum time or with minimum fuel consumption. The driver may use the brakes severaltimes in sequence before using the accelerator, or the accelerator for some time without braking.
9Modigliani and Miller also assumed perfect capital markets, with investors able to borrow and lend at the same interest rate and zerotransactions costs. This created arbitrage opportunities not found in the real world.
Cost of Capital 37
After–Tax %Cost of Capital
20
18
16
14
12
10
8
6
4
2
0 10 20 30 40 50 60Leverage(%:Debt /Assets)
ke
kd
k
Figure 4.1 Cost of capital schedule for a firm
(Unlike debt and equity, however, we may surmise that brake and accelerator will never beused simultaneously.)
The following example assumes constant capital market conditions and investor attitudesfor purposes of illustration. Figure 4.l illustrates the cost of capital schedule for a hypotheticalfirm. For simplicity, it is assumed that debt and equity may be raised in arbitrarily smallincrements, although this abstracts from real world considerations that militate against smallissues of either debt or equity.
Note that the cost of equity capital, ke, rises continuously as leverage increases. This reflectsthe increasing risk to the common shareholders as financial leverage increases: shareholdersrequire a greater return as variability in earnings allocated to them increases. The cost of debt,in contrast, begins at 4.68 percent pre-tax × (1 – 0.48 percent marginal tax rate)] and does notrise until leverage goes beyond 20 percent. Beyond this amount creditors become increasinglysensitive to the risk of firm insolvency as earnings become less and less a multiple of theinterest that must be paid.
Lowest overall cost of capital is reached at 40 percent leverage, even though at this leveragethe component costs of equity and debt are not at their lowest levels. Beyond 40 percentleverage the overall cost of capital rises for additional leverage at a faster rate than it declinedprior to the optimum, thereby reflecting the rapidly rising debt and equity schedules.
If one could find an industry in which the firms were similar except for leverage, a schedulelike that in Figure 4.l could be produced. Each set of observations for a given leverage wouldcorrespond to one firm, or the average of several if multiple observations at the same leveragewere obtained. However, it is difficult to find an industry in which the firms are truly similar,because most firms today are in varying degree diversified in their operations, and are normallyclassified on the basis of their major activity. Furthermore, size disparity between firms would
38 Capital Asset Investment: Strategy, Tactics & Tools
present problems to the extent that there may be differences in access to capital markets,industry dominance, product brand differentiation, and so on.
INTERACTION OF FINANCING AND INVESTMENT
The foregoing discussion of the firm’s cost of capital assumed that cost of capital was inde-pendent of the firm’s capital investment decisions. In practice this view may be unrealistic.
If the firm consistently follows the practice of investing in capital projects that yield a returnequal to or greater than its existing cost of capital, and do not affect the riskiness of overallreturns to the firm on its investments, then we may assume independence of cost of capital andinvestment. However, if the firm adopts a policy of adopting investments that alter its overallprofitability or variability in earnings, we must recognize that there are likely to be resultantchanges in the cost of capital components and thus the firm’s overall cost of capital.
Changes in the firm’s cost of capital through pursuit of investment policy that alters thecharacteristics of risk (that is, variability in return and probability of insolvency) are notnecessarily adverse. The firm may reduce its cost of capital by reducing risk.10 On the otherhand, the firm may increase its cost of capital if it consistently adopts investments that, althoughoffering high expected returns, at the same time contain commensurately high risk. This willnot necessarily be so, but discussion of this point is deferred to Chapter 22, after portfolioeffects have been considered. Depending upon the correlation between project returns on newinvestments and the existing capital assets of the firm, risk may actually be reduced overall byadopting a very risky new investment.
A simple example with an intuitive interpretation will serve to illustrate this point for now.Assume that the firm undertakes a very risky capital investment project, but has returns that areexpected to be highly negatively correlated with the firm’s existing assets. Acceptance of theproject will reduce the overall riskiness of the firm. If, in a particular year, the existing assetsprovide high cash returns, the new investment will provide low returns. However, if the cashreturns on existing assets were low, then the new investment would provide high returns. Theoverall result will be to smooth out variability in earnings and thus reduce risk.
CAUTIONARY NOTE
Estimation of the cost of capital is fraught with practical problems that militate against assigningsuch work to employees who are not aware of them. The dividend capitalization approach, forinstance, requires that the firm pay dividends and that there be a market-determined price and asense of the market’s expectation for future dividend growth. Other approaches are developing,but present other problems requiring assumptions that may not adequately reflect reality. Noris the capital asset pricing model (CAPM) approach free of practical difficulties. In order touse the CAPM to estimate the cost of equity capital, one must have both market return data andreturn data for the firm and the risk-free rate. For a firm whose shares are not actively tradedin the market it may be a stretch to find a sensible measure of returns. And then there remainquestions of which market index to use, etc. For instance, should a large-capitalization indexlike the Standard & Poor’s 500 be used when calculating cost of capital for a small or mediumsize firm, or an index of small and medium-size firms?
10Some portfolio-approach advocates would argue that, since investors can diversify their portfolios to reduce risk, it is unnecessary,and perhaps even detrimental to stockholder interests, for the firm to diversify its investments.
5Traditional Methods that Ignore
Time-value of Money
This chapter discusses some methods that are often encountered in practice which do not takethe time-value of money into account. In most circumstances these methods should be replacedby better ones that integrate the time-value of money into their rationale.
PAYBACK AND NAIVE RATE OF RETURN
Payback
The payback period criterion has consistently been demonstrated to be the single most popularmeasure of project merit used in practice. Possibly, the payback criterion is the oldest ofcapital-budgeting measures as well.
The payback method is extremely simple to employ and intuitively appealing. To apply themethod to a project costing amount C with uniform cash flows of amount R each period, oneneed only take the ratio of C to R:
Payback ≡ C
R(5.1)
when R is uniform each period. In cases in which cash flows are not expected to be uniform,the method is somewhat more complicated:
Payback ≡ P +C −
P∑t=1
Rt
Rt+1(5.2)
where
C −P∑
t=1
Rt > 0 and C −P+1∑t=1
Rt ≤ 0
Two illustrations will serve to make the procedure clear. We consider first a project designatedas project A (Figure 5.1). This investment requires an outlay today (at t = 0) of $5000, and willyield uniform net cash flows of $2500 each year over its economic, or useful life, of 10 years.
Applying formula (5.1), since the cash flows are uniform, we obtain a payback for projectA of
P(A) = C
R= 5000
2500= 2.0 years
The parenthetical A with P serves to distinguish this from the payback on other projects.Next let us consider project B (Figure 5.2), which, although requiring the same initial outlay
in year t = 0 of $5000, has a nonuniform net cash flow sequence. Cash flows for B in years t = 1through t = 4 are $500, $1000, $2000, and $4000, respectively. In years t = 5 through t = 10,
40 Capital Asset Investment: Strategy, Tactics & Tools
0 1 2 3 4 5 6 7 8 9 10Year
4
2
0
−2
−4
−6
$ T
hous
and
Figure 5.1 Project A
0 1 2 3 4 5 6 7 8 9 10Year
4
6
8
2
0
−2
−4
−6
$ T
hous
and
Figure 5.2 Project B
the cash flows are uniformly $8000. Since project B has nonuniform cash flows, formula (5.1)is not applicable. We must instead use (5.2). To do this we proceed as follows, taking theabsolute value of C, and successively subtracting the net cash flows:
C = $5000−R(1) = 500
4500 P = 1−R(2) = 1000
3500 P = 2−R(3) = 2000
1500 P = 3
The remaining $1500 is less than the cash flow in year 4. It represents the unrecovered portion ofthe initial outlay. The question now is how far into year 4 we must go to recover this remainingamount. Implicit in the payback method is the assumption that cash flows are uniform overa particular period even if nonuniform from period to period. Therefore, we must go intoyear 4:
$1500
R(4)= $1500
$4000= 3
8or 0.375
Combining results, we obtain the payback period for project B of:
P(B) = P + 0.375 = 3.375
Traditional Methods that Ignore Time-value of Money 41
This procedure for calculating payback is in the form of an algorithm. An algorithm is asystematic, multiple step procedure for obtaining a solution to a problem. Other, more compli-cated algorithms will be introduced in later chapters. Many may be converted into computerprograms in order to reduce human effort and with it the chance of calculation errors.
The Naive Rate of Return
When a manager who relies on the payback criterion speaks of rate of return, this normallyrefers to something other than time-adjusted return on investment. For example, a project thatpromises a two-year payback will be said to offer a 50 percent per year rate of return. This werefer to as naive rate of return (NROR), since it ignores the effects of cash flows beyond thepayback period as well as the effects of compounding from period to period.
NROR ≡ 1/Payback (in years) (5.3)
The criticisms of payback are thus equally applicable to naive rate of return.
Strong Points of Payback
1. It is easily understood.2. It favors projects that offer large immediate cash flows.3. It offers a means of coping with risk due to increasing unreliability of forecasted cash flows
as the time horizon increases.4. It provides a powerful tool for capital rationing when the organization has a critical need to
do so.5. Because it is so simple to understand, it provides a means for decentralizing capital-
budgeting decisions by having non-specialists screen proposals at lower levels in the orga-nization.
Weak Points of Payback
1. It ignores all cash flows beyond the payback period.2. It ignores the time-value of funds.3. It does not distinguish between projects of different size in terms of investment required.4. It can be made shorter by postponing replacement of worn and deteriorating capital until a
later period.5. It emphasizes short-run profitability to the exclusion of long-run profitability.
To illustrate, let us consider along with projects A and B another project, C (Figure 5.3). Thisproject also requires an investment of $5000, and yields net cash flows of $5000, $1000, and$500 in years 1, 2, and 3, respectively. It offers no cash returns beyond year 3. Payback forproject C is easily seen to be 1.0 year.
If we rank the three projects now in terms of payback, we obtain:
Project Payback RankingC 1.0 1A 2.0 2B 3.375 3
42 Capital Asset Investment: Strategy, Tactics & Tools
0 1 2 3 4 5 6 7 8 9 10Year
4
6
2
0
−2
−4
−6
$ T
hous
and
Figure 5.3 Project C
If we assume that A, B, and C are mutually exclusive, the reliance on payback alone impliesthat project C will be selected as the best project because it has the shortest payback of thethree. In the absence of project C, A would be selected as preferable to B. But which projectwould management prefer to have at the end of four years? Clearly, project B is preferable atthat point, if there is any reliability in the estimates of project characteristics, and if we canassume reasonable certainty of these estimates.
Let us next consider the effects of a large negative salvage value for project C in year 4. Hasthe payback measure for this project been changed? No, it has not, because cash flows beyondpayback are ignored.
Terborgh is [151, p. 207] has provided perhaps the most clever and effective statement ofthe way payback favors delaying replacement of an already deteriorated capital project untilit has deteriorated still further. Payback treats relief from losses caused by undue delay inreplacements as return on the new investment. Consider Terborgh’s analogy:
A corporation has a president 70 years of age who in the judgment of the directors can be retired andreplaced at a net annual advantage to the company of $10,000. Someone points out, however, that if heis kept to age 75, and if he suffers in the interval the increasing decrepitude normally to be expected,the gain from replacing him at that time will be $50,000 a year, while it should be substantially higherstill, say $100,000 at the age of 80. It is urged, therefore, that his retirement should be deferred. Thegenius advancing this proposal is recognized at once as a candidate for the booby hatch, yet it is notdifferent in principle from the rate-of-return requirement.
No one can deny that the advantage of $100,000 a year (if such it is) from retiring the presidentat 80 is a real advantage, given the situation then prevailing. The question is whether this situationshould be deliberately created for the sake of reaping this gain. Similarly, the question is whethera machine should be retained beyond its proper service life in order to get a larger benefit from itsreplacement. The answer in both cases is obvious. The executive who knowingly and wilfully followsthis practice should sleep on a spike bed to enjoy the relief of getting up in the morning.
Considering the strong and weak points of payback all together, is there any merit on balancein using this method? Absolutely not, unless payback is not the sole criterion employed. As asingle criterion, payback may be worse than useless because its implicit assumptions disregardimportant information about events beyond payback. As a tie-breaker to supplement othermethods, payback has considerable merit if we assume that, other things equal, rapid invest-ment recovery in the early years of an investment’s life is preferable to the later years. This viewis reinforced if one considers that the further from the present we attempt to estimate anything,the less reliable our estimates become. In other words, once we introduce risk into our consid-erations, we will ceteris paribus prefer early return of our investment to later recovery of it.
Traditional Methods that Ignore Time-value of Money 43
Unrecovered Investment
A concept related to payback, but taking the time-value of money into account, is unrecoveredinvestment [35, pp. 200–219]. Again taking C to be the cost or investment committed (at t = 0)to a capital project, and taking rate k as the firm’s per period opportunity cost of the funds tiedup in the project, unrecovered investment is defined by
U (k, t) = C(1 + k)t −t∑
t=1
Ri (1 + k)t−i (5.4)
Note that if we set k = 0 for values of U ≥ 0, and k = ∞ for U < 0, we find that U = 0defines payback as in the previous section.
If for all periods we take k as the firm’s cost of capital, then it follows that the value of t forwhich U = 0 provides us with a time-adjusted payback period. The value of t for which thisis true will not necessarily be an integer. The t value thus obtained suggests how long it willtake for the firm to recover its investment in the project plus the cost of the funds committedto the project. If k is in fact a precise measure of the enterprise’s opportunity cost, then the tfor which U (k, t) = 0 corresponds to the time at which the firm will be no worse off than if ithad never undertaken the project. Since we assume cash flows occur at the end of each period,this relationship will be approximate but nevertheless useful.
The subject of unrecovered investment is treated further in Chapter 14. The concept of time-adjusted payback is used in the method of analysis discussed in the appendix to Chapter 11.
ACCOUNTING METHOD: ALIAS AVERAGE RETURNON AVERAGE INVESTMENT
The accounting method [116] is based on calculation of an average net cash return on theaverage investment in a project. It is not as popular a method of capital-budgeting projectanalysis as payback, but it is nonetheless still sometimes encountered today. It is easy to apply:it requires only that one:
1. Calculate average accounting profit by(1/N )∑N
t=1 At where A is accounting profit in timet . (Note that this is at odds with our stated use of only net, after-tax cash flows for capitalinvestment analysis.)
2. Calculate average investment by (C + S)/2, where S is estimated salvage value.3. Divide average return by average investment, and express as a percentage.
Thus the average return on average investment (ARAI) is defined as:
ARAI ≡
N∑t=1
At
N
(C + S)/2=
N∑t=1
Rt
N
(C + S)/2(5.5)
Note that if A were uniform, with straight-line depreciation and S = 0, ARAI would be similarto the naive rate of return (NROR). This is because the At terms include depreciation, whereasthe Rt terms include the cash flow dollars shielded from taxes by depreciation.
Strong Points of Accounting Method
1. It is easily understood.2. It does not ignore any periods in the project life.
44 Capital Asset Investment: Strategy, Tactics & Tools
3. It is, in a sense, more conservative than payback and naive rate of return.4. It explicitly recognizes salvage value.5. Because it is easy to understand, it (like payback) may provide a means for decentralizing
the process of preliminary screening of proposals.
Weak Points of Accounting Method
1. Like payback, it ignores the time-value of funds.2. It assumes that capital recovery is linear over time.3. It does not distinguish between projects of different sizes in terms of the investment required.4. It conveys the impression of greater precision than payback since it requires more calculation
effort, while suffering from faults as serious as that method.5. It does not favor early returns over later returns.6. The method violates the criterion that we consider only net cash flows. We can only pay
out dividends in cash and reinvest cash; not book profits.
COMPREHENSIVE EXAMPLE
Consider the following two projects. They have net after-tax cash flows and depreciationcharges as shown. It is assumed that the class life asset depreciation range system is used, witha switch from DDB to SYD method in year two. Project D has zero salvage, E has salvage of$20,000. The depreciable lifetime for D is five years, for E it is eight years. Note that projectD’s economic life is a year longer than the depreciation life used.
Project D Project E
Initial outlay $100,000 $100,000Cash flow/depreciation for year:1 $25,000/$40,000 $40,000/$25,0002 35,000/24,000 30,000/18,7503 40,000/18,000 20,000/16,0714 40,000/12,000 20,000/13,3935 40,000/6,000 20,000/6,7866 40,000/0 20,000/07 0 20,000/08 0 20,000/0
Payback periods are calculated as follows:
Project D
$100,000−25,000
75,000 P = 1−35,000
40,000 P = 2−40,000
0 P = 3
Traditional Methods that Ignore Time-value of Money 45
The payback for project D is exactly 3 years.
Project E
$100,000−40,000
60,000 P = 1−30,000
30,000 P = 2−20,000
10,000 P = 3
and 10,000/20,000 = 1/2, so the payback for project E is 31/2 years.We calculate ARAI on the basis of accounting profits, not cash flows. If we assume that the
timing of accounting profits and cash flows is approximately the same, the accounting profitin any year t , At , will be equal to the net, after-tax cash flow, Rt , less the depreciation chargedin that year: At = Rt − Dt .
Then ARAI for the investment projects is:
Project D
1/6 ($25,000 − $40,000 +35,000 − 24,000 +40,000 − 18,000 +40,000 − 12,000 +
40,000 − 6,000 +40,000 − 0)÷
1/2($100,000 + 0)= $20,000/$50,000= 40.00 percent
Project E
1/8 ($40,000 − $25,000 +30,000 − 18,750 +20,000 − 16,071 +20,000 − 13,393 +
20,000 − 6,786 +20,000 − 0 +20,000 − 0 +
20,000 − 0)÷
1/2($100,000 + 20,000 )= $13,750/$60,000= 22.92%
46 Capital Asset Investment: Strategy, Tactics & Tools
If we assume the firm has a 10 percent annual cost of capital, the unrecovered investment atthe end of year 3 for each project is:
Project D
$100,000 (1.10)3 −$25,000 (1.10)2
−35,000 (1.10)1
−40,000 (1.10)0
= $133,100 − $108,750= $24,350
Project E
$100,000 (1.10)3 −$40,000 (1.10)2
−30,000 (1.10)1
−20,000 (1.10)0
= $133,100 −$101,400= $31,700
6Traditional Methods that Recognize
Time-value of Money: the Net Present Value
The internal rate of return considered in the next chapter involves finding a unique real rootto a polynomial equation with real coefficients. Prior to modern calculators and computers,that required tedious calculations, particularly for projects with unequal cash flows over manyperiods. And the internal rate of return is fraught with danger for the unaware and careless. Thenet present value (NPV) is considered in the present chapter. The NPV calculation requires themuch simpler task of evaluating that polynomial equation with real coefficients for a given dis-count rate. That rate usually will not be a root to the polynomial. NPV is defined in the equation:
NPV ≡ C +N∑
t=1
Rt
(1 + k)t orN∑
t=0
Rt
(1 + k)t (6.1)
where C is the installed cost, k the enterprise’s cost of capital, Rt the net, after-tax cash flow atthe end of time t, and N the years in the project’s economic life. We can denote C = R0 as theright-hand side of equation (6.1) shows. Thus we have the compact form:
NPV ≡N∑
t=0
Rt (1 + k)t (6.2)
This equation appears to be identical to the basic valuation model considered in Chapter 2.However, there is a subtle difference between them that the notation does not reflect. In thebasic valuation model the Rt are the total or aggregated net, after-tax cash flows in each period.In equation (6.1), on the other hand, the Rt are the net, after-tax cash flows of the particular,individual project under analysis. The same symbols are used in them, but the meaning is thussomewhat different.
The NPV of a particular project provides a measure that is compatible with the valuationmodel of the firm. This much is perhaps obvious because of the mathematical form of theequations. Since we take as given the overarching goal of maximization of the enterprise’s value,we must recognize that acceptance of individual projects with a positive NPV will contributeto the increase of that value. In the absence of capital rationing, in other words, if there is noshortage of money to accept all projects with positive NPVs, the enterprise should do so.
In reference to the calculation of NPV, the conventional approach has been to calculate firstthe gross present value, which is the present value of all the net, after-tax cash inflows, andthen subtract the initial outlay that is assumed to be at present value since it is incurred at t = 0.For projects requiring net outlays beyond the initial period, the outlays are brought to presentvalue in the same manner as the cash receipts.
In the calculation of NPV we take the discount rate, k, as given. The rate used is generally1
the organization’s cost of capital; more particularly, it is the marginal cost of capital. If an
1In the case of risky projects, which we are not considering in this section of the text, a “risk-adjusted discount rate” or “hurdle rate”is sometimes used as an alternative to other methods for dealing with risk.
48 Capital Asset Investment: Strategy, Tactics & Tools
Table 6.1 The NPV calculations for project A
Year Present worth factor × Cash flow = Present value
1 0.86957 $2,500 $2,173.922 0.75614 2,500 1,890.353 0.65752 2,500 1,643.804 0.57175 2,500 1,429.385 0.49718 2,500 1,242.956 0.43233 2,500 1,080.827 0.37594 2,500 939.858 0.32690 2,500 817.259 0.28426 2,500 710.65
10 0.24718 2,500 617.955.01877 $12,546.92
investment yields a discounted return greater than its discounted cost, it will have NPV > 0.Conversely, if the discounted cost exceeds the discounted expected returns, it will have NPV <
0. Therefore the rule for project adoption under the NPV criterion is
If NPV ≤ 0, reject
> 0, accept
The discounting process employed simply allows cash flows to be judged after they have beenadjusted for the time-value of money. The time reference we use is immaterial: we could justas easily have used net future value by adjusting the cash flows to their compounded (ratherthan discounted) value at t = N rather than at t = 0. However, it is a traditional convention thatNPV rather than net future value be used. And it should be noted that NPV is unambiguouswhile, in contrast, there exists an unlimited spectrum of future dates at which future value canbe calculated.
The NPV method can be further clarified by means of an example. Let us assume that k,the discount rate, is 15 percent per annum and find the NPV for project A, considered in theprevious chapter. This project requires an initial outlay of $5000 and returns $2500 each yearover a 10-year useful life, net after taxes. Table 6.1 illustrates the NPV calculations. Noticethat because the cash flows for this project are a uniform $2500, we could have summed thepresent worth factors and then multiplied the sum once by $2500 to get the NPV. However,this summation has already been done: the results are listed in Appendix Table A.4 for annuitypresent worth factors.2
Now, to obtain net present value, we need to remove only the $5000 project cost that, sinceit occurs at t = 0, is already at present value. (That is, the present value factor for t = 0 is1.00000.) Therefore, the NPV of project A at k = 15 percent is
$7546.92 = $12,546.92 − $5000
NPV > 0, so the project is acceptable for investment.
2Alternatively, if we have a modern calculator, we can find the present worth of annuity factor for N periods and rate k from therelationship
aN k
= 1 − (1 + k)−N
k
The Net Present Value 49
Table 6.2 The NPV calculations for project B
Year Present worth factor ×Cash flow = Present value
1 0.86957 $ 500 $ 434.782 0.75614 1,000 756.143 0.65752 2,000 1,315.044 0.57175 4,000 2,287.00
−2.85498 $4,792.965 0.49718 8,000 $3,977.446 0.43233 8,000 3,458.647 0.37594 8,000 3,007.528 0.32690 8,000 2,615.209 0.28426 8,000 2,274.08
10 0.24718 8,000 1,977.445.018772.16379 × 8,000 = 17,310.32
Gross PV $22,103.28Less C at t = 0 −5,000.00
NPV = $17,103.28
Next, let us find the NPV for project B (also considered in the previous chapter). Thecalculations are shown in Table 6.2. Note that since cash flow in years 5–10 is a uniform $8000,the present value of these flows can be obtained by multiplying by the present worth of annuityfactor for 10 years less the present worth of annuity factor for four years: $8000(5.01877 −2.85498).
If we were considering A and B as mutually exclusive projects, B would clearly be preferredbecause its NPV of $17,103.28 is more than 21/2 times larger, and the required $5000 investmentis the same for both projects.
The internal rate of return (IRR), which the next chapter covers, is a special case of NPV.The IRR is that particular discount rate for which the NPV is equal to zero. In other words, theIRR is that rate of discount for which the present value of net cash inflows equals the presentvalue of net cash outflows.
Figure 6.1 illustrates the NPV functions for projects A and B for various discount rates.Negative rates are for purposes of illustration only, not because of any economically meaningfulinterpretation of such rates as cost of capital. Note that the IRR of each project is at that rate ofdiscount where the NPV of that project intersects the horizontal axis, and that NPV = 0 at thesepoints. It can be shown that for all investments with R0 = C < 0, and Rt ≥ 0 for all 0 < t ≤ N ,that the NPV function is concave from above. For projects having some Rt ≥ 0 for 0 < t ≤ Nthis will not necessarily be true.
Let us consider an investment project that has some negative cash flows after t = 0. We willcall this project AA. The cash flows over a four-year economic life are:
R0 R1 R2 R3 R4
−1000 1200 600 300 −1000
This project has two “IRRs”: −8.14 percent and +42.27 percent. Quotation marks are placed
50 Capital Asset Investment: Strategy, Tactics & Tools
−5−10
010 20 30 40 50 60
%
IRRA
IRRB
10
20
30
40
50
B
A
NPV ($000)
Figure 6.1 NPV functions for projects A and B
–10 –5 5 10 15 20 25 30 35 555040 60%
–8.14%15
30
45
60
105
120
135
$NPV
42.27%
0
Figure 6.2 NPV curve for project AA
on the IRR because, as we shall see in Chapter 10, neither rate is internal to the project andneither rate measures the return on investment for this project. Figure 6.2 contains the graphof the NPV function for this project. Note that NPV is positive until the discount rate reaches42.27 percent, but that the NPV is concave from below up to that point.
We shall consider such projects in more detail in Chapter 10. For now it will be useful tomake a mental note that they are neither purely investments nor purely financing projects (loansto the enterprise) but a mixture of these. Because many interesting projects are of this nature,
The Net Present Value 51
it is important that we have means for their proper analysis, although we shall defer this untilChapter 10.
UNEQUAL PROJECT SIZE
A common difficulty that arises with NPV is that marginally valuable projects may show ahigher NPV than more desirable projects simply because they are larger. For example, considerprojects D and E:
Project D Project E
Cost = R0 $100,000 $10,000Cash flows, R1 through R10 30,000 10,000R11. . . . . . . . . . . . 0 0
Again using k = 15 percent, we see that the NPV for project D is $50,570, whereas that forproject E is only $40,190. Therefore, D would seem preferable to E. However, D costs 10times as much as E. The extra NPV is only 11.5 percent of the additional $90,000 requiredinvestment — an amount less than the 15 percent cost of capital.
This problem is a serious one, especially when investment funds are limited and, as aconsequence, there is capital rationing within the organization. Fortunately, this problem iseasily corrected.
The Profitability Index
The problem of unequal project size with the NPV is easily corrected by using what is calledthe profitability index (PI). The PI may be defined in two ways; these are identical except fora constant of 1.0. The more common way is by the ratio of gross present value to project cost.Under this definition an acceptable project, one with NPV < 0, will have a PI > 1.0.
This author’s preference is to define PI as the ratio of net present value to present value ofproject cost:
PI ≡ NPV
Cor
NPV
R0(6.3)
Thus, if NPV > 0, PI > 0. On the other hand, if NPV ≤ 0, PI < 0. It is immaterial whichdefinition is used, provided that it is used consistently. They are identical except that definition(6.3) yields a PI that is 1.0 less than that obtained under the alternative definition.
Let us look again at projects D and E in terms of their PI:
PI: D = 0.5057 E = 4.019
The results obtained strongly favor project E because the return on the investment is muchhigher proportionately than that on project D.
UNEQUAL PROJECT LIVES
Another problem can exist with NPV and the corresponding PI. This is due to the effect ofunequal project lives. For example, let us assume that a given manufacturing operation is
52 Capital Asset Investment: Strategy, Tactics & Tools
expected to be required for an indefinite time, and that two mutually exclusive projects, F andG, both have acceptable NPVs and IRRs. However, project F is expected to last for 10 yearswhile project G is expected to last for only five, after which it will have to be replaced. Canwe determine which should be accepted on the basis of PI as defined earlier? No. We musttake into account the effect of the difference in project lives. As a focus for defining how thisshould be done, let us consider the specific projects F and G.
Net cash flows
Year Project F Project G
0 $100,000 $100,0001 30,000 40,0002 30,000 40,0003 30,000 40,0004 30,000 40,0005 30,000 40,0006 30,000 07 30,000 08 30,000 09 30,000 0
10 30,000 0
Letting k = 15 percent, we obtain for the projects:
F G
NPV $50,563 $34,086PI 0.506 0.341
Since project G will have to be replaced at the end of year 5, we need to take this into accountsystematically. (We will assume it can be replaced for $100,000 at that time, ignoring the un-certain effects of inflation.) One way is to assume that if it is adopted, project G will be replacedby an identical project at the end of year 5 and then treat the cash flows over the entire 10-yearperiod explicitly. In other words, we calculate NPV and PI for project G that now has cash flows:
Net cash flows
FirstTime Project G Replacement Combined
0 $−100,000 $−100,0001 40,000 40,0002 40,000 40,0003 40,000 40,0004 40,000 40,0005 40,000 $−100,000 −60,0006 40,000 40,0007 40,000 40,0008 40,000 40,0009 40,000 40,000
10 40,000 40,000
The Net Present Value 53
The values obtained now for project G are:
NPV = $51,033
PI = 0.510
so that, contrary to the unadjusted results previously obtained, project G is preferable to projectF. The method just illustrated for adjusting for unequal lives can be tedious to apply, sinceit requires us to calculate NPV over the least common denominator number of years of theproject lives. In this example project F lasts exactly twice as long as project G, so this presentsno problem. However, what if we had one project lasting 11 years and another lasting 13 years(both prime numbers)? In this case we would have to evaluate 143 cash flows for each project!Fortunately, there is a better way.
Level Annuities
A much easier equivalent method is to calculate the time-adjusted annual average (that is, thelevel annuity) for each project. This is done by multiplying the unadjusted NPV by the capitalrecovery factor (also called the annuity with present value of 1.0) for the expected number ofyears of useful life in the project. The capital recovery factor (CRF) is simply the reciprocalof the present value of annuity factor.3
For projects F and G, the time-adjusted annual averages are:
Project F: $50,563 × .019925 (10-year CRF) = $10,075
Project G: $34,086 × .029832 (5-year CRF) = $10,169
It is apparent that project G offers a somewhat higher NPV on an annual basis than does F.Now, to convert these results to adjusted NPVs for the projects over the longer-lived project, inthis case 10 years, we need only multiply by the present value of annuity factor correspondingto this number of years. Thus, we obtain adjusted NPVs of:4
Project F: $10,075 × 5.0188 = $50,564
Project G: $10,169 × 5.0188 = $51,036
And the corresponding profitability indexes are 0.506 and 0.510.
SUMMARY AND CONCLUSION
Now that we have seen how two serious difficulties with NPV may be surmounted with theprofitability index and uniform annual equivalents, what can we say in summary about NPV?
Strong Points of NPV
1. It is conceptually superior to both the payback and accounting methods.2. It does not ignore any periods in the project life nor any cash flows.3. It takes into account the time-value of money.
3The CRF is defined by the equation k/[1 − (1 + k)−N ] for rate k and N periods. The CRF enables us to spread out any given presentvalue over a specified number of years as well as providing a means for finding a time-value-adjusted average present value.4Minor rounding errors make the calculated result different by a small amount.
54 Capital Asset Investment: Strategy, Tactics & Tools
4. It is consistent with the basic valuation model of modern finance.5. It is easier to apply than the IRR since it involves evaluating a polynomial rather than finding
a polynomial root.6. It favors early cash flows over later ones.
Weak Points of NPV
1. Like the IRR, it requires that we have an estimate of the organization’s cost of capital, k.Also, the given k is embedded in the NPV, whereas with the IRR the internal rate can bejudged by management. Management will determine whether it is reasonable that the IRRis greater or less than k when k is not known with confidence. In other words, IRR may bemore intuitive.
2. It is more difficult to apply NPV than payback or the accounting method, and thus lesssuitable for use by lower levels in the organization without proper training in its application.That may not be feasible.
3. Unless modified by conversion to uniform annual equivalents and converted to a profitabilityindex, NPV will give distorted comparison between projects of unequal size and/or unequaleconomic lives.
4. NPV depends on forecasts, estimates of future cash flows that become increasingly lessreliable and more nebulous the further into the future they are expected to occur.
7Traditional Methods that Recognize Time-value
of Money: the Internal Rate of Return
DEFINITION OF THE IRR
The internal rate of return (IRR) is defined as the rate of interest which exactly equates theNPV of all net cash flows to the required investment outlay. Equivalently, the IRR is the ratefor which the NPV of the cash flows is equal to zero. Thus IRR is the rate of discount r, whichsatisfies the relationship
C = R1
(1 + r )1 + R2
(1 + r )2 + R3
(1 + r )3 + · · · + RN
(1 + r )N (7.1)
If we rename C to R0, this can be restated as
0 = R0
(1 + r )0 + R1
(1 + r )1 + R2
(1 + r )2 + · · · + RN
(1 + r )N (7.2)
or in the more compact equivalent forms
0 =N∑
t=0
Rt
(1 + r )t =N∑
t=0
Rt (1 + r )−t (7.3)
Equations (7.1), (7.2), and (7.3) define a polynomial equation with real coefficients R0,R1, . . . , RN . The IRR is thus the root of a polynomial. Unfortunately, under some circum-stances there may be two or more real roots, or an economically misleading root, and thiscan create some significant problems. The general case of multiple roots will be dealt withlater. For now we will consider only investment projects that have just one sign change inthe coefficients, and for which we are assured by Descartes’ rule of signs, can have but onereal, positive root. Cost C, or as we will normally call it from now on, R0, we assume tobe negative, and R1, R2, R3, . . . , RN to be positive. In later chapters this assumption may berelaxed.
In the past, before the personal computer and spreadsheet software, finding the IRR couldoften be a troublesome task, especially when the cash flows after R0 were nonuniform. We shalldeal here first with a project having uniform expected cash flows. To illustrate the calculationof IRR when cash flows are uniform, and we have only a calculator, let us consider project Afrom Chapter 5 again, and use equation (7.1):
$5000 =$2500
1 + r+ $2500
(1 + r )2 + · · · + $2500
(1 + r )10 (7.4)
56 Capital Asset Investment: Strategy, Tactics & Tools
which, since the Rt are uniform, can be rewritten as
$5000 = $250010∑
t=1
1
(1 + r )t = $250010∑
t=1
(1 + r )−t (7.5)
so that we have
2.0 = $5000
$2500=
10∑t=1
(1 + r )−t = 1 − (1 + r )−10
r≡ a
10 r(7.6)
Since the last term on the right here is the summation representing the present value of anannuity of 1.0 for 10 periods at r percent,1 we look in Appendix Table A.4.2 Moving acrossthe row that corresponds to the 10 periods, we find the factor 2.003 under 49 percent. Sincethis is very close to 2.0, we might conclude that the IRR for project A is essentially 49 percent(the actual rate is 49.1 percent to the nearest one-tenth percent).
Now let us see what must be done when cash flows are not uniform for all periods beyondt = 0. Finding the IRR for project B is not so easy, because the cash flows are not uniform.To find r we use the Appendix tables for single-amount present worth and for annuity presentworth.
First, let us estimate that r is 60 percent. We then proceed as follows:
Present worth PresentYear factor × Cash flow = value
1 0.62500 $500 $312.502 0.39063 1000 390.633 0.24414 2000 488.284 0.15259 4000 610.365 • •6 • •7 0.2391 } 8000 } 1912.808 (1.6515 − 1.4124) •9 • •
10 • •Project present value at 60% = $3714.57
1The term is derived from the formula for the summation of a geometric progression, which is
Sum = αρN − 1
ρ − 1
where ρ is the common ratio (in the case above, ρ = (1 + r )−1), α the first term in the progression, and N the number of periods. Foran ordinary annuity this becomes
(1 + r )−1 (1 + r )−N − 1
(1 + r )−1 − 1= (1 + r )−1 (1 + r )−N − 1
−r (1 + r )−1 = 1 − (1 + r )−N
r≡ a
N r
2Modern hand-held calculators provide a better approach: calculate the factor for a trial r, and revise the estimate until the result issatisfactorily close. If a table is readily available, it provides a good first guess, of course. Many reasonably priced calculators areavailable with circuitry preprogrammed to provide very precise solutions for r.
The Internal Rate of Return 57
Present worth PresentYear factor × Cash flow = value
1 0.66667 $500 $333.332 0.44444 1000 444.443 0.29630 2000 592.604 0.19753 4000 790.125 • •6 • •7 0.3604 } 8000 } 2883.208 (1.9653 – 1.6049) •9 • •
10 • •Project present value at 50% = $5043.69
Since the cash flows in years 5–10 are a uniform $8000, we need multiply this amount onlyonce by the present worth of annuity factor for 10 years at 60 percent less the present worthof annuity factor for four years at 60 percent. Notice that the latter is equal to the sum of thesingle amount present worth factors for years 1–4. The project present value at 60 percent is$3714.57, which is less than the $5000 investment outlay. Therefore we know that 60 percentis too high a discount rate, and that IRR is less than this. Let us next try 50 percent.
This is just slightly above the $5000 investment cost. We have bracketed the IRR, and nowknow that it is very close to 50 percent. We can now refine our result by interpolation.
60% 3714.57
r50%
] [5000
5043.69
−10
50 − r= 1329.12
43.69
r = 50 + 436.90
1329.12= 50.329% or 50.3%
This interpolation assumes a linear relationship, whereas we have an exponential one, so ourresult is only approximate. However, for capital-budgeting applications it will normally beprecise enough. We should generally give interpolated answers to only one decimal or thenearest percent or tenth of a percent in order not to convey the impression of greater precisionthan we in fact have. The actual rate, correct to the nearest tenth of a percent, is 50.3 percentin this case. In general, the smaller the range over which we bracket the rate, the more preciseour interpolated result will be.
Nowadays one should seldom if ever have to resort to interpolation to find an IRR. Allpopularly used spreadsheet programs provide built-in functions to find IRR. Still, it may beuseful to know how to find a result by interpolation should one be caught without access to acomputer spreadsheet program on rare occasions.
By this point it should be clear that calculation of IRR can be a tedious task when cash flowsare not uniform. In fact, to ease the computational burden, computer spreadsheet programshave built-in functions to calculate IRR, and some hand-held calculators are programmed to
58 Capital Asset Investment: Strategy, Tactics & Tools
solve for IRR.3 Programs to calculate IRR are not difficult to write; the Newton–Raphson orinterval bisection methods, in conjunction with Horner’s method of polynomial evaluation,both yield good results efficiently.
A CAUTION AND A RULE FOR IRR
Assume we have a project costing $4000 that will last only two years and provides cash flowsof $1000 and $5000 in years 1 and 2. To find the IRR for this project we may use the quadratic
formula −b ±√
b2 − 4ac/2a, since (dropping the $ signs):
4000 = 1000
1 + r+ 5000
(1 + r )2 (7.7)
is equivalent to
−4 + 1
1 + r+ 5
(1 + r )2 = 0 (7.8)
Multiplying by (1 +r)2, we get
−4(1 + r )2 + (1 + r ) + 5 = 0 (7.9)
Letting x = 1 +r , we obtain the quadratic equation
−4x2 + x + 5 = 0 (7.10)
which has roots given by
−1 ± √1 + 80
−8
of x = −1.0 and +1.25. Converting to r, we have IRR of −200 percent and +25 percent!Which do we take as the IRR for this project? The rule to follow in such cases is:
In the case of one positive or zero root and one negative root, choose the negative root only if theproject cost is strictly greater than the undiscounted sum of the cash flows in periods 1 through N .
In this case 4000 < 1000 + 5000, so we choose + 25 percent for IRR. A negative IRR makesno economic sense in cases in which the total cash return is greater than or equal to the projectcost. Conversely, for cases in which the cost exceeds the undiscounted cash flows, a positiveIRR makes no economic sense. A negative IRR < −100 percent makes no economic sensebecause it is not possible to lose more than all of what is lost on a bad investment when allcash flows attributable to the project are included.
PAYBACK AND IRR RELATIONSHIP
In Chapter 5 the naive rate of return (NROR) was defined as the reciprocal of payback. Is thereany relationship between NROR and IRR? Yes. Provided that certain conditions are met, theNROR may approximate the IRR.
3Texas Instruments, Hewlett-Packard, and others make such calculators.
The Internal Rate of Return 59
Assuming uniform cash flows, R = R1 = R2 = · · · = RN . Then payback = C/R, NROR =1/payback = R/C , and IRR is the r such that
C =N∑
t=1
R
(1 + r )t = RN∑
t=1
(1 + r )−t (7.11)
C
R=
N∑t=1
(1 + r )−t (7.12)
so that
R
C= 1
N∑t=1
(1 + r )−t
= r (1 + r )N
(1 + r )N − 1= r
1 − (1 + r )−N (7.13)
which is defined as the capital recovery factor for N periods at rate r per period. Now, takingthe limit as N increases, we obtain
R
C= lim
N→∞
[r
1 − (1 + r )−N
]= r (7.14)
which is the same result as the NROR provides.Therefore, under some conditions, the NROR may provide a useful approximation to IRR.
What are the conditions? The first condition is that the project life should be at least twice thepayback period. The second is that the cash flows be at least approximately uniform:
1. N ≥ 2C/R and2. uniform R.
Financial calculators make the task of finding the IRR of a uniform cash flow series easy,as long as there are not a great many cash flow amounts to enter on the keypad. However,for projects with nonuniform cash flows, it is useful to employ a computer, especially if thereare many cash flow periods, or if it is desired to deal with the problem of mixed cash flowsaccording to methods described in Chapter 10 and those following it. Occasionally one mightwish to write a computer program that will be used in large-scale production runs, as at a bank.Thus, it is worthwhile to have a general understanding of ways to efficiently find the IRR.
MATHEMATICAL LOGIC FOR FINDING IRR
For projects that we know will have only one real, positive root, by Descartes’ rule of signs,4
several methods can be easily programmed for computer solution. The most straightforwardare interval bisection and Newton–Raphson. A capital investment with one real positive IRRyields an NPV polynomial function that looks like the one in Figure 7.1, that is, concave fromabove.
Note that the IRR is the discount rate for which the discounted value of the project, includingthe cost C (i.e. R0), is zero. In other words, the IRR is the discount rate for which the projectNPV is zero. Thus, to find the IRR we must find the rate for which the polynomial intersectsthe horizontal axis.
4Descartes’ rule states that the number of unique real roots to a polynomial with real coefficients cannot exceed the number of changesin the sign of the coefficients and that complex roots must be in conjugate pairs. That is, if x + ci is a root, x − ci is also a root.
60 Capital Asset Investment: Strategy, Tactics & Tools
0r
Rate
+$
−$
Figure 7.1 IRR polynomial
$NPV
0m
r
hRate
Figure 7.2 Interval bisection
Interval Bisection
With the interval bisection method, we select a high value and a low value that we think willbracket the IRR. If we are wrong, we will try higher values the next time. Assume we firstselect 0 as our low value and h as our high value. This is shown in Figure 7.2. We evaluate theNPV polynomial at rate 0 and at rate h. Since the NPV is positive for rate 0 and negative forrate h, we know the function has a root (crosses the horizontal axis) between these rates.
Next we improve our results by bisecting the interval between 0 and h and testing to determinein which subinterval the function crosses. We also test for the possibility that the midpoint ofour earlier range, m, may actually be the IRR, even though unlikely. If we determine that theIRR lies in the interval between m and h, we bisect this range again, and repeat the process.We determine at the outset to stop when the difference between the low and high intervalvalues is less than or equal to an arbitrarily small value. This error tolerance cannot be toosmall, however, because all digital computers have intrinsic round-off errors in computationand different capability for precision calculation.5 In general, however, a tolerance of 0.0001
5This is potentially a very serious problem, especially as the number of cash flows becomes large. Logically correct programs mayyield totally incorrect results for which the program user is unprepared after testing the program with small problems having few cashflows and getting correct results.
The Internal Rate of Return 61
0
$NPV
rnr
rn + 1
Rate
Figure 7.3 Newton–Raphson method
or 0.01 percent should cause no problems. If it does, a change to double-precision calculationmay remedy the difficulty. The manual method of IRR solution is similar to this, except thatthe final IRR estimate is made by interpolation rather than continued iteration, once it is knownthat the IRR value has been bracketed.
Newton–Raphson Method
This method is somewhat more sophisticated than the interval bisection approach, and in somecases marginally more efficient.6 Any increase in efficiency it yields will likely result in verysmall savings in computer time, however.
The approach with this method is to modify the original “guess” for the IRR by using theintersection of the tangent line to the NPV curve with the rate axis as the improved “guess.”The process is repeated until the NPV is sufficiently close to zero. This might be expressedas an error less than or equal to some small percentage of the project cost, for example, oralternatively an NPV less than some small money amount.
The Newton–Raphson method requires that the numerical value of the derivative with respectto rate of the NPV be obtained at the current rate estimate. If we refer to rn as the current IRRestimate and rn+1 as the revised estimate, f () as the NPV function, and f ’( ) its derivative, then:
rn+1 = rn − f (rn)
f ′(rn)
The value of rn+1 is our revised estimate of the IRR. Graphically, the Newton–Raphsonapproach may be visualized as shown in Figure 7.3.
For the NPV function at rate i, the polynomial is
NPV = f (i) = R0(1 + i)−0 + R1(1 + i)−1 + · · · + Rn(1 + i)−n
and the derivative:
f ′(i) = −R1(1 + i)−2 − 2R2(1 + i)−3 − · · · − n Rn(1 + i)−(n−1)
6Marginal is used here in the sense that one or two seconds of computer time is not usually significant unless a program is run veryfrequently in a commercial environment.
62 Capital Asset Investment: Strategy, Tactics & Tools
Computer evaluation of f ′(i) involves modification of f (i) by dropping the first term contain-ing R0, negating all the subsequent cash flow terms, multiplying each by the correspondingexponent, and then decrementing each of the exponent powers by one.
Strong Points of IRR
1. It is conceptually superior to the payback and accounting methods.2. It does not ignore any periods in the project life nor any cash flows.3. It takes into account the time-value of funds.4. It is consistent with the basic valuation model of modern finance.5. It yields a percentage that management can examine and make judgment about when k is
not known with confidence. That is, it yields an intuitive figure to management.6. It favors early cash flows over later ones.
Weak Points of IRR
1. It requires an estimate of the organization’s cost of capital, or at least a range of values inwhich this is likely to be found.
2. It is much more difficult to apply without a computer than the payback or accountingmethods, and when cash flows are nonuniform, much more difficult to apply than the NPVmethod.
3. It does not distinguish between projects of different size and/or different economic lives.However, adjustment for this may be made along lines similar to such adjustments for NPV.
4. It often yields multiple, and thus ambiguous, results when there is more than one signchange in the cash flows.
5. Some have criticized the method on the basis that it implicitly assumes that cash flows maybe reinvested at a return equal to the IRR.
6. Like the NPV, the IRR requires forecast estimates of future cash flows, and thus suffersfrom whatever error to which those forecasts are subject.
A DIGRESSION ON NOMINAL AND EFFECTIVE RATES
Investment–Financing Relationship
The mathematics of the IRR are identical to those required for finding the effective interest rateon a loan or the yield to maturity or call on a bond. To the lender, a loan or a bond purchase isan investment. Symmetry of the loan or bond relationship requires that the effective cost rateto the borrower (bond issuer) be the same as the rate of return to the lender (bond purchaser)on a pre-tax basis. Mathematically, the only difference in a loan or bond from the perspectivesof borrower and lender is that the signs of the cash flows will be reversed.
Consider a bond that has a face value of $1000, a nominal interest yield or coupon rate of7 percent, and a 20-year maturity. It sells when first issued for $935. Interest will be calculatedsemiannually and we assume the bond will not be called prior to its maturity date. To thepurchaser of the bond who buys at issue, it is an investment with cash flows in periods zerothrough 40 (semiannual payments over 20 years) and the IRR equation is, dropping the $ signs,
−935 + 35(1 + r s)−1 + 35(1 + r s)
−2 + · · · + 35(1 + r s)−40 + 1000(1 + r s)
−40 = 0
(7.15)
The Internal Rate of Return 63
To the issuer of the bond, the equation to calculate the effective cost is
+935 − 35(1 + r s)−1 − 35(1 + r s)
−2 − · · · − 35(1 + r s)−40 − 1000(1 + r s)
−40 = 0(7.16)
The solution to the real, positive root of equation (7.15) is the same as that to equation (7.16).
Nominal Rate and Effective Rate
Note that the rate in equations (7.15) and (7.16) is denoted by rs. This is to flag it as a semiannualrate. The corresponding nominal annual yield or cost rate is two times rs. The effective annualrate is that rate that would be equivalent if interest payments were made annually at the end ofeach year instead of at the end of each semiannual period.
For the bond in question the (pre-tax) rates are:
Semiannual effective yield rs = 3.81964%Nominal annual yield rn = 2rs = 7.63928%Effective annual yield r = (1 + rs)2 − 1 = 7.78518%
It is usual financial practice in the United States to report yields in terms of nominal annualrates rather than effective annual rates. If the investor has the opportunity to reinvest interestpayments as they are received at the effective per-period rate, however, the period-to-periodcompounding implicit in this should be recognized. The effective annual yield recognizes suchcompounding; the nominal annual rate does not.
Clarification of Nominal and Effective Rates
Let us examine here a problem that may help to clarify the relationship between, and meaningof, the various rates. In the case of a bond, the coupon rate is the rate applied to the face or parvalue of the bond to determine the dollar interest to be paid. The coupon rate is a nominal annualrate. The rate advertised by thrift institutions, which is compounded so many times per year,is also a nominal annual rate. For example, 53/4 percent compounded quarterly is a nominalannual rate. If we divide it by the number of compounding periods, we obtain the effective rateper quarter, which is 1.4375 percent per quarter. Money left on deposit will thus earn 1.4375percent every three months. If $1000 is deposited at the beginning of a year under these terms,it will grow to $1058.75 at year end. This is thus an effective increase of 5.875 percent, not53/4 percent.
In general, nominal annual (coupon) rate, per period effective rate, and effective annual rateare related as follows:
1. Nominal annual rate (rn) ÷ number of compounding periods in a year (M) = effective perperiod rate (rp).
2. The quantity one plus the effective per period rate raised to the power corresponding to thenumber of compounding periods in a year = one plus the effective annual rate.
rn
M= rp
(1 + rp)M − 1 = r
Or, comprehensively in one equation:
r = (1 + rn
M
)M− 1
64 Capital Asset Investment: Strategy, Tactics & Tools
Figure 7.4 Quarterly versus annual cash flows for IRR calculation
Unless it is perfectly clear from the context in which the term is used, rn should never becalled just the “annual rate,” but should be called the “nominal annual rate” or “annual ratecompounded M times per year.” To do otherwise is imprecise, confusing, and misleading.
IRR with Quarterly Cash Flows
Consider an investment project costing $5000 that will last an estimated five years and providesnet, after-tax cash flows at the end of each quarter of $300 in all but in the final quarter which,with salvage, amounts to $1000.
The quarterly, per-period IRR is rp = 2.701 percent. The nominal annual IRR, rn is 10.804percent compounded quarterly. The effective annual IRR rate, r = 11.249 percent. If we wereto ignore the fact that the cash flows occur quarterly, and instead treat the flows for each yearas if they fell on the last day of the year, we would get an IRR of r = 9.880 percent. Note thatthis is significantly less than if the cash flows were treated as falling at the end of every threemonths, because the cash flows, on average, are pushed into the future some one-and-one-halfquarters or some four-and-one-half months (see Figure 7.4). One may take comfort in the factthat such differences in the IRR will seldom cause a worthwhile investment to be shunned ora poor one to be accepted.
8Reinvestment Rate Assumptions for NPV
and IRR and Conflicting Rankings
This chapter discusses some unsettled conceptual problems that have figured prominently inthe literature of capital budgeting. Warning: the material presented here may be somewhatcontroversial to other writers in the field of capital investments. My intent is that this chapterwill motivate readers who are already familiar with the standard lore of capital budgeting tothink critically of some things that may have been taken for granted, and judge for themselvesthe truth of the matter.
REINVESTMENT RATE ASSUMPTIONS FOR NPV AND IRR
Comparison of equations (7.3) with (6.1) and (6.2) shows that the internal rate of return (IRR)is nothing more nor less than a special case of net present value (NPV). The IRR is defined asthe rate for which NPV is zero. With this in mind, let us examine a general equation combining(7.3) and (6.1):
X = R0
(1 + d)0 + R1
(1 + d)1 + · · · + RN
(1 + d)N (8.1)
or
X =N∑
t=0
Rt
(1 + d)t =N∑
t=0
Rt (1 + d)−t (8.2)
If the discount rate d is such that X = 0, we say that d is the IRR. If X = 0, we say that X isthe NPV for cost of capital k = d. Therefore, by employing equations (8.1) and (8.2) we candeal simultaneously with both IRR and NPV.
So far, all mathematical formulations for IRR and NPV have been in terms of present value.We can easily convert present value to future value by multiplying both sides of the equationby (1 + d)N , where d is either the IRR or cost of capital k, as the case may be. If we do thiswith equations (8.1) and (8.2), we obtain
X (1 + d)N = R0(1 + d)N + R1(1 + d)N−1 + · · · + RN (8.3)
X (1 + d)N =N∑
t=0
Rt
(1 + d)t−N =N∑
t=0
Rt (1 + d)N−t (8.4)
Note that if d is the IRR, then X = 0 and X (1 + d)N is zero. Therefore, IRR could be as easilyobtained from a future value formulation as from the conventional present value formulation.Also, if X were not zero, that is, if X were an NPV for k = d , the effect of multiplying by(1 + d)N simply moves the reference point from t = 0 to t = N .
In other words, looking at equation (8.3), we obtain the same results except for a constantof (1 + d)N times the NPV by assuming the cash flows are reinvested at earning rate d, rather
66 Capital Asset Investment: Strategy, Tactics & Tools
than being discounted at that rate. The R0, which for most projects will be the cost, or initialoutlay, is “invested” for N periods at rate d compounded each period. For the initial outlaythis may be interpreted as the opportunity cost of committing funds to this project insteadof to an alternative purpose in which rate d could be earned. Or, in the case where d = k,the opportunity cost may arise from the decision to undertake a project requiring funds to beraised, whereas without the project no new funds would need to be raised. The mathematicalsymmetry between these formulations has been a cause for concern among finance theorists.Let us consider the kernel of their concern.
The implication of future value formulation is that the project return, whether measured byIRR or by NPV, will depend on the rate at which cash flows can be reinvested. For a firm ina growth situation, in which profitable investment opportunities abound, the IRR assumptionthat cash flows may be reinvested at a rate of earning equal to the IRR may thus be reasonable.For other firms, and government institutions, some analysts think it is more realistic to assumethat the cash flows can be reinvested at a rate equal to the cost of capital k. This is the usualformulation of the reinvestment rate assumption.
Let us now consider the IRR and reinvestment rate in another light. Consider a loan of$100,000 that is made by a bank to an individual business proprietor for a period of five years.The loan is to be repaid in equal installments of $33,438 (to the nearest whole dollar). Fromthe bank’s viewpoint this is an investment, with cash flows:
t = 0 t = 1 t = 2 t = 3 t = 4 t = 5−$100,000 $33,438 $33,438 $33,438 $33,438 $33,438
and yield (that is, IRR) of 20 percent.From the borrower’s viewpoint the cash flows are identical except that the signs are reversed.
The cash flows of the borrower are precisely the cash flows of the lender multiplied by minusone. Therefore, the borrower has cash flows:
t = 0 t = 1 t = 2 t = 3 t = 4 t = 5+$100,000 −$33,438 −$33,438 −$33,438 −$33,438 −$33,438
and his effective cost is 20 percent on the loan. The borrower must earn at least 20 percent perperiod on the loan just to be able to repay it. The pre-tax return to the lender on the investment(loan) cannot be less than the cost to the borrower. Even if the bank does not reinvest the cashflows as the loan is repaid, its implicit return will still be 20 percent. The return is measuredas a time-adjusted percentage of the principal amount outstanding, and is independent of whatdisposition is made of the cash flows as they are received. This is not to say that the uses to whichthe cash flows are put will have no effect on the organization, for they will. However, althoughthe yield on the funds originally invested may be increased by such uses, it cannot be reduced bylack of such investment opportunities. This is a strong position to take, and requires explanation.
The payments made by the borrower, once given over to the lender, can earn nothing forthe borrower. The borrower must, in the absence of other sources of funds, be able to earn20 percent per period on the remaining loan principal. If the borrower is unable to earn anythingon the remaining loan principal, he or she must still make the required periodical payments.The payments, even if made from other sources of funds, will be the same as those required ifthe loan were to generate funds at 20 percent per period. If funds must be diverted from otherprojects to repay the loan, the opportunity cost to the borrower may be more than 20 percent,if the funds could have earned more than this percentage in other uses. The cost internal to theloan itself, however, is 20 percent.
Reinvestment Rate Assumptions for NPV and IRR 67
Table 8.1 Component breakdown of cash flows (amountsrounded to nearest $)
Beginning Interest on Principalt principal principal repayment
1 100,000 20,000 13,438a
2 86,562 17,312 16,1263 70,437 14,087 19,3514 51,086 10,217 23,2215 27,865 5,573 27,865
a Assumes that end-of-year payment of $33,438 is composed of interestof 20% on the beginning balance plus a repayment of principal (that is,$20,000 + $13,438).
Table 8.1 provides a breakdown of the loan payments into the principal and interest com-ponents implicit in the IRR method of rate calculation. Note that the interest is computed at20 percent per year on the beginning-of-period principal balance. The excess of payment overthis amount is used to reduce the principal.
The following loan (to the borrower) will have an identical cost of 20 percent. However, theprincipal is not amortized but is paid in full at the very end of the loan:
t = 0 t = 1 t = 2 t = 3 t = 4 t = 5+$100,000 −$20,000 −$20,000 −$20,000 −$20,000 −$120,000
The borrower may place the loan principal in a (hypothetical) bank account that pays exactly20 percent annually on the deposit. At the end of each year the borrower withdraws the interestand pays it to the lender. At the termination of the loan the borrower withdraws the principalplus interest and repays the loan. Since the interest is paid to the lender as soon as it is earned,the borrower does not earn interest on interest.
The bank pays the borrower exactly 20 percent annually on the deposit, which he or sheimmediately turns over to the lender. The loan costs are exactly equal to the 20 percent annualinterest the bank pays the borrower for the deposit, so there is no net gain to him or her. (Wehave ignored transactions costs to simplify discussion.)
For the lender the loan also yields exactly 20 percent. However, there is an important dif-ference: the lender may reinvest the interest payments if desired and thus increase the gain.Such increase, significantly, does not depend on the loan itself, but on reinvestment oppor-tunities available for the loan interest when it is received. The 20 percent return is thus aminimum return on the loan, and this minimum is independent of reinvestment opportuni-ties. The reinvestment rate could be zero and still the lender would earn 20 percent on theloan.
The only difference between these two loans is the handling of principal repayment: in thefirst, the principal is amortized over the life of the loan. In the second loan, the entire principalrepayment is made at the loan maturity date. The first loan does provide better reinvestmentopportunities to the lender since larger payments are received in all but the last year, and thelender may be able to reinvest them and so raise the return on the loan. Once again, however,20 percent is the minimum return to be expected, even if the reinvestment opportunity rate wereto be zero. The lender earns exactly 20 percent on the principal amount still in the hands of theborrower. If the lender can earn 20 percent or more on the recovered principal, so much the
68 Capital Asset Investment: Strategy, Tactics & Tools
Table 8.2 Per period return on remaining principal(constant amortization of $20,000 per period)
Principalt remaining % Return
1 $100,000 13.4382 80,000 16.7983 60,000 22.3974 40,000 33.5955 20,000 67.190
better for the lender. If he or she cannot, the lender nevertheless continues to earn 20 percenton the still unrecovered principal.
Let us assume a zero reinvestment rate. With the first loan, let us consider that the $100,000principal is returned in equal annual installments of $20,000 over the five-year loan maturity.This means that $13,438 over and above the principal repayment is earned on the remainingprincipal. The percentage return on the remaining principal in each year is then as shown inTable 8.2. The geometric mean return is 29.4 percent, the arithmetic mean return 30 percent, andthe median return 22.4 percent. This treatment differs from the IRR formulation in assuming afixed allocation of periodical payments to principal amortization (straight-line amortization ofthe principal) rather than a gradually increasing amortization payment. Therefore, the principalis more quickly reduced, consequently yielding a higher return on that which remains.
These results do not require any reinvestment rate other than zero. They show that thepercentage return on an investment does not depend on the available reinvestment rate. Theactual gain to the investor (or lender) may, of course, be higher than this minimum amountif the available reinvestment rate is greater than zero, but that is a condition external to theinvestment. The IRR is concerned with the internal characteristics only, and therefore providesa measure of the minimum return on the investment.
In summary, the conceptual difficulty with the reinvestment rate assumption arises from fo-cusing on the superficial aspects of the mathematics of the IRR while neglecting the economicinterpretation of the initial investment and the subsequent cash flows. It is important to remem-ber that mathematics is a tool in finance, economics the master. The reinvestment rate problemarises from confusion of this hierarchy — from trying to make the economics conform to themathematics. The IRR might be called more properly return on invested capital to make clearits economic assumptions. However, this term has another specific meaning that is coveredlater, and is thus reserved for that.
Treatment of the reinvestment rate with the NPV, along lines similar to that of the IRR, isleft as an exercise.
CONFLICTING RANKINGS AND FISHER’S INTERSECTION[1]
The rules for project acceptance once again are:
IRR NPVAccept if r > k NPV > 0Reject if r ≤ k NPV ≤ 0
Now it would seem that if we are comparing two acceptable, but mutually exclusive projects weshould get the same preference ranking by NPV as we do by IRR. However, this will not always
Reinvestment Rate Assumptions for NPV and IRR 69
10
$(1000)
8
6
4
2
0 10 20 30 40% Rate ofDiscount
H
I
rf
Figure 8.1 Fisher’s rate, rf, and conflicting IRR–NPV project rankings
be the case. Conflicting rankings may arise because one or both projects have nonuniform cashflows.
To begin, let us consider two projects each costing $7000 at t = 0, and each having aneconomic life of five years, in order that we do not need to correct for unequal project size orlife. The projects have the following cash flows (dropping the $ signs):
Project H Project ICost, R0 7,000 7,000
Cash flowt = l 6,000 250t = 2 3,000 500t = 3 1,500 750t = 4 750 4,000t = 5 375 10,000
The internal rates of return are IRRH = 35.120 percent, IRRI = 19.745 percent. As shown inFigure 8.1, the IRR values are fixed. However, the NPV values change, since they depend onthe value of d that is used. For d = 0 (that is, zero discount) they are for projects H and I $4625and $8500, respectively.
Notice that the two NPV curves intersect at a discount rate of approximately 11 percent(11.408 percent), which we denote as rf. For smaller discount rates, project I has an NPVgreater than that of project H. However, IRRH > IRRI always. Therefore, for discount ratesless than rf, there will always be a conflict between the NPV ranking and the IRR ranking, but,for discount rates greater than rf, both IRR and NPV will yield the same ranking.
Let us examine these projects further by treating them each as loans and inspecting theirrepayment schedules, as in Table 8.3. We assume that each net cash flow is composed of twocomponents: an interest payment and a principal repayment. First, we will use as discount rateFisher’s rate of 11.408 percent.
Referring to Table 8.3, the project NPVs at t = 0 can be found by discounting the amountsin column 4. For project H we have
$996.26(1.11408)−2 + 1500(1.11408)−3 + 750(1.11408)−4 + 375(1.11408)−5
equaling $2592.80. For project I we have $4449.93(1.11408)−5 or $2592.80. As we should
70 Capital Asset Investment: Strategy, Tactics & Tools
Table 8.3 Component treatment of projects H and I at Fisher’s rate of 11.408%(amounts in $)
(1) (2) (3) (4) (5)Beginning Interest on Principal Excess over Total net cash
t principal principal repayment (2) + (3) flow
Project Hl 7,000.00 798.56 5,201.44 0 6,000.002 1,798.56 205.18 1,798.56 996.26 3,000.003 0 0 0 1,500.00 1,500.004 0 0 0 750.00 750.005 0 0 0 375.00 375.00
Project Il 7,000.00 798.56 −548.56 0 250.002 7,548.56 861.14 −361.14 0 500.003 7,909.70 902.34 −152.34 0 750.004 8,062.04 919.72 3,080.28 0 4,000.005 4,981.75 568.32 4,981.75 4,449.93 10,000.00
have expected, at Fisher’s rate the NPVs of projects H and I are equal, for that is how Fisher’srate is defined.
Notice an important difference between the two projects with respect to column 1, thebeginning-of-period principal remaining. The net cash flows for project H were sufficientlyhigh at the end of periods 1 and 2 to pay off the entire principal after paying the requiredinterest. In fact, at the end of period 2 the net cash flow was $996.26 in excess of what wasrequired to pay off the interest and remaining principal. This is the meaning of NPV, and whysome authors choose to call it excess present value. The NPV is the amount, at discount, bywhich the net cash flows of an investment project exceed what is required for payment ofinterest on remaining principal and principal repayment. Note that although project H paid offthe entire principal by the end of period 2, project I did not pay off its principal until the endof period 5.
We know that projects H and I are not loans, but are capital investment projects. The principalrepayments represent recovery of the initial investment, whereas the interest payments representthe opportunity cost of funds committed to the projects and as yet unrecovered. Which projectis preferable? Both have the same NPV of $2592.80 at Fisher’s rate. Does this mean that weshould feel indifferent about the two projects?
Although projects H and I have identical NPVs at Fisher’s rate of 11.408 percent, project His clearly preferable. The reason is that project H provides for faster recovery of the fundsinvested in it than does project I. In fact, project I is incapable of paying the interest onbeginning-of-period principal in any of periods 1–3. Therefore, instead of giving us back ouroriginal investment with interest from the start, project I requires that we wait until the endof period 4 before any reduction in principal plus accumulated interest can be made. On theprinciple that a bird in the hand is worth two in the bush, or rather that a dollar today is betterthan a dollar plus interest tomorrow (if there is any doubt about getting the dollar tomorrow),we prefer project H to project I. The fact that the NPVs of the two projects are identical shouldnot cause us to be indifferent between projects such as H and I, for the NPV is only one criterionby which project merit may be gauged. If the NPVs are equal, we should, of course, prefer theproject that offers the more rapid capital recovery. Invested funds recovered early are subject
Reinvestment Rate Assumptions for NPV and IRR 71
Table 8.4 Component treatment of projects H and I at 5%, a rate less than Fisher’srate ($ amounts)
(1) (2) (3) (4) (5)Beginning Interest on Principal Excess over Total net cash
t principal principal repayment (2) + (3) flow
Project Hl 7,000.00 350.00 5,650.00 0 6,000.002 1,350.00 67.50 1,350.00 1,582.50 3,000.003 0 0 0 1,500.00 1,500.004 0 0 0 750.00 750.005 0 0 0 375.00 375.00
Project Il 7,000.00 350.00 −100.00 0 250.002 7,100.00 355.00 145.00 0 500.003 6,955.00 347.75 402.25 0 750.004 6,552.75 327.64 3,672.36 0 4,000.005 2,880.39 144.02 2,880.39 6,975.59 10,000.00
to less uncertainty of receipt than funds to be recovered later, and funds recovered early canbe reinvested for a longer period to enhance the earnings of the firm.
We see that at Fisher’s rate project H is preferable to project I even though the NPVs areidentical. This preference is consistent with what we would have obtained by choosing betweenthe projects on the basis of their IRRs. Now let us examine these same projects at a rate ofdiscount substantially less than rf, Fisher’s rate. For illustration we will use a discount rate of5 percent. Table 8.4 contains the results of our calculations.
We may again determine the NPVs by bringing the column 4 amounts to t = 0 at a discountrate of 5 percent per period. For project H we obtain an NPV of $3641.98 and for project I anNPV of $5465.56. Since NPV of I > NPV of H, which we would expect for a discount rateless than Fisher’s rate, the NPV criterion favors project I. The profitability indexes (PIs) wouldnot suggest a different relationship, because both projects require the same initial investmentoutlay.
At a 5 percent cost of capital we find that the NPV (or PI) criterion favors project I. TheIRR criterion, however, always favors project H. Which criterion should we use? For a costof capital (discount rate) greater than or equal to Fisher’s rate we would have no problem,because both the IRR and the NPV would favor project H. But we have under consideration arate of 5 percent and a conflict in ranking that must be resolved.
To help us decide which project is preferable, let us again examine columns 1 and 4 ofTable 8.4. Notice that project H (column 1) allows us to recover our entire investment plusinterest (opportunity cost on the funds committed) and yields an excess of $1582.50 at theend of the second year (column 4). Project I, in contrast, cannot even compensate for ouropportunity cost in year 1. It does not yield recovery of our investment until the end of thefifth year. In fact, the entire NPV of project I depends on the large cash flow in period 5. Ifunanticipated events in year 3, 4, or 5 caused all cash flows to be zero, project H would stillhave a positive NPV of $1582.50(1.05)−2 = $1435.37, but project I would have a negativeNPV of −$6308.39. Not only does project H have a higher IRR (35.120 percent) than project I(IRR = 19.751 percent), but also it does not depend on the accuracy of our cash flow estimatesbeyond year 2 to be acceptable.
72 Capital Asset Investment: Strategy, Tactics & Tools
The foregoing analysis illustrates the danger of relying on any single measure of investmentmerit. This applies to the NPV and the PI as it does to other measures. Some have emphasizedthe NPV–PI criterion to an extent tantamount to recommending it as a universal prescriptionfor capital-budgeting analysis. But we have just seen that the NPV alone does not providesufficient information to choose between two projects that have the same required investmentand useful lives. A formal treatment of risk in capital budgeting is deferred until the latterchapters. However, we must realize that risk is our constant companion whether we choose todeal with it formally or to ignore it.
On the premise that point estimates are subject to error, and that the expected error becomesgreater the further from the present the event we estimate will occur, ceteris paribus weprefer an investment that promises early recovery of funds committed, and early receipt offunds above this amount. In the applied world of capital investments, no single measureadequately captures the multifaceted character of capital-budgeting projects. For this reasoncompanies often examine several measures of a capital investment in their decision-makingprocesses.
RELATIONSHIP OF IRR AND NPV
Mao [98, p. 196] in his classical book has explicitly shown the relationship of IRR to NPV inan interesting way. Let us again examine equations (7.3) and (6.2).
0 =N∑
t=0
Rt (1 + r )−t (7.3)
NPV =N∑
t=0
Rt (1 + k)−t (6.2)
If we subtract (7.3) from (6.2), we obtain
NPV =N∑
t=0
Rt [(1 + k)−t − (1 + r )−t ] (8.5)
Now, taking any term beyond t = 0, we assume that k and r are both positive and the Rt
all nonnegative. For NPV to be positive it is necessary only for the relation r > k to be true.Conversely, for NPV to be negative, it is necessary that the relation r < k be true. This showsthe equivalence of the NPV and IRR criteria for projects that have Rt ≥ 0 for t > 0, in termsof the accept/reject decision.
ADJUSTED, OR MODIFIED, IRR
The controversy over whether or not the IRR should be used when there is doubt that theproject’s cash flows can be reinvested at the IRR led to the development of the adjusted IRR.The idea behind the adjusted IRR is to assume that all cash flows after the initial outlay oroutlays are invested to earn at the firm’s cost of capital or some other conservative reinvestmentrate. Thus all cash flows except for the initial outlay are taken out to a future value at the terminalyear of the project’s life, and zeroes are used to replace them in the adjusted series. After this
Reinvestment Rate Assumptions for NPV and IRR 73
Using cost of capital/reinvestment rate of 12.00%:
Project H: t � 0 t � 1 t � 2 t � 3 t � 4 t � 5Original flows:
Adjusted flows:
Adjusted IRR � 19.07% � [(16,752.50/7,000)(1/5)� 1] � 100%
Adjusted IRR � 18.73% � [(16,516.64/7,000)(1/5)� 1] � 100%
($7,000)
($7,000)
6,000 3,000 1,500 750 375
840.001,881.604,214.789,441.12
16,752.500 0 0 0
Project I: t � 0 t � 1 t � 2 t � 3 t � 4 t � 5Original flows:
Adjusted flows:
($7,000)
($7,000)
250 500 750 4,000 10,000
4,480.00940.80702.46393.38
16,516.640 0 0 0
Figure 8.2 Comparison using modified IRR
adjustment it is easy to calculate the implicit rate of return that would take the initial investmentto the terminal amount that contains the compounded sum of all cash flows from t = 1 throught = N .
The adjusted, or modified, IRR is analogous to the rate of return on a zero coupon bond.Since there are no cash flows to be reinvested, the rate at which the firm can reinvest cash is nolonger material to the rate of return. The procedure for calculating an adjusted IRR is clearlya type of sinking fund method. Sinking fund methods in general are covered in Chapter 11.
The modified IRR seems to have been more widely accepted by engineering economiststhan by finance writers. However, that situation may be changing.
The application of the adjusted IRR to projects H and I will serve to clarify the procedureinvolved. Figure 8.2 illustrates the method.
SUMMARY AND CONCLUSION
In this chapter we examined (1) the reinvestment assumptions implicit in the discounted cashflow methods, IRR and NPV, (2) the reasons for conflicting rankings between IRR and NPV,and (3) Mao’s treatment of the IRR–NPV relationship.
The reinvestment rate assumptions are seen to arise from the mathematical relationshipbetween present value and future value formulations of the discounted cash flow methods. Itis shown that the return on investment, that is, the remaining unrecovered initial investment, isnot dependent on reinvestment opportunities. To do this, the net cash flows are separated intotwo components: a payment of “interest” (return) on remaining “principal” (investment), anda repayment of “principal” (recovery of investment).
A treatment similar to that used to deal with the reinvestment assumptions is employed toanalyze, in conjunction with Fisher’s rate (rf), the reason for contradictory rankings between
74 Capital Asset Investment: Strategy, Tactics & Tools
IRR and NPV. It is seen that for rates less than rf the NPV alone is inadequate to judge which oftwo projects is better, since the NPV does not distinguish between the timing of cash receiptsto recover the investment and those that provide a net return.
Mao’s mathematical connection of IRR and NPV inside a single equation serves to illustratethe conditions under which the NPV is positive or negative, and to focus more clearly on theIRR–NPV relationship.
9
The MAPI Method
George Terborgh, Research Director of the Machinery and Allied Products Institute1 (MAPI),authored the classical book, Dynamic Equipment Policy. This work still provides perhapsthe most theoretically sound and practical means for analyzing component projects. About adecade after that treatise was published, a simplified and streamlined version of the method-ology was published in Business Investment Policy, many points of which were contained inAn Introduction to Business Investment Analysis, based on an address delivered in 1958 byTerborgh. That these publications are out of print and no longer available from the originalsource can be confirmed by a visit to the Manufacturers’ Alliance/MAPI website, which nolonger contains references to Terborgh or his works.2
In order to appreciate how Terborgh’s method relates to the other methods that employdiscounted cash flow, it will be useful if first the concept of duality is understood. Those familiarwith mathematical programming may skip the following section without loss of continuity.
THE CONCEPT OF DUALITY
One of the important contributions of mathematical programming is that of duality. Simplystated, duality means that, for every maximization problem, there corresponds a minimizationproblem which yields identical solution values. If the problem at hand, called the primalproblem, is one of maximization, the dual problem will always be a minimization problem.The converse is also true: if the primal problem is a minimization formulation, the dual willbe a maximization formulation. Furthermore, the dual of the dual will be the primal.
Let us examine the general linear programming problem:
Maximize p1q1 + p2q2 + · · · + pnqn
Subject to: a11q1 + a12q2 + · · · + a1nqn ≤ b1
a21q1 + a22q2 + · · · + a2nqn ≤ b2
··
am1q1 + am2q2 + · · · + amnqn ≤ bm
and for all iqi ≥ 0
(9.1)
1The name of the association has been changed, though the acronym MAPI still applies. “The Manufacturers Alliance/MAPI is anexecutive education and policy research organization serving the needs of industry. Founded in 1933 by capital goods manufactur-ers and known then as the Machinery and Allied Products Institute (MAPI), the corporate membership of 450 companies has beenbroadened over the years to encompass the full range of manufacturing industries, such as automotive, aerospace, computer, elec-tronics, chemical, machinery, and pharmaceutical, including manufacturers of a wide range of consumer products and businesses thatprovide related services such as telecommunications, power distribution, and software services.” From the Alliance’s web page athttp://www.mapi.net/html/research.cfm2I am sure many readers would agree that it is a sad commentary on today’s world that such remarkable and lucid works are relegatedto oblivion, but will not belabor the point here. The interested reader may find a copy to borrow or useful information on the methodat http://www.mises.org/wardlibrary detail.asp?control=6480 and http://www.albany.edu/∼renshaw/leading/ess05.html
76 Capital Asset Investment: Strategy, Tactics & Tools
which in the shorthand notation of matrix algebra becomes
Maximize p · q
Subject to A · q ≤ b
qi ≥ 0 for all i
(9.2)
This particular linear programming problem may be considered one of maximizing total firmprofit (price times quantity for each product) subject to constraints on the output of each productassociated with limitations on machine time, labor, and so on, and the relative requirements ofeach product for these limited resources.
If the above maximization problem has a feasible solution, the dual formulation will alsohave a feasible solution, and the dual solution will yield the same amounts of each product tobe produced. The corresponding dual problem is
Minimize b1u1 + b2u2 + · · · + bmum
Subject to: a11u1 + a21u2 + · · · + am1um ≥ p1
a12u1 + a22u2 + · · · + am2um ≥ p2
··
a1nu1 + a2nu2 + · · · + amnum ≥ pn
and for all iui ≥ 0
(9.3)
which in matrix algebra notation corresponding to the primal becomes
Minimize b′u
Subject to A′u ≥ p′
ui ≥ 0 for all i
(9.4)
where b′ is the transpose of the vector of constraint constants in the primal problem, and soon. The u variables are called shadow prices and represent the opportunity costs of unutilizedresources. Thus, if we minimize the opportunity costs of nonoptimal employment of ourmachine, labor, and other resources, we achieve a lowest cost solution.
The important point of all this is that the primal and the dual formulations both lead tothe same allocation of available resources. Therefore, it is not important which of the twoformulations we solve. For reasons of ease of formulation, calculation, or computer efficiency,we may choose to work with either the primal or the dual formulation.
Now, getting back to the MAPI method of capital-budgeting project evaluation, we may saythat the method is analogous to a dual approach to the goal of maximization of the value ofthe firm expressed in equation (2.1). It will be useful to again state equation (2.1) here as
Maximize V =∞∑
t=0
Rt
(1 + k)t(9.5)
If we recognize that the net cash flows Rt are composed of gross cash profits Pt as well ascash costs Ct , then
Rt = Pt − Ct (9.6)
For component projects we will assume that the Pt are fixed: we cannot directly associatecash inflows with the project. Even for major projects we may have to take the cash inflowsas given, since to some extent they are beyond our control, at least as far as finance and
The MAPI Method 77
production are concerned. Marketing staff may influence gross revenues through advertising,salesmanship, and marketing logistics, but still they will be subject to the state of the generaleconomy, the activities of our competition, and “acts of God.”
If we take the Pt as given, or fixed, and is thus independent of the capital equipment usedin production, then we may reformulate our objective as
Minimize cost =∞∑
t=0
Ct
(1 + k)t(9.7)
which may be made operational by accepting the lowest cost projects that can satisfactorilyperform a given task. This is the basic idea behind the MAPI method, which we will nowexamine.
THE MAPI FRAMEWORK
The methodology of the MAPI method involves calculating the time-adjusted annual averagecost of the project or projects under consideration. However, several concepts vital to intelligentapplication of the method must first be understood.
Challenger and Defender
In Terborgh’s colorful terminology, the capital equipment currently in use is referred to asthe defender, and the alternative that may be considered for replacement as the challenger.The MAPI method as originally developed emphasized capital equipment replacement, butis applicable to nonreplacement decisions as well since, in such cases, the status quo may beconsidered the defender. Various potential challengers may be compared against one anotherin a winnowing process, with the project promising the lowest time-adjusted annual averagecost selected as the challenger.
If the challenger is superior to the defender as well as to presently available rivals, it maystill be inferior to future alternatives. The current challenger is the best replacement for thedefender only when there is no future challenger worth delaying for. This requires that aseries of capital equipment not currently in existence be appraised, and this presents somedifficulties:
Now obviously it is impossible as a rule for mere mortals to foresee the form and character of machinesnot yet in existence. In some cases, no doubt, closely impending developments may be more or lessdimly discerned and so may be weighted, after a fashion, in the replacement analysis, but in no casecan the future be penetrated more than a fraction of the distance that is theoretically necessary for anexact, or even a close solution of the problem. What then is the answer? Since the machines of thefuture cannot be foreseen, their character must be assumed [151, pp. 57–58].
The exact nature of the necessary assumptions is dependent on some additional terminology,which will be introduced at this point.
Capital Cost
The mechanics of the MAPI method require that we obtain the time-adjusted annual averagecost of the project under consideration. The two components that determine what the averagecost will be are: (1) capital cost and (2) operating inferiority. Capital cost must not be confused
78 Capital Asset Investment: Strategy, Tactics & Tools
with the firm’s cost of capital, because they are quite different things despite the similarity inlabels.
Capital cost in the MAPI framework is the uniform annual dollar amount, including theopportunity cost of funds, that are tied up in the project and must be recovered if the projectis retained for either one year, or for two years, or for three years, and so on. To clarifythis concept, assume we are examining a project that costs $10,000, and that our firm has a15 percent annual cost of capital. Our opportunity cost for funds committed to the project is atleast 15 percent annually since, if funds were not committed to it, our cost of funds would belower by this amount. That is, our firm would require $10,000 less, for which it is incurring anannual 15 percent cost.
Now, if we were to accept this project, and then at the end of one year abandon it, whatis the capital cost? It is $10,000 plus the opportunity cost at 15 percent, or $11,500. What ifthe project is abandoned at the end of the second year? In this case the total capital cost is$10,000(1.15)2 or $13,225. However, since the project will be kept for two years, the annualcapital cost will be much less, not even one-half of the $13,225. The reason for this is thatfunds recovered in the first year do not incur opportunity cost during the second year. At a15 percent annual cost of capital, the time-adjusted annual average capital cost in each of thetwo years will be $6151. If the project were retained for three years, the time-adjusted annualaverage cost will be $4380. These amounts are obtained by multiplying the initial investmentby the capital recovery factor corresponding to the annual percentage cost of capital and thenumber of years the project is retained. The capital recovery factor is the reciprocal of theordinary annuity (uniform series) present worth factor.
The longer a project is kept in service, the smaller the amount of investment that must berecovered in each individual year of the project’s life. A project that is retained only one ortwo years must therefore yield a larger cash flow each year to allow recovery of the initialinvestment, plus opportunity cost of the committed funds, than the same project if kept formany years.
In the present value and internal rate of return methods for capital-budgeting project eval-uation, the initial investment is considered only at time period zero. In the MAPI method, theinitial investment is spread over the years of the project’s life. With the NPV method, since thediscount rate is assumed to be known, that is, it is the firm’s cost of capital, it is possible thatthe initial investment could be treated in the same way as in MAPI. However, in practice it isnot treated that way.
Operating Inferiority
Operating inferiority is defined as the deficiency of the defender, the incumbent, existing projector the status quo, relative to the best available alternative for performing the same functions.Operating inferiority is considered to be composed of two components: physical deteriorationand technological obsolescence. In the MAPI method we measure operating inferiority usingthe best capital equipment as a benchmark:
In the firmament of mechanical alternatives there is but one fixed star: the best machine for the job.This is base point and the standard for evaluating all others. What an operator can afford to pay forany rival or competitor of this machine must therefore be derived by a top-down measurement. Butthe process is not reversible. He cannot properly compute what he can afford to pay for the best bymeasurement upward from its inferiors [151, p. 35].
The MAPI Method 79
Physical Deterioration
Physical deterioration of capital can be determined by comparing the equipment in service withthe same equipment when new and undegraded by past operation. Physical deterioration, then,will be the excess of the operating cost of the old machine over its new replica’s operating cost.We normally would expect rapid physical deterioration in the first few years of a project’s life,then for it to accumulate more slowly, perhaps reaching a steady-state equilibrium, with repaircosts at a relatively constant level per period, keeping the quality of service approximatelyconstant. The comparison of the equipment in service with its new replica should be in termsof operating costs, including maintenance and repair, additional direct labor required, extraindirect labor required (such as for quality control inspection), and the cost of higher scrapoutput.
Technological Obsolescence
Although physical deterioration may be considered an internal, age-related aspect of capitaldegradation, technological obsolescence is external to it and not necessarily related to age.Obsolescence consists of the sum of the excess operating cost of the same capital that is newover that of the best alternative now available plus the deficiency in the value of service relativeto the best alternative. Physical deterioration is degradation of the firm’s existing capital relativeto new, identical capital. Technological obsolescence is the inferiority of the existing capitalrelative to the latest generation of capital for doing the job.
Two Basic Assumptions
Unfortunately, although the concept of operating inferiority is not difficult to grasp, imple-mentation poses some problems. Although we may be able to estimate the cost of operatinginferiority for this year, and perhaps the next as well, the task becomes increasingly difficultand tenuous as we attempt to carry the process into the future. Physical deterioration may notoccur uniformly: it may be substantial in the early years, tapering off later, or it may be just theopposite. Technological developments tend to occur randomly over time. Although some maybe anticipated in advance in rapidly developing fields, such as computer technology, predictionof when they will be brought to market is still a somewhat uncertain enterprise.
In addition to the problems inherent in estimating operating inferiority, there is yet anotherobstacle. This is related to the characteristics of future challengers:
It is true that the challenger has eliminated all presently available rivals. But it has not eliminatedfuture rivals. The latter, though at present mere potentialities, are important figures in the contest. Forthe current challenger can make good its claim to succeed the defender only when there is no futurechallenger worth waiting for. It must engage, as it were, in a two-front war, attacking on one side theaged machine it hopes to dislodge and on the other an array of rivals still unborn who also hope todislodge the same aged machine, but later [151, p. 55].
The MAPI analysis emphasizes the importance of future capital equipment:
For since the choice between living machines can be made only by reference to the machines oftomorrow, the latter remain, whether we like it or not, an indispensable element in the calculation. Itmay be said . . . that the appraisal of the ghosts involved is the heart of the . . . analysis. No replacementtheory, no formula, no rule of thumb that fails to take cognizance of these ghosts and to assess theirrole in the play can lay claim to rational justification [151, p. 57].
80 Capital Asset Investment: Strategy, Tactics & Tools
In order to deal with these problems Terborgh proposed two “standard” assumptions on thebasis that “The best the analyst can do is to start with a set of standard assumptions and shadethe results of their application as his judgment dictates [151, p. 60].” In other words, if we haveinformation we will use it; if not, we will employ reasonable standard assumptions.
Adverse Minimum
The key to the MAPI method is the “adverse minimum” for the capital project. This is definedas the lowest combined time-adjusted average of capital cost and operating inferiority thatcan be obtained by keeping the project in service the number of years necessary to reach thisminimum, and no longer.
First Standard Assumption
Future challengers will have the same adverse minimum as the present one [151, p. 64].
Second Standard Assumption
The present challenger will accumulate operating inferiority at a constant rate over its servicelife [151, p. 65].
The first standard assumption is justified on the basis that there is no alternative that is morereasonable. In the absence of information to the contrary, what compulsion is there for us toassume that future challengers will have either higher or lower adverse minima? If we haveinformation that leads us to believe that future challengers will have different adverse minima,then we may modify the standard assumption. Furthermore, this standard assumption facilitatesdeveloping a simpler replacement formula than would be otherwise possible.
The second standard assumption again is justified on the basis of methodological necessity,since the analyst typically does not have data on a sufficient sample size of similar equipmentto make a more reasonable assumption. In the absence of information to the contrary, the bestwe can do is predict the future by extrapolation from the present and past experiences on thebasis that there are elements of continuity and recurrence that will be repeated into the future.If we were to reject entirely this continuity and recurrence over time, we would be utterlyincapable of dealing with the future in all but those situations in which change is at least dimlyvisible on the horizon.
APPLICATION OF THE MAPI METHOD
In order to apply the MAPI method to a potential challenger, we require an estimate of the firm’scost of capital, the cost of the project, and an estimate of the project’s first-year accumulationof operating inferiority. From this information we derive the adverse minima of potentialchallengers, thereby selecting the project with the lowest adverse minimum as the challenger.The adverse minimum of the defender, if there is existing capital equipment that may bereplaced by the challenger, is determined similarly. In many cases the defender will be foundto have already passed the point in time at which its adverse minimum occurs. In such casesTerborgh has recommended that the next-year total of capital cost and operating inferiority beused as the defender’s adverse minimum.
The MAPI Method 81
Table 9.1 Derivation of the adverse minimum of a challenger costing $100,000 with inferioritygradient of $7000 a year. Assumes no capital additions and no salvage value. Cost of capital is 15%
(4)(2) (3) Present (5)
(1) Present Present worth of CapitalOperating worth worth of operating recovery (6) (7) (8)
Year inferiority factor operating inferiority, factor Operating Cost Bothof for year for year inferiority accumulated for year inferiority (5) × combinedservice indicated indicated (1) × (2) (3) accumulated indicated (4) × (5) $100,000 (6) + (7)
1 $0 0.86957 $0 $0 1.15000 $0 $115,000 $115,0002 7,000 0.75614 5,293 5,293 0.61512 3,256 61,512 64,7683 14,000 0.65752 9,205 14,498 0.43798 6,350 43,798 50,1484 21,000 0.57175 12,007 26,505 0.35027 9,284 35,027 44,3115 28,000 0.49718 13,921 40,426 0.29832 12,060 29,832 41,8926 35,000 0.43233 15,132 55,558 0.26424 14,681 26,424 41,105a
7 42,000 0.37594 15,789 71,346 0.24036 17,149 24,036 41,1858 49,000 0.32690 16,018 87,364 0.22285 19,469 22,285 41,7549 56,000 0.28426 15,919 103,283 0.20957 21,645 20,957 42,602
10 63,000 0.24718 15,572 118,855 0.19925 23,682 19,925 43,60711 70,000 0.21494 15,046 133,901 0.19107 25,584 19,107 44,69112 77,000 0.18691 14,392 148,293 0.18448 27,357 18,448 45,805
a Adverse minimum.
Let us assume the firm’s cost of capital is 15 percent per annum, and that we have anexisting machine (defender) that will have a combined capital cost and operating inferiority of$70,000. There is only one potential challenger. It costs $100,000 and is estimated to accumulateoperating inferiority during the first year of service amounting to $7000. Application of thesecond standard assumption means we assume operating inferiority will be accumulated at$7000 each year the challenger would be in service. We ignore salvage value for now tosimplify the exposition. Table 9.1 illustrates the technique of finding the challenger’s adverseminimum.
Since the adverse minimum of the challenger of $41,105 per year if held six years issubstantially lower than the next-year operating inferiority of the defender, we would replacethe defender this year. In fact, even if the challenger were to be replaced itself at the end of thesecond year of its service by a still better but yet unbuilt new challenger, the decision wouldstill be correct. If kept in service two years, the challenger costs $64,768,whereas the defenderwill cost $70,000 next year and, if it continues to accumulate operating inferiority, still morethe following year. The figures in column 8 of Table 9.1 are average annual costs, adjusted forthe time-value of money. The lowest of these, as stated earlier, is the adverse minimum.
The time-adjusted annual average costs are composed of a capital cost component thatdeclines over time, and an operating inferiority component that increases over time. For theproject analyzed in Table 9.1, capital cost declines rapidly in the early years of the project’s life,whereas operating inferiority rises rapidly. Therefore, this challenger will obtain its adverseminimum in only six years. Some projects may not reach their adverse minima for many years.For instance, management may have a policy of abandoning equipment at the end of, say,15 years regardless of its condition. In such an environment the adverse minimum may notbe reached. However, if the time-adjusted annual average cost declines constantly, we may
82 Capital Asset Investment: Strategy, Tactics & Tools
Table 9.2 Comparison of alternatives
Adverse Annual productionMachine minimum capability Cost per unit
Defender $50,000 10,000 $5.00Alternative A 500,000 150,000 3.33Alternative B 1,000,000 400,000 2.50
120,000
100,000
80,000
60,000
40,000
20,000
01 2 3 4 5 6 7 8 9 10 11 12
Period of Service (Years)
Dol
lars
Capital Cost Operating Inferiority Both Combined
Adverse Minimum
Figure 9.1 Graphic of MAPI method (using data from Table 9.1)
then take the last year of service’s value as our adverse minimum. Of course, as we go intothe future further and further, the reliability of the two standard assumptions employed in thebeginning begins to wane. Thus, other things being equal, we would prefer a challenger thatreached its adverse minimum within just a few years to one that took 15 or 20 years to reach it.
Figure 9.1 illustrates graphically how the adverse minimum is determined as the minimumpoint on the total cost function, which is the vertical sum of the individual costs of the componentfunctions for capital cost and for operating inferiority. Readers who are familiar with thederivation of the basic economic order quantity model will note some similarity of the graphs.
The Problem of Capacity Disparities
The existence of alternatives that provide for different production capacities requires somemodification to the MAPI method. Assume that the defender, having a next-year operatinginferiority of $50,000, is capable of producing 10,000 units of output annually. Assume furtherthat potential challengers A and B have respective adverse minima of $500,000 and $1 millionand annual production capabilities of 150,000 and 400,000 units.
At first glance it would appear that the defender should not be replaced, for it has the lowestadverse minimum (next-year operating inferiority). However, if we express the adverse minimain terms of the units of annual production, we find that alternative B promises the lowest costper unit (see Table 9.2).
The MAPI Method 83
Now we see that the ordering of the per unit costs is the opposite of the ordering of theadverse minima for the three alternatives. Which should be selected? The answer depends onthe firm’s annual production requirements. For instance, if the firm requires that 35,000 unitsbe produced each year, then the cost per unit will be $14.28 with alternative A and $28.57 withalternative B. The defender alone cannot produce more than 10,000 units annually. However, ifexact duplicates of the defender could be acquired (which will have the same adverse minimumas the defender), then we would need to add three machines. This would raise capacity to 40,000units in total — 5000 more than required. The cost per unit would be $5.71, which is muchless than the per unit costs of machines A and B.
Therefore, adverse minima in themselves are meaningless unless the alternative machineshave the same annual productive capacity. If they do not, it is necessary to make adjustments.Cost should be expressed in terms of cost per unit, since to do otherwise may lead to improperselection of the challenger and to a wrong replacement decision. Furthermore, the cost per unitmust be based on the firm’s requirements, not on the rated machine capacities, which may belesser or greater than the production required by the firm.
CONCLUSION
The Terborgh method of capital equipment analysis provides an alternative means of investmentevaluation that is based on minimization of costs. It is thus, in a sense, a dual formulation ofmethods based on maximization of some measure of investment return such as the discountedcash flow (DCF) measures.
Because the Terborgh method is based on cost minimization, it is suitable for analysis ofprojects where the customary DCF measures are much more difficult, if not impossible, toapply. Such projects are those this author has defined as component projects — they do nothave cash revenues directly attributable to them alone. For these projects the cash inflows maybe assumed invariant with respect to the production equipment employed, whereas cash costswill vary directly with respect to the choice of equipment. The Terborgh method, unlike theDCF methods, requires no estimates of cash inflows. Instead it requires cost estimates thatoften may be provided more easily, and provided by those personnel whose experience inproduction promises they may be the best obtainable estimates. Conversely, revenue estimatesfor component projects are likely to be based on tenuous premises if not pure guesses.
Proper application of the Terborgh method requires that the adverse minima of alternativeprojects be adjusted to reflect production capacity differences or the firm’s production require-ments. Otherwise the per unit cost of one alternative may be greater than that of another eventhough it may have a lower adverse minimum.
10
The Problem of Mixed Cash Flows: I
When the internal rate of return (IRR) was discussed a restriction was placed on the cashflows that assured there would be only one real IRR in the range of −100 percent to +∞. Therestriction was that there would be only one change in the sign of the flows. In this chapterthat restriction is removed, the consequences examined, and a method analyzed that providesa unique measure of return on investment.
INTERNAL RATE OF RETURN DEFICIENCIES
Under certain circumstances the IRR is not unique, and thus we have to decide which, if any,of the IRRs is a correct measure of return on investment. As we shall see, when there are twoor more IRRs, none of them is a true measure of return on investment.
This difficulty with IRR arises because of mixed cash flows. We define mixed cash flowsas a project cash flow series that has more than one change in arithmetic sign. There must, ofcourse, be one change in sign for us to find any IRR. However, when there is more than onechange in sign, the IRR, even if unique, sometimes will not measure the return on investment.
Example 10.1 Let us consider the following capital-budgeting project, for which we havemixed cash flows:
t = 0 t = 1 t = 2
−$100 +$320 −$240
This project has two IRRs: 20 percent and 100 percent. The net present value (NPV) is positivefor any cost of capital greater than 20 percent but less than 100 percent. Figure 10.1 illustratesthe NPV function for this project. The NPV reaches a maximum of $6.25 for a cost of capitalof 60 percent. The profitability index (PI) of 0.0625 (or 1.0625 by the common, alternativedefinition of PI) is not likely to cause much enthusiasm, but let us retain this example forfurther analysis.
DESCARTES’ RULE
Descartes’ rule of signs states that the number of unique, positive, real roots to a polynomialequation with real coefficients (such as the equation for IRR) must be less than or equal to thenumber of sign changes between the coefficients. If less than the number of sign changes, thenumber of positive,1 real roots must be less by an even number, since complex roots come inconjugate pairs, and an nth degree polynomial will have n roots, not necessarily distinct.
1We are interested in positive, real roots because, in solving any given polynomial, we will find the values of x = 1 + r, where r is theIRR value in decimal form. Therefore, after solving for the roots of the IRR polynomial, we must make the transformation r j = x j − 1for each root, j = 1, . . . , n. Since we work with all net cash flows attributable to the particular project, we therefore cannot lose morethan 100 percent of our investment. Thus limiting the r j to ≥ −1.00 restricts the x j to ≥ 0.
86 Capital Asset Investment: Strategy, Tactics & Tools
k
4
6
8
2
0
−2
−4
−6
−8−10
$NPV
0% 20% 40% 60% 80% 100% 120% 140% 160%
Figure 10.1 Net present value
Example 10.2 The following cash flows yield a third-degree IRR polynomial equation:
t = 0 t = 1 t = 2 t = 3
−$1000 +$3800 −$4730 +$1936
The IRR equation is (dropping $):
−1000 + 3800(1 + r )−1 − 4730(1 + r )−2 + 1936(1 + r )−3 = 0 (10.1)
or, alternatively, by multiplying by (1 + r )3 to put into future value form, then dividing by 1000and letting x = 1 + r:
1x3 − 3.8x2 + 4.73x − 1.936 = 0 (10.2)
This equation has three real roots, one with a multiplicity of two (double root). The rootsare 10 percent, 10 percent, and 60 percent, which we can verify by generating an equationby multiplication and then comparing it with equation (10.2). If r = 10 percent = 0.10, thenx = 1.1 and x − 1.1 is a zero (root) to the equation. Similarly, x = 1.6 is a zero to the equation.Multiplying, we obtain
x − 1.1
x − 1.1
x2 − 2.2x + 1.21
x − 1.6
x3 − 2.2x2 + 1.21x
− 1.6x2 + 3.52x − 1.936
x3 − 3.8x2 + 4.73x − 1.936
which is identical to equation (10.2).
The Problem of Mixed Cash Flows: I 87
We shall leave the matter of multiple roots now, because, when a project has mixed cashflows, even a unique, real IRR is no assurance that the IRR is a correct measure of investmentreturn.
THE TEICHROEW, ROBICHEK, AND MONTALBANO(TRM) ANALYSIS
Teichroew, and later Teichroew, Robichek, and Montalbano (TRM) [148–150] formally ex-plained the existence of multiple IRRs and proposed an algorithm for determining a uniquemeasure of the return on invested capital (RIC). James C. T. Mao was among the first to offera lucid summary of the main points of the TRM work [97, 98], and his work is still a valuablereference on the topic.
The analysis by TRM demonstrated that projects with mixed cash flows may often be neitherclearly investments nor clearly financing projects. For example, if the firm makes a loan thecash flow sequence will be − + + · · · + + if the loan is amortized with periodic payments.This is identical to a capital investment with the same net cash flows. From the viewpoint of theborrower (or the capital asset, if we can attribute a viewpoint to it), the cash flow sequence willbe the negative of our firm’s: + − − · · · − −. Depending on whether we view the cash flowsequence through the firm’s eyes or those of the borrower, we have what is unambiguouslyeither an investment or a financing project. Since there is only one change in sign between thecash flows, we know that the IRR will be unique.
Now, what if we have instead a cash flow sequence of − + + − + or + − − + −+?Can we say a priori that we have an investment or a financing project based only on examina-tion of the signs of the cash flows? The answer is no, we cannot. The TRM analysis recognizesthat some projects with mixed cash flows have attributes of both investment and financingprojects, while others are purely investments. The returns on projects that have characteristicsof both investment and financing projects are not, as the IRR method assumes, independentof the firm’s cost of capital. To understand the TRM analysis, we need to define some terms.
Let
a0, a1, . . . , an
denote the project cash flows. And let
st (r ) =t∑
i=0
ai (1 + r )t−i , 0 ≤ t < n
and
st (r ) = (1 + r )st−1 + at
denote the project balance equations, and
sn(r ) =n∑
i= 0
ai (1 + r )n−i
denote the future value of the project.The minimum rate is rmin for which all the project balance equations are less than or equal
to zero: st (rmin) ≤ 0 for 0 < t < n.
88 Capital Asset Investment: Strategy, Tactics & Tools
The project balance at the end of period t, at rate r, is interpreted as the future value of (1) theamount the firm has invested in the project or (2) the firm has received from the project fromperiod zero to the end of period t. By using TRM’s classifications, at rate r, the following occur:
1. If st (r ) ≤ 0 for 0 ≤ t < n, we have a pure investment project.2. If st (r ) ≥ 0 for 0 < t < n, we have a pure financing project.3. If st (r ) ≤ 0 for some t, st (r ) > 0 for some t, and 0 ≤ t < n, we have a mixed project.
A simple project is one in which the sign of a0 is different from the sign of ai for all i > 0. Ina mixed project, the firm has money invested in the project during some periods, and “owes”the project money during some other periods.
It can be shown that all simple investments are pure investments. However, the converse isnot true: not all pure investments are simple investments.
Let us follow TRM’s notation in using PFR to denote the project financing rate, the rateapplied for periods in which the project can be viewed as providing funds to the firm; that is,as a net financing source, with positive project balance. We use k, the firm’s cost of capital forPFR, and PIR to denote the project investment rate, r∗ (TRM use r for this), the rate that theproject yields when the project balance is negative. We also refer to the PIR as the RIC, thereturn on invested capital.
To determine the PIR, or RIC, we proceed as follows, first negating all cash flows if a0 > 0:
s0(r, k) = a0
s1(r, k) = (1 + r )s0 + a1 if s0 < 0
= (1 + k)s0 + a1 if s0 ≥ 0
s2(r, k) = (1 + r )s1 + a2 if s1 < 0
= (1 + k)s1 + a2 if s1 ≥ 0···
sn(r, k) = (1 + r )sn−1 + an if sn−1 < 0
= (1 + k)sn−1 + an if sn−1 ≥ 0
In order to find whether, for any j (0 < j ≤ n), s j (r, k) < 0 or ≥ 0, we substitute rmin for r inevaluating it. Since we will use rmin an estimate of the firm’s cost of capital for k, the onlyunknown in sn(r, k) is r. We solve this equation for r, and since the solution is a particularvalue, the RIC, we refer to it as r∗.
Note that because rmin is defined as the smallest real root for which all the project balanceequations, st (rmin), are ≤ 0, with 0 j t < n , it is an which determines whether the project ispure or mixed. If tsn(rmin) ≥ 0, a greater discount rate r, one for which sn(rmin) = 0, willretain the condition that st (r) ≤ 0 for 0 ≤ t < n. In this case, the r for which sn(r) = 0 willbe the r∗ of the project. It will also be the IRR of the project. From this it follows that theIRR is found by assuming that the project financing rate equals the project investment rate, sothat k does not enter the equation for IRR. In general, this will not be a correct assumption.However, it does not affect our results when sn(r, k) = sn(r), a condition we have for all pureinvestments. In other words, for pure investments only, r∗ = IRR, and k does not affect r∗: theIRR is “internal” to the project.
The Problem of Mixed Cash Flows: I 89
THE TRM ALGORITHM
The foregoing leads to an algorithm for determining the RIC on an investment, either simpleor with mixed cash flows. The steps of the algorithm are:
1. If a0 > 0, negate all cash flows before beginning. Find rmin, the minimum real rate for whichall the project balance equations, st (rmin) are ≤ 0, for 0 ≤ t < n.
2. Evaluate sn(rmin).(i) If sn(rmin) ≥ 0, then st (r, k) = st (r ) and r∗ equals the unique IRR as traditionally
found.(ii) If sn(rmin) < 0, then we proceed to step (3).
3. Let k be the firm’s cost of capital.
s0 = a0
s1 = (1 + r )s0 + a1 if s0 < 0
= (1 + k)s0 + a1 if s0 ≥ 0
s2 = (1 + r )s1 + a2 if s1 < 0
= (1 + k)s1 + a2 if s1 ≥ 0···
sn = (1 + r )sn−1 + an if sn−1 < 0
= (1 + k)sn−1 + an if sn−1 ≥ 0
In every st (r, k), use rmin for r to determine whether the project balance is less than zeroor greater than or equal to zero.
4. Solve sn(r, k) for unique r. Call this r∗ the return on invested capital.
Note that the return on invested capital r∗ may be the IRR, but in general will not be.
Example 10.3 Let us now take up the project discussed earlier, which had cash flows:
t = 0 t = 1 t = 2
−$100 +$320 −$240
and apply the TRM algorithm.
1. Find rmin:
−100(1 + r ) + 320 = 0
r = 3.2 − 1 = 2.2 or 220%
rmin = r
(In this example there is only one project balance equation.)2. Evaluate
sn(rmin) = −100(1 + r )2 + 320(1 + r ) − 240
= −240 and −240 < 0
So we have a mixed investment.
90 Capital Asset Investment: Strategy, Tactics & Tools
3. Let k be the firm’s cost of capital:
s0 = a0 = −100 < 0
s1 = s0(1 + r ) + a1 since s0 < 0
= 0 when evaluated for r = rmin
s2 = s1(1 + k) + a2 since s1 ≥ 0
= −100(1 + r )(1 + k) + 320(1 + k) − 240
4. Solve for r∗ = r:
1 + r = 320(1 + k) − 240
100(1 + k)
r∗ = r = 2.2 − 2.4
1 + k
Under the IRR assumption of r = k, we find that
r = 2.2 − 2.4
1 + r
r2 − 1.2r + .2 = 0
and, using the quadratic formula,
r = 1.2 ± √1.44 − .8
2= 1.2 ± .8
2= 1.0, 0.20
= 100 percent, 20 percent
What if r = k? Figure 10.2 shows the function for r∗ in terms of k. The figure shows clearlythat for only two values of k will r∗ = k for this project. Under the rule that we accept aproject if r∗ > k and reject if r∗ < k, this project is acceptable for the same values of k thatwe found with the NPV criterion.
k
160.00%
140.00%
120.00%
100.00%
80.00%
60.00%
40.00%
20.00%
0.00%
−20.00%
r *
0.00% 50.00% 100.00% 150.00% 200.00% 250.00%
Figure 10.2 r∗ as function of k for Example 10.3
The Problem of Mixed Cash Flows: I 91
Example 10.4 For a second application of the TRM algorithm let us take a project consideredearlier that had cash flows:
t = 0 t = 1 t = 2 t = 3
−$1000 +$3800 −$4730 +$1936
1. Find rmin:(i)
−1000 (1 + r ) + 3800 = 0
r1 = 3.8 − 1 = 2.8 or 280%
(ii)
−1000(1 + r )2 + 3800(1 + r ) − 4730 = 0
r2 = −3.8 ±√
(3.8)2 − (4)(4.73)
−2
has complex roots. Therefore, rmin = 2.8 = r1.2. Evaluate sn(rmin):
sn(rmin) = −1000(3.8)3 + 3800(3.8)2 − 4730(3.8) + 1936 = −16,038 < 0
Hence this is a mixed project.3. Let k be the firm’s cost of capital:
s0 = a0 = −1000 < 0
s1 = s0(1 + r ) + a1 = 0
= −1000(1 + r ) + 3800 = 0 at r = rmin
s2 = s1(1 + k) + a2
= −1000(1 + r )(1 + k) + 3800(1 + k) − 4730 < 0 at r = rmin
s3 = −1000(1 + r )2(1 + k) + 3800(1 + r )(1 + k) − 4730(1 + r ) + 1936
4. Solve for r∗ = r .This yields a fairly complicated expression for r in terms of k, although for a specific valueof k, the solution can be easily accomplished with the quadratic formula. The expressionfor k in terms of r is
k = 1.936 − 4.73(1 + r )
(1 + r )2 − 3.8(1 + r )− 1
from which we may generate values of k corresponding to various r∗ = r and plot thefunction (see Figure 10.3).
Note that r∗ is a double-valued function of k. Since we cannot lose more than we haveinvested in the project,2 values of r < −100 percent are not economically meaningful andmay thus be ignored. We cannot lose more than 100 percent of what we invest in the project,because the cash flows reflect the total effect of the project on the firm; all costs and revenuesattributable to the project are incorporated in the cash flows. At 10 percent, a double root to
2We are still assuming project independence in this chapter.
92 Capital Asset Investment: Strategy, Tactics & Tools
k
150.00%
100.00%
50.00%
0.00%
−50.00%
−100.00%
r *
−150.00% −100.00% −50.00% 0.00% 50.00% 100.00% 150.00%
Figure 10.3 r∗ as function of k for Example 10.4
k
15.00
10.00
5.00
0.00
−5.00
−10.00
−15.00
$NPV
0.00%150.00% 20.00% 40.00% 60.00% 80.00%
Figure 10.4 NPV function for Example 10.4
the IRR equation, it is interesting that the function touches, but does not cross, the r∗ = k-axis. This project is acceptable for 0 ≤ k < 10 percent and 10% < k < 60%, the same asby the NPV criterion. The NPV function is shown in Figure 10.4.
Example 10.5 Let us now solve for the RIC of a project having cash flows in six periods:
t = 0 t = 1 t = 2 t = 3 t = 4 t = 5
−$100 +$600 −$1509 +$2027 +$1436 +$418
This project has real IRR values of 0 percent, 10 percent, and 90 percent. The IRR equationalso has complex roots of 0 + i and 0 − i .
1. Find rmin:(i) −100(1 + r ) + 600 = 0
r1 = r = 5 or 500 percent
The Problem of Mixed Cash Flows: I 93
(ii)
−100(1 + r )2 + 600(1 + r ) = 1509 = −100(1 + r )2 + 600(1 + r ) − 1509 = 0
r = −600 ±√
(600)2 − (4)(−100)(−1509)
−200
Since r is a complex root, skip r2.(iii) We can approach this in two ways. First, we could solve the equation
−100(1 + r )3 + 600(1 + r )2 − 1509(1 + r ) + 2027
after setting it equal to zero. This is a laborious process, conducive to errors in calcu-lations, unless we use a computer program. Even in this case there may be time lost inaccessing a computer and waiting for results. Once we did find r we would set r3 = r.The second approach may save us this trouble.
Let us substitute the rmin thus far obtained: r1 = 500 percent. If this equation valueis less than zero for r1 , we do not need to solve for the value of r3. Using this approach,we find the equation value is − 7027, so we need not solve for r3.
(iv) As in (iii), we could solve for r in the equation
−100(1 + r )4 + 600(1 + r )3 − 1509(1 + r )2 + 2027(1 + r ) − 1436
after setting it equal to zero. But again, let us first try the second approach. We obtaina value of −43,595; so again we need not solve for r4.
We have found that rmin = r1 = 500 percent, since for this rate, and no lesser rate, allproject balance equations are less than or equal to zero.
2. Evaluate sn(rmin):
sn(rmin) = −100(6)5 + 600(6)4 − 1509(6)3 + 2027(6)2 − 1436(6) + 418
= −261,152 < 0
So this is a mixed project.3. Let k be the firm’s cost of capital:
s0 = a0 = −100
s1 = −100(1 + r ) + 600 = 0 at r = rmin
s2 = −100(1 + r )(1 + k) + 600(1 + k) − 1509 < 0
s3 = −100(1 + r )2(1 + k) + 600(1 + r )(1 + k) − 1509(1 + r ) + 2027 < 0
s4 = −100(1 + r )3(1 + k) + 600(1 + r )2(1 + k) − 1509(1 + r )2 + 2027(1 + r )
−1436 < 0
s5 = −100(1 + r )4(1 + k) + 600(1 + r )3(1 + k) − 1509(1 + r )3 + 2027(1 + r )2
−1436(1 + r ) + 418
4. Solve for r∗ = r .Because this involves solution of a fourth-degree equation, we will present the solutions
for k = 15 percent and k = 25 percent. At k = 15 percent, r∗ = r = 15.08 percent, and theproject is marginally acceptable. At k = 25 percent, r∗ = r = 25.28 percent, and againthe project is marginally acceptable.
94 Capital Asset Investment: Strategy, Tactics & Tools
THE UNIQUE, REAL INTERNAL RATE OF RETURN:CAVEAT EMPTOR!3
The internal rate of return (IRR), even when unique and real, may nevertheless be an incorrectmeasure of the return on investment. All projects characterized by negative flows occurringonly at the beginning and the end will be mixed investments for which the IRR, whether uniqueand real or not, is not a correct measure of investment return. Several years ago, W. H. Jean[76] proved that for capital budgeting projects in which only the first and last cash flows werenegative that there would be a unique, real, positive internal rate of return or no positive IRR.J. Hirschleifer [69] subsequently showed that if the sum of cash flows beyond the first was lessthan or equal to the first cash flow, then multiple IRRs can exist for the project. This promptedProfessor Jean [77] to extend his treatment, and further specify the conditions for unique IRRfor such cases as Hirschleifer cited. Although Jean’s results are mathematically interesting, theydo not take into account the way in which such projects violate the assumption of independencebetween the IRR and the firm’s cost of capital, which destroys the economic meaning of theresulting IRR. The conditions under which the IRR of a project is not independent of the firm’scost of capital have been widely ignored in the literature, one noteworthy exception beingMao’s out-of-print text. Jean’s article and examples are cited here for purposes of illustration,since his article is mathematically rigorous.
Mathematical uniqueness of a real root to the traditional IRR equation, although a necessarycondition, is not sufficient to insure that one has obtained a rate independent of the firm’s costof capital, and thus a measure of investment return “internal” to the cash flow of the project.In fact, we shall prove that in the case of a project with negative flows only at beginning andend, for which Professor Jean proved a unique, positive IRR can always be found, the IRR willnever (excluding rare or contrived cases in which the firm’s cost of capital is the same as therate rmin, which is discussed later) be independent of the firm’s cost of capital.
A theorem will be proved later that has two corollaries relating to the discussion of professorsJean and Hirschleifer. First, however, the examples provided by Jean and Hirschleifer will beanalyzed within the framework provided by TRM. A cost of capital k = 10 percent will beassumed for all cases.
Case 1
Cash flows are −1, 5, −6.
IRR = −100%, 200%
rmin = 400%
sn(rmin) = −6 < 0
Hence this is a mixed investment.
s0 = a0 = −1 < 0
s1 = s0(1 + r∗) + a1
= −1(1 + r∗) + 5 = 0 (using rmin for r∗)
s2 = s1(1 + k) + a2
= −1(1 + r∗)(1 + k) + 5(1 + k) − 6
3Reprinted with permission of Journal of Financial and Quantitative Analysis, Copyright c© 1978. With corrections.
The Problem of Mixed Cash Flows: I 95
Therefore r∗ = −145.45%, the rate for which s2 = 0.
Case 2
First m inflows are negative. Cash flows are −5, −1, 2, 2.
IRR = −15.9%
rmin = −45.9%
sn(rmin) = +2 > 0
Hence this is a pure investment and the IRR is a unique, real measure of project return,independent of k.
Case 3
In middle life, m inflows are negative. Cash flows are −1, 2, −4, 2.
IRR = −36.0%
rmin = 100%
sn(rmin) = −6 < 0
Hence this is a mixed project.
s0 = a0 = −1 < 0
s1 = s0(1 + r∗) + a1
= −1(1 + r∗) + 2 = 0 (using rminfor r∗)
s2 = −1(1 + r∗)(1 + k) + 2(1 + k) − 4 < 0
s3 = s2(1 + r∗) + a3
= −1(1 + r∗)2(1 + k) + 2(1 + r∗)(1 + k) − 4(1 + r∗) + 2
So r∗ = −24.09%, the rate for which s3 = 0.Of the three cases considered, only case 2 has a return “internal” to the project. Case 1 has
two IRRs and thus the IRR is not only an incorrect measure of investment return, but alsoambiguous. Case 3 has a unique, real IRR. However, it is not a proper measure of return oninvestment. This is a crucial criticism of the IRR — even though it may be unique and real inthe mathematical sense, this in itself is not a sufficient condition for it to be a correct measureof return on investment.
An example presented by Professor Mao vividly emphasizes this point in case 4.
Case 4
Cash flows are −10, + 40, −40.
IRR = 100%
rmin = 300%
sn(rmin) = −40 < 0
96 Capital Asset Investment: Strategy, Tactics & Tools
Hence this is a mixed project.
s0 = a0 = −10 < 0
s1 = s0(1 + r∗) + a1
= −10(1 + r∗) + 40 = 0 (using rmin for r∗)
s2 = s1(1 + k) + a2
= −10(1 + r∗)(1 + k) + 40(1 + k) − 40
So r∗ = −63.64%, the rate for which s2 = 0.This is a mixed investment with return on invested capital of minus 63.64 percent, even
though the project has a unique, real, positive IRR of 100 percent. Thus, use of the IRR wouldlead to acceptance of the project for any cost of capital k < 100 percent — a very undesirableconsequence for a firm with normal financial management goals.
A NEW THEOREM
To generalize our findings, we now present a theorem that has significant implications on theclass of investment projects with negative cash flows only at the beginning and the end.
Theorem Given that a0 < 0, an < 0, at > 0 for t = 1, . . . , n − 1 with some at > 0 for aproject, the project will always be a mixed investment (a mixed financing project 4 if a0 > 0,an < 0, and at ≤ 0 for t = 1, . . . , n − l with some at < 0).
Proof Since a0 < 0, and at > 0 for t = 1 to n − 1, if we look only at the cash flows a0 throughan−1, they form, in themselves a simple investment that TRM have proved has a unique, realrate r for which all the st < 0 for t = 1, 2, . . . , n − 1. In particular, sn−1 < 0. Therefore,this r is the rmin of the project, for any larger r would cause all st to be less than zero fort ≤ n − 1.
In evaluating sn at rate r = rmin , we simply add an , which is less than zero, to (1 + rmin)sn−1, which is equal to zero. Then, since sn < 0, we have a mixed investment as defined byTRM, and the project rate r∗ is not independent of the firm’s cost of capital.
Corollary I If
at < 0 for m < n − 1
and
an < 0
with
at ≥ 0 for m ≤ t ≤ n and some at > 0
then the project will be a mixed investment.
4In this case we negate all cash flows before applying the TRM algorithm.
11
The Problem of Mixed Cash Flows: II
In Chapter 10 the problem of mixed cash flows was introduced, and a particular method ofanalysis, that of Teichroew, Robichek, and Montalbano (TRM), was discussed. Although thetheoretical merits of the rigorously developed TRM approach may be superior, it is computa-tionally very demanding. So other methods are more commonly used in practice to deal withprojects having mixed cash flows. The question of which, if any, of the existing methods ofanalysis is universally “best” may be unresolvable. The appropriateness of any of the methodsto a given investment depends on the extent to which the method’s underlying assumptionsmatch the particular situation and the goals of the enterprise’s management. Different circum-stances may require different analytical assumptions, or desired emphasis from investment,and these in turn may imply different methods of analysis.
In this chapter the methods examined are the Wiar method and the sinking fund fam-ily of methods. Because it is fundamentally different from the others, the Wiar method isdiscussed first.
THE WIAR METHOD
Robert Wiar [171] developed this method for analysis of the investment returns on leases,1
for which he asserted it is inappropriate to analyze directly the net cash flows to equity. Hisapproach was to employ an aspect of the Keynesian theory which states that the supply costof funds cannot exceed the imputed income stream yield. In other words, analysis must behandled by simultaneous treatment of three components:
1. positive cash inflows;2. mortgage bond amortization flows; and3. the required equity investment.
This analysis can be stated in two equivalent ways:
E0(1 + re)t =t∑
i=1
Ri(1 + r∗)t−1 −t∑
i=1
Mi (1 + rb)t−1 (11.1)
or
R0(1 + r∗)t = M0(1 + rb)t + E0(1 + re)t (11.2)
where
M0 = the amount financed by bondsMi = the fixed amortization paymentrb = the effective yield to the bond holdersE0 = the equity investment
1Leveraged leases are covered in detail in Chapter 13. The characteristic of leveraged leases of concern to us now is that they usuallyhave mixed net, after-tax cash flows to equity.
98 Capital Asset Investment: Strategy, Tactics & Tools
re = the return on equity, ignoring bond serviceR0 = the initial outflow — investment — equity plus bond financingr∗ = the overall return on aggregate investmentRi = the future income stream
In the case of capital-budgeting projects it will normally be appropriate to consider the supplycost of funds as the firm’s overall marginal cost of capital k.2 Let us examine equation (11.2),letting M0 = 0, k = re, and R0 = E0. Then
R0(1 + r∗)t = R0(1 + k)t (11.3)
and it is clear that r∗ = k. This means that the imputed income stream yield, r∗, equals thesupply cost of funds k. This is what we would expect, in equilibrium, at the cut-off point,for a firm not constrained by capital rationing, and it is consistent with the IRR criterion orKeynesian marginal efficiency of capital (MEC).3
Example 11.1 An application of the Wiar method4 Consider a project costing $10 millionthat is expected to yield net, after-tax cash flows of $1.5 million at the end of each year of itsuseful lifetime of 10 years. There is expected to be a salvage value of zero.
The firm’s existing capital structure is 25 percent debt, 75 percent equity, and is consideredoptimal. The project, if accepted, will be financed by a private placement of $2.5 million inbonds yielding 10 percent and maturing in 10 years, and the balance by equity.
Assuming the bonds are sold at par value, the payment (assumed to be made at year end)will be $250,000. The entire $2.5 million must be retired in the tenth year, the year of maturityfor the bond.
The overall cash flow stream is composed of two component streams, as shown in Table11.1. It is the return on equity we are interested in. The equity cash flow stream has mixed cashflows. Applying the Wiar method, we obtain from (11.2) the equation to be solved for re:
0.75(1 + re)10 = 1.0(1.0814)10 − 0.25(1.1000)10
and re = 7.45 percent
which is greater than the IRR of 7.03 percent on the equity stream. The ordinary IRR on themixed equity stream is unique in this particular problem, but it need not be so. The previouschapter showed that, when mixed cash flows are considered, even a unique IRR does notmeasure return on investment.
The re obtained is then compared with the required return on equity. If it is greater than therequired rate, the project will be acceptable.
The Wiar method will be discussed again in Chapter 13 that deals with leveraged leases.For now we recognize that for capital budgeting projects as a class, the method reduces to theIRR method already treated in detail, and that it is inadequate for analyzing projects havingmixed cash flows.
2This is because of risk considerations. No individual capital-budgeting project, unless independent of the firm’s existing assets, canbe properly considered in isolation and without regard to its effect on the firm’s risk characteristics that affect the firm’s cost of capital.3The Keynesian term “marginal efficiency of capital” or “MEC” is normally used in macroeconomic discussions concerning theaggregate investment return curve for an entire national economy, whereas IRR is normally used to refer to the return on individualcapital investment projects.4A nice ExcelTMspreadsheet that illustrates the Wiar method and others can be found at the URL http://www.acst.mq.edu.au/unit info/ACST827/levlease.xls at the Department of Actuarial Studies of Macquarie University, Australia.
The Problem of Mixed Cash Flows: II 99
Table 11.1 Cash flows for Wiar method example
Project For bond Net cash flowYear cash flow service to equity
0 − $10,000,000 $2,500,000 −$7,500,0001 1,500,000 −250,000 1,250,0002 1,500,000 −250,000 1,250,0003 1,500,000 −250,000 1,250,0004 1,500,000 −250,000 1,250,0005 1,500,000 −250,000 1,250,0006 1,500,000 −250,000 1,250,0007 1,500,000 −250,000 1,250,0008 1,500,000 −250,000 1,250,0009 1,500,000 −250,000 1,250,000
10 1,500,000 −2,750,000 −1,250,000IRR of cash flow streams = 8.14% 10.00% 7.03%
SINKING FUND METHODS
Sinking fund methods, as a class, are characterized by some adjustment being made to theoriginal cash flow series, aimed at making the adjusted cash flow series amenable to treatmentby IRR analysis. To apply any of them, we first follow some procedure for systematicallymodifying the cash flow series to remove all negative flows except those at the beginning,5
which represent the initial cash outlay or outlays. All negative cash flows beyond the initialoutlay sequence are forced to zero. Next, because the adjusted cash flows have only one signchange, the IRR procedure is applied. Under this definition, the Teichroew, or Teichroew,Robichek, and Montalbano method discussed in the previous chapter can also be considered asinking fund method.
To avoid, or perhaps clear up, some semantic difficulties, let us state here that (at least) threesinking fund methods have been used in practice. One is the initial investment method (IIM),another is the sinking fund method (SFM), the third is the multiple investment sinking fundmethod (MISFM). A possible semantic problem exists because the SFM, although a particularmethod, carries the name of the entire class of methods. In other words, the sinking fund methodis really only one of several methods, all of which can be characterized as sinking fund methods.As a class they include all of these methods as well as the TRM method. To avoid confusion,we shall refer to “the” sinking fund method as the traditional sinking fund method (TSFM).
The sinking fund earnings rate refers to the assumed rate at which funds that are set asidein a (hypothetical) sinking fund will accrue interest earnings. The sinking fund rate, sinkingfund rate of return, or sinking fund return on investment refers to the IRR on the adjusted cashflow sequence. Similarly, the initial investment rate, and so on, refers to the analogous IRR onthe adjusted cash flow series with the initial investment method.
To avoid complicating matters, we rule out the possibility of early project abandonmentin this chapter. That is, we assume all projects will be held until the end of their economiclives. Furthermore, a uniform period-to-period sinking fund earnings rate is assumed in orderto simplify and streamline exposition.
5The multiple investment sinking fund method is an exception to this because not all negative cash flows are removed beyond theinitial negative flow or flows.
100 Capital Asset Investment: Strategy, Tactics & Tools
The Initial Investment Method
The initial investment method (IIM) is a type of sinking fund method. In the standard, conser-vative IIM it is assumed that an additional amount is invested, at the beginning of the project,specifically for the purpose of accumulating funds exactly sufficient to cover all negative cashflows occurring after the first positive flow.6 Such initial investment is assumed to earn atsome rate of return compounded from period to period. The rate of return is calculated on therevised initial investment and the subsequent positive cash flows. Negative flows are zeroedout because they are assumed to be exactly offset by the additional initial investment growingat a compound rate. Another way of applying the IIM is to assume that the earliest positivecash flows are set aside into a sinking fund earning just sufficient to match exactly the laternegative cash flows.
To gain an understanding of the IIM, it will help to analyze an example. However, in orderthat the two similar methods may be compared together, this will be postponed until after thefollowing discussion of the traditional sinking fund method.
The Traditional Sinking Fund Method
The traditional sinking fund method (TSFM), like the IIM, assumes that positive cash flows canbe invested at some nonnegative rate of return so that later negative cash flows may be exactlycovered. Unlike the IIM, however, the TSFM assumes that the most proximate positive cashflows preceding the negative flow will be put into a sinking fund to the extent required to offsetthe negative flow or flows. The TSFM, therefore, can be considered a much less conservativemethod than the IIM, to the extent it delays investment for what may be a considerable timeand thus does not provide the benefit of having at least some accumulated earnings should theactual available earnings rate decline later in the project life.
Initial Investment and Traditional Sinking Fund Methods
The initial investment and traditional sinking fund methods are based on similar assumptions.Both the IIM and the TSFM are based on a technique that modifies the cash flows of a projecthaving mixed cash flows (and thus often a mixed project in the TRM sense) to produce asimple, pure project that has zeros where the original had negative flows, in all but the initialoutlays (the nonpositive cash flows preceding the first positive cash flow). In fact, the TRMalgorithm of the preceding chapter is closely related, with the earning rate on a “sinking fund”equal to k, the firm’s cost of capital. However, with the TRM method the timing and the amountof investment in a “sinking fund” are perhaps obscured by the nature and the complexity ofthe algorithm, with the cash flows “compressed”7 prior to the solution for return on investedcapital. With the IIM and TSFM, cash flows that had been negative are zeroed prior to solutionfor rate of return. Thus, the order of the polynomial that will be solved for a unique, real rootwill be less for the TRM than for the IIM or TSFM. This will be clarified by using two of thesame examples that were used in the previous chapter with the TRM algorithm, but this timewith the IIM and TSFM.
6A series of m negative cash flows followed by a series of positive flows would cause no problem in determining return, for we coulduse the IRR directly in such cases.7Compressed in the sense that the polynomial that must be solved for the return on invested capital, r∗, is of lesser degree than theIRR polynomial would be for the same project.
The Problem of Mixed Cash Flows: II 101
Original Investment +$320
t � 0
−$100
Initial Investment Assumption
$198.35(1.10)2
t � 0
−$198.35
Combined
1
1
+$320
t � 0
−$298.35
2
2
2
0Time
Time
Time
−$240
+$240
1
Figure 11.1 Initial investment method applied to Example 11.2 cash flows
Example 11.2 This example is the same as Example 10.1 used in Chapter 10. The cash flow is:
t = 0 t = 1 t = 2−$100 +$320 −$240
As stated in Chapter 10, this project has two IRRs: 20 percent and 100 percent. The NPV reachesa maximum of $6.67 at 50 percent cost of capital. The return on invested capital for this project,for k = 10 percent, is 1.82 percent. Figure 11.1 indicates the procedure used in applying theinitial investment method to the cash flows. The IIM rate of return on investment is 7.26 percentfor this example. Figure 11.2 suggests the procedure followed in applying the traditional sinkingfund method. It is assumed that the applicable sinking fund’s earning rate is 10 percent.
The essence of application of the TSFM to this project is to set aside sufficient cash, atsome assumed earnings rate, to accumulate to an amount exactly sufficient, one period later,to match the negative cash flow.
Thus, for any positive cost of capital above 1.82 percent, the project would be unacceptablefor investment under either method. Like the IRR method or that of TRM, we compare theproject yield to the enterprise’s cost of capital and reject the project if the yield rate is less thanthe cost of capital. For this project, if the earnings rate on the funds set aside8 is 10 percent, andthe cost of capital is k = 10 percent, then rTSFM = 1.82 percent — the same as that obtainedwith the TRM method in Chapter 10. In general, they will not be the same. However, it isindicative of the relationship between the methods.
8In practice, it will be rare to find an investor actually depositing or setting aside cash to meet outflows later in the life of a project.Rather, the firm will employ the positive cash flows in its operations. To be conservative, we could assume that a zero earnings rate isapplicable. However, it would be normally appropriate to assume that funds earn at the firm’s cost of capital.
102 Capital Asset Investment: Strategy, Tactics & Tools
Original Investment $320
2Time
−$240
+$240
Time
Time
2
$218.18(1.10)
−$218.18
$101.82Combined
−$100
t � 0
t � 0
t � 0
−$100
Sinking Fund Assumption
1 2
0
1
1
Figure 11.2 Traditional sinking fund method applied to Example 11.2 cash flows
$3800
t � 0
x(1 + i)2
2
1
−$4730−($1000 + x)
Time
+$1936
3
Figure 11.3 Initial investment method: application to Example 11.3
Next we examine another example, this the same as Example 10.2.
Example 11.3 Figure 11.3 displays the net cash flows for this project and indicates theapplication of the IIM.
Note that positive cash flows are compounded forward in time. They are not discountedback to an earlier point in time with the various sinking fund methods.9 Otherwise, the $1936
9An exception to this general rule is to be found in what is referred to as the modified sinking fund method (MSFM). We shall notdiscuss this method here, since it is a straightforward variation of the TSFM in which later positive cash flows may be discounted topay off earlier negative flows
The Problem of Mixed Cash Flows: II 103
$3800−x$3800−x
(1 + i)
2
$1936
3t � 0
−$10001
Time
−$4730
Figure 11.4 Traditional sinking fund method: application to Example 11.3
t � 0
−$100
+$70+$90
−$22
3
21Time
Figure 11.5 Original cash flow series for Example 11.4
positive cash flow at the end of year 3 could have been discounted back one year to offsetpartially the negative flow of $4730 at the end of year 2.
The cash flow sequence that must be solved for rIIM is: −($1000 + x) at t = 0; $3800 att = 1; 0 at t = 2; and $1936 at t = 3. The value of x depends on the assumed earnings rate onthe sinking fund. It is equal to $4730/(1 + i )2 where i is the applicable rate. For i = 10 percent,we obtain x = $3909.10 and rIIM = 10 percent.
Figure 11.4 illustrates application of the TSFM to the same project. The adjusted cash flowsfrom which we find the rTSFM are: −$1000 at t = 0; $3800 − x at t = 1; 0 at t = 2; and $1936at t = 3. The value of x is $4730/(1 + i ), where i is the sinking fund earnings rate. For i = 10percent, we obtain rTSFM = 10 percent.
Example 11.4 The cash flows for this project are illustrated in the time diagram of Figure11.5. This project has two positive cash flows, one at the end of each year preceding the finalcash flow of minus $22 at the end of year 3.
Figure 11.6 shows the procedure involved in applying the TSFM to Example 11.4. At anassumed earnings rate of 10 percent on the sinking fund, a set-aside of $20 (out of the $90
t � 0
−$100
+$70+$90
321Time
+$20(1.10)+$70
+ $22− 22
$0
Figure 11.6 Traditional sinking fund method, adjusted cash flows for Example 11.4 with i = 10 percent
104 Capital Asset Investment: Strategy, Tactics & Tools
t � 0
+$70
−$100.00− 16.53 (1.10)3
+$90
1 2 3Time
−$116.53
−$22 22
$0
Figure 11.7 Initial investment method: adjusted cash flows for Example 11.4 with i = 10 percent
$70.00−18.18 (1.10)2
$51.82 $90$22−22
t � 0
−$100
1 2 3Time
$0
Figure 11.8 Initial investment method: adjusted cash flows for Example 11.4 with i = 10 percent
cash flow) at the end of year 2 will increase to $22 a year later. The sinking fund amount of $20plus $2 interest will exactly offset the negative cash flow at the end of year 3. After set-aside of$20 at end of year 2, $70 remains for other uses. The yield on the adjusted cash flows is 25.7percent = rTSFM.
Application of the initial investment method to Example 11.4 is illustrated in Figure 11.7.Again, the assumed earnings rate on the funds set aside is 10 percent. With this method wemust invest an extra $16.53 at t = 0 to offset the negative $22 cash flow at t = 3. The rIIM onthe adjusted cash flow series is 22.91 percent.
The IIM, a special case of the sinking fund method, may be considered more conservativethan the TSFM. This difference lies in the manner of selecting the timing and amounts to beset aside.
One might argue that it is too extreme a conservatism to assume that sufficient extra fundsmust be put into a sinking fund at t = 0 to cover the later negative cash flows. The IIM sofar discussed is the limiting case on the conservative end of the spectrum (especially so ifthe assumed earnings rate were to be zero). As an alternative initial investment approach, wecould assume that sufficient funds are set aside from the earliest positive cash flows to offsetlater negative flows. Figure 11.8 illustrates this variation of the IIM. The rIIM in this case is24.25 percent.
Example 11.5 Now let us consider a somewhat more complex project than those in theprevious examples of this chapter. Consider, for instance, a replacement chain of one capitalinvestment following another. Let us assume we have a mine, which will cost $100 initially todevelop and which will provide net, after-tax cash flows of $150 at the end of each year of itsthree-year economic life. Furthermore, assume that undertaking this mining project commitsour organization to a project for secondary mineral recovery costing $350 to initiate andproviding net, after-tax cash flows of $100 at the end of each year of its three-year useful life.Finally, at the end of the secondary recovery, our organization will have to pay out $350 to
The Problem of Mixed Cash Flows: II 105
$150$100 $100
$150
t � 0
−$100
−$200
1 2
3
4 5
6Time
−$250
Figure 11.9 Original cash flow series for Example 11.5
$NPV
605040302010
−10−20−30−40−50−60
%
10.23%
Figure 11.10 Net present value as a function of cost of capital for Example
close down operations and rehabilitate the land on which the mine is situated to comply withenvironmental legislation. Since we have some positive cash flows from the projects at the endof years 3 and 6, the overall net outlays in those years are not $350, but are $200 and $250,respectively. Figure 11.9 contains a time diagram illustrating the cash flows for this example.
This project has two IRRs: 10.23 percent and 85.47 percent. From Chapter 10 we know thatneither is a correct indication of the return on investment. The NPV function for the projectis plotted in Figure 11.10. Note that under the NPV criterion the project would be consideredacceptable for values of cost of capital k, such that 10.23 percent < k < 85.47 percent.
Since this project is more complex than those considered previously in this chapter, appli-cation of the TSFM and IIM is illustrated in the tables. Table 11.2 shows, for a 10 percenttraditional sinking fund earnings rate, the adjustment to the cash flows that will be made em-ploying the TSFM. The TSFM yield rate can be seen by inspecting the adjusted cash flows tobe 9.27 percent. Figure 11.11 contains a time diagram showing the adjusted cash flows.
Table 11.3 shows, for a 10 percent sinking fund earnings rate, the procedure employed foradjusting the cash flows with the less conservative variation of the IIM. Figure 11.12 contains
106 Capital Asset Investment: Strategy, Tactics & Tools
Table 11.2 Traditional sinking fund method adjustment to original cash flows forExample 11.5
Time periodoriginal t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6
cash flows −$100 $150 $150 −$200 $100 $100 −$250
−100Step 1 ×(1.1)———–→ 110
−100Step 2 ×(1.1)2————————→ 121
−12.98Step 3 ×(1.1)4——————————————————→19
−137.02Step 4 ×(1.1) → 150.72
−40.73Step 5 ×(1.1)2——————→49.28Sinking fund
method cash −100 109.27 0 0 0 0 0flow
Table 11.3 Modified initial investment method adjustment to original cash flows forExample 11.5
Time periodoriginal t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6
cash flows −$100 $150 $150 −$200 $100 $100 −$250
−150Step 1 ×(1.1)2—————-→181.50
16.82Step 2 ×(1.1)→ 18.50
−133.18Step 3 ×(1.1)4———————————————→ 194.99
−45.46Step 4 ×(1.1)2————————→ 55.01Initial investment
methodadjusted cashflows −100 0 0 0 54.54 100 0
Time0 0 0 0 0
1 2 3 4 5 6t � 0
−$100
$109.27
Figure 11.11 Traditional sinking fund method
The Problem of Mixed Cash Flows: II 107
−$100
1 2
00
3
0
4
$54.54
$100
5 6Time
0t � 0
Figure 11.12 Modified initial investment method: adjusted cash flows for Example 11.5 with i = 10percent
100
80
60
40
20
20 40 60 80 100
TRM (2)
TRM (1)SFM
IIM
k%0
−20
r% r � k
Figure 11.13 Sensitivity analysis of Example 11.5 project
the time diagram corresponding to the adjusted cash flows. The IIM yield rate cannot beobtained by inspection (in contrast to the sinking fund yield rate). By calculation we find it tobe 9.84 percent.
Figure 11.13 and Table 11.4 contain sensitivity analyses of the (less conservative) initialinvestment, traditional sinking fund, and TRM rates of return associated with various cost ofcapital percentages (traditional sinking fund earnings rates). Note that for k = 0 there is noreal solution to the TRM return on investment equation.
THE MULTIPLE INVESTMENT SINKING FUND METHOD
The final sinking fund variant we will discuss in this chapter is the multiple investment sinkingfund method (MISFM). The idea underlying the MISFM is to adjust the cash flow sequence toobtain two distinct, nonoverlapping investment sequences having identical IRRs. For example,the project having a five-year useful life can be decomposed as follows into adjusted cash flowsplus sinking fund. We assume the sinking fund earns at 15 percent per period.
108 Capital Asset Investment: Strategy, Tactics & Tools
Table 11.4 Sensitivity analysis of Example 11.5project
k% TRM(1)% TRM(2)% IIM% TSFM%
0 3.19 50.0010 51.01 10.11 0.61 0.8720 60.74 2.80 23.10 39.0830 67.07 −2.40 35.81 54.8540 71.94 −6.40 47.50 64.2550 75.87 −9.66 58.01 70.8960 79.14 −12.40 67.25 76.0870 81.90 −14.77 75.26 80.2880 84.29 −16.86 82.14 83.7990 86.38 −18.72 88.02 86.76
100 88.21 −20.39 93.02 89.31110 89.85 −21.91 97.29 91.53120 91.31 −23.30 100.94 93.49130 92.63 −24.58 104.06 95.22
The adjusted cash flows can be considered to be two nonoverlapping investments, eachhaving a unique, positive internal rate of return of 50.0 percent. For this particular project therIIM is 28.93 percent, rTSFM is 40.95 percent, and the RIC is 52.24 percent.
t = 0 t = 1 t = 2 t = 3 t = 4 t = 5Original cash flows −$1000 765 2500 −3500 1000 3000Adjusted cash flows −$1000 500 1500 −2000 1000 3000Sinking fund 265 1000 −1500
The MISFM is much more difficult to apply10 than the IIM or TSFM, because it re-quires that the original cash flows be adjusted so that one, two, or several nonoverlap-ping subsequences remain, all of these having the same unique, real IRR, which is therMISFM. The method is used by some major organizations in their leveraged lease analysis.Limited experience with results of the MISFM suggests that the rate is generally close tothe TRM RIC rate. This suggests that the RIC could be used as a first approximation tothe rMISFM.
Strong Points of the Methods
1. Both the TSFM and IIM are based on theoretically defensible assumptions.2. The methods take into account the time-value of money and do not exclude any cash flows.3. The methods are much easier to apply than the Teichroew, Robichek, and Montalbano
algorithm, except for the MISFM.4. Both methods assure a unique, real measure of return on investment.5. Application of the methods is not especially difficult to learn.
10The method is relatively easy to employ today with modern spreadsheet programs, as illustrated in the ExcelTMworkbook available atthis writing from Actuarial Studies at Macquarie University in Australia: http://www.acst.mq.edu.au/unit info/ACST827/levlease.xls
The Problem of Mixed Cash Flows: II 109
Weak Points of the Methods
1. Both the TSFM and IIM may require, in addition to an estimate of the organization’s cost ofcapital, an estimate of the sinking fund rate at which cash may be invested. For reinvestmentof funds within the firm this rate could be the same as the cost of capital.
2. Both methods are more difficult to apply without a computer than the NPV method.3. The methods do not distinguish between projects of different size and/or different economic
lives. However, adjustment may be made for this.4. In general, the methods may not adequately reflect the interrelated nature of investment and
financing imbedded within mixed projects.
11AAppendix: The Problem of Mixed Cash
Flows III — a Two-stage Method of Analysis
The two preceding chapters present the Teichroew, Robichek, and Montalbano method andother methods classifiable as sinking fund approaches to obtaining a unique, real rate of returnmeasure on those investments having mixed cash flows. It was shown in Chapter 10 that evena mathematically unique, real IRR is no assurance, under the TRM assumptions, of a measureof return on investment independent of the enterprise’s cost of capital. This chapter presentsan alternative, developed by the author, to the various sinking fund methods.
The method of analysis presented here is designed to explicitly separate the analysis ofinvestments with mixed cash flows into two separate but related decision stages. The methodyields two measures by which investments with mixed cash flows, such as leveraged leases,may be evaluated:
1. The time required to recover the initial investment plus the opportunity cost associated withthe funds committed to the project.
2. The implicit “borrowing” rate contained within the cash flows occurring beyond the capitalrecovery time.
Together, the two measures provide a decision rule: if the capital recovery period is acceptableand the implicit borrowing rate less than the rate at which the firm can acquire funds (the firm’scost of capital), or reinvest the second stage cash flows, the project is acceptable.
RELATIONSHIP TO OTHER METHOD
The net present value (NPV) method assumes that the enterprise cost of capital is the appropriaterate at which to discount the cash flows, whether they be positive or negative. The NPV itself is ameasure of monetary return over and above the investment outlay. Although it is implicit in theNPV, the method and the measure provide no information pertaining to the timing of recoveryof the funds invested in the project. The NPV method relates directly to the basic valuationmodel of modern financial management and is generally preferred by academic writers to othercapital budgeting methods. It will be shown in this chapter that the two-stage method is, unlikethe internal rate of return (IRR), perfectly compatible with the NPV. The two-stage methodyields identical accept/reject decisions to those obtained with the NPV, when the acceptabletime span to recover the invested funds is unconstrained.
The two-stage method, although yielding results identical to NPV, provides decision mea-sures in the separate stages that are in terms of time and percentage rate. Thus, the first stage,by providing a time measure, relates to the ubiquitous payback method. As shall be seen,however, it does not suffer from the well-known shortcomings of the payback method. Thesecond stage relates to the IRR, except that the rate found will be a cost rate rather than a returnrate.
The Problem of Mixed Cash Flows III 111
THE TWO-STAGE METHOD
Studies have shown that many managers employ the payback method very heavily, much to thedismay of those academics who have dwelt upon and publicized its shortcomings. The majorshortcomings of the payback method are that it ignores the time-value of money and ignorescash flows beyond the end of the payback period. Or, equivalently, a zero opportunity cost rateis assigned during the payback period, with an infinite rate thereafter.
In stage one of the method proposed here, a payback period for the initial investment isdetermined. However, this payback period takes the time-value of money into account byrequiring not only that the initial investment be recovered but, in addition, the opportunitycost of the unrecovered funds remaining invested in the project. This is related to Durand’sunrecovered investment: at this payback, and not before, unrecovered investment is zero.Unrecovered investment is discussed in Chapter 14. Because the time-value of funds is takeninto account, this payback measure is economically justifiable.1
In stage two, the cash flows remaining beyond the payback are analyzed, in the “negativeinvestment” or “loan” phase of the project. Thus the second major objection to the traditionalpayback method is of no consequence. In the second stage, an imputed rate is determined thatcan be compared to the enterprise’s cost of capital or the investment opportunity rate availableto the firm over the periods remaining in the project life from payback to the end of the projectlife.2 If the “loan” rate is less than the firm’s cost of capital, or the rate the firm expects to beable to earn on the cash flows from the project, it is acceptable on the rate basis. Since thefirm’s cost of capital would be less than or equal to the reinvestment rate offered by futureacceptable projects,3 we shall assume that comparison will be with the cost of capital.
Otherwise, it could be possible to accept a project with an implicit cost greater than theenterprise cost of capital because the reinvestment rate of return is greater.
The two-stage method can perhaps be best explained by detailed treatment of examples.
Example 11A.1 We shall analyze the cash flows to equity on a leveraged lease discussedby Childs and Gridley [22]. Chapter 13 examines leveraged leases in detail, and this sameleasing project is examined further. For now it will suffice to take the net cash flows to equity asgiven and note that our purpose here is to illustrate the two-stage method of project analysisand not to discuss leveraged leases. Table 11A.1 contains the original net cash flows and theadjusted cash flow series. We assume a cost of capital k = 10 percent.
The project (lease) requires an initial equity outlay of $20. With k = 10 percent the enterprisemust recover $20 plus 10 percent of $20, $20(1 + k), at the end of the following year. Theenterprise will be no better off nor worse off if it can recover the capital committed to theproject along with the cost of those funds; its capital will still be preserved. Because the cashflow at t = 1 is only $13.74, with $8.26 of the $22 remaining to be recovered a year later,which with opportunity cost amounts to $9.09. The cash flow of $5.89 at t = 2 cannot quitesuffice, and therefore $3.52 must be recovered at t = 3. Figure 11A.1 illustrates the procedure.
1For the remainder of this chapter “payback” will refer to this time-value-of-money-adjusted payback — “traditional payback” to themethod as it is usually applied.2Or, if abandonment is to be considered, rates can be found corresponding to abandonment at any time prior to the end of the project’seconomic life.3If the “loan” rate is less than the cost of capital, the project is providing funds at a cost less than the combined total of other sources.Such funds can be invested as they are received to earn the opportunity rate available to the firm. In the idealized construct of a worldwithout capital rationing, the enterprise would be expected to invest funds up to the point at which the marginal rate of return equaledthe marginal cost of capital.
112 Capital Asset Investment: Strategy, Tactics & Tools
Table 11A.1 Assumed per-annum opportunity cost of funds(cost of capital) is 10%
Original Originalinvestment net after-tax Adjusted
plus opportunity cash cash flowt cost flows seriesa
0 0 $−20.00 $−20.00 2.6981 $−22.00 13.74 −8.26 Year2 −9.09 5.89 −3.20 Payback
3 −3.52 5.04 1.524 4.19 4.195 3.32 3.326 2.43 2.437 1.54 1.548 0.63 0.639 −0.51 −0.51
10 −2.68 −2.6811 −2.90 −2.9012 −3.16 −3.1613 −3.41 −3.4114 −3.70 −3.7015 −3.90 −1.7216 +2.40 0
a The payback is in 2.7 years. The $1.52 remaining after payback in year 3 is assumed to occurat the end of the period, just as the original flow of $5.04 is.
Before solving for the implicit loan rate, the last flow of +$2.40 is discounted back at theopportunity cost rate, using a sinking fund approach until absorbed by the negative flows. Thismay be interpreted as assigning, at discount, the last positive flow to the project — in otherwords, using it to prepay a portion of the loan.
Original Cash Flows
t � 0 t � 1 t � 2 t � 3
5.89
5.04
−$20(1 + k) −22.00− 8.26(1 + k) −9.09
−3.20(1 + k) −3.52+1.52
Adjusted Cash Flows
t � 0 t � 1 t � 2 t � 3+1.52
The payback is thus 2 + 3.52/5.04 years, or 2.7 years.
0 0 0
+$13.74
Figure 11A.1 Stage I analysis
The Problem of Mixed Cash Flows III 113
Original Cash Flows
t � 15 t � 16
t � 15 t � 160
t � 14
t � 14
+2.18
−3.70
−1.72
−3.70
−3.90
−1.72
Adjusted Cash Flows
+ 2.40 ÷ (1 + k)
Figure 11A.2 Preparation for stage II analysis
By allowing a noninteger value for payback, we violate the assumption that cash flows occuronly at the end of each period. To be consistent with this assumption we would take three yearsas the payback period required to recover the initial investment plus opportunity cost of funds.However, if we recognize that the assumption of end-of-period cash flows is only to facilitatecalculations, and that cash flows do, in fact, occur more or less uniformly over time, we willbe comfortable with the fractional result as calculated.
The residual $1.52 after investment recovery, or capital recovery, is the first nonzero cashflow in the adjusted series. If it were not for the final cash flow of +$2.40 at t = 16 we wouldhave, in the remaining cash flows, the series of a simple financing project with only onesign change in the cash flows. Before applying stage II of the method, we must first get ridof this last cash flow. One approach, a conservative one, would be to ignore the +$2.40, toassume it will not be received. A better approach, in the author’s opinion, is to assume thatfunds can be borrowed at rate k, and the loan proceeds used to offset one or more of theimmediately preceding negative cash flows. The +$2.40 would be used at t = 16 to repay theloan. Equivalently, we may assume the +$2.40 is assigned (at discount of k) to a creditor. Thislatter approach is the one that will be used. Figure 11A.2 illustrates the method.
The enterprise has recovered its initial investment and the associated cost of the unrecoveredfunds remaining committed by the end of the first 2.7 years. The remaining adjusted cash flowsoccurring subsequent to that time are gratuitous to capital recovery of the initial investment.They are characteristic of a loan, and therefore we are interested in the rate implied by thesecash flows. This “loan” rate is determined to be 3.9 percent, much less than our assumed 10percent cost of capital. If we can obtain funds at a rate lower than our cost of capital, weshould do so,4 and the project is acceptable on this basis.
As will become apparent in the formal development of the two-stage method, if a projectis not rejected on the basis of an unacceptable payback, the accept/reject decision obtainedfrom stage II will be identical to that obtained with the NPV method. This is an important
4Even if we recognize that risk considerations may alter the conclusion, the effect of the project itself on k could have been incorporatedinto the rate itself prior to this analysis.
114 Capital Asset Investment: Strategy, Tactics & Tools
result. Although the stage II results are perfectly compatible with the NPV, the stage Ipayback may mark a project as unacceptable because capital recovery takes longer than isconsidered acceptable.
Before getting into the formal, mathematical treatment of the two-stage method, it will beuseful to analyze one more example. This is the same as Example 10.2 from Chapter 10.
Example 11A.2 This project has cash flows:
t = 0 t = 1 t = 2 t = 3−$1000 +$3800 −$4730 +$1936
Again assuming k = 10%, the $1000 initial investment plus cost of the funds committed (atotal of $1100) is fully absorbed by the large positive cash flow at t = 1. The stage I paybackfor this project is 1100/3800 = 0.289 or 0.3. The adjusted cash flows are:
t = 0 t = 1 t = 2 t = 30 +$2700 −$4730 +$1936
Before finding the stage II rate, we assume the $1936 is assigned to a creditor, at a discountof k percent, and the proceeds received at t = 2. Thus, the adjusted cash flow at t = 2 is−$4730 + ($1936/1.10) = −$2970. To find the stage II rate, rB, we solve for the IRR of thecash flow series:
t = 0 t = 1 t = 2 t = 30 +$2700 −$2970 0
And $2700(1 + rB) = $2970 so that rB = 10 percent. Since rB = k the project is not accept-able. This project, not coincidentally, as will be shown, has a zero NPV for k = 10 percent.Figure 11A.3 contains a plot of the two-stage method results and NPV for various values of k.
FORMAL DEFINITION AND RELATIONSHIP TO NPV
It was stated earlier in this chapter that the two-stage method yields results identical to thoseobtained with the NPV provided that time for full capital recovery is unimportant. To makethe two-stage NPV relationship explicit and at the same time provide a formal definition ofthe two-stage method, we first write the formula for NPV as
NPV =n∑
t=0
Rt (1 + rr)n−t
(1 + k)n(11A.1)
where Rt are the net, after-tax cash flows, and rr the reinvestment rate.
NPV =n∑
t=0
Rt
(1 + k)t=
n∑t=0
Rt (1 + k)−t (11A.2)
also by reduction of equation (11A.1)Similarly, the two-stage method may be written as the following.
The Problem of Mixed Cash Flows III 115
$NPV
30
20
10
0
−10
−20
−30
−40
Years
1.50
1.00
.50
0.00−10 10 20 30 40 50 60 70 80 90 100
%k
%rB
NPV � f(k)
70
60
50
40
30
20
10
−10
−20
rB > k rB > krB < k
rB � f (k)
rB � k
Figure 11A.3 NPV and two-stage measures for Example 11A.2
Payback Stage
P∑t=0
Rt (1 + rr)P−t
(1 + k)P= 0 ⇒ P (11A.3)
P∑t=0
Rt
(1 + k)t=
P∑t=0
Rt (1 + k)−t = 0 ⇒ P (11A.4)
Multiplying equation (11A.4) by (1 + k)n yields the equivalent form actually employed in theproblem treated earlier:
P∑t=0
Rt (1 + k)n−t = 0 ⇒ P (11A.5)
116 Capital Asset Investment: Strategy, Tactics & Tools
“Borrowing” Rate
n∑t=P+1
Rt (1 + rB)n−t
(1 + rB)n−P=
n∑t=P+1
Rt (1 + rB)−t = 0 ⇒ rB (11A.6)
where rB is the implicit cost of funds inherent in the flows remaining after payback.5 Formula(11A.6) assumes the opportunity cost rate equals the implicit cost of funds. If necessary, thelast cash flow at the end of the project can be forced to zero as illustrated earlier.
Formula (11A.6) can be modified to incorporate the firm’s cost of capital and to obtain apresent value formulation:
n∑t=P+1
Rt (1 + rB)n−t
(1 + rB)n−P= present value (11A.7)
Now, adding (11A.3) and (l1A.7), we obtain
n∑t=0
Rt (1 + rr)P−t
(1 + k)P=
n∑t=P+1
Rt (1 + rB)n−t
(1 + k)n−P(11A.8)
which, for reinvestment rate rr = “borrowing” rate rB = r , reduces to
n∑t=0
Rt (1 + r )n−t
(1 + k)n(11A.9)
which is identical to the NPV formulation of (11A.1).In this chapter we have so far considered projects with mixed cash flows. What if we now
apply the two-stage method to a simple investment? Consider the cash flows in the followingexample.
Example 11A.3
t = 0 t = 1 t = 2 t = 3 t = 4−$1000 1000 1000 1000 1000
If we again let k = 10 percent, the stage I payback is 1.11 periods and the revised cash flowseries is
t = 0 t = 1 t = 2 t = 3 t = 40 0 +$890 1000 1000
There is no real rate that satisfies the second-stage rate equation. However, because theremaining cash flows are all positive, they constitute in themselves a “loan” that does nothave to be repaid, or a gift to the firm, and the project is acceptable on this basis. The two-stage method may be used for simple investments as well as those with mixed cash flows. It isgenerally applicable, which was to be expected from what is basically a special formulationof the NPV.
5The rates rr, rB, and k are assumed to be greater than or equal to zero in order that they have a meaningful economic interpretation.
The Problem of Mixed Cash Flows III 117
CONCLUSION
The NPV method of analysis has been largely ignored by those decision-makers who haveshown continuing preference for the traditional payback method and to a lesser extent the(internal) rate of return. The two-stage method discussed in this chapter presents the NPV interms decision-makers are accustomed to: payback and percentage rate. The payback, however,takes into account the time-value of funds at the enterprise’s cost of capital; the percentage rate,for nonsimple investments, is a cost rate implicit in the cash flows after payback. If paybackis not constrained, the two-stage method will always yield the same accept/reject decision asthe NPV method.
Because decision-makers have shown long-standing tenacity for the traditional paybackmethod, the two-stage algorithm may find better acceptance by practitioners than the NPVmethod has received. And the two-stage method makes explicit, in the payback measure, thetime required to recover the capital committed to a project. This is something the NPV methoddoes not do, as it is usually stated.
A BRIEF DIGRESSION ON UNCERTAINTY
Up to this point we have considered the environment in which capital investment decisions areto be made one of certainty. If we relax this assumption, as we do in the next section, we arecompelled to admit that, to the extent that cash flow estimates become increasingly tenuousand subject to error the further they occur from the present, a project that returns the initialinvestment early is to be preferred to one that does not, ceteris paribus. This is particularlyso during times of economic, political, and social instability, the combined effects of whichmay cause cumulative exogenous effects to the enterprise that are impossible to predict far inadvance.
Some time ago a distinction was often made between risk and uncertainty. Today it seems thedistinction is often ignored, perhaps because the theory of finance in general, and investmentsin particular, have been developed to their present state by assuming risk rather than the moreintractable uncertainty. The distinction is this: with risk we take as known the probabilitydistributions of the variables; with uncertainty we assume ignorance of the distributions of thevariables.
The two-stage algorithm for investment analysis provides, in its payback measure, a meansof addressing uncertainty. Two investments with identical NPV may have substantially differentcapital recovery payback and, in a world characterized by uncertainty, the investment with theshorter payback is to be preferred. Because the two-stage method provides a capital recoverymeasure, it allows management to determine whether or not the capital recovery is swiftenough. Because capital preservation may be a goal that overrides possible investment returns,the payback should be of interest.
12
Leasing
A lease is a contract under which the user (lessee) receives use of an asset from its owner(lessor) in return for promising to make a series of periodic payments over the life of thelease. A lease separates use from ownership. The two basic types of leases are operating andfinancial. Operating leases have relatively short terms, provide less than full payout,1 and maybe canceled by the lessee. A hotel room, or home telephone, water or electrical service maythus be considered forms of operating lease. In contrast, a financial lease is for a long term,provides for full payout, and cannot be canceled without penalty by the lessee. We shall notbe concerned with operating leases, but instead focus on financial leases.
Financial leases may be separated into two main categories: ordinary and leveraged. Thischapter is concerned with ordinary financial leases. Chapter 13 considers leveraged leases andtheir unique attributes and problems. Both kinds of financial leases have assumed increasingimportance in recent years and we may expect growth in leasing to continue over the nextdecade, barring major changes in tax laws that apply to them.
ALLEGED ADVANTAGES TO LEASING
Many advantages over conventional financing have been attributed to leasing. Although somehave genuine value, other may have advantages only to certain firms in particular circumstances,and still others may have dubious value altogether. Among the claimed advantages are thefollowing:
1. Off-balance-sheet financing This is of dubious value, since the existence of financialleases must be footnoted and analysts will treat a lease as if its capitalized value were alisted liability.
2. Provides 100 percent financing This may be advantageous when other financing is notavailable or available only under unacceptable terms.
3. Longer maturity than debt For a long-lived asset this may be a significant advantage.Financial leases generally run for the life of the asset. Loan terms, on the other hand, aregenerally set by the policy of the lender and maturity may be much shorter than the asset life.
4. Entire lease payment tax deductible This can be advantageous if land is involved sinceit is not depreciable if owned.
5. Level of required authorization Leases may sometimes be authorized by plant managers,whereas purchase of the same asset may require approval higher in the organization.
6. Avoids underwriting and flotation expense Leasing also avoids the public disclosureassociated with sale of securities.
7. Front-end costs reduced Delivery and installation costs are spread over the life of the lease.
1Full payout for a lease requires that the total of payments be sufficient for the lessor to recover, in addition to the capital investment,the cost of funds and profit.
120 Capital Asset Investment: Strategy, Tactics & Tools
8. Lease payments fixed over time Both the lessee and the lessor know the costs over thelife of the lease.
9. Less restrictive, quicker, more flexible10. May conserve available credit Possibly, but consider comment under (1).11. Lease may “sell” depreciation The lessee, if unable to use depreciation and investment
tax credit directly because of losses, in essence “sells” them to the lessor for more favorableleasing terms and thus gains from what would otherwise be lost.
12. Leased assets provide own collateral The lessee does not have to pledge other assets thatmight have to be pledged to secure debt financing for the same leased equipment. Becausethe lessor owns the leased asset, he can recover it in the event the terms of the lease arebroken.
ANALYSIS OF LEASES
The analysis of ordinary financial leases in the literature has focused almost exclusively onlease evaluation from the viewpoint of the lessee, the user of the equipment. Very little hasbeen written on lease analysis from the lessor’s view until recently. Evaluation by the lessor isin itself a capital-budgeting problem that, depending on the terms of the lease and the qualityof the lessee, may approach a certainty environment in many respects. In this chapter theanalysis of leases will be considered from both the lessee and the lessor viewpoints. First, thetraditional analysis from the lessee’s position will be considered. Then an integrated treatmentof the lessee’s and the lessor’s positions will be discussed.
It should be made clear at the outset that lease analysis itself does not address the questionof whether a particular asset should be acquired or not. Rather, lease analysis starts with thepremise that the asset should be acquired by the lessee.2 The question that lease analysistries to answer is whether the asset in question should be purchased or leased. This is oftenexpressed as “lease or buy” or “lease or borrow.” The traditional analysis, through the lessee’seyes, involves finding the least cost alternative to acquiring an asset: the minimum of the leasecost and the alternative financing cost. The alternative financing is generally assumed to be100 percent debt financing, since leasing commits the lessee to making periodic payments justas a fully amortized bond would do. And contrary to alleged advantages (1) and (10) above, ithas become widely recognized that leases do displace debt.
Traditional Analysis
Many approaches to the valuation of leases have been proposed. The one proposed by Bower[17] is representative of a broad class of net present value (NPV) models, and thus will bediscussed first. The Bower model (in this author’s notation) is:
NAL = C −H∑
t=1
Lt
(1 + r1)t+
H∑t=1
T Lt
(1 + r2)t−
H∑t=1
T Dt
(1 + r3)t
(12.1)−
H∑t=1
T It
(1 + r4)t+
H∑t=1
Ot (1 − T )
(1 + r5)t− SH
(1 + r6)H
2This question has been answered by the capital-budgeting methods generally applied to determine project acceptance.
Leasing 121
where
NAL = net advantage to leasingC = asset cost if purchasedH = life of the leaseLt = periodic lease paymentT = marginal tax rate on ordinary incomeDt = depreciation charged in period tIt = interest portion of loan paymentOt = operating maintenance cost in period tSH = realized after-tax salvage valuert = applicable discount rate
This model allows for discount rates that are different for each of the terms. However, Bowerconcludes that the appropriate discount rate is the firm’s cost of capital. With this in mind wedrop the term containing It since the interest tax shelter is implicitly contained in the cost ofcapital. The model then becomes
NAL = C −H∑
t=1
Lt
(1 + k)t+
H∑t=1
T Lt
(1 + k)t−
H∑t=1
T Dt
(1 + k)t
(12.2)+
H∑t=1
Ot (1 − T )
(1 + k)t− SH
(1 + k)H
or, by combining terms:
NAL = C −H∑
t=1
Lt (1 − T ) + T Dt − Ot (1 − T )
(1 + k)t− SH
(1 + k)H(12.3)
At this point let us consider a numerical example.
Example 12.1 A firm with 12 percent overall marginal cost of capital has decided to acquirean asset that, if purchased, would cost $100. This same asset may be leased for five years at anannual lease payment of $30. Operating maintenance is expected to be $1 a year, and straight-line depreciation to a zero salvage value would be used if the asset were to be purchased. Theprospective lessee is in the 48 percent marginal tax category. Applying equation (12.3) weobtain
NAL = 100 −5∑
t=1
30(1 − 0.48) + 20(0.48) − (1 − 0.48)
(1.12)t
= 100 −5∑
t=1
15.60 + 9.60 − 0.52
(1.12)t
= 100 −5∑
t=1
24.68
(1.12)t
= $11.03
122 Capital Asset Investment: Strategy, Tactics & Tools
Since $11.03 > 0, the leasing alternative is preferable to purchase of the asset. But, what ifthe net, realized after-tax salvage were estimated to be $20? In this case the NAL would beonly $6.61 and the lease would be less attractive.
Alternative Analysis3
The discussion so far has been limited to the case of the lessee. Also, the impact of the leaseon the lessee’s debt capacity has not yet been considered. Myers, Dill, and Bautista (MDB)developed a model that allowed for the impact of the lease on the lessee’s debt capacity [114].The MDB model assumes that the lessee borrows 100 percent of the tax shields created byinterest payments, lease payments, and depreciation. This debt constraint is used to eliminatethe debt displacement term normally used in the lease valuation equation, assuming that adollar of debt is displaced by a dollar of lease. Myers, Dill, and Bautista generalize theirmodel by removing the constraint that a dollar of lease displaces a dollar of debt. However,they constrain the proportion of debt displaced by a dollar of the lease, λ, to be equal to theproportion of the tax shields the lessee borrows against, γ .
Here the effects of allowing λ to vary from 1.0 are examined. At the same time it is as-sumed that the lessee borrows 100 percent of the tax shields (γ = 1.0) in order to main-tain an optimal capital structure. The generalized model, using MDB’s model as the startingpoint, is:4
V0 = 1 −H∑
t=1
Pt (1 − T ) + T bt
(1 + r − γ rT )t+
H−1∑t=0
H∑τ=t+1
rT Pτ (γ − λ)
(1 + r − γ rT )t+1(1 + r )τ−t(12.4)
where
V0 = value of the lease to the lesseePt = lease payment in period t (normalized by dividing by the purchase price of the asset
leased)bt = normalized depreciation forgone in period t if the asset is leased instead of purchasedr = lessee’s borrowing rateT = lessee’s marginal tax rate on incomeH = life of the leaseλ = proportion of debt displaced by a dollar of the leaseγ = proportion of the tax shields the lessee borrows against
Equation (12.4) follows MDB’s notation except for the inclusion of γ . Salvage value andforgone investment tax credit are assumed to be zero to simplify the model, and operatingmaintenance expenses absorbed by the lessor are also assumed to be zero.
The valuation model in (12.4), once again, is for the lessee. To determine the value of thelease to the lessor may be somewhat more controversial if for no other reason than little workin this area has been published. The claim of MDB is that the lessor’s valuation is the lessee’svaluation model multiplied by −1.0 to reflect the reverse direction of the cash flow. And forthe lessor they claim that λ is the proportion of debt supported by the lease, because the leaseis an investment to the lessor. (Remember that for the lessee, λ represents the proportion of
3This section is based on the extension to the MDB work by Perg and Herbst [124].4The derivation is contained in the appendix to this chapter.
Leasing 123
debt displaced by the lease.) The lessor’s λ will very likely be different from the lessee’s; somay the tax rate, T.
If λ were to be the proportion of the lessor’s debt supported by the lease (in (12.4) multipliedby −1.0 and with λ = γ ), however, then r would be the lessor’s borrowing rate, not the lessee’sborrowing rate. This presents a problem. If the lessor acts as financial intermediary, thenthe lessor’s borrowing rate will generally be less than the lessee’s borrowing rate, becausethe debt obligations of the lessor are less risky. This lower risk is due to the lessor’s equitycushion, the likely more liquid nature of the lessor’s obligations, and the diversification throughholding many different leases. Financial intermediaries also tend to keep the maturity of theirobligations shorter than their assets in order to take advantage of a yield curve, that is, anaverage, upward sloping. Myers, Dill, and Bautista made a valuable contribution to the literatureon leasing. The problem, however, of two different discount rates (the lessee’s and the lessor’s)make the MDB approach to determining the lessor’s valuation of the lease unsuitable. We willnow look at an alternative model for the lessor.
We base our approach to determining the value of a lease to the lessor on the fact that, tothe lessor, the lease is an investment. The NPV of the lease is equal to the present value ofits after-tax cash flows, valued at the after-tax discount rate appropriate for the level of riskassociated with investment in the lease, less the purchase price of the asset to be leased. It isassumed that the lessor is also a lender, a share value maximizer, and financial markets arecompetitive. From this we can say that the lessor will invest in bonds, including those of thelessee, until their after-tax return is equal to the lessor’s after-tax cost of capital appropriate tothe risk associated with holding the bonds.
If the lease is equivalent in risk5 to the lessee’s bonds, then the cost of capital associatedwith investing in the lease is equal to the cost of capital associated with investing in thosebonds: the after-tax borrowing rate of the lessee. If the risk is not equal, then the lessor’s costof capital for the lease equals χ times the after-tax borrowing rate of the lessee, where χ > 1.0if the lease is riskier than the lessee’s bonds, and χ < 1 if the lease is less risky than the bond.Recognizing this we obtain the model for the value of the lease to the lessor:
V0 =H∑
t=1
Pt (1 − T ) + T bt
[1 + Xr (1 − T )]t− 1 (12.5)
Here Pt , bt , and r are the same as in (12.4). But T now represents the lessor’s marginal tax rate,not the lessee’s, and X is a risk adjustment factor. The factor X can be reasonably expectedto be related to the debt displacement factor λ in (12.4). For example, if leases have financialcharacteristics similar to subordinated debt, then the lessor’s cost of capital for the lease willexceed his cost of capital for the bonds (χ > 1), and a dollar of lease will displace less thana dollar of bonds (λ < 1). On the other hand, if leases possess financial characteristics thatmake them senior to the firm’s bonds, we would expect χ < 1 and λ > 1. These possiblerelationships are discussed in the following analysis.
5Strictly speaking, this section deals with a certainty environment. However, in discussing analysis of leases it is necessary to bringrisk into consideration. The awkward alternative would be either to deal with leases under certainty here and bring in risk in a laterchapter or to postpone treatment of leasing until later, and out of this author’s desired sequence of topics. Risk treatment in a formalsense is deferred, however, until later chapters.
124 Capital Asset Investment: Strategy, Tactics & Tools
Table 12.1 H = 15, r = 0.10, straight-line depreciation γ = 1
Lessee Lessor
Lessee’s Lessee’s Value of leasemarginal break-eventax rate lease payment
to lessorχ = 1
λ = 1 T = 0 T = 0.25 T = 0.50T = 0 0.1314738 0 0.0175207 0.0283163T = 0.25 0.1288274 −0.0201286 0 0.0145813T = 0.50 0.1260179 −0.0414979 −0.0185998 0
χ = 1.2
λ = 0.8 T = 0 T = 0.25 T = 0.50T = 0 0.1314738 −0.1045497 −0.0708275 −0.0378043T = 0.25 0.1337616 −0.0889678 −0.0569969 −0.0266946T = 0.50 0.1410994 −0.0389911 −0.0126358 0.0089385
χ = 1.0
λ = 0.8 T = 0 T = 0.25 T = 0.50T = 0 0.1314738 0 0.0175207 0.0283163T = 0.25 0.1337616 0.0174014 0.0326661 0.0401897T = 0.50 0.1410994 0.0732132 0.0812453 0.0782708
χ = 0.8
λ = 1.2 T = 0 T = 0.25 T = 0.50T = 0 0.1314738 0.1253474 0.1195513 0.1015017T = 0.25 0.1242443 0.0634667 0.0668896 0.0613131T = 0.50 0.1138490 −0.255119 −0.0088307 0.0035225
χ = 1
λ = 1.2 T = 0 T = 0.25 T = 0.50T = 0 0.1314738 0 0.0175207 0.0283163T = 0.25 0.1242443 −0.0549879 −0.0303416 −0.0092031T = 0.50 0.1138490 −0.1340555 −0.0991614 −0.0631539
Analysis
The factors that enter into the possible superiority of leasing over conventional financing are(1) the marginal tax rates (T ) of the lessor and the lessee and (2) the relationship betweenλ and χ . In order to explore the effects of the interactions of these variables on the value ofa lease, we compute the break-even lease payment of the lessee for various combinations ofthe lessee’s tax rate and λ. The break-even lease payment is then used to compute the value of thelease to the lessor for various combinations of the lessor’s tax rate and χ . The patterns of theresults are of major interest. They are not affected by varying H, r, or the use of accelerateddepreciation since these factors are common to both the lessee’s and the lessor’s valuationmodels. Therefore, only the results for H = 15, r = 10%, and straight-line depreciation arepresented in Table 12.1. In calculating the values for Table 12.1 the value of γ is kept equal to1.0 because it is thought the results will be more meaningful if the lessee always maintains anoptimal capital structure.
The topmost section of Table 12.1 confirms the conventional results that hold when leasesare financially equivalent to loans. Thus, if a dollar of lease displaces a dollar of debt (that is,
Leasing 125
λ = 1) and leases have the same risk as loans (χ = 1), then leasing is advantageous if, and onlyif, the lessee’s marginal tax rate is less than that of the lessor. The advantage occurs becausethe lease is tantamount to the sale of depreciation tax shields by the lessee to the lessor. If thelessor has a higher tax rate, then the value of the tax shields is greater to the lessor than thecost to the lessee for giving up the tax shields. The lower the lessee’s tax rate, and the higherthe lessor’s, the more mutually advantageous leasing becomes.
Results change dramatically once the assumption of financial equivalency is dropped. Thesecond and third panels of Table 12.1 illustrate what happens if a dollar of lease displaces only80 cents of debt (λ = 0.8). With this value of lambda the lessee’s break-even lease paymentactually increases as the tax rate increases, rather than decreasing as we might have expectedfrom the standard result at the top of the table. To understand this we refer to an importantpoint made by MDB. The lessee’s lease valuation rests upon the well-known Modigliani andMiller assumptions, which imply (among other things) that the only advantage to borrowingis the interest tax shield. If λ = 1.0, then leasing decreases the present value of the availabletax shields because the depreciation tax shield is surrendered. Therefore, as the lessee’s taxrate rises, his break-even lease payment falls. But if λ is sufficiently less than 1 (λ = 0.8 issufficiently less), then leasing actually increases the present value of the tax shields availablesince the increase in debt capacity — and therefore interest tax shields — more than outweighsthe loss of the depreciation tax shield. Because of this the lessee’s break-even lease paymentrises as the tax rate rises.
It follows that if λ is sufficiently less than 1, the value of leasing is a positive function of thetax rates of both the lessor and the lessee. How high the tax rates must be for lessor and lesseeto make leasing advantageous depends on how much riskier these debt-capacity-increasingleases are than the lending/borrowing alternative. If they are no more risky than lending (thatis, χ = 1), leasing is then advantageous for all positive tax rates, regardless of whether thelessee’s or lessor’s tax rate is higher.
The two panels at the bottom of Table 12.1 show the case in which $1 of lease displaces$1.20 of debt. In this case the lessee’s break-even lease payment falls even faster as his or hertax rate rises than it does when λ = 1. The reason this happens is that the lessee gives up thepresent value of reduced interest tax shields as the lessee’s debt capacity falls, in addition to thepresent value of the depreciation tax shield.6 Here the lessee’s marginal tax rate must be as lowas possible if leasing is to be advantageous. However, it does not follow that it is necessarilybest for the lessor’s tax rate to be as high as possible to make the lease most advantageous.When the debt-capacity-reducing effect of the lease (perhaps as a result of its senior claim onthe leased asset or assets) is reflected in lower risk and a lower cost of capital for the lessor (forexample, χ = 0.8), the greatest gain to leasing occurs if both the lessee and the lessor were tohave zero tax rates.
Implications
From the foregoing it is clear that the conventional condition for leasing to be advantageous —that the lessee’s tax rate be less than the lessor’s — applies only if leasing and borrowingare financially equivalent (that is, both λ = 1.0 and χ = 1.0). In the literature it has becometraditional to argue that financial equivalency should hold. However, for such equivalency toprevail would require that product differentiation of financial instruments not influence their
6Note that for T = 0 the lessee’s break-even lease payment is not affected by λ since the value of the tax shields is zero.
126 Capital Asset Investment: Strategy, Tactics & Tools
sales. Whether or not product differentiation can affect sales may not be settled to everyone’ssatisfaction. But until empirical evidence refuting parallel effects to those among consumergoods are published and confirmed, it would seem reasonable to expect differentiation toapply. If there really were no marketable differences between debentures, mortgage bonds,leases, and so on, one would be hard-pressed to explain why different financial instrumentsand arrangements persist.
If λ and χ may differ from 1.0, we may ask how they may do so. First, let us consider a lesseein sound financial condition; all debt is on an unsecured basis. A lease, however, is a highlysecured form of debt.7 This extra security — actual ownership of the physical asset — makesthe lease senior to the firm’s debt and impairs the security of the other debt.8 It follows thatif the riskiness of this debt is not going to increase (and it cannot if γ , the cost of capital,is to stay constant), total borrowing — including the obligation on the lease — must fall, andthis means λ > 1. The senior claim of the lease makes it somewhat less risky than the lessee’sdebt. To the extent that proceeds from the sale or re-lease of the asset, if it were to be seizedfrom the delinquent lessee, would be insufficient to cover the unpaid rentals and expensescaused by the default, the lessor becomes a general creditor. Altogether, this implies thatχ < 1.
Now, in contrast, let us consider a highly leveraged firm whose debt is virtually all secured.In such a case existing creditors would not perceive themselves as being harmed by leasing,relative to financing the asset with debt, because the new debt would have been secured inany case. But prospective creditors might find themselves more willing to make new leasesthan new secured loans because of their superior position with respect to recovering the assetthey hold title to if the lessee were to encounter financial difficulty. This enhanced recoveryability may cause such prospective creditors to be more willing to make leases rather thanloans (λ < 1.0) even though risk were to be increased (χ > 1.0).
Practical Perspective
It is probably reasonable to assume that much, if not most, leasing is done in situations whereλ = 1.0 and χ = 1.0. For high-quality, low-risk lessees it is likely that λ > 1.0 and χ < 1.0.From the bottom two panels of Table 12.1 we see that a low tax rate for the lessee is of thegreatest importance in this situation. It is also in the lessee’s vital interest to negotiate terms forthe lease that reflect the low risk associated with the lease. Because most standardized leasesare set up to protect the lessor from the higher-risk lessees, this would tend to limit low-riskprospective lessees to large, negotiated, and (usually) leveraged leases. Leveraged leases arediscussed in the following chapter.
In high-risk leases, λ < 1.0 and χ > 1.0. As shown by the middle two panels of Table 12.1,it may make leasing advantageous if the lessee were to have a high tax rate, provided thatλ is sufficiently less than 1. In this case expansion of his or her debt capacity is of primeconcern to the lessee; the foregoing analysis may understate the value of leasing becausethe value of additional debt capacity to the lessee may well exceed the present value ofthe tax shields since the future of the firm as a viable, going concern may well lie in thebalance.
7The lessor holds title to the asset and thus may recover it more easily than if title were held by the lessee.8If the asset were purchased, the title would be held by the firm that otherwise would be the lessee and it would serve to secure all thatfirm’s debt along with all its other assets.
Leasing 127
SUMMARY AND CONCLUSION
A financial lease is a financing arrangement in which the lessee purchases the use of an assetowned by the lessor. Asset use is separate from asset ownership. Lease analysis does notdetermine whether or not an asset should be acquired; it starts with the premise that the assetshould be acquired and attempts to determine if leasing provides an attractive alternative forfinancing the asset. The associated question is normally phrased as “lease or buy” or “lease orborrow.”
Traditional analysis of leases has focused almost exclusively on the problem of the lessee.However, to the lessor the lease is even more of a capital-budgeting decision; it is an investment.The lessor’s decision is not simply the mirror image of the lessee’s decision, because of differenttax rates and different risk implications of the same lease to each of them. Traditional analysishas assumed that leases were substitutes for debt, but it did not address the impact of leasingon the lessee’s debt capacity, or the measure of substitutability.
Myers, Dill, and Bautista extended the traditional analysis of leases to take into account theeffect of leasing on the debt capacity of the lessee. Perg and Herbst extended the analysis tothe lessor and integrated the analysis of lessee and lessor.
Appendix
The derivation of the generalized lease valuation equation for the lessee uses MDB’s notationexcept for introducing γ to represent the proportion of the tax shields that the lessee borrowsagainst. The starting point of the derivation is the debt constraint of the lessee. ϒt is the totaldebt of the firm in period t, L is the initial dollar value of the leased asset, Dt is the debt displacedin t per dollar of asset leased (Dt ≡ ∂Yt/∂L), St is the lessee’s total tax shield due to bookdepreciation on all assets owned in t, and Z is optimal borrowing excluding any contributionto debt capacity made by depreciation and interest tax shields.
Yt = λ
H∑τ = t+1
Pτ L
(1 + r )τ−t
(12A.1)= Z + γ
{ ∞∑τ = t+1
Sτ + rT Yτ
(1 + r )τ−t+
H∑τ = t+1
T Pτ L
(1 + r )τ−t
}
Differentiating ϒH−1 with respect to L and solving for DH−1:
DH−1 = −PH (λ − γ T ) − γ T bH
1 + r − γ rT(12A.2)
Substituting DH−1 into the expression for VH−1 and simplifying, yields the expression
VH−1 = −PH (1 − T ) − T bH
1 + r − γ rT+ rT PH (γ − λ)
(1 + r )(1 + r − γ rT )(12A.3)
Going back to the debt constraint, (12A.1) for YH−2, and differentiating with respect to L, andsolving for DH−2 we get
DH−2 = −PH−1(λ − γ T ) − γ T bH−1
1 + r − γ rT+ DH−1
(1 + r − γ rT )(12A.4)
Substituting (12A.2) into (12A.4) and then substituting (12A.4) into the expression for VH−2
and simplifying gives
VH−2 = −PH−1(1 − T ) − T bH−1 + VH−1
1 + r − γ rT+
H∑τ=H−1
rT Pτ (γ − λ)
(1 + r − γ rT )(1 + r )τ−H+2
(12A.5)
Leasing: Appendix 129
Clearly, this reasoning repeats, and in general (except for t = 0):
Vt = −Pt+1(1 − T ) − T bt+1 + Vt+1
1 + r − γ rT+
H∑τ = t+1
rT Pτ (γ − λ)
(1 + r − γ rT )(1 + r )τ−t(12A.6)
For t = 0:
V0 = −P1(1 − T ) − T b1 + V1
1 + r − γ rT+
H∑τ=1
rT Pτ (γ − λ)
(1 + r − γ rT )(1 + r )τ(12A.7)
By successive substitution, V1, V2, . . . , VH−1 are eliminated to obtain
V0 = 1 −H∑
t=1
−Pt (1 − T ) − T bt
(1 + r − γ rT )t+
H−1∑τ=0
H∑τ=t+1
rT Pτ (γ − λ)
(1 + r − γ rT )t+1(1 + r )τ−t(12A.8)
13
Leveraged Leases
DEFINITION AND CHARACTERISTICS
Analytical techniques discussed in previous chapters are applicable to analysis of the leaseversus purchase decision and to lease analysis in general. This chapter is concerned with theanalysis of a particular category of financial lease termed leveraged leases.
Leveraged leases are tax-sheltered financial leases in the sense that some of the returns tothe lessor are attributable to tax legislation intended to encourage capital investment. The USinvestment tax credit, for example, had its raison d’etre in encouraging capital investment.However, as Campanella [20] points out, companies that could benefit from new capital oftencannot benefit from the investment tax credit. Such firms might not be producing taxableincome, for instance, or carryover from previous tax shelter may make the credit useless.These firms find it advantageous to find a lessor who can use the tax benefits, and then arrangea lease in return for a lease cost that is lowered by those tax benefits. Leveraged leasing grewrapidly during the 1960s and is still significant. Changes in the tax laws have not, as somethought they might, negated the advantages of leveraged lease arrangements.
A leveraged lease is typically done through a trust arrangement. The lessor contributes asmall percentage (typically 20–40%) of the capital equipment cost, and the trust then borrowsthe balance from institutional investors on a nonrecourse basis to the owner. The loans tothe trust are secured by a first lien on the equipment, along with an assignment of the leaseand the lease payments. According to Campanella “a leveraged lease is a direct lease whereinthe lessor, through a trust, has borrowed a portion of the equipment cost to help finance thetransaction. Under the ‘true’ lease concept, the lessor retains a material equity and ownershipin the leased property and no option to purchase, other than a fair market value option at theexpiration of the lease term, is given to the lessee.” This is an important point because, withthe similar railroad trust certificate leases, the lessee is treated as the owner for tax purposes,and no tax advantage exists.
The leveraged lease trust is usually administered by a commercial bank, called the ownertrustee. This trustee takes title to the capital equipment and enters into the lease arrangementwith the actual user, the lessee. The lease is a long-term, full-payment “true” lease. With atrue lease the lessor retains a material equity ownership interest in the capital equipment andno option to purchase the equipment is provided to the lessee, at any price other than “fairmarket value.” The tax treatment on which leveraged leases are predicated rests on their beingtrue leases. After first servicing principal and interest on debt, the trustee remits any remainingfunds, pro rata, to the equity investors. It is the profitability of the leasing arrangement to theequity investors that we are primarily concerned with in this chapter.
Although the returns to the creditors of the trust normally may be straightforwardly calcu-lated as the yield on a fully amortized note, and the cost to the lessee may similarly be found,calculating the return to equity is much more troublesome. The reason for this difficulty is inthe mixed cash flows to equity that is characteristic of leveraged leases. A number of methodsfor computing the attractiveness of returns to equity on leveraged leases have been proposed.
132 Capital Asset Investment: Strategy, Tactics & Tools
Unfortunately, instead of resolving the issue, the various proposals themselves may have addedto the confusion. When one applies the methods to the typical leveraged lease, the results mayseemingly be contradictory. Leveraged leases provide us with an interesting application ofanalytical techniques that have been proposed for dealing with mixed cash flows. Furthermore,the assumption of certainty is not so unreasonable as it may be with most capital investmentprojects, because the returns are governed by the contractual terms of the lease.
METHODS: LEVERAGED LEASE ANALYSIS
The methods of analysis we shall consider are the net present value, the Wiar method, the sink-ing fund methods (including the Teichroew, Robichek, and Montalbano (TRM) approach),and the two-stage method. Since these methods have been introduced in previous chapters,it is assumed the reader is already familiar with the basic application of each. We shalltherefore analyze several example problems to provide a structure for comparing the variousmethods.
APPLICATION OF THE METHODS
Example 13.1 This example was proposed by Bierman [11] to illustrate the superiority ofthe net present value (NPV) method of analysis. The project has cash flows of −$400, $1100,and −$700 at the end of years t = 0, 1, and 2. Figure 13.1 contains plotted results of NPV fordifferent values of k, the cost of capital. The NPV function reaches a maximum at 27.0 percent.The project has IRR at 0 and 75 percent, and between these values the NPV is positive. Thissuggests that for 0 percent < k < 75 percent the project is acceptable.
That this is a mixed project is apparent if we decompose the cash flows into two subsequencesas follows: subprojects (a) and (c) are investments, whereas (b) and (d) are financing projects.In the particular subproject above, the investment returns are exactly offset by the correspond-ing financing costs. At rates between 0 and 75 percent the investment return dominates the
−.40 −.20 .00 .20 .40 .60 .80 1.00 1.20 1.40k–Axis−60
−40
−20
0
40
20
60
NPV Axis
NPV Function
Figure 13.1 Bierman example. Plot of investment characteristics
Leveraged Leases 133
RIC Function
RIC–Axis
k–Axis−.40 −.20
−.20
.00
.20
.40
.60
.80
1
.00 .20 .40 .60 .80 1.00 1.20 1.40
r � k
Figure 13.2 Bierman example. Plot of investment characteristics
financing cost. Consider the decomposition corresponding to 10 percent cost of capital, forexample:
t = 0 t = 1 t = 2(a) −400 +400 Return = 0 percent(b) +700 −700 Cost = 0 percent
or
t = 0 t = 1 t = 2(c) −400 700 Return = 75 percent(d) 400 −700 Cost = 75 percent
t = 0 t = 1 t = 2
(e) −400 463.64(f) 636.36 −700
The “loan” implicit in cash flow sequence (f) is at 10 percent. The NPV of sequence (e) is21.49 and has an IRR of 15.91 percent.
The TRM return on invested capital (RIC = r∗) is plotted against k in Figure 13.2. Ther∗ values for r∗ > k are for 0 percent < k < 75 percent — the same as obtained with theNPV approach. The initial investment method (IIM) yields rIIM values of 0 and 75 percent forcorresponding cost of capital percentages. Furthermore, the traditional sinking fund method(TSFM) yields rTSFM values of 0 and 75 percent. The two-stage method yields results consistentwith these. Next let us consider a project that has two more cash flows.
Example 13.2 This project has cash flows at end of years 0, 1, 2, 3, and 4 of −$1000, $2000,$2000, $2000, and −$5000, respectively. The NPV is positive between 0 and 400 percent. (TheNPV versus k graph in Figure 13.3 extends only to 140 percent.) The corresponding RIC or r ∗
plot is contained in Figure 13.4. Note that the r∗ is a double-valued function of k. In this casethe plot of interest is for positive values of r∗. We reject the negative values on the basis thatthe adjusted cash flow series from which we find r∗ for a given k value sums to greater than
134 Capital Asset Investment: Strategy, Tactics & Tools
NPV Axis � 101
NPV Function160
120
40
−40
−.40 −.20 .00 .20 .40 .60 .80
80
−80
0
1 1.20 1.40k−Axis
Figure 13.3 NPV–RIC conflict. Plot of investment characteristics
225
200
175
150
125
0
−25
−50
−75
−100
−125
−150
20 40 60 80 100 120 140k%
r � −100%
r%
RIC Function
Figure 13.4 NPV–RIC conflict
zero. In other words, the simple sum of the adjusted cash flows is positive, so we do not admitnegative returns even though mathematically they are acceptable.
Table 13.1 contains an outline of the TRM solution and some selected dual values ofr∗ for corresponding ki . As we have noted before, returns of less than −100 percent haveno economic significance because it is not possible to “lose more than we lose” with aproject.
In contrast to the Bierman project in Example 13.1, for this case the NPV and the RIC arenot totally in agreement. In fact, the RIC conflicts with the other, NPV-compatible methodsalso, as is to be expected. The NPV is greater than zero between the IRR boundary values of0 and 400 percent. The r∗ (RIC) is greater than k over much of the same range but not all
Leveraged Leases 135
Table 13.1 TRM analysis of Example 13.2
Solution to projectOriginal cash flows balance equation
−1000 100.00000%2000 173.205082000 191.96396 = rmin
2000−5000
Some solution values are:
k r∗ r∗%
−5 −90.93% 125.500 −100.00 130.255 −107.75 135.25
10 −114.55 139.2520 −126.10 145.5030 −135.73 150.2540 −144.00 154.3450 −151.29 157.55
100 −178.96 167.73400 — 183.12
of it. At a k value of 200 percent, for instance, the r∗ is 176.65 percent whereas the NPV isstill positive. One could dismiss this conflict as irrelevant in a world where k cannot be sogreat, since the r∗ and NPV do agree over the range of k values that firms could, in fact, beexpected to have. However, because they are different over part of the range of k, we shouldabandon any complacency generated by the Bierman example and other examples involvingonly two cash flows beyond the initial outlay. The Bierman example underscores the hazardslurking in the shadows for those who would try to generalize from the three-cash-flow case tothe N-cash-flow case.
The conflict between the NPV and the RIC for large values of k is attributable in this caseto the assumption implicit in the NPV that the investment and financing rates are both equalto the cost of capital, whereas the RIC requires only that the financing rate be equal to ki .
Let us next apply the Wiar method (WM), the initial investment method (IIM), traditionalsinking fund method (TSFM), and the two-stage method (TSM) to this example and summarizethe results obtained in Table 13.2.
The various methods are in agreement that for k values of 10, 20, and 30 percent the projectis acceptable. The Wiar method could not properly be applied because it is predicated onoverall debt service and equity-return cash flow streams. Since no debt was assumed for thisexample, the method could not yield results other than the IRR values of 0 and 400 percent.This would not be a proper test of the method.
This is a mixed project, for which, to find r∗ for a given F , we must solve the equation
−(1 + k)(1 + r∗)3 + 2(1 + k)(1 + r∗)2 + 2(1 + k)(1 + r∗) + [2(1 + k) − 5] = 0
136 Capital Asset Investment: Strategy, Tactics & Tools
Table 13.2 Summary of analytical results for Example 13.2
For k = 10%NPV method $558.64 AcceptTRM RIC 139.25% AcceptWiar method equity return InapplicableIIM return 17.06% AcceptTSFM return 71.45% AcceptTwo-stage measures 0.55 years, 4.58% Accepta
For k = 20%NPV method $801.70 AcceptTRM RIC 145.50% AcceptWiar method equity return InapplicableIIM return 34.57% AcceptTSFM return 109.09% AcceptTwo-stage measures 0.60 years, 9.05% Accept
For k = 30%NPV method $881.59 AcceptTRM RIC 150.25% AcceptWiar method equity return InapplicableIIM return 52.01% AcceptTSFM return 125.70% AcceptTwo-stage measures 0.65 years, 13.47% Accepta
a Assumes management requirement for capital recovery is not less thanthese fractions of a year.
Table 13.3 Childs and Gridley leveraged lease example — a la Wiar’s analysis, borrow $80 at 8% for15 Years, invest $20 equity, tax rate 48%
NetMortgage Effect Effect Effect on Mortgage after-tax
Lease interest on earnings on earnings principal cash flowt Year payment Depreciation payment before taxes taxes after taxes payment (to equity)
0 1 0 0 0 0 0 0 0 −20.001 2 9.64 16.67 6.40 −13.43 −13.45a 0.02 2.95 13.74a
2 3 9.64 15.15 6.16 −11.67 −5.60 −6.07 3.19 5.893 4 9.64 13.64 5.91 −9.91 −4.75 −5.16 3.44 5.044 5 9.64 12.12 5.63 −8.11 −3.90 −4.21 3.72 4.195 6 9.64 10.61 5.34 −6.31 −3.03 −3.28 4.01 3.326 7 9.64 9.09 5.02 −4.47 −2.14 −2.33 4.33 2.437 8 9.64 7.57 4.67 −2.60 −1.25 −1.35 4.68 1.548 9 9.64 6.06 4.29 −0.71 −0.34 −0.37 5.06 0.639 10 9.64 4.09 3.89 1.66 0.80 0.86 5.46 −0.51
10 11 9.64 0 3.45 6.19 2.97 3.22 5.90 −2.6811 12 9.64 0 2.98 6.66 3.19 3.47 6.37 −2.9012 13 9.64 0 2.47 7.17 3.45 3.72 6.88 −3.1613 14 9.64 0 1.92 7.72 3.70 4.02 7.43 −3.4114 15 9.64 0 1.33 8.31 3.99 4.32 8.02 −3.7015 16 9.64 0 0.68 8.96 4.30 4.66 8.56 −3.90
Total 144.60 95.00 60.14 −10.54 −12.06 1.52 80.00 −3.4816 Residual 0 5.00 0 −5.00 −2.40 −2.60 0 2.40
Total 144.60 100.00 60.14 −15.54 −14.46 −1.08 80.00 −1.08
a Includes 7% investment tax credit on the $100 investment.
Leveraged Leases 137
ANALYSIS OF A TYPICAL LEVERAGED LEASE(CHILDS AND GRIDLEY)
Example 13.3 This example, first proposed by Childs and Gridley [22], has since beendiscussed extensively in the literature. Table 13.3 contains the salient data for the Childsand Gridley leveraged lease. A 7 percent investment tax credit applies to the equipment.Depreciation is calculated using sum-of-the-years’ digits method and an 11-year lifetime. Theequipment is depreciated to a salvage-for-tax amount of 5 percent of the original depreciablevalue. The depreciation for t = 9 (year 10) is shown as $4.09; the calculated amount is $4.55.This difference is due to the requirement that $5 remain for write-off at t = 15 (year 16). Thecolumn of data of most concern to the equity investor is the “Net-after-tax cash flow”column.It is this column on which our analysis will focus, although the Wiar method incorporatesadditional information.
This series of cash flows is a mixed project in the TRM sense. The rmin can be calculated fromthe series by taking the cash flows for t = 0 through t = 8 and solving for the IRR. Verificationthat the result obtained, 28.5038 percent, is, in fact, the project rmin can be obtained byevaluating the project balance equations corresponding to the preceding cash flows at thisrate; they will all be less than or equal to zero.
The Wiar analysis [ ] can be carried out by evaluating the equation
20(1 + rw)16 − 100(1.045220)16 − 80(1.0416)16
to find that rw = 5.80736 percent after tax (or 11.16799 percent pre-tax). The rate of 4.52220percent is the return on the overall cash flow to the leveraged lease, whereas the 4.16 percent
Table 13.4 Analysis of Childs and Gridley leveraged lease example — a la Wiar
NetEffect on Effect Effect on after-tax
Lease earnings on earnings cash flowt Year payment Depreciation before taxes taxes after taxes (overall lease)
0 1 0 0 0 0 0 −100.001 2 9.64 16.67 −7.03 −10.37a −3.34a 20.01a
2 3 9.64 15.15 −5.51 −2.64 −2.87 12.283 4 9.64 13.64 −4.00 −1.92 −2.08 11.564 5 9.64 12.12 −2.48 −1.19 −1.29 10.835 6 9.64 10.61 −0.97 −0.47 −0.50 10.116 7 9.64 9.09 0.55 0.26 0.29 9.387 8 9.64 7.57 2.07 0.99 1.08 8.658 9 9.64 6.06 3.58 1.72 1.86 7.929 10 9.64 4.09 5.55 2.66 2.89 6.98
10 11 9.64 0 9.64 4.63 5.01 5.0111 12 9.64 0 9.64 4.63 5.01 5.0112 13 9.64 0 9.64 4.63 5.01 5.0113 14 9.64 0 9.64 4.63 5.01 5.0114 15 9.64 0 9.64 4.63 5.01 5.0115 16 9.64 0 9.64 4.63 5.01 5.01
Total 144.60 95.00 49.60 16.82 32.78 7.7816 Residual 0 5.00 −5.00 −2.40 2.60 2.40
Total 144.60 100.00 44.60 14.42 35.38 10.18
a Includes 7 percent investment tax credit on the $100 investment.
138 Capital Asset Investment: Strategy, Tactics & Tools
30
25
20
15
10
0
−5
−10
5.81
Two−stage Payback
0
5
10
Years
k%
TRM r∗
5
10
15 20 25
Wiar Rate
TRM r∗
r% Two−stage “Loan” Rate
r � k
Trad
ition
alSi
nkin
gFu
nd
Meth
od Rate
Initi
alIn
vestm
ent M
ethod
Rate
Figure 13.5 Several measures on the Childs and Gridley leveraged lease returns
rate is the after-tax interest on the debt financing. The overall cash flows to this lease aredisplayed in Table 13.4. Note that the Wiar rate does not depend on the lessor’s cost of capital.The decision to accept the leasing project as an investment or not is made by comparing therw with the lessor’s cut-off or hurdle rate.
The TRM–RIC function for the Childs and Gridley leveraged lease is plotted in Figure13.5, along with the initial investment, traditional sinking fund, and two-stage measures.The functions are plotted on the same graph to illustrate the consistency of the methods: allbut the Wiar equity return would mark the project acceptable for values of k < 25 percent,and unacceptable for k > 25 percent. The apparent symmetry of the “loan” phase mea-sure of the two-stage method with the traditional sinking fund method is interesting. Thelease is acceptable for k < 25 percent with the two-stage method, provided that the pay-back (recovery of investment plus opportunity cost) is acceptable, since the implicit loanembedded in the later cash flows is at a rate less than k in this range. The fact that thetwo-stage payback goes to infinite value after k reaches 28.5 percent is noteworthy, forthat is the value of rmin in the TRM solution. An infinite value of the two-stage paybackmeans that the initial investment and opportunity cost of the funds committed will neverbe fully recovered. Any such payback beyond the project’s economic life may be similarlyinterpreted.
Leveraged Leases 139
CONCLUSION
Leveraged leases provide perhaps the closest “real-world” approximation to the certainty envi-ronment we have considered to exist for capital-budgeting analysis to this point. Furthermore,growth in the use of leveraged leasing and the magnitude of funds involved make leveragedlease profitability analysis a significant area for investigation.
In this chapter we examined leveraged leases as an interesting and practical application ofanalytical methods designed to deal with mixed cash flows. The results obtained suggest that anaccept/reject decision may be correctly reached by several methods: NPV; two-stage method;initial investment method; traditional sinking fund method; and the TRM approach.
The Wiar method would reject projects acceptable under the other criteria. In fact, the Wiarrate is truly internal to the project — the cost of capital does not enter into its calculation at all.It is solely a function of the IRR to the overall lease cash flows and the interest rate on the debtcomponent of the lease. Because of this the Wiar rate, rw, is plotted on a straight line parallelto the k-axis.
14
Alternative Investment Measures
This chapter introduces several additional measures that may be applied to capital-budgetingprojects. They include the geometric mean rate of return, the average discounted rate of return,Boulding’s time spread, and Macaulay’s duration.
In order to simplify the exposition, in this chapter capital outlays will be restricted to theinitial outlay, and subsequent net cash flows will be assumed to be nonnegative. All cash flowsare assumed to occur at the end of the corresponding time periods. In other words, only simpleinvestments in the sense defined in Chapter 10 will be considered.
ADDITIONAL RATE OF RETURN MEASURES
Geometric Mean Rate of Return
In contrast to the more generally known arithmetic mean, the geometric mean is obtained bytaking the nth root of the product of the n items, rather than dividing their sum by the number n.When considering the average of interest or growth rates over a period of time, the geometricmean is considered more appropriate than the arithmetic mean because it takes into accountthe effects of period-to-period compounding that the arithmetic mean ignores. Because of thetime-value of money, it is, for example, not correct to say that a deposit of $1000 that earns4 percent the first year, 8 percent the second year, and 9 percent the third year has earned anaverage rate of 7 percent over the three-year period. The amount earned is $224.29 on the $1000principal. The uniform, average annual rate for which $1000 will grow to $1224.29 over threeyears is not 7 percent but 6.98 percent. For a three-year period, this is not a great difference, butthe error of the arithmetic mean over the geometric mean becomes greater the longer the timespan covered. For example, if a principal amount can earn 4 percent for 10 periods, 8 percentfor the next 10 periods, and 9 percent for the last 10 periods of a 30-year investment life, thegeometric mean rate is 6.16 percent, whereas the arithmetic mean is 7 percent.
The geometric mean rate of return on an investment, rg, may be defined by
rg =[
n∏t=1
(1 + yt )
]1/n
− 1 (14.1)
where
yt = Et − Et−1 + Rt
Et−1(14.2)
with Rt the cash flow at end of period t and the Et representing the market value of the investmentat end of period t. Note that this formulation measures the period-to-period changes in equityvalue over the life of the investment, even though the overall change in equity value is notrealized until disposition of the asset at the end of its useful life.
142 Capital Asset Investment: Strategy, Tactics & Tools
Average Discounted Rate of Return
The average discounted rate of return, ra, may be defined as
ra =
n∑t=1
yt (1 + ra)−t
n∑t=1
(1 + ra)−t
(14.3)
where yt is as defined in equation (14.2). Since ra is defined recursively, it would appear difficultand time consuming to compute. However, existing computer programs for finding IRR canbe used. Equation (14.3) is equivalent to
1 − (1 + ra)−n =n∑
t=1
yt (1 + ra)−t
sincen∑
t=1
(1 + ra)−t = 1 − (1 + ra)−n
ra
which is the equation an¬ra , for the present value of an annuity of one for n periods at rate ra.Rearranging the terms, we obtain
1 =n−1∑t=1
yt (1 + ra)−t + (1 + yn)(1 + ra)−n (14.4)
To solve for ra in equation (14.4), we need only recognize that this is the equation for an“investment” of $1, returning y, . . . yn−1, and a final return of 1 + yn at time n, and applyany computer program for finding IRR. This equation is the same as that for finding yield tomaturity for a variable-payment bond that sells for its par value of $1, and at maturity returnsthe $1 with the final payment of interest.
Robert R. Trippi illustrated the application and usefulness of geometric mean return andaverage discounted rate of return for measuring the returns on investments that undergo changesin market value over their useful lives [162]. Trippi employed 12 illustrative examples in hisexposition. Here we will examine several similar examples to demonstrate the techniques andcompare rg with ra and the IRR, which is denoted by rc for “conventional” rate of return inTrippi’s notation.
To illustrate the meaning and calculation of rg and ra, let us consider the following example,which requires a $1000 initial investment and returns the net, after-tax cash flows indicated.In addition, the market value of the asset changes from period to period.
Example 14.1
t = 1 t = 2 t = 3 t = 4 t = 5Cash flow $200 200 300 400 200Market value $1100 1200 1250 1300 1325
Assuming the asset will be disposed of at the end of year 5, the $1325 market value willbe realized as a positive salvage value and the total cash flow increased by this amount to$1525. For simplicity we assume that the market values are the after-tax proceeds that would
Alternative Investment Measures 143
be realized if the asset were to be disposed of at the end of any indicated year. The IRR (heredenoted rc) is 28.5044 percent, on the cash flow series −$1000, $200, $200, $300, $400, and$1525.
To calculate rg we employ equation (14.2) n = 5 times, substituting into equation (14.1).Thus
y1 = E1 − E0 + R1
E0= 1100 − 1000 + 200
1000= 0.3000
y2 = E2 − E1 + R2
E1= 1200 − 1100 + 200
1100= 0.2727
and so on. Applying equation (14.1), we obtain
rg = [(1.3)(1.27273)(1.29166)(1.36)(1.17308)]1/5 − 1 = 27.8024 percent
Now, to find ra we employ equation (14.4). This results in the following series to which weapply the procedure for finding the IRR
−1 0.3 0.27273 0.29167 0.36 1.17308
Note that this is identical to the series for a hypothetical bond that sells for $1, yields thevarious amounts defined by equation (14.2), and returns the original investment in a balloonpayment of one at t = 5. Thus we find that
ra = 28.6193 percent
For this example, ra > rc > rg. This will not always be the case. In fact, several differentcases were identified by Trippi. Table 14.1 displays the relationship between rg, ra, and rc forsome different patterns of cash flow and market value. It is important to note that the IRR, rc,depends only on the pattern of cash flows, whereas both rg and ra depend on the pattern ofchanges in market value in addition to cash flows. Thus many projects having the same IRRwill have different rg and still different ra.
Trippi proposed the average discounted rate of return as an alternative to the geometricmean rate of return for incorporating the change in equity value and its pattern in measuringthe return on investment. This is something that had received more attention in the area ofsecurities analysis and portfolio management than in capital budgeting. However, the conceptis as applicable to capital investment projects as it is to investments in securities.
Shortly after the appearance of Trippi’s paper, Peter Bacon, Robert Haessler, and RichardWilliams [4] found an interesting counterintuitive example, an asset that produces no cashflow prior to its sale, doubles in value in the first year, and then in the second year declinesto its original value. To quote Bacon, Haessler, and Williams: “The marginal return in year 1is 100% and year 2 is −50%. Calculating the geometric mean and the internal rate of returnor just relying on intuition all indicate that the true return is zero. . . . However, when . . . ra
is computed, the result is 36% [4].” As they go on to point out, the problem with Trippi’sra is that it discounts the marginal returns in addition to averaging them. Since percentageincreases are weighted the same as percentage decreases, the problem is not readily apparentwith those investments that only increase or decrease in market value. However, with assetsthat first increase and then decrease, or vice versa, the investment return is misrepresented byra. The fact that early marginal percentage changes are discounted less than those occurringlater further contributes to the problem.
144 Capital Asset Investment: Strategy, Tactics & Tools
Table 14.1 Examples of rg, ra and rc for various cases, each requiring $1000
Case no. 1 2 3 4 5 rg% ra% rc (IRR)%
1 Cash flow 0 100 200 300 400Market value 1000 1000 1000 1000 1000 19.1596 16.9082 16.9082
2 Cash flow 200 200 200 200 200Market value 1000 1000 1000 1000 1000 20.0000 20.0000 20.0000
3 Cash flow 400 300 200 100 0Market value 1000 1000 1000 1000 1000 19.1596 24.2573 24.2573
4 Cash flow 0 0 0 0 0Market value 1000 1100 1300 1600 2000 14.8698 13.6208 14.8698
5 Cash flow 0 0 0 0 0Market value 1200 1400 1600 1800 2000 14.8698 15.5507 14.8698
6 Cash flow 0 0 0 0 0Market value 1400 1700 1900 2000 2000 14.8698 19.1003 14.8698
7 Cash flow 200 200 200 200 200Market value 1000 900 700 400 0 −10.7682 −10.5832 0.0000
8 Cash flow 200 200 200 200 200Market value 800 600 400 200 0 0.0000 0.0000 0.0000
9 Cash flow 200 200 200 200 200Market value 1000 600 300 100 0 9.8561 12.7813 0.0000
10 Cash flow 200 200 200 200 200Market value 1000 1100 1200 1300 1400 25.0228 24.9019 24.8848
11 Cash flow 200 200 200 200 200Market value 1000 1400 1400 1400 1400 23.4399 27.0023 24.8848
12 Cash flow 200 200 200 200 200Market value 1000 1000 1000 1000 1400 27.1069 24.8848 24.8848
In his reply to Bacon, Haessler, and Williams, Trippi emphasized [163] that his “primaryintent . . . was not to advocate universal acceptance of one measure over the others, but ratherto demonstrate the general difficulties and frequent lack of conformity of each of the mea-sures. . . . Clearly some non-unity marginal rate of substitution of present for future undis-tributed wealth is likely to apply . . . this phenomenon being totally lost with the conventionalmeasures.”
Thus, although caution must be exercised, the geometric mean rate of return and averagediscounted rate of return may be considered adjuncts in the process of investment evaluation,particularly where there is unrealized (monotonic) increase or decrease over the life of theinvestment. The methods yielding rg and ra should perhaps not be used at all for investmentscharacterized by nonmonotonically changing market value.
Chapter 15 takes up the topic of abandonment value in capital budgeting. There it may beseen that an alternative to ra exists. For now, we shall go on to finish this chapter with the topicsof unrecovered investment, duration, and time spread.
TIME-RELATED MEASURES IN INVESTMENT ANALYSIS
Although the literature has generally concentrated on other aspects of the topic of capitalbudgeting, time-related measures are useful and may provide additional insight. Here we shallconsider two time-related measures: Macaulay’s duration and Boulding’s time spread. Thelatter, although identical to the modern actuaries’ equated time, was proposed in 1936.
Alternative Investment Measures 145
Boulding’s Time Spread
Kenneth E. Boulding [15] proposed time spread (TS) as a measure of the average time intervalelapsing between sets of capital outlays and returns. For investments having a single initialoutlay at time t = 0, Boulding’s time spread is defined by
N∑t=1
Rt =N∑
t=1
Rt (1 + r )TS−t (14.5)
Since (1 + r )TS is constant for a given r and TS,
N∑t=1
Rt = (1 + r )TSN∑
t=1
Rt (1 + r )−t (14.5a)
so that
(1 + r )TS =
N∑t=1
Rt
N∑t=1
Rt (1 + r )−t
(14.5b)
and
TS = log
N∑t=1
Rt
N∑t=1
Rt (1 + r )−t
÷ log(1 + r ) (14.6)
In the case of r = r∗ (the IRR), it can be shown (as Boulding did) that
TS = log
N∑t=1
Rt
R0
÷ log(1 + r ) (14.7)
where R0 is the initial (and only) capital outlay for the project. The proof of this involves therecognition that, for the IRR, the initial outlay is equal to the sum of the discounted cash flowsat discount rate r∗.
When used with the IRR, time spread shows how long the initial investment remains investedon average at rate r∗ (the IRR). Time spread provides the point in time at which a single amount,equal to the undiscounted sum of cash flows, would be equivalent to the individual cash flowsat intervals over the life of the investment, at a given rate of interest. It is a measure of theaverage time between capital outlays and net cash receipts. In cases for which the only cashoutlay occurs at t = 0, time spread therefore measures the average time elapsed to receive thenet cash flows over the interval t = 1 through t = N . The following example will clarify thisand set the concept.
Consider case 1 in Table 14.1. This project, costing $1000, yields total cash flows of $2000over its life, including $1000 realized on disposition of the asset at t = 5. Time spread for thisproject is TS = 4.4370 years, and the IRR is rc = 16.9082 percent. The individual net cashflows could be replaced by a single cash flow of $2000 (equal to the undiscounted sum of cash
146 Capital Asset Investment: Strategy, Tactics & Tools
inflows) at t = TS: the equation
R0 =N∑
t=1
Rt (1 + r∗)−TS = (1 + r∗)−TSN∑
t=1
Rt (14.8)
follows directly from equation (14.7). Substituting the parameters of case 1, we obtain
$1000 = ((1.169082)−4.4370)$2000 = $1000
Thus the entire cash flow series beyond the initial outlay may be replaced by a single amountequal to its sum at t = TS. Similarly, rates other than the IRR may be used, with the sameinterpretation, although for different rates different values for TS will be obtained.
Macaulay’s Duration
Following soon after Boulding, Frederick R. Macaulay [94] developed the concept of durationas an alternative to the conventional time measure for bonds — the term to maturity. For thoseinvestments with a single cash outlay at t = 0, duration is a weighted average of repaymenttimes (or dates) with weights equal to the present values of the cash flows at their respectivedates. Equation (14.9) defines duration:
D =
N∑t=1
t Rt (1 + r )−t
N∑t=1
Rt (1 + r )−t
(14.9)
For r = r∗, the IRR, the denominator, by definition of the IRR, is equal to R0, the initial outlay.Therefore, for r∗,
D =
N∑t=1
t Rt (1 + r∗)−t
R0(14.10)
Calculation of duration is straightforward, even if tedious. As Durand has pointed out [35],even though different, D converges to the same value as TS when N is finite and the discountrate approaches zero. It might be added that when the only cash flow is at t = N , D = TS = N .
The history of development and application of duration is very well described by Weil, who,among other things, points out that Hicks’s elasticity of capital with respect to discount factorsis equivalent to duration [168], although apparently developed independently and somewhatlater. Weil also mentions Tjalling C. Koopmans’s (1942, unpublished) paper on matching lifeinsurance assets and liabilities to “immunize” the company against effects of interest ratechanges. At the time he wrote it Koopmans was employed by Penn Mutual Life InsuranceCompany [168, p. 591]. Credit for the seminal idea on immunization is often awarded toRedington [128] for his later contribution, which appears to be the earliest published paper onthe topic although appearing a decade after Koopmans’s paper was written.
Like time spread, duration provides a useful adjunct measure to be used in capital budgetingalthough developed for another purpose. For case 1 of Table 14.1, D = 4.3696. This providesthe “average” time that elapses for a dollar of present value to be received from this project.This is somewhat less than the time spread value. It may be shown that D ≤ TS.
Alternative Investment Measures 147
$2,000
1,500
500
–500
1,000
0
0
–1,000
1 2 3 4 5Time
r � 0%
r � 0%
r � 30%
r � 30%
r � r∗
r � r∗
Figure 14.1 Unrecovered investment at end of time period indicated
Unrecovered Investment
The unrecovered investment of a capital-budgeting project is defined by
U = R0(1 + r )T −T∑
t=1
Rt (1 + r )T −t (14.11)
With the IRR it is implicitly assumed that the IRR rate, r∗, is earned on the unrecoveredprincipal as measured at the beginning of each period over the project life. This was made ex-plicit in the treatment of conflicting IRR–NPV rankings in Chapter 8. With r∗, the unrecoveredinvestment will be exactly zero at t = N . For t < N and with r = r∗, U > 0; the unrecov-ered investment will be positive prior to the end of the project life. For t < N and r < r∗, Ubecomes negative prior to t = N . Figure 14.1 shows the graphs for U as a function of t andr . Since it is assumed that cash flows occur only at the end of each period, discontinuitiesoccur at these points. For r = r∗ the unrecovered investment becomes zero after the end ofperiod 4. At a higher rate, r = 30 percent, for example, the cash flows were not even adequatefor paying the “interest” on the unrecovered investment by t = 5, so that the investment isnot fully recovered by that time. In fact, at t = 5 the unrecovered investment is greater thanthe initial investment at t = 0. For rates less than r∗ the investment is fully recovered priorto t = 5.
Table 14.2, which employs the same type of component breakdown used in Chapter 8,illustrates the concept of unrecovered investment. The table displays unrecovered investmentunder the heading “Ending principal” for the beginning-of-period points. Table 14.2 treats theinvestment case 1 of Table 14.1 as though it were a loan, and unrecovered investment is seento be the “loan principal.”
148 Capital Asset Investment: Strategy, Tactics & Tools
Table 14.2 Component breakdown of cash flows (amounts rounded to nearestcents) for r = r∗
Interest onBeginning beginning Principal Total Endingprincipal principal repayment payment principal
1 $1000.00 $169.08 $0 $0 $1169.082 1169.08 197.67 0 100 1266.753 1266.75 214.18 0 200 1280.934 1280.93 216.58 83.42 300 1197.515 1197.51 202.48 1400.00 1400 0
Note: It is assumed that end-of-period payments are composed of interest at r∗ = 16.9082% onbeginning-of-period principal plus (if there is an excess over the interest) principal repayment.
Unrecovered investment, U , is related to payback period P. The point at which U becomeszero corresponds to the time at which the investment has been fully recovered. For r = 0,the conventionally defined payback period is obtained. (With the conventional calculation ofpayback, the assumption that cash flows occur only at the end of each period is violated; oncethe payback period has been bracketed, the end-of-period cash flow at the further time periodis treated as if it occurred uniformly over the period. The formula for U does not violate theassumption of end-of-period cash receipts, and to this extent there is a difference with paybackcalculation.)
The concept and measure of unrecovered investment are useful as an adjunct to other capital-budgeting measures. It serves to focus attention on the nature of the process of investmentrecovery implicit within other measures.
SUMMARY AND CONCLUSION
The additional investment measures presented in this chapter provide useful adjuncts to othercapital-budgeting measures. They illustrate that factors other than cash flow alone may be ofinterest; as, for instance, Trippi’s average rate of return that, to some extent, incorporates theappreciation in asset value that is not realized until the end of the project life. Used alonethey are not especially useful; but used with other measurements of capital-budgeting projectcharacteristics they can provide additional insight, thereby facilitating better decision-making.
Perhaps Durand states the case as well as anyone when he says:
From all this I conclude that we need to take a far broader view of capital budgeting than we have in thepast. We have squandered altogether too much effort on a futile search for that elusive will-o’-the-wispthe one and only index of profitability; and we have lost valuable perspective thereby [35, p. 191].
15
Project Abandonment Analysis
Up to this point it has been implicitly assumed that capital-budgeting projects, if accepted forinvestment, would (for better or for worse) be held tenaciously until t = N . This is undulyrestrictive and unrealistic. It violates the realities of capital-budgeting practice; capital invest-ments are often abandoned prior to termination of their theoretical maximum useful lives. Inthis chapter we formally consider abandonment prior to the end of project life.
THE ROBICHEK–VAN HORNE ANALYSIS
Alexander A. Robichek and James C. Van Horne (R–VH) presented an algorithm for determin-ing if and when a capital investment project should be abandoned prior to the end of its usefullife at t = N [130]. The original procedure was modified somewhat [131] after Edward A.Dyl and Hugh W. Long [38] showed that the original algorithm could, in some circumstances,break down.
The R–VH paper became widely known and cited, perhaps because it was the first paper ina major journal to have dealt with the subject of abandonment value. It provided an importantprod in the process of awakening academics to a problem that in practice has always been afactor considered by practitioners, but for which little mention was to be found in the literature.In order to facilitate their analysis R–VH assumed, that (1) an adequate estimate of the firm’scost of capital exists; (2) there is no capital rationing; and (3) a unique IRR exists for theprojects considered. Assumption (3) may be satisfied by considering only simple investments,in the sense defined in Chapter 10.
The R–VH algorithm (corrected to satisfy the Dyl–Long critique) is stated as:
(A) Compute PVτ ·a for a = n, where
PVτ ·a =a∑
t=τ+1
ECt ·τ(1 + k)(t−τ )
+ AVa·τ(1 + k)(a−τ )
(B) If PVτ ·n > AVτ , continue to hold project and evaluate it again at time τ + 1, based uponexpectation at that time.
(C) If PVτ ·n < AVτ , compute PVτ ·a for a = n − 1.(D) Compare PVτ ·n−i with AVτ as in (B) and (C) above. Continue this procedure until either
the decision to hold is reached or a = τ + 1.(E) If PVτ ·a ≤ AV for all τ + 1 ≤ a ≤ n, then abandon project at time τ . . . .
where
ECt ·τ = expected cash flow in year t as of year τ .AVt = abandonment value in year t.ACt = “actual” simulated cash flow in year t [131, p. 96].
150 Capital Asset Investment: Strategy, Tactics & Tools
This algorithm as it is stated appears to have been written to deal with the timing of abandonmentfor a project after it had been accepted. However, R–VH suggest that their rule might beextended to ex ante (prior to acceptance) project analysis. This chapter is primarily concernedwith such ex ante capital investment project analysis. This emphasis, however, should not beconstrued to imply that continued project review, such as that suggested in the R–VH algorithm,is any less important.
Step (A) of the R–VH algorithm defines the present value at time reference point τ as thediscounted sum of all cash flows occurring from the period immediately following τ to theend of the project life plus the present value at τ of the expected salvage value to be receivedat the end of the project life.
Step (B) states that we should keep the project if the present value of continuing to do so asdefined in step (A) is greater than the present value of salvage. Note that these present valuesare at time = τ . Present values are normally calculated for t = 0.
Step (C) requires that we perform additional calculation and analysis before abandoningthe project, even though the salvage at time τ is greater than or equal to the present value ofexpected cash flows from time τ + 1 to the end of the project’s maximum useful life at t = N .This step is necessary in order to avoid premature abandonment of the project; the NPV maypossibly be increased by holding the project for one or more additional time periods eventhough it will not be held all the way to t = N .
Step (D) specifies that the analysis in steps (A) through (C) inclusive be repeated until aperiod is found for which, in light of the expected returns as of today, the decision is to holdon to the project, or else we get to a = τ + 1. In the latter case the decision is to hold on forthe current period, then abandon.
Finally, step (E) prescribes that if the salvage value at any point in time τ exceeds the presentvalues that are potentially to be obtained by holding on to the project, it should be abandonedat time τ .
As specified above, the R–VH algorithm seems designed particularly for ongoing, periodicanalysis of capital investments with a view to whether they should be abandoned or kept inservice for another time period. However, although it may not be clear from the wording ofthe procedure, the R–VH approach is suitable for analyzing capital-budgeting projects ex anteas well as the ex post, which the R–VH paper appears to stress. In such cases it could beused to help answer these questions. (1) What is the optimal period to keep the project if it isaccepted? (2) What is the expected present value if the project were to be accepted and heldfor the optimal period and no longer?
The R–VH paper was useful in calling attention to the problem of project abandonment.However, it is equivalent to a “dual” formulation of the Terborgh–MAPI method discussed inChapter 9. Such formulation yields, instead of the MAPI “adverse minimum,” a “propitiousmaximum” NPV, if NPV is the measure of project acceptability or desirability employed.Associated with this maximum is the optimum number of years over which the project, ifaccepted, would be held. To develop the methodology, we need, in addition to the R–VHassumptions, the assumption that we have or can obtain reliable estimates of salvage valuesfor time periods between the adoption of the project and the end of its useful life at t = N .For those fairly standard types of equipment for which there is a well-developed secondarymarket, this should be a reasonable assumption.1 On the other hand, for plant and for custom-made equipment this assumption will in general lack the reliability associated with the former
1For example, we could develop estimates of salvage value deterioration gradients by careful, systematic analysis of trade publicationscarrying advertisements for used equipment and by consulting with dealers specializing in such equipment.
Project Abandonment Analysis 151
category. For a thorough treatment of the problems associated with extraction of such estimates,the reader is referred to Terborgh’s Dynamic Equipment Policy [151].
The following two sections illustrate application of the modified MAPI procedure to threecapital-budgeting projects and compare results with the R–VH method.
AN ALTERNATIVE METHOD: A PARABLE2
To highlight the points presented above an example will be employed. The capital projectcommittee of Typical Manufacturing Company (TMC) is considering which of three mutuallyexclusive production machines it should purchase to perform certain operations on a newproduct that the company has decided to add to its line. The machines are all of standard design,and hundreds of various vintages are in use across the nation. The following information hasbeen presented to the committee:
Purchase For the year indicatedPrice net after-tax cash benefits
Machine A $2000 $600 $600 $600 $600 $100 $100 $100Machine B 2000 700 600 500 400 300 200 100Machine C 1000 100 200 300 400 300 200 100Year 1 2 3 4 5 6 7
All three machines have useful lives of seven years. Machine A has an estimated salvagevalue of $419.43, B $419.43, and C $478.30 at the end of seven years.
The recommendation provided to the committee is that machine C be purchased because, atthe firm’s 10 percent cost of capital, it has an NPV of $350.69, while A and B have respectivelyNPVs of $286.99 and $346.75. Since most members of the committee are well versed in thetraditional finance literature concerned with capital budgeting, C is chosen for purchase withlittle discussion. Of course, there is some argument over the significance of the slight edge inNPV that C has over B, but A is out of the running from the beginning.
Has the committee selected wisely? No! “But,” the reply will be, “by selecting the projectwith the highest net present value we are assuring the maximum increase in the value of equity.”However, there is more to the story. In the approach to project selection that was followed,no attention was paid to salvage value prior to the end of each machine’s useful life. In theexample presented here the salvage values for A and B represent 20 percent per annum declinesfrom the prior year’s value, beginning with the purchase price paid for each, while that for Crepresents a 10 percent per annum decline.
Tables 15.1(a)–(c) present alternative calculations that might have been performed for ma-chines A, B, and C. Readers familiar with the MAPI method for replacement evaluationpresented by Terborgh will note some similarity in that his method also considers interme-diate salvage values. However, the MAPI method was developed primarily for replacementdecisions, and is based on minimum cost (adverse minima) rather than maximum benefit con-siderations. The method presented here might be considered the dual to Terborgh’s method.Like the MAPI method, that shown in Tables 15.1(a)–(c) uses the concept of time-adjustedannual averages, or level annuities. However, instead of finding adverse minima, we insteadfind what might be called “propitious maxima.” In this instance these are employed to present
2The following pages are reprinted with permission from The Engineering Economist, Vol. 22, No. 1 (Fall 1976). Copyright c©American Institute of Industrial Engineers, Inc., 25 Technology Park/Atlanta, Norcross, GA. 30092.
152 Capital Asset Investment: Strategy, Tactics & Tools
Tabl
e15
.1(a
)C
apita
l-bu
dget
ing
proj
ectw
ithsa
lvag
eva
lue
aten
dof
each
year
(ini
tialo
utla
y=
$200
0;es
timat
edus
eful
life
=7
year
s;de
clin
ein
salv
age
valu
efr
ombe
ginn
ing
ofpe
riod
valu
e=
20%
year
)
23
45
67
Pres
ent
PVof
Acc
umul
ated
Cap
ital
Lev
elL
evel
1va
lue
retu
rn=
PV=
reco
very
annu
ity=
annu
ity=
Yea
rR
etur
nfa
ctor
—10
%(1
)×
(2)
(3)
Acc
umul
ated
fact
or—
10%
(4)×
(5)
cost
$200
0×
(5)
1$6
000.
9091
545.
4654
5.46
1.10
0060
0.00
−220
0.00
260
00.
8264
495.
8410
41.3
00.
5762
600.
00−1
152.
403
600
0.75
1345
0.78
1492
.08
0.40
2160
0.00
−804
.20
460
00.
6830
409.
8019
01.8
80.
3155
600.
00−6
31.0
05
100
0.62
0962
.09
1963
.97
0.26
3851
8.10
−527
.60
610
00.
5645
56.4
520
20.4
20.
2296
463.
89−4
59.2
07
100
0.51
3251
.32
2071
.74
0.20
5442
5.54
−410
.80
89
1012
13a
14sa
lvag
ePV
ofL
evel
11PV
ofN
PVIn
tern
alva
lue
atsa
lvag
e=
annu
ity=
(6)+
(7)
annu
ity—
of(1
1)=
rate
ofY
ear
year
end
(2)×
(8)
(5)×
(9)
+(1
0)10
%(1
1)×
(12)
retu
rn
116
00.0
014
54.5
616
00.0
00.
00.
909
0.0
0.10
002
1280
.00
1057
.79
609.
5057
.10
1.73
699
.13
0.13
113
1024
.00
769.
3330
9.35
105.
152.
487
261.
510.
1609
481
9.20
559.
5117
6.53
145.
533.
170
461.
330.
1881
565
5.36
406.
9110
7.34
97.8
43.
791
370.
890.
1680
652
4.28
295.
9667
.95
72.6
44.
355
316.
350.
1565
741
9.43
215.
2544
.21
58.9
54.
868
286.
970.
1502
aC
olum
n13
will
gene
rally
beve
rysl
ight
lydi
ffer
entt
han
ifN
PVw
ere
calc
ulat
eddi
rect
ly,d
ueto
roun
ding
.
Project Abandonment Analysis 153
Tabl
e15
.1(b
)C
apita
l-bu
dget
ing
proj
ectw
ithsa
lvag
eva
lue
aten
dof
each
year
(ini
tialo
utla
y=
$200
0;es
timat
edus
eful
life
=7
year
s;de
clin
ein
salv
age
valu
efr
ombe
ginn
ing
ofpe
riod
=20
%ye
ar)
23
45
67
Pres
ent
PVof
Acc
umul
ated
Cap
ital
Lev
elL
evel
1va
lue
retu
rn=
PV=
reco
very
annu
ity=
annu
ity=
Yea
rR
etur
nfa
ctor
—10
%(1
)×
(2)
(3)
Acc
umul
ated
fact
or—
10%
(4)×
(5)
cost
$200
0×
(5)
1$7
000.
9091
636.
3763
6.37
1.10
0070
0.00
−220
0.00
260
00.
8264
495.
8411
32.2
10.
5762
652.
38−1
152.
403
500
0.75
1337
5.65
1507
.86
0.40
2160
6.31
−804
.20
440
00.
6830
273.
2017
81.0
60.
3155
561.
92−6
31.0
05
300
0.62
0918
6.27
1967
.33
0.26
3851
8.98
−527
.60
620
00.
5645
112.
9020
80.2
30.
2296
477.
62−4
59.2
07
100
0.51
3251
.32
2131
.55
0.20
5443
7.82
−410
.80
89
1012
13a
14sa
lvag
ePV
ofL
evel
11PV
ofN
PVIn
tern
alva
lue
atsa
lvag
e=
annu
ity=
(6)+
(7)
annu
ityof
(11)
=ra
teof
Yea
rye
aren
d(2
)×
(8)
(5)×
(9)
+(1
0)fa
ctor
—10
%(1
1)×
(12)
retu
rn
116
00.0
014
54.5
616
00.0
010
0.00
0.90
990
.90
0.15
002
1280
.00
1057
.79
609.
5010
9.48
1.73
619
0.06
0.16
023
1024
.00
769.
3330
9.35
111.
462.
487
277.
200.
1668
481
9.20
559.
5117
6.53
107.
453.
170
340.
620.
1699
565
5.36
406.
9110
7.34
98.7
23.
791
374.
250.
1696
652
4.28
295.
9667
.95
86.3
74.
355
376.
140.
1662
741
9.43
215.
2544
.21
71.2
34.
868
346.
750.
1599
aC
olum
n13
will
gene
rally
beve
rysl
ight
lydi
ffer
entt
han
ifN
PVw
ere
calc
ulat
eddi
rect
lydu
eto
roun
ding
.
154 Capital Asset Investment: Strategy, Tactics & Tools
Tabl
e15
.1(c
)C
apita
l-bu
dget
ing
proj
ectw
ithsa
lvag
eva
lue
aten
dof
each
year
(ini
tialo
utla
y=
$200
0;es
timat
edus
eful
life
=7
year
s;de
clin
ein
salv
age
valu
efr
ombe
ginn
ing
ofpe
riod
=20
%Y
ear)
23
45
67
Pres
ent
PVof
Acc
umul
ated
Cap
ital
Lev
elL
evel
1va
lue
retu
rn=
PV=
reco
very
annu
ity=
annu
ity=
Yea
rR
etur
nfa
ctor
—10
%(1
)×
(2)
(3)
Acc
umul
ated
fact
or—
10%
(4)×
(5)
cost
$200
0×
(5)
1$1
000.
9091
90.9
190
.91
1.10
0010
0.00
−110
0.00
220
00.
8264
165.
2825
6.19
0.57
6214
7.62
−576
.20
330
00.
7513
225.
3948
1.58
0.40
2119
3.64
−402
.10
440
00.
6830
273.
2075
4.78
0.31
5523
8.13
−315
.50
530
00.
6209
186.
2794
1.05
0.26
3824
8.25
−263
.80
620
00.
5645
112.
9010
53.9
50.
2296
241.
99−2
29.6
07
100
0.51
3251
.32
1105
.27
0.20
5422
7.02
−205
.40
89
1012
13a
14sa
lvag
ePV
ofL
evel
11PV
ofN
PVIn
tern
alva
lue
atsa
lvag
e=
annu
ity=
(6)+
(7)
annu
ityof
(11)
=ra
teof
Yea
rye
aren
d(2
)×
(8)
(5)×
(9)
+(1
0)fa
ctor
—10
%(1
1)×
(12)
retu
rn
190
0.00
818.
1990
0.00
−100
.00
0.90
9−9
0.90
0.00
002
810.
0066
9.38
385.
70−4
2.88
1.73
6−7
4.44
0.05
623
729.
0054
7.70
220.
2311
.77
2.48
729
.27
0.11
204
656.
1044
8.12
141.
3864
.01
3.17
020
2.91
0.16
375
590.
4936
6.64
96.7
255
.76
3.79
121
1.39
0.18
216
531.
4430
0.00
68.8
881
.27
4.35
535
3.93
0.18
587
478.
3024
5.46
50.4
272
.04
4.86
835
0.69
0.18
12
aC
olum
n13
will
gene
rally
beve
rysl
ight
lydi
ffer
entt
han
ifN
PVw
ere
calc
ulat
eddi
rect
ly,d
ueto
roun
ding
.
Project Abandonment Analysis 155
matrices representing the continuum of NPV opportunities of each project, assuming that theprojects may be abandoned at the end of years 1, 2, 3, . . . , n where n represents the last yearin the useful life of the project.
Column 6 in the tables gives the level annuity having the same accumulated value at theend of the indicated year as the values in column 4. Column 7 gives the level annuities for thenumber of years indicated that are equivalent to the initial outlay. Column 10 gives the uniformannual equivalent to the salvage value in each indicated year. By summing across columns 6,7, and 10 and then multiplying by the appropriate factors for the present value of an annuity,the NPVs of column 13 are obtained. The figures in column 13 are the NPVs of the projects ifthey are abandoned and sold for salvage at the end of the year indicated. Column 13 could havebeen calculated directly, of course, and to a somewhat greater precision in the trailing digits.
Armed with the information contained in Tables 15.1(a)–(c) the committee probably wouldhave selected project A and not project C. If project A were selected, and then abandoned atthe end of the fourth year of service, it would provide an NPV of $461.33. This is higher thanthat of B, which reaches a peak of $376.14 in the sixth year, and C, which must also be keptfor six years if its peak NPV of $353.93 is to be realized.
Since the projects reach maximum NPV with different timing, we have a situation tantamountto that of projects with unequal economic lives. Thus, it may appear necessary to adjust theTable 15.1 figures to reflect the different timing of optimal abandonment for each project.Ordinarily, the easiest means for adjusting for different economic lives is to find the uniformannual equivalent of the NPVs by multiplying them by the corresponding capital recoveryfactor. However, we already have these results in column 11.
Over a 12-year time horizon, the least common denominator of four- and six-year lives,projects A, B, and C have NPVs of $991.78, $588.46, and $553.71, respectively. These figuresmay be obtained by multiplying the uniform annual equivalents for optimal abandonment bythe present value of annuity factor for 12 periods. Note that since the same present value ofannuity factor is used, the comparison could have been directly between the uniform annualequivalents.
Comparison to R–VH3
This section is concerned with comparison of the method illustrated in the foregoing sectionwith the revised Robichek–Van Horne algorithm.
Let us begin by applying the R–VH rule to the machines that were considered by TMC.Employing the R–VH procedure with fixed, point estimates of cash returns in each periodresults in the values displayed in Table 15.2. The values in row 1 for projects A, B, and Care identical (except for rounding errors in the trailing digits) to the values in column 13 ofTable 15.1. In row 5 of Table 15.2(a) we see that the first figure is negative. The interpretationof this value is that, if at the end of year 4 project A is not abandoned, the company will incuran opportunity cost with present value as of the end of year 4, of $132.51 during year 5.
The figure of −$212.35 that follows in row 5 is the present value as of the end of year 4, of thecumulative opportunity cost if project A is held through years 5 and 6. The figure of −$255.39in the last column in row 5 is the present value, as of the end of year 4, of the cumulativeopportunity cost that will be suffered if project A is held from the end of year 4 through theend of year 7.
3Ibid., pp. 63–71.
156 Capital Asset Investment: Strategy, Tactics & Tools
Table 15.2 Present value in row year I, of salvage in column year J, plus cumulative returns throughcolumn year J, less salvage value at start of column year J
1 2 3 4 5 6 7
(A)1 0.00 99.17 261.46 461.45 370.94 316.41 287.012 109.09 287.60 507.59 408.03 348.05 315.713 196.36 438.35 328.84 262.86 227.284 266.18 145.72 73.14 34.015 −132.51 −212.35 −255.396 −87.82 −135.177 −52.08
(B)1 90.91 190.08 277.24 340.62 374.30 376.22 346.822 109.09 204.96 274.68 311.73 313.84 281.503 105.46 182.15 222.00 225.22 189.654 84.36 129.19 131.74 92.625 49.31 52.12 9.086 3.09 −44.267 −52.08
(C)1 −90.91 −74.38 29.30 202.92 307.72 353.96 350.732 18.18 132.23 323.22 438.50 489.35 485.803 125.45 335.54 462.34 518.29 514.384 231.09 370.58 432.11 427.825 153.44 221.12 216.406 74.46 69.267 −5.72
Thus, in terms of the figures of Table 15.2, each project should be abandoned at the end ofthe year prior to that corresponding to the row in which the figures become negative. In termsof opportunity cost, the interpretation is that a project should be abandoned when continuedretention results in an opportunity cost, from loss in salvage value, greater than revenues insubsequent periods.
The figures in Table 15.2 could have been generated entirely by the procedure implicit inTable 15.1, simply by shifting the time reference point forward one period for each new rowof figures generated. However, in Table 15.2, for project A the values in column 4 are greaterthan any values in their corresponding rows. Therefore, at t = 0 there is no need to generateany more than the first row of values for each project. However, once a particular project hasbeen selected, it may be useful to reevaluate it at the end of subsequent periods to determineif the optimal time of abandonment has shifted under changing estimates of cash flow andabandonment value. Such a procedure is equivalent to the R–VH approach.
A DYNAMIC PROGRAMMING APPROACH4
The solutions shown in Table 15.3 were obtained by using a dynamic programming approach,which provides a useful alternative to that described in the preceding section and also to theR–VH algorithm. Note that this dynamic programming formulation and solution employ the
4The solutions to the preceding examples, which are shown in Table 15.1, were offered by an anonymous referee, who reviewed thepaper for The Engineering Economist.
Project Abandonment Analysis 157
Table 15.3 Calculation of optimal abandonment decision and present value for three examplemachines (dynamic programming approach)
End of Return Abandonment Discounted return (DRt ) =year t Rt value (AVt ) max{AVt ; (0.9091)DRt+1} + Rt Decision
Project A (Machine A)7 $100 $419.43 $519.43 –6 100 524.28 624.28 Abandon5 100 655.36 755.36 Abandon4 600 819.20 1419.20 Abandon3 600 1024.00 1890.19 Keep2 600 1280.00 2318.37 Keep1 600 1600.00 2707.63 Keep0 – – 2461.51 –
Project B (Machine B)7 $100 $419.43 $519.43 –6 200 524.28 724.28 Abandon5 300 655.36 958.44 Keep4 400 819.20 1271.32 Keep3 500 1024.00 1655.76 Keep2 600 1280.00 2105.25 Keep1 700 1600.00 2613.88 Keep0 – – 2376.28 Keep
Project C (Machine C)7 $100 $478.30 $578.30 –6 200 531.44 731.44 Abandon5 300 590.49 964.95 Keep4 400 656.10 1277.24 Keep3 300 729.00 1461.14 Keep2 200 810.00 1548.32 Keep1 100 900.00 1489.40 Keep0 – – 1354.00 –
“backward searching” algorithm. (This is described in the chapter on dynamic programming byDaniel Teichroew [148, pp. 610–621].) This is what James L. Pappas used in his contributionon project abandonment [120], which was published simultaneously with the paper from whichthe preceding section was extracted.
Application of dynamic programming to equipment repair and replacement problems is cov-ered in most texts on management science/operations research. The problem of abandonmentvalue reduces to a special case of replacement, one in which an existing asset may be replacedby a hypothetical asset that does not exist, and therefore has a value of zero for the parametersof cost, cash flows, and so on, associated with it.
Since the methodology may not be familiar to many readers, a few words about Table 15.3are in order. Starting with the year 7 values for project A, the value $519.43 is the sum ofthat year’s return and salvage value. Subsequent returns do not have to be considered sincethe machine lasts only seven years. For year 6, the value $624.28 is obtained by addingthe $100 return in year 6 to $524.28, which is the year 6’s abandonment value. Since theabandonment value is $524.28, which is a greater amount than the discounted future returns($519.43 × 0.9091 = $472.21), the decision is to abandon. The decision to abandon holds untilyear 3, where the $1024 abandonment value is less than $1290.19 (= $1419.20 × 0.9091).
158 Capital Asset Investment: Strategy, Tactics & Tools
SUMMARY AND CONCLUSION
It has been shown that consideration of abandonment values can change the selection fromamong alternatives that would otherwise be made if only final salvage values were considered.
The possibility of abandoning a capital investment at a point in time prior to the estimateduseful or economic life has important implications for capital budgeting. Although we have notyet considered the effects of risk and uncertainty, the possibility of abandonment expands theoptions available to management and subsequently reduces the risk associated with decisionsbased on holding assets to the end of their lives. We must recognize that in a world cloudedwith great economic and political uncertainties, abandonment analysis synchronizes with thearray of techniques that fall under the topic broadly termed contingency planning. To neglectthe meaning and impact of abandonment and intermediate salvage values would be to refuse amost valuable instrument for gaining additional insight into the process of capital investmentevaluation.
The origins of abandonment analysis are implicit in writings going back at least as early asTerborgh’s Dynamic Equipment Policy. Actually, the adverse minimum of the Terborgh–MAPImethod does identify the optimal project life. The methods illustrated in this chapter can nodoubt be supplemented, modified, argued, and discussed much further. Some may prefer theR–VH algorithm, some the tabularized procedure, others the dynamic programming technique.Since the methods presented in this chapter yield equivalent results, the question of which oneshould be employed is largely a matter of personal preference.
16
Multiple Project Capital Budgeting
Preceding chapters considered various means for measuring the acceptability of individualcapital-budgeting projects under conditions of certainty and no risk. Ranking of projects waslimited to the problem of choosing which one project from a set of mutually exclusive projectsshould be selected when all the candidate invesments meet at least the minimum criteria foradoption. The problem of capital rationing was not previously considered, although actually nofirm has unlimited capital, and most have funds limitations, at least periodically, that precludeinvestment in the entire set of projects meeting their minimum criteria. Neither, to this point,were the effects of other constraints considered, whether economic, technical, or managerialpolicy.
BUDGET AND OTHER CONSTRAINTS
In this chapter we will continue to assume a world of certainty in order not to let considera-tions relating to risk and uncertainty obscure exposition of basic principles and methodologies.However, we shall deal explicitly with the implications of those factors that constrain man-agement to choose a subset of the total array of projects that would individually be acceptablein the absence of restrictions.
To simplify exposition, a single measure of investment worth will be employed throughoutthis chapter. The net present value (NPV) will be used provide the measure of individual projectdesirability. The NPV chosen as the single measure to be used primarily because it relates moredirectly and unambiguously to the basic valuation model of financial management, which wasintroduced at the outset of this book, than other measures do. Alternatively, if preferred, theprofitability index, internal rate of return, payback, or a composite function of measures canbe used. (We would hope, however, that the payback measure would not be adopted as the solecriterion by anyone who has read this far.)
Consider the three capital investment projects in Table 16.1. If the projects are not mutuallyexclusive and there are no limits on funds that may be invested, all three projects will beundertaken by the firm. However, once we begin to consider capital rationing it becomes clearthat a method is needed for selecting a subset from among candidate projects. For example,various budget limitations yield differing selections and total NPV for the capital budget(see Table 16.2). Here we have only three projects, and the only constraint is the one onfunds available for investments — capital rationing. Other constraints are common and furthercomplicate the selection process.
GENERAL LINEAR PROGRAMMING APPROACH
Additional constraints may take several forms. For example, suppose that one project is toconstruct a new assembly facility a short distance from our existing plant, and another projectis to build an overhead conveyor from our existing plant to the new facility. Obviously, we should
160 Capital Asset Investment: Strategy, Tactics & Tools
Table 16.1
Project Cost NPV
A $60,000 $30,000B 30,000 20,000C 40,000 25,000
Table 16.2
Budget Accepted Total NPV
≥$130,000 A, B, C $75,000100,000 A, C 55,000
90,000 A, B 50,00070,000 B, C 45,00060,000 A 30,000
not even consider building the conveyor unless the construction project has first been accepted.Another type of constraint is that of mutual exclusivity. For example the two projects: (1) repairthe existing facility now, and the mutually exclusive alternative (2) destroy the existing facilitynow and replace it with something new. Still another type of constraint is the requirement that iftwo projects are both accepted, a third project will also be accepted. Depending upon whetherour objective is to maximize a value (such as NPV) or minimize a value (such as cost), thegeneral linear programming problem may be specified as:
Maximize p1x1 + p2x2 + · · · + pn xn
Subject to a11x1 + a12x2 + · · · + a1n xn ≤ b1
a21x1 + a22x2 + · · · + a2n xn ≤ b2
...am1x1 + am2x2 + · · · + amn xn ≤ bm
and for all i, xi ≥ 0
or (16.1)
Minimize b1u1 + b2u2 + · · · + bmum
Subject to a11u1 + a21u2 + · · · + amum ≥ p1
a12u1 + a22u2 + · · · + am2um ≥ p2...a1nu1 + am2u2 + · · · + amnum ≥ pn
and for all i, ui ≥ 0
which, in matrix algebra notation, becomes
Maximize p · x Minimize b′uSubject to A · x ≤ b or Subject to A′u ≥ p′
xi ≥ 0 for all i ui ≥ 0 for all i
(16.2)
Multiple Project Capital Budgeting 161
By adding the requirement that xi be integer-valued for all i, we have made this into a linearinteger programming problem. Further restriction on the xi , specifically the requirement thatthey take on only the values zero or one, produces a zero–one integer programming problem.We formally specify the following two frequently encountered and important constraints.
Mutual Exclusivity
A set of n projects, from which at most one may be selected, yields the constraintn∑
i=1
xi ≤ 1 (16.3)
Since we already have a nonnegativity constraint on all the xi , this means that only one of thexi may have a nonzero value, namely a value of one. However, the constraint allows for nonebeing accepted, since a zero value for every one of the project xi satisfies the constraint.
Contingent Projects
Project B is said to be contingent on project A if it can be accepted only if A is accepted. Thisyields the constraint
xb ≤ xa (16.4)
which is equivalent to
xb − xa ≤ 0 (16.5)
or
xa − xb ≥ 0
This last constraint form allows project A to be accepted, yet does not force its acceptance.However, an attempt to accept B with A not already accepted produces a value of −1, whichis less than zero and violates the constraint. Therefore this constraint accomplishes what wewant and no more.
As the number of projects increases, the difficulty of selecting a subset that is in some sense“best” increases. When constraints in addition to budget limitations apply to the selectionproblem, things becomes unmanageable without a systematic procedure for carrying out theselection process.
We have used the method of linear programming to illustrate how a subset of projects maybe selected. Linear programming facilitates the handling of constraints, but its use in capitalbudgeting is limited. Capital investments are not finely divisible. We either accept a projectcompletely or reject it; for example, we do not choose to invest in 0.763 or 1.917 of a project.This means that we invest in 0, 1, 2, or some other integer number of projects of the particulartype. In fact, it often will be the case that we have a unique project, so that the relevant values are0 (reject the project) or 1 (accept the project). For such projects there is no option of acceptinga second, a third, and so on, because they do not exist. When there are multiples of a particularproject, for example, construction of one warehouse, construction of a second warehouse, andso on, then each may be considered to be a separate project, identical to the others. Linearprogramming may be made appropriate to capital-budgeting applications by modifying, for
162 Capital Asset Investment: Strategy, Tactics & Tools
example, the well-known simplex method,1 to incorporate Gomory’s cutting-plane approach[55]. The result of such modification is an integer linear programming algorithm. However,because most capital-budgeting proposals involve one or a few projects of a particular type,and because the problem setup and constraint system are similar, the approach that will betaken here is that of 0–1 (zero–one) integer programming. The 0–1 terminology is useful sincea project is either selected (1) or rejected (0) and thus can be assigned a numerical value ofeither 0 or 1 as the algorithm proceeds.
ZERO–ONE INTEGER PROGRAMMING2
Two similar approaches to 0–1 integer programming are those of Balas’ [5] and of Lawler andBell [83]. Both rest on the concept of vector partial ordering as a heuristic for systematicallysearching out the optimal solution (if one exists) to an array of n projects in m constraints,without the necessity of evaluating each and every one of the 2n possible combinations ofprojects.3 The discussion that follows is patterned on that of Lawler and Bell. Developmentof new zero–one solution algorithms and refinements on existing methods continue becausemethods developed so far have their individual idiosyncracies and none is clearly superior to theothers for all problems. The Lawler and Bell algorithm, for purposes of illustration, is as usefulas any. However, in terms of computational efficiency, there seem to be marked differencesbetween alternative solution algorithms. For instance, Pettway examined the efficiency ofseveral, and found wide differences among them [127]. So has Horvath [71], who developeda computer program for using the algorithm that he found the most efficient and reliable. Avector x is said to be a binary vector if each element is either 0 or 1. The vector may then belooked upon as a binary number. For two binary vectors, x and y, vector partial ordering meansthat x ≤ y if, and only if, xi ≤ yi for all i. For example, x ≤ y for x = (0 0 1 0 1) and y =(0 0 1 1 1). For a particular vector x, there may or may not exist a vector or vectors x ′ suchthat x < x ′.
Lawler and Bell denote by x ∗ the vector following x in numerical (binary number) orderingfor which x = x∗. The vector x ∗ can be calculated by treating the vector x as a binary number.There are three steps involved. First, subtract 1 from x. Next, apply the logical “or” operation4
to x and x −1 to obtain x∗ − 1. Finally, add 1 to x∗ = 1. The alert reader will note that thesethree steps are equivalent to binary addition of “1” to the rightmost “1” in x. Applying thenotion of vector partial ordering to the problem of:
Minimize g0(x)
Subject to −g1(x) ≥ 0
g2(x) ≥ 0
where x = (x1, x2, . . . , xn) and x1 = 0 or 1, Lawler and Bell developed an optimizationalgorithm containing only three decision rules. The vector i denotes the best solution so farobtained.
1For applications of the simplex method and a nicely done FORTRAN implementation, see Hans G. Daellenbach and Earl J. Bell [27].2This section is provided for historical perspective and to provide a look “at the engine room.” It may be skipped by those not concernedwith how the 0–1 algorithms work.3The number 2n rapidly becomes very large as n increases. For example, 225 = 33,554,432, and 235 = 34,359,738,368.4The logical “or” means here that if both the jth element of x and the jth element of x −1 are 0, then the jth element of x∗ −1 is set to0, or else it is set to 1.
Multiple Project Capital Budgeting 163
Figure 16.1 Flow chart for Lawler and Bell 0–1 integer algorithm
Rule 1. If g0(x) ≥ g0(i), replace x by x∗.Rule 2. If x is feasible, replace x by x∗. Feasibility means that −g1(x) ≥ 0 and g2(x) ≥ 0, orthat gi1(x) − gi2(x) < 0.Rule 3. If for any i, gi1(x∗ − 1) = gi2(x) = 0, replace x by x∗.
If no rule is applicable, replace x by x + 1 and continue. (It is strongly recommended thatthe interested reader refer to the original article by Lawler and Bell for a much more detailedexplanation of the algorithm, and for examples of how problems involving nonbinary integercoefficients and quadratic objective functions may be handled.)
A flow chart for the algorithm with a sample problem and solution is shown in Figure 16.1.In using the algorithm to solve problems, it is suggested, based on the experience of Lawlerand Bell, that those variables that a priori would seem to be least significant be placed so asto occupy the rightmost positions in the solution vector. During so will reduce the number ofiterations required, and hence the requisite solution time.
This algorithm is minimizing, and is predicated on monotonically nondecreasing functionsfor the objective equation and the constraints. However, maximization problems can be handledby negating the objective function and the constraint equations and objective function made
164 Capital Asset Investment: Strategy, Tactics & Tools
monotonically nondecreasing by substituting x ′ = 1 − x in the original problem. After theoptimal solution to the minimization problem has been obtained, reverse substitution for thex ′ will provide the optimal array of projects for the maximum problem.
Since zero–one programming problems of practical scale do not lend themselves to manualcalculation, and since computer programs are available for application of zero–one integerprogramming algorithms, the steps involved in solving a specific problem will not be illustratedhere. Even small problems, although easily solvable on a computer, involve too many iterationsfor the methods to be considered suitable for manual application. The important thing, sinceproblem setup is similar or identical with all the methods, is an understanding and mastery ofthe problem setup and constraint specification. This is vital, because an incorrectly specifiedconstraint will often cause an incorrect problem solution. In practice, it may be useful to havethe same problem set up independently by two or more persons or teams to provide a checkon results.
Example 16.1 To illustrate the type of problem amenable to solution by zero–one integeralgorithms and the formulation of the constraints, consider the following problem faced bythe management of Tangent Manufacturing Company. Tangent is a small firm engaged in lightmanufacturing. The company has recently experienced a substantial increase in demand that isexpected to continue for several years. The company currently owns a dilapidated warehouse,and a small building for assembling its products. The condition of the existing warehouse issuch that it cannot be used much longer in its current condition.
The executive committee consider it likely that the company will require at least 3000 squarefeet of new warehouse floor space next year and, in the three following years, 7000, 8000, and11,000 square feet of new space. In none of these years, however, do they want more than20,000 square feet of warehouse space. (The old warehouse should be ignored in determininghow much space the firm has.)
The company also wants to expand its assembly operations by either renovating its currentplant or by constructing another building for its assembly operations. Outlays for capitalimprovements are to be limited to $55,000 the first year, and to $45,000, $35,000, and $20,000in the following three years. The treasurer of the company has developed information on thepresent value of returns and on the required outlays for alternatives available to the firm. Thisinformation is presented in Table 16.3.
Management will not allow both project 1 and project 2 to be undertaken. Furthermore,adoption of project 7 is dependent on the prior adoption of project 3; project 8 is dependent onthe prior adoption of project 4. Finally, management requires that there be some expansion inassembly capacity, so either project 5, or project 6, or both must be adopted. Table 16.4 containsthe system of constraints. Notice that the last constraint may be stated in two alternative ways.Also, constraints 5, 6, 9, and 10 are redundant in the presence of constraints 7, 8, 11, and 12,respectively, and can be eliminated.
Since the Lawler and Bell algorithm is predicated on monotonically decreasing func-tions, we must substitute x ′ = 1 − x for in the objective function and in the constraints asfollows:
Case Original constraint Modified constraintI. ai xi ≤ b ai x ′
i ≥ + ai − bII. ai xi ≥ b −ai x ′
i ≥ − ai + b
Multiple Project Capital Budgeting 165
Table 16.3
$ Thousands required outlays
ProjectPresentvalue Year 1 Year 2 Year 3 Year 4
1 Construct new warehouse (10,000 $50 $25 $7 $0 $0sq. ft) in year 1
2 Renovate existing warehouse 27 10 5 5 5(7000 sq. ft) in year 1
3 Lease warehouse for 2 years (3000 15 6 6 0 0sq. ft) in year 1
4 Lease second warehouse for 2 15 6 6 0 0years (3000 sq. ft) in year 1
5 Construct new assembly plant in 35 20 5 0 0year 1 (3000 unit capacity)
6 Renovate existing plant for 25 12 3 0 0expanded production (1500 unitincrease in capacity
7 Lease warehouse for 2 years (3000 13 0 0 6 6sq. ft) in year 3
8 Lease second warehouse for 2 13 0 0 6 6years (3000 sq. ft) in year 3
9 Construct new warehouse (10,000 40 0 0 25 7sq. ft) in year 3Outlay constraints ≤ 55 ≤ 45 ≤ 35 ≤ 20Warehouse space constraints ≤ 20 ≤ 20 ≤ 20 ≤ 20(thousands of sq. ft) > 3 > 7 > 8 > 11
After substitution, the modified constraint coefficients are ready to be submitted for solution bythe program. The modified constraint system is contained in Table 16.5. Note that all constraintsare now in the “≥” form.
Acceptance of all but projects 1, 7, and 8 yields an objective function NPV value of of $157.All problem constraints are satisfied.
The appendix to this chapter contains computer solutions achieved with ExcelTM and QuattroProTM. For problems that are not truly large these spreadsheet programs are very easy to use andyield quick, accurate solutions. On modern personal computers these spreadsheet programsare capable of providing results that formerly would have required specialized programs onmainframe computers.
GOAL PROGRAMMING
Early on, this book specified maximization of shareholder wealth as the goal of modernfinancial management. Using NPV as the criterion for project selection serves to move the firmtoward this goal when capital markets are approximately perfect and there is certainty withrespect to project parameters. These, however, are abstractions from the reality that typicallyprevails; capital markets are less than perfect and uncertainty does prevail. Situations are oftenencountered in which management has more than one objective. In fact, this is probably thenorm rather than an exception. Management recognizes that the market takes into account more
166 Capital Asset Investment: Strategy, Tactics & Tools
Table 16.4
Constraint x1 x2 x3 x4 x5 x6 x7 x8 x9 b
Financial constraints 1 25 10 6 6 20 12 0 0 0 ≤ 552 7 5 6 6 5 3 0 0 0 ≤ 453 0 5 0 0 0 0 6 6 25 ≤ 354 0 5 0 0 0 0 6 6 7 ≤ 20
Warehouse spaceconstraints
5 10 7 3 3 0 0 0 0 0 ≤ 20Yr 1
{6 10 7 3 3 0 0 0 0 0 > 3
7 10 7 3 3 0 0 0 0 0 ≤ 20Yr 2
{8 10 7 3 3 0 0 0 0 0 > 7
9 10 7 0 0 0 0 3 3 10 ≤ 20Yr 3
{10 10 7 0 0 0 0 3 3 10 > 8
11 10 7 0 0 0 0 3 3 10 ≤ 20Yr 4
{12 10 10 0 0 0 0 3 3 10 > 11
Projects 1 and 2 13 1 1 0 0 0 0 0 0 0 ≤ 1mutually exclusive
Project 7 dependent 14 0 0 1 0 0 0 −1 0 0 ≥ 0on prior adoptionof project 3
Project 8 dependent 15 0 0 0 1 0 0 0 −1 0 ≥ 0on prior adoptionof project 4
Assembly facilities 16 0 0 0 0 3 1.5 0 0 0 ≥ 1.5must be expanded or
16′ 0 0 0 0 1 1 0 0 0 ≥ 1
Table 16.5 Constraints for Lawler and Bell algorithm solution
Constraint x1 x2 x3 x4 x5 x6 x7 x8 x9 ≥ b
1 25 10 6 6 20 12 0 0 0 242 7 5 6 6 5 3 0 0 0 −133 0 5 0 0 0 0 6 6 25 74 0 5 0 0 0 0 6 6 7 45 10 7 3 3 0 0 0 0 0 36 −10 −7 −3 −3 0 0 0 0 0 −167 10 7 0 0 0 0 3 3 10 138 −10 −7 0 0 0 0 −3 −3 −10 −259 1 1 0 0 0 0 0 0 0 1
10 0 0 −1 0 0 0 1 0 0 011 0 0 0 −1 0 0 0 1 0 012 0 0 0 0 −1 −1 0 0 0 −1
information than the NPV of accepted capital investment projects could provide. Accountingprofits, earnings, and dividend stability and growth, market share, total assets, and so on, affectthe value of shareholder wealth.
Management may have compatible goals, goals for which the achievement of one does notprevent achievement of the others. On the other hand, goals may be incompatible; steps to
Multiple Project Capital Budgeting 167
reach one goal may require moving further from other goals. Ordinary linear programmingcan lead to results that are less satisfactory than can be obtained with what is known as goalprogramming. And in cases for which there is no solution when target values are treated asconstraints in ordinary linear programming, goal programming can provide feasible solutions.
Goal programming is an extension of ordinary linear programming. In fact, the basic simplexalgorithm and its computerized implementations are suitable in those instances in which integersolutions are not required. The general goal programming problem may be specified as
Minimize f = (M1 y+1 + N1 y−
1 ) + (M1 y+2 + N2 y−
2 ) + · · ·+ (Mn y+
n + Nn y−n )
Subject to a11x1 + a12x2 + · · · + a1n xn − y+1 + y−
1 = b1
a21x1 + a22x2 + · · · + a2n xn − y+2 + y−
2 = b2...am1x1 + am2x2 + · · · + amn xn − y+
m + y−m = bm
and for all i, xi , y+i , y−
i ≥ 0
(16.6)
In the more compact form obtained by using matrix notation for the constraint system thisbecomes
Minimize f =m∑
t=1
(Mi y+i + Ni y−
i )
Subject to A · x − y+ + y− = b
(16.7)
Several things should be noted about the goal programming model. First, many of the y+
and y− will be unimportant; in this case they will not appear in the objective function and maybe ignored. Second, the normal linear programming constraints are present. These representtechnological, economic, legal, or other requirements that must not be violated. Third, sincethe y+
i and y−i represent underachieving or overachieving the same goal, one of them must be
zero. The Mi and Ni provide for different weights to be assigned to under- or overachieving agoal. If either is zero it means that no importance is attached to that particular deviation fromgoal. The Mi and Ni also allow priority levels to be established. For instance, if goal i must beachieved before goal j can be considered, this may be specified by defining the priority levelcoefficients. The relationship
Mi > > > M j
denotes Mi to be a very large value in comparison with M j , so large that goal i will be givenabsolute preference to goal j. If Mi is to take absolute preference over M j , but M j is onlytwice as important as Mk , this can be stated as: Mi >>> M j = 2Mk . Thus, we can (1) definea hierarchical structure of goals, in which each level is fixed in relation to the others and(2) define trade-off functions between goals within a particular hierarchical stratum. When westate that goal j is twice as important as goal k, we are defining a trade-off function. Absolutepriority is accomplished by making the trade-off too costly to be considered.
A goal programming problem formulation requires three main items:
1. An objective function for which the weighted deviations from the target or goal levels areminimized according to specified priority rankings. This is in contrast to the ordinary linear
168 Capital Asset Investment: Strategy, Tactics & Tools
Table 16.6 Goal specification
Reach at least a specified minimum level: Minimize y−
Do not exceed a specified maximum level: Minimize y+
Approach specified target as closely as possible: Minimize (y+ + y−)Achieve specified minimum level, then move as
far above as possible: Minimize (−y+ + y−)Achieve specified maximum level, then move as
far below as possible: Minimize (y+ −y−)
programming formulation, in which an aggregate value objective function is maximized or,equivalently, the opportunity costs (shadow prices) are minimized.
2. The normal linear programming constraints reflecting economic, technological, legal, andother constraints.
3. The goal constraints that incorporate one or both of y+i and y−
i , the deviational variables.
The objective function must specify (1) the priority level of each goal, (2) the relativeweight of each goal, and (3) the appropriate deviational variables. The deviational variables areviewed as penalty costs associated with under- or overachieving a particular target, or goal. Thespecification of deviational variables in the objective function determines whether a particulargoal is to be reached as exactly as possible, whether either under- or overachievement is to beavoided, or whether it is desired to move as far from some target level as possible. Specificationof these goals is summarized in Table 16.6. A great advantage of goal programming is that itcan handle both complementary and conflicting goals as long as a trade-off function is specifiedthat links the conflicting goals. Since management must make decisions in a world of risk andimperfect capital markets, project characteristics beyond NPV may have to be factored intothe decision process. Interactions between the value of the firm and capital investments mayhave to be recognized. For example, large capital projects may require borrowing that couldaffect the firm’s capital structure and its cost of capital for several years. Also, accountingprofits may be important to the extent they influence the markets for the firm’s shares anddebt instruments. Goal programming is particularly useful, because it allows managementperceptions and policies, and some interrelationships between the firm and the prospectivecapital investments to be handled simultaneously. It enables management to obtain insight intothe implicit costs of its goals and trade-off functions if sensitivity analysis is carried out toshow the effects of changing goal and trade-off parameters.
Goal programming allows the objective of maximizing shareholder wealth — enterprisevalue — to be approached by setting lesser goals that, if reached, contribute to the major ob-jective. In other words, goal programming provides a means for disaggregating a strategicobjective into a series of tactical goals that, taken together, move the firm toward that objec-tive. And the tactical goals themselves may be interrelated by trade-off relationships. Threedifficulties affect the use of goal programming, particularly in capital budgeting:
1. In capital-budgeting projects indivisibility is the rule rather than the exception. This meansthat ordinary linear programming computer codes are generally not suitable and one mustresort instead to integer programming algorithms that are not as generally available.
2. Specification of goals is often based on conjecture or “hunches” about empirical questions.Thus, different managers will generally make different subjective assessments of reality,based on individual experience, perception, and bias. This means that solutions obtained
Multiple Project Capital Budgeting 169
will depend on whose goals are achieved in the goal programming formulation. Thereis nothing necessarily wrong with this. In fact, since goals must be clearly specified, anadded benefit may result when managers must articulate goals in a form amenable toprogramming solution. What goal programming can provide is an objective procedurefor systematically and accurately reaching goals — goals that themselves may have beensubjectively determined.
3. Conflicting goals require that trade-off relationships be defined. This often requires thatnoncommensurable items be compared, that exchange rates between “apples and oranges”be defined even though the trade-off function may involve less tangible factors than these. Afurther difficulty in this vein is that trade-off functions may not be linear, but may change overa range of values. This third difficulty means that one of the strengths of goal programming,its ability to deal with conflicting goals, is also potentially one of its greatest weaknesses ifnot approached with care.
SUMMARY AND CONCLUSION
Selection of which (if any) capital investments out of an array of candidates should be acceptedrequires a systematic approach. The indivisible nature of capital investment projects means thatordinary linear programming may not produce correct results. Integer linear programming maybe used, but requires constraints that reflect the number of each project available for adoption.Zero–one integer programming provides a very useful means of selection, and requires fewerexplicit constraints.
Several methods of zero-one integer programming have been developed, and there willlikely be further refinements and new developments. Experimentation has suggested that somemethods are more generally useful than others. However, since problem specification will beidentical, or at least very similar, for all existing zero–one integer programming algorithms, andprobably for new developments as well, we have concentrated on specification of the problemand constraints.
A programming solution to multiple project selection is particularly useful because manyconstraints beyond that imposed by capital rationing may be handled easily. Great care mustbe exercised, however, in specifying all constraints. A seemingly minor error in one constraintmay cause an entirely incorrect solution to be obtained.
The evolution of add-in programs in the ubiquitous spreadsheet programs now found onvirtually every personal computer, along with abundant computer memory and high-speedcentral processors, makes it not only possible, but desirable to solve many optimization prob-lems without resort to specialized programs. Only the larger problems require that one resortto those formerly necessary remedies. The appendix to this chapter illustrates this.
16AAppendix to Multiple Project
Capital Budgeting
Although methods developed for mainframe computers are still useful, especially for large-scale problems with hundreds of variables and constraints, for many problems personal com-puters and popular spreadsheet programs can be used. One must be careful to ensure that theproblems are accurately specified, and that constraints are correct, or spurious answers willlikely be found. An advantage of using spreadsheet programs instead of mainframe computerprograms is that one may easily examine the effects of changing constraints or variable costsand their payoffs.
Although ExcelTM and Quattro ProTM are used for illustration here, other programs will yieldsimilar results and are used similarly. So users of LotusTM, ApplixwareTM, or Star OfficeTM
need not be at a disadvantage.1 The setup for ExcelTM is contained in Figures 16A.1, 16A.2.A rose by another name is still a rose — Figures 16A.3 and 16A.4 contain the Quattro ProTM
optimizer setup and options. It is obvious that there is no significant difference between them,indicating that either program can be used for this purpose.
Figure 16A.5 contains the ExcelTM spreadsheet for solving the zero–one integer problemof Example 16.1. It should be noted that the lower set of cells contains values obtained bymultiplying the contents in the set above by the 0 and 1 x solution values in cells Q6 throughQ14. The sums of the column contents in the lower set then serve for most of the constraintsin the problem. The Quattro ProTM spreadsheet is so similar in appearance to the ExcelTM thatit is not shown separately.
Figure 16A.1 ExcelTM solver parameters
1Some of these other programs at this writing do not have the add-ins required for these programming applications. However, it is onlya matter of time before they too will include them integrally, or as third-party add-in applications.
Appendix to Multiple Project Capital Budgeting 171
Figure 16A.2 ExcelTM solver options
Figure 16A.3 Quattro ProTM optimizer setup
Figure 16A.4 Quattro ProTM optimizer options
Example 16-1Excel 97 Setup
ProjectConstruct New Warehouse in Year 1Renovate Existing Warehouse in Year 1Lease Warehouse for 2 Years in Year 2Lease 2nd Warehouse for 2 Years in Year 2Construct New Assembly Plant in Year 1Renovate Existing PlantLease Warehouse for 2 Years in Year 3Lease 2nd Warehouse for 2 Years in Year 3Construct New Warehouse in Year 3
Sums =
x1x2x3x4x5x6x7x8x9
Projects 1 and 2 mutually exclusiveProject 7 depends on prior adoption of 3Project 8 depends on prior adoption of 4Assembly facilities must be expanded
(i.e, project 5 or 6 must be accepted)
Sol’nValue
1-1-12
from
NPV$157
SqFt SqFt SqFt SqFt Units Units Units Units
x1 + x2 <= 1x7 - x3 <= 0x8 - x3 <= 0x5 + x6 >= 1
13 13 17 17 4.5 4.5 4.5 4.5 $54 $25 $30 $12
4000
2535151527
0 0 0
0000
10
0
00
7
00000
10
0
07
00000
0733
00000
07
33
7
0
000000
50
000000
5
25
0566
35
000
66
010
2012
000
000
00003
000
00003
1.5 1.5
000
00003
1.5
000
00003
1.5
Capacity Increases Available in Each YearYear 1 Year 1Year2 Year 2 Year 3 Year 4Year3Year4 Investment Outlays
Year 1 Year 2 Year 3 Year 4NPV SqFt SqFt SqFt SqFt Units Units Units Units
$233 23 23 33 33 4.5 4.5 4.5 4.5 79 32 42 24
3 333
3 33 3
NPVs$50
2715153525131340 10 10
7 7 7 710 10 10 10
1.5 1.5 1.5 1.53 3 3 3
SqFt SqFt SqFt SqFt Units Units Units Units
Capacity Increases Available in Each Year
00
000
0000
0000
0000
2510
62012 3
56665 5 57
6 666
25 70
0
001
1
11
11
x1x2x3x4x5x6x7x8x9
Sol’nXjValue ProjectYear 1 Year 2Year 3 Year 4
Req’d OutlaysYear 1 Year2 Year3Year4 Year 1 Year 2 Year 3 Year 4
Figure 16A.5 ExcelTM spreadsheet for Example 16.1.
Appendix to Multiple Project Capital Budgeting 173
The solution model from the ExcelTM setup as:
Not using “= binary” Model using x 1 . . . X 9 = binary
=MAX($D$33) =MAX($D$33)
=COUNT($Q$6:$Q$14) =COUNT($Q$6:$Q$14)
=$D$38<=1 =$D$38<=1
=$D$39<=0 =$D$39<=0
=$D$40<=0 =$D$40<=0
=$D$41>=1 =$D$41>=1
=$E$33<=20 =$E$33<=20
=$F$33<=20 =$F$33<=20
=$F$33>=7 =$F$33>=7
=$H$33<=20 =$H$33<=20
=$H$33>=11 =$H$33>=11
=$M$33<=55 =$M$33<=55
=$N$33<=45 =$N$33<=45
=$O$33<=35 =$O$33<=35
=$P$33<=20 =$P$33<=20
=$Q$6<=1 =($Q$8=0)+($Q$8=1)=1
=$Q$6=INT($Q$6) =($Q$7=0)+($Q$7=1)=1
=$Q$6>=0 =($Q$6=0)+($Q$6=1)=1
=$Q$7<=1 =($Q$10=0)+($Q$10=1)=1
=$Q$7=INT($Q$7) =($Q$13=0)+($Q$13=1)=1
=$Q$7>=0 =($Q$12=0)+($Q$12=1)=1
=$Q$8<=1 =($Q$11=0)+($Q$11=1)=1
=$Q$8=INT($Q$8) =($Q$14=0)+($Q$14=1)=1
=$Q$8>=0 =($Q$9=0)+($Q$9=1)=1
=$Q$9<=1 ={100,1500,0.000001,0.01,TRUE,FALSE,FALSE,2,2,2,0.001,FALSE}=$Q$9 = INT($Q$9)
=$Q$10<=1 Note how much more compact this alternative model specification is.=$Q$10=INT($Q$10)
=$Q$10>=0
=$Q$11<=1
=$Q$11=INT($Q$11)
=$Q$11>=0
=$Q$12<=1
=$Q$12=INT($Q$12)
=$Q$12>=0
=$Q$13<=1
=$Q$13=INT($Q$13)
=$Q$13>=0
=$Q$14<=1
=$Q$14=INT($Q$14)
=$Q$14>=0
= {100,1000,0.000001,0.01,True,False,False,1,1,1,0.0001,False}
174 Capital Asset Investment: Strategy, Tactics & Tools
It should be mentioned that cells Q6 through Q14 are named x 1 through x 9 in order tofacilitate entering the constraints in the model. Thus x 1 can be substituted for Q6, x 2 for Q7,etc., in any of the formulae.
The solution from the ExcelTM solver is given as:
Microsoft Excel Answer ReportWorksheet: [Example 16-1.xls]16-1 SetupReport Created: 3/21/2002 10:20:27 PM
Target Cell (Max)
Cell Name Original Value Final Value
$D$33 xj*NPV $0 $157
Adjustable Cells
Cell Name Original Value Final Value
$Q$6 x 1 0 0
$Q$7 x 2 0 1
$Q$8 x 3 0 1
$Q$9 x 4 0 1
$Q$10 x 5 0 1
$Q$11 x 6 0 1
$Q$12 x 7 0 0
$Q$13 x 8 0 0
$Q$14 x 9 0 1
Appendix to Multiple Project Capital Budgeting 175
And the system of constraints after optimization is given by
Constraints
Cell Name Cell Value Formula Status Slack
$D$38 Projects 1 and 2 mutually exclusive Value 1 $D$38<=1 Binding 0
$D$39 Project 7 depends on prior adoption of 3 Value -1 $D$39<=0 Not Binding 1
$D$40 Project 8 depends on prior adoption of 4 Value -1 $D$40<=0 Not Binding 1
$D$41 Assembly facilities must be expanded Value 2 $D$41>=1 Not Binding 1
$E$33 SqFt 13 $E$33<=20 Not Binding 7
$F$33 SqFt 13 $F$33<=20 Not Binding 7
$F$33 SqFt 13 $F$33>=7 Not Binding 6
$H$33 SqFt 17 $H$33<=20 Not Binding 3
$H$33 SqFt 17 $H$33>=11 Not Binding 6
$M$33 Year 1 $54 $M$33<=55 Not Binding 1
$N$33 Year 2 $25 $N$33<=45 Not Binding 20
$O$33 Year 3 $30 $O$33<=35 Not Binding 5
$P$33 Year 4 $12 $P$33<=20 Not Binding 8
$Q$6 x 1 0 $Q$6<=1 Not Binding 1
$Q$6 x 1 0 $Q$6=integer Binding 0
$Q$6 x 1 0 $Q$6>=0 Binding 0
$Q$7 x 2 1 $Q$7<=1 Binding 0
$Q$7 x 2 1 $Q$7=integer Binding 0
$Q$7 x 2 1 $Q$7=0 Not Binding 1
$Q$8 x 3 1 $Q$8<=1 Binding 0
$Q$8 x 3 1 $Q$8<=integer Binding 0
$Q$8 x 3 1 $Q$8>=0 Not Binding 1
$Q$9 x 4 1 $Q$9<=1 Binding 0
$Q$9 x 4 1 $Q$9=integer Binding 0
$Q$10 x 5 1 $Q$10<=1 Binding 0
$Q$10 x 5 1 $Q$10=integer Binding 0
$Q$10 x 5 1 $Q$10>=0 Not Binding 1
$Q$11 x 6 1 $Q$11<=1 Binding 0
$Q$11 x 6 1 $Q$11=integer Binding 0
$Q$11 x 6 1 $Q$11>=0 Not Binding 1
$Q$12 x 7 0 $Q$12<=1 Not Binding 1
$Q$12 x 7 0 $Q$12=integer Binding 0
$Q$12 x 7 0 $Q$12>=0 Binding 0
$Q$13 x 8 0 $Q$13<=1 Not Binding 1
$Q$13 x 8 0 $Q$13=integer Binding 0
$Q$13 x 8 0 $Q$13>=0 Binding 0
$Q$14 x 9 1 $Q$14<=1 Binding 0
$Q$14 x 9 1 $Q$14=integer Binding 0
$Q$14 x 9 1 $Q$14>=0 Not Binding 1
176 Capital Asset Investment: Strategy, Tactics & Tools
The alternate solution, using x 1 through x 9 = binary, is given in
Microsoft Excel 10.0 Answer Report
Worksheet: [Example 16-1-Binary.xls]16-1 SetupReport Created: 4/19/2002 12:32:10 PM
Target Cell (Max)Cell Name Original Value Final Value
$D$33 xj*NPV $0 $157
Adjustable CellsCell Name Original Value Final Value
$Q$6 x 1 0 0$Q$7 x 2 0 1$Q$8 x 3 0 1$Q$9 x 4 0 1$Q$10 x 5 0 1$Q$11 x 6 0 1$Q$12 x 7 0 0$Q$13 x 8 0 0$Q$14 9 0 1
ConstraintsCell Name Cell Value Formula Status Slack
$D$38 Projects 1 and 2 mutually exclusive Value 1 $D$38<=1 Binding 0$D$39 Project 7 depends on prior adoption of 3 Value −1 $D$39<=0 Not Binding 1$D$40 Project 8 depends on prior adoption of 4 Value −1 $D$40<=0 Not Binding 1$D$41 Assembly facilities must be expanded Value 2 $D$41>=1 Not Binding 1$E$33 SqFt 13 $E$33<=20 Not Binding 7$F$33 SqFt 13 $F$33<=20 Not Binding 7$F$33 SqFt 13 $F$33>=7 Not Binding 6$H$33 SqFt 17 $H$33<=20 Not Binding 3$H$33 SqFt 17 $H$33>=11 Not Binding 6$M$33 Year 1 $54 $M$33<=55 Not Binding 1$N$33 Year 2 $25 $N$33<=45 Not Binding 20$O$33 Year 3 $30 $O$33<=35 Not Binding 5$P$33 Year 4 $12 $P$33<=20 Not Binding 8$Q$8 x 3 1 $Q$8=binary Binding 0$Q$7 x 2 1 $Q$7=binary Binding 0$Q$6 x 1 0 $Q$6=binary Binding 0$Q$10 x 5 1 $Q$10=binary Binding 0$Q$13 x 8 0 $Q$13=binary Binding 0$Q$12 x 7 0 $Q$12=binary Binding 0$Q$11 x 6 1 $Q$11=binary Binding 0$Q$14 x 9 1 $Q$14=binary Binding 0$Q$9 x 4 1 $Q$9=binary Binding 0
x
17
Utility and Risk Aversion
In this chapter, the first one to explicitly consider risk, we examine the concept of utilityand how a utility function may be calculated, various ways of estimating risk, and problemsassociated with the probability of ruin. This last topic is of some importance to the enterprise(perhaps a small firm) for which candidate capital investment projects are large in relation toits financial resources.
The concept of utility is essential if we are to develop a rationale by which a decision may bereached for a capital investment that has a range of possible outcomes, each with an associatedprobability. Some of the possible outcomes may be large losses, whereas others are large gains.
In a world of certainty we would be able to say whether or not a project is acceptable basedon objective criteria, such as the profitability index. Once risk is introduced, however, we musttake into account the decision-maker’s1 attitude toward risk in order to reach an economicallyrational decision. Two different persons are likely to disagree about accepting risky projectsif they have different attitudes toward risk. One may take risk in stride, whereas the other isconsiderably troubled by it.
THE CONCEPT OF UTILITY
Economists developed the concept of utility long ago, and it is the foundation upon whichmicroeconomic analysis rests. Utility is a reflection of personal satisfaction. Something thatprovides more feeling of pleasure than something else (or, equivalently, less pain) is said tohave greater utility.
In this work we measure investment project traits in units of currency — dollars. Becauseof this our task is much easier than it would be otherwise. We do not have to compare therespective utility of oranges and that of apples — only that of more dollars versus fewer.
The utility of more dollars is assumed to be greater than the utility of fewer dollars, sothat the utility function we will be working with is monotonically nondecreasing in dollars,at least over the range of values we will be considering. A second dollar may have the sameutility value as the first (constant marginal utility), greater value (increasing marginal utility),or lesser value (decreasing marginal utility).
When risk is introduced, we recognize that there is a trade-off between possible dollar gain(and thus utility) and risk. Projects promising the greatest return are also normally the riskiestprojects. If it were otherwise, they would have been snapped up before now.
The classic article on risk as it relates to utility is that of Tobin [159]. The following sectionfollows his terminology.
1We shall avoid the problem of whose attitude specifically is to be taken into account by using the term “decision-maker” or“management.”
178 Capital Asset Investment: Strategy, Tactics & Tools
Figure 17.1 Relationship of money gain to utility for a risk averter
Attitudes toward Risk
Individual attitudes toward risk are revealed in the curvature of utility functions. Before goinginto the determination of an individual utility function, it will be useful to examine theseattitudes toward risk. First, in Figure 17.1 we have a utility function corresponding to a risk-averse individual, a risk averter. Here the utility function is concave from below. This meansthat each additional monetary gain, while contributing to increased utility, contributes lessthan the same amount when the individual has less to begin with. In other words, the functiondepicts diminishing marginal utility. An extra dollar offers less additional utility to given risk-averse individuals when they have $100,000 than it does when they have $10,000. Diminishingmarginal utility means that the loss of a dollar causes more disutility (negative utility, or lossof utility) than the gain of a dollar offers. For example, consider a person with $10,000 in cashsavings, which may have taken many years to accumulate. An extra $1000 would increaseutility, but not as much as a loss of $1000 would decrease it. If this person were offered a “fairbet,” one with zero expected value based on equal probabilities of gaining or losing a largeamount, the bet would be rejected. It would be rejected because the expected utility is negative,even though the expected monetary gain is zero.
For small amounts of monetary gain or loss, a risk-averse individual may take “fair bets” oreven bets where the expected monetary gain is negative. This may be explained on the basis thatthe negative utility of the small loss is dominated by the utility provided by the entertainmentprovided, or the utility provided by the hope, however remote, of a large monetary gain (as ina lottery).
Figure 17.2 corresponds to a risk-neutral, that is, a risk-indifferent individual. Each ad-ditional dollar, no matter at what point on the function, offers the same utility. The utilityfunction of a risk-indifferent individual is a straight line. The slope is constant and thereforethe marginal utility of monetary gain does not change over the curve. A large loss would reducethis person’s utility by the same amount of utility as would be gained by winning an identicaldollar amount. A risk-indifferent individual could be expected to take fair bets, even thoseinvolving large amounts of possible gain or loss.
The final utility curve is that of a risk lover. This is illustrated in Figure 17.3. Tobin concludedthat there are no 100 percent risk lovers. Such individuals, on reflection, would have to have
Utility and Risk Aversion 179
Figure 17.2 Relationship of money gain to utility for a risk-neutral person
Figure 17.3 Relationship of money gain to utility for a risk lover
Figure 17.4 Overall utility of money for an individual
a self-destructive compulsion to prefer the riskier of any two propositions. The utility curvehas the property of increasing marginal utility, suggesting greed. The important thing aboutsuch utility function is that most individuals have utility functions that exhibit properties ofrisk aversion, risk neutrality, or risk seeking for different ranges of monetary values. This isillustrated in Figure 17.4.
180 Capital Asset Investment: Strategy, Tactics & Tools
For the individual whose utility function is shown in Figure 17.4 we can observe that theperson is risk averse for amounts less than m1 and greater than m3. This individual is essentiallyrisk neutral for amounts between m2 and m3, and is a risk lover or risk seeker for values betweenm1 and m2. The next section will show you how to estimate your own utility function or thatof someone else. Do not be surprised if it is similar to the one in Figure 17.4, nor shouldyou be bothered if it is not. The benefit you derive from plotting points along your utilityfunction is that you gain a better understanding of how you react to risk. Your attitude towardrisk influences your performance as a decision-maker evaluating capital investments. And itis likely that your attitude toward risk changes as the dollar magnitude of the potential gain orloss changes.
CALCULATING PERSONAL UTILITY
This section presents a method for estimating an individual’s utility function. The methodcould also be used to try to determine a composite utility curve for a group.
The treatment here parallels the presentation contained in Teweles, Harlow, and Stone [158].Their purpose was to illustrate how the individual commodity futures trader could obtain his orher personal utility function for the risks found in such activity. The principle is also applicablein the present context. Consider that you are asked to put yourself in the role of managerhere, and not one who is risking personal funds. That this is a distinction that a sole proprietorcannot make will not be argued. Because most managers in various enterprises do have differentattitudes about their own money vis-a-vis that of their employers, try to make this distinctionto the extent it is possible.
After assuming the “managerial frame of mind,” the next step is to determine the largestdollar gains your capital investment decisions have regularly made and the largest dollar lossesthat have similarly resulted. This may be impossible for many readers. Thus students andmanagers who have not made such decisions will have to imagine what the amounts would be,and try to be perfectly honest about it. This is not a test, there are no right or wrong answers.It is not necessary to do any calculations in reaching your answers, although you may if youprefer. Calculation of the reader’s personal utility function for monetary gains may be done asan exercise. For now we shall only illustrate the procedure.
Let us begin by developing the utility function for an entrepreneur who tells us that he(individually) has regularly made decisions that have resulted in gains of as much as $2 millionand losses of as much as $500,000. We shall not concern ourselves here with whether these aregains and losses over the entire project life, or with other details of the timing of the amounts.The procedure would not be materially different anyway. Having established the largest regulargains and losses, we write them down as shown in Table 17.1, column 4. The gain of $2 millionis associated with utility of 1.0 and the loss of $500,000 with utility 0.0. Other values couldhave been used for the utilities, such as +1.0 and −1.0, but the scaling is unimportant and byassigning zero utility to the worst outcome the calculations are a little easier.
Having established utilities of 1.0 for a gain of $2 million with probability 1.0, and 0.0for a loss of $500,000 with probability 1.0,we now need to find intermediate values. Supposenow we ask our entrepreneur to tell us if he would accept an investment offering a gain of$2 million with probability 0.9 and loss of $500,000 with probability 0.1; in other words,9 chances in 10 of gaining $2 million and one chance in 10 of losing $500,000. He answers“yes,” certainly he would accept such an investment. Now we ask if he would pay us $1 millionfor the opportunity to make such an investment. Yes, he would. $1.5 million? No, not that much.
Utility and Risk Aversion 181
Table 17.1 Computation of decision-maker’s utilities for variousmonetary gains
Probability of:
(1) (2) (3) (4)Best Worst Computed Dollar cashresult result utilitya equivalent
1.0 0.0 1.0 $2,000,000 Best result0.9 0.1 0.9 1,200,0000.8 0.2 0.8 950,0000.7 0.3 0.7 750,0000.6 0.4 0.6 550,0000.5 0.5 0.5 400,0000.4 0.6 0.4 −100,0000.3 0.7 0.3 −300,0000.2 0.8 0.2 −400,0000.1 0.9 0.1 −450,0000.0 1.0 0.0 −500,000 Worst result
a 1.0 × column 1 + 0.0 × column 2. The utility of the best result is arbitrarilyassigned a value of 1.0, the worst result, a value of 0.0. Other values could beused if desired.
Figure 17.5 Plotted utility function of entrepreneur. From Table 17.1 dollar equivalents
How about $1.2 million? Maybe. At $1.2 million he is not sure. For $50,000 more he will nottake the investment, for $50,000 less he will. Thus in Table 17.1 we write in the second row,rightmost column, the amount $1,200,000.
We repeat the process for gain of $2 million with probability 0.8 and loss of $500,000with probability 0.2. Our entrepreneur will pay up to $950,000 but no more than that foran investment offering these prospects. For each missing value we repeat the process untilcolumn 4 is completed. Then we can plot the results obtained, as shown in Figure 17.5, and fitan approximate curve to the points.
182 Capital Asset Investment: Strategy, Tactics & Tools
Table 17.2 Computation of refined utility of decision-maker’smonetary gain
Best outcome Worst outcome
(1) (2) (3) (4)Utility Probability Utility Probability
(5)Computed
utilitya
(6)Dollarcash
equivalent
0.5 1.0 0.4 0.0 0.50 $400,0000.5 0.9 0.4 0.1 0.49 360,0000.5 0.8 0.4 0.2 0.48 350,0000.5 0.7 0.4 0.3 0.47 325,0000.5 0.6 0.4 0.4 0.46 280,0000.5 0.5 0.4 0.5 0.45 250,0000.5 0.4 0.4 0.6 0.44 180,0000.5 0.3 0.4 0.7 0.43 160,0000.5 0.2 0.4 0.8 0.42 50,0000.5 0.1 0.4 0.9 0.41 00.5 0.0 0.4 1.0 0.40 −100,000
a Computed utility is obtained as best outcome times best outcome probability plusworst outcome times worst outcome probability.
In constructing his or her own utility curve for monetary gains and losses, the reader canperform a self-interview or work with someone who will perform the function of interviewer.Further insight into the process is found in Teweles et al., cited earlier [158].
The curve obtained and plotted in the graph of Figure 17.5 is reasonably satisfactory, exceptthat we do not have enough data points between −$100,000 and +$400,000 to be confidentin the shape of the curve in that range. We can obtain more information by continuing theinterview process. Let us begin by constructing Table 17.2 with the best result now $400,000and the worst result −$100,000. The associated utilities are, respectively, 0.5 and 0.4.
Values between $400,000 gain and $100,000 loss are obtained as before. The decision-makeris asked if he would accept an investment project offering $400,000 gain with 0.9 probabilityand $100,000 loss with 0.1 probability. Yes. Would he pay $380,000? No. $370,000? No.$360,000? Maybe. Don’t know. $350,000? Yes. We write $360,000 in column 6 of Table 17.2.The process is repeated until column 6 is filled and then the results are plotted. Figure 17.6contains the graph for the section of utility curve between −$100,000 and +$400,000. Thescale is enlarged from that used in Figure 17.5. After combining the information contained inTable 17.2 with that of Table 17.1, and plotting the results, we obtain Figure 17.7. The additionaldetail for utility between −$100,000 and +$400,000 enables a more refined approximation.The tentative judgment that our entrepreneur is a risk seeker for monetary gains seems tobe vindicated by additional information over the range of about $0 to $400,000. Because ofthe rather wide spread between utility values for dollar amounts in the ranges −$300,000 to−$100,000 and $1.2 million to $2 million, we might want to repeat the procedure for obtainingdetail over these ranges, if our entrepreneur’s patience has not been exhausted. We shall notdo this here, however, since the procedure has been illustrated.
The plotted results obtained from our entrepreneur do not deviate very widely from thefitted curve. In fact, they are very close to it, thus indicating a high degree of consistency inevaluating alternatives over the range of values regularly experienced by the decision-maker.We should not have been surprised if the points were much more scattered about the fitted
Utility and Risk Aversion 183
Figure 17.6 Plot of refined utility of decision-maker’s monetary gains from Table 17.2
Figure 17.7 Revised decision-maker’s utility of monetary gain
curve. What happens if we now attempt to obtain utility values for dollar gains and lossesbeyond the range of the decision-maker’s regular experience? If we attempt to do this, we willmost likely find that the decision-maker becomes increasingly inconsistent as the monetaryvalues become further and further removed from the domain of his experience.
To compute the utility of larger monetary gains, the formula used is
U (Gain) × Pr (Gain) + U (Loss) × Pr (Loss) = U (Cash equivalent) (17.1)
which is equivalent to
U (G)p + U (L)(1 − p) = U (C) (17.2)
184 Capital Asset Investment: Strategy, Tactics & Tools
By rearranging the terms, we obtain formulas for computing the utilities of gains and lossesoutside the range of the decision-maker’s experience:
U (G) = U (C) − U (L)(1 − p)
p(17.3)
and
U (L) = U (C) − U (G)p
1 − p(17.4)
In calculating the utility of gains and losses for extended amounts for commodity traders,Teweles, Harlow, and Stone suggest computing them three times. The reason behind repeatingthe procedure is to find out how consistent the decision-maker is in making decisions outsidethe range of his or her experience. The reader is referred to the excellent discussion by Teweleset al. for a detailed treatment [158].
The main reason for concern about the decision-maker’s consistency for larger gains andlosses than he or she has regularly experienced is this: if judgments do become increasinglyinconsistent for larger gains and losses, then the decision-maker should either exercise greatercaution (something the person will probably do anyway) in evaluating such prospects, or elseavoid them entirely if possible. Otherwise it is likely that decisions will be made for whichthe perceived a priori and a posteriori utilities are different and the decision-maker regrets thedecision after making it, even before the results are in.
In the next chapter we shall examine capital investments with stochastically determinedcharacteristics. The returns from such projects cannot be known ahead of time with certainty.It will be shown, however, that the probable outcomes can be estimated, and such estimatesevaluated by comparison with the decision-maker’s utility function.
Whose Utility?
In the foregoing discussion of utility it was assumed that the utility function of interest is that ofan individual decision-maker. An entrepreneur with sole responsibility for capital investmentdecisions in a proprietorship provides a clear example. For the entrepreneur it is reasonable toassume that the utility function for monetary gains and losses to the enterprise is none otherthan that of the entrepreneur.
But what is to be done for a partnership, in which each partner shares personally in the gainsand the losses? Can we still obtain a meaningful estimate of the utility of monetary gain? Wecan certainly repeat the procedures illustrated earlier, and in so doing obtain a curve relatingcash equivalents to risky alternatives. This author chooses to dodge the question of whether thecurve obtained is really a utility curve or whether an aggregate utility function exists. Instead,let us refer to the curve obtained for all partners in a partnership as the enterprise’s investmentcurve.
More complex conceptual considerations enter the scene when we begin to examine thesituation for enterprises in which ownership is largely separated from management. For suchenterprises (in which we would include public sector undertakings), the utilities of money gainand loss on investments may cover a wide spectrum just among the owners. Those who makethe capital investment decisions may (and most likely do) have personal utility curves that aresubstantially different from what their decisions as managers would indicate. What managersmust do is surmise the enterprise’s investment curve, the composite of the aggregated owners’
Utility and Risk Aversion 185
preferences. This may be linked to the personal utilities of individual high-ranking managersinasmuch as their job security, bonuses, reputations, and the like may reflect how well or poorlythey serve the owners. Thus, monetary gains and losses to the enterprise may be associatedwith a manager’s personal utility.
To conclude this section, we adopt the operational guideline that the required utility func-tion or investment curve is that of the person or persons who, individually or collectively,are responsible for making investment decisions of the magnitude with which we are con-cerned. In corporations this may mean the board of directors itself for large projects. Oncethe individual or the group responsible for decisions on capital investments has been iden-tified it is theoretically possible to derive an investment curve along the lines illustratedfor an individual. However, in practice there would likely be considerable difficulty in get-ting a group to cooperate fully over a long enough time to obtain enough data points to beuseful.
MEASURES OF RISK
As mentioned earlier, in a risk environment variables are not known with certainty but followprobability distributions. With risk, the outcome of a capital investment decision is a particularvalue, but the value cannot be known a priori. When risk is present, we may be able to predictwith statistical confidence the range of values within which the outcome may be expected tooccur.
For experiments that may be repeated a number of times we can generally estimate theempirical risk parameters. However, capital investment projects are usually one-of-a-kindundertakings; we will not have the opportunity to repeat them. Therefore, empirical estimationof the parameters is not possible and we must instead make subjective judgments about thegoverning probability distributions. Past experience with similar projects may be helpful. Forexample, we may have determined that the useful, economic life of capital investments is, ingeneral, Poisson distributed, or that it is negative exponentially distributed.
When “risk” is discussed in relation to the overall return on an investment, the word hascome to mean the potential variability of the actual return from its expected value. Other thingsbeing equal, the investment with the greatest range of possible results is said to be the riskiest.The most widely used measure for risk is the variance. Figure 17.8 illustrates the concept.Projects RA and RB are both risky; both have a range of possible outcomes associated withthem. The possible outcomes are distributed around the most likely values V A and V B, whichare defined as the expected values of returns on the projects. Project RB is the riskier projectbecause the range of possible outcomes is much wider than for RA. If we were able to say with95 percent (statistical) confidence that project RA would have a value in the range RAL to RAH
Figure 17.8 Comparative riskiness
186 Capital Asset Investment: Strategy, Tactics & Tools
and project RB a value in the range RBL to RBH, then RAH − RAL < RBH − RBL. Therefore,RB is riskier. Since the range of values, for any particular statistical level of confidence, isa function of the distribution’s variance (or standard deviation),we can say that for projectswhose outcomes follow the same type of probability distribution, the project having the largervariance of outcome is the riskier.
There is a problem with using the variance alone as risk measure, however, although it iseasily adjusted for: the variance does not distinguish between projects of different size. A largeinvestment undertaking may have a larger variance than a smaller undertaking simply becauseit is larger. This problem is readily resolved by substituting for the variance the coefficientof variation. The coefficient of variation is a “risk relative” measure defined as the ratio ofstandard deviation to expected value. The investment having coefficient of variation 2.3 isrelatively riskier than the investment with coefficient 1.9.
Occasionally, we may want to analyze the pattern of net, after-tax cash flows for pastprojects in order to estimate the cash flows for similar projects under consideration now. Suchan endeavor is predicated on a stable relationship of the cash flows to one or more variables thatpresumably can be accurately forecasted. If we fit a regression for cash flows to the independentvariables, one of the pieces of information obtained is the standard error of the estimate. Thismeasures the variability of actual values from predicted values of the regression equation.In other words, the standard error of estimate is analogous to the standard deviation, but iscalculated from deviations of actual values around the computed regression line. Estimationof values from a regression equation with large standard error is riskier than from a regressionwith small standard error.
The question has been posed as to why we use variance to measure risk when people do notfeel the same about bad results as they do about better-than-expected results. This asymmetryof attitude toward deviation above and below expected values can be explained in terms ofdiminishing marginal utility; the increase in utility from a gain above the expected value is lessthan the utility loss from an outcome with the same dollar amount below the expected value.Consider Figure 17.9.
In Figure 17.9 outcomes less than the value V T are considered undesirable. On the other hand,outcomes above V T are considered fortuitous. Yet the variance does not distinguish betweendeviations above the expected value and those below it. Mao [99] and others have suggestedthat decision-makers are sensitive to, and influenced much more by, outcomes with values lessthan the expected value. In other words, when decision-makers speak of risk, they mean thedistribution of outcomes that are worse than anticipated. This is not measured by the variancebut by the semivariance. The semivariance is calculated like the variance, but only for values
Figure 17.9 Risk perception for results better and worse than expected value
Utility and Risk Aversion 187
less than the targeted value.2 For an empirical distribution this means summing the squares ofthe deviations equal to or below the target and dividing by the number of observations. For atheoretical distribution it means integrating or summing the density function up to the targetvalue. Above this value deviations are ignored.3
A conceptual problem with the semivariance is that fortuitous outcomes are usually ignoredin risk considerations (although they may nevertheless enter determination of the target value).We might expect that instead of ignoring better-than-expected results, they should to someextent counteract the worse-than-average outcomes that are possible. If we were to use util-ity weights instead of dollar NPV or other value weights, this problem might be alleviated.Although the variance may leave much to be desired as a risk measure, the semivariance offersa debatable advantage to it, and a generalized utility-based distribution may not be obtainable.
In subsequent chapters the variances and covariances of returns are used to illustrate port-folio selection approaches to investment and the capital asset pricing model. Although thesemivariance could, in principle, be substituted, it is not. The reason for this is that the mathe-matical calculations are much less tractable for the semivariance. This suggests a reason whydevelopment of the literature on investments has employed the variance or coefficient of vari-ation almost exclusively. To the extent efforts have been made to use the semivariance, it hasbeen by those who have attempted to modify results obtained originally with variance as therisk measure for a specific application. General substitution of the semivariance has not beensuccessful.
J. C. T. Mao’s Survey Results
Mao’s survey was based on personal interviews with managers of eight medium and largecompanies in electronics, aerospace, petroleum, household equipment, and office equipmentfirms. He found that decision-makers do indeed think of risk as being related to outcomesthat are worse than expected, and that the possibility of a bad outcome significantly influencesthe capital investment decision. To verify the relevance of semivariance he designed a testin which the decision-makers were asked to choose between two (hypothetical) mutuallyexclusive investments, each costing the same amount and returning after one year the cost plusprofit or loss. The probability distributions of the returns were as shown in Figure 17.10.
The reader may readily verify an interesting property of the distributions of projects A andB: they both have expected value of 3 and variance of 4. Since they require the same investmentoutlay, their coefficients of variation are identical, and therefore their risk as conventionallymeasured. But are they equally risky? Mao’s respondents did not think so. Their answers wereconsistent with the semivariances (using zero in each as the critical value) that were 0.2 forproject A and 0 for project B. Note also that there is a nonzero probability that project A couldactually lose money.
To summarize Mao’s findings, they are that: executives were more likely to choose A if theirbusinesses accustomed them to that degree of risk, if they personally preferred risky ventures,and if they could control the loss possibility in A through diversification. When these conditions
21t is necessary to say “targeted value” rather than “expected value” because an arbitrary value may be used in calculating the semi-variance. The arbitrary value could be the expected value of the distribution of outcomes, as is the case with the variance, but it mayjust as well be some critical value established by management policy.3Mao [99, p. 353] describes the semi-variance as follows: consider investment return R, as a random variable with known probabilitydistribution. If h stands for a critical value against which the actual values of R are compared, and (R − h)− stands for (R − h) if(R − h) ≤ 0 and for zero if (R − h) > 0, then Sh (semi-variance with h the reference point) is given by the formula: Sh = E[(R − h)−]2
where E is the expectation operator.
188 Capital Asset Investment: Strategy, Tactics & Tools
Figure 17.10 Mao’s test distributions
are absent, the executive is more likely to pick B because the absence of loss possibility makesit a more secure investment.
The previous analysis was concerned with a choice between two individual investments. Itis equally important to examine the risk concept from the portfolio viewpoint. The executiveswere asked to imagine the same alternatives on a larger scale, where X represented the totalcompany investment, and questioned as to which portfolio they would select. The answerwas unanimously B, and their reasons were most succinctly voiced by the statement of oneexecutive:
The key is survival. We take a chance on evaluating individual projects rather optimistically, but wewill not take a chance on the main company. One of our obligations is to sustain this company in lifeand every time we put it in a minus position, it dies a little bit if not in total. This evidence is consistentwith semi-variance as a concept of risk. . . . [99, p. 355.]
RISK OF RUIN
If all capital investments were small relative to the enterprise’s financial resources, this sectionwould not be necessary. However, capital investments are often large relative to the enterprise’ssize as well as absolutely. Because of this an adverse investment result may impair the firm’scapital; several such adverse investments with one right after another might destroy the firm.Therefore, in addition to analysis of individual projects, it will be useful to examine therelationship of project size relative to firm size within the context of the risk affecting theprobabilities of return on investment.
Risk of ruin, like many useful statistical concepts, has its origins in gambling. It is applicableto other situations in which a series of losses (adverse outcomes) can result in ruin (loss ofequity). Teweles, Harlow, and Stone illustrated application of the concept to an individual whoundertakes the trading of commodity futures. This can be a very risky venture because of thesmall margin requirements, usually around 5–10 percent of the commodity contract value.Their discussion is highly recommended to readers who wish to reinforce their understandingof the concept and its application. Here we shall focus attention on risk of ruin as it applies tocapital investment decisions.
The probability of eventual ruin (loss of equity, bankruptcy) is given by
R =(
1 − A
1 + A
)C
(17.5)
where A is the investor’s advantage in decimal form and C is the number of investment units withwhich the investor starts [48]. For capital investments we may define the investor’s advantageas the probability of a favorable outcome minus the probability of an unfavorable outcome
Utility and Risk Aversion 189
resulting in monetary loss.4 If P is the probability of a favorable outcome, this means that theadvantage is P − (1 − P) or 2P − 1. To obtain C we divide the required investment outlayinto the enterprise’s total equity. (If desired, a lesser amount may be used.)
To illustrate, let us take the case of a company considering an investment of $40 million in aproject that is thought to offer 0.60 chance of returning 100 percent and a 0.40 chance of lossof the entire investment. The firm has equity of $160 million. Translating these numbers intorisk of ruin, we get
R =(
1 − 0.2
1 + 0.2
)4
= 0.198 (17.6)
or about one chance in five of eventual ruin.This one project cannot by itself ruin the firm. However, there is 0.40 chance that the entire
investment will be lost. The significance of the risk of ruin is this: if the firm makes a habit ofundertaking investments of similar risk, cost, and return, there is one chance in five that the firmwill go bankrupt as a result. Thus, risk of ruin relates to the firm’s ongoing policy with respectto investments. Should the firm regularly undertake investments for which the probability ofeventual ruin is one-fifth? Is this an exceptional case? If so, how does it relate to the firm’sinvestment policy? Examination of the formula for risk of ruin yields some principles thatmay be useful in policy formulation. Of major significance is the fact that by reducing the sizeof investments that may be considered, the risk of ruin may be greatly reduced, other thingsbeing constant. In the example just considered, reducing the size of the investment to $20million reduces risk of eventual ruin to 0.039 or about four chances in 100. Halving the sizeof investment outlay in this case reduced the risk of ruin by 80 percent. Although one chancein five would be considered an unacceptable risk of ruin by many executives, four chances in100 might well be considered quite good odds.
Risk of ruin as presented above is somewhat an abstraction. Firms do not have continuingopportunity to undertake a series of identical capital investments over the time continuum.Furthermore, enterprise policy is not immutable; it tends to change to adjust to circumstances.If a firm has a run of losses on its investments, sooner or later it will review its investmentpolicies, objectives, and procedures in order to correct its practices.
Nevertheless, risk of eventual ruin provides a gauge for measuring the enterprise’s approachto risky investments. It indicates management’s attitude toward the ultimate investment risk —that of financial ruin, bankruptcy. The measure itself could be incorporated into the decisionprocess as a constraint. For instance, a policy might be adopted constraining investments torisk of ruin of X chances in 100 (based on the calculation above, which assumes the sameinvestment outlay and success/failure probabilities to be repeated over time).
SUMMARY
Risk, in the context of this chapter, was considered without regard to portfolio effects. Thischapter introduced the subject of risk and the notion that risk cannot be adequately handledwithout reference to utility. We saw that the utility function governs the investor’s attitudetoward risky investments, and that diminishing marginal utility of monetary gain leads to riskaversion.
4Ruin, in the sense we are using the word here, rests upon the probability of actual out-of-pocket money losses, if ruin would beconsidered to result from its loss.
190 Capital Asset Investment: Strategy, Tactics & Tools
Risk is normally measured by the variance or coefficient of variation of the outcome variable.Professor Mao’s interviews with executives who make capital investment decisions yielded at-titudes congruent with semivariance as a measure of risk actually representative of managementthought.
Even though capital investments are typically one of a kind, the concept of risk of eventualruin was introduced. By calculating the probability of ruin it is possible to assess more fullythan otherwise managements’ attitudes toward risky investments. Risk of ruin also ties in wellwith one of Mao’s survey respondents who put survival of the enterprise ahead of any profitaspirations.
18
Single Project Analysis under Risk
This chapter considers methods that are used to analyze capital investment projects when oneor more project characteristics are random variables. Independence of projects (with respectto the existing assets of the firm as well as one another) is assumed throughout this chapterin order to concentrate on risk considerations without the added complexities introduced byportfolio effects. Project analysis with diversification and project interdependence is coveredin Chapter, 20.
Chapter 17 considered those factors that influence our attitudes toward risk and introducedthe concept of risk of ruin. In this chapter we examine means for measuring risk and dealingwith it. Among the matters to be covered here are payback as a risk-coping method, the certaintyequivalent approach, the risk-adjusted discount rate approach, and computer simulation.
THE PAYBACK METHOD
Payback has been rationalized as a method for coping with risk. If we recognize that con-siderable importance may be attached to capital preservation, it is understandable that somemeasure of solace may be gained from the knowledge that a particular investment projectpromises a quick return of the funds invested in it. The longer the time required to recover aproject’s investment cost, the more uncertain we are about the recovery. The more time thatpasses between the present and an anticipated event, the more the unanticipated influences thatcan spring up to render our forecast results worthless.
The shortcomings of the payback criterion are well known; Chapter 5 discusses them. As ameans of dealing with risk or uncertainty, payback does favor those projects that promise earlyrecovery of investment. It does not make much sense to use payback as the sole criterion or evenmajor criterion, but, used as an adjunct to the discounted cash flows, it provides an unequivocal,if crude, means of screening projects to eliminate those that take what management considerstoo long to recover their investment and, as a consequence, are all too likely to fail to helppreserve the enterprise’s capital.
To an extent, reliance on payback may be tantamount to an admission of forecasting impo-tence. That is, if we feel it is not at all possible to make usefully reliable forecasts, then wemay adopt the project that exposes our capital to the ravages of time the least. After havinggot back our investment, if the project then offers further returns, that is all to the good; ifnot, we still have our funds to invest in something else. In relatively stable economic timespayback may have little to recommend it. In times like the present, with forecasting madeall the more difficult by one crisis after another communicated instantly around the world,compounded with periodic surges of inflation, preservation of capital is not a trivial matter, ifit ever was. It is understandable that payback holds a prominent position among the methodsof capital asset selection used by those who must make decisions with their firm’s resourcesand be held accountable for results. And if a project selected on the basis of payback does notperform according to expectations, it will be recognized early, within the payback period, and
192 Capital Asset Investment: Strategy, Tactics & Tools
appropriate measures taken — no need to wait years for a return only to find it is a will-o-the-wisp that vanishes as approached.
CERTAINTY EQUIVALENTS: METHOD I
In the previous chapter it was shown how a utility curve can be estimated by determining the(certain) cash amount that is considered equivalent to two different risky amounts, each witha probability attached to it. The cash was considered equivalent to the risky investment, whichcould yield either more or less than the cash equivalent depending on the outcome. This isthe essence of the certainty equivalent approach to capital investment analysis. If amount Amay be received with probability P(A) and amount B with probability P(B) = 1 − P(A),and the events are mutually exclusive,1 then amount C is said to be a certainty equivalent tothe investment that could result in outcome A or outcome B, and P(C) = 1.0. This can begeneralized to investments with more than two outcome possibilities.
For investments with more than two outcome possibilities, the certainty equivalent, C ,may be defined as the (certain) cash amount for which the investor is indifferent to the riskyinvestment for which the dollar returns are described by the probability distribution P(I ),where I is a vector. It is important to note that C is not the expected value of the investment,E[I ]. Rather, C is the certain cash amount with utility that is equivalent to the investment. Inother words, if U denotes utility, then
U (C) = E[U (I )] (18.1)
Because it is the utility of the returns vector I that determines investor attitudes toward theinvestment, it is unimportant in determining certainty equivalents whether I is the vector of netpresent values (NPVs), internal rates of return (IRR), or other measure of investment returns.In this chapter we use the NPV as a matter of convenience only, noting that the certaintyequivalent may be based on any measure for which the investor can make consistent choices.
Although utility itself is a personal matter, we may, as suggested in the previous chapter,estimate a composite investment curve for the decision-makers of a particular organization.2
To illustrate the use of certainty equivalents, in Figure 18.1 a risk–return trade-off curve fora given enterprise is illustrated. All points of the curve have equal utility, so we could referto it as an iso-utility curve. Different organizations may be expected to have different curves.Figure 18.1 is consistent with risk aversion because expected returns must rise more rapidlythan risk (however risk is measured) for the utility to remain constant.
Example 18.1 The Ajax Company is evaluating a production mixer that costs $25,000. Themixer is expected to produce cash flows of $10,000, $15,000, $20,000, and $20,000 at the endof each year of its four-year economic life. The firm’s management believes that the cash flowsbecome riskier the further into the future they are expected to occur. Certainty equivalentsare calculated at 0.982, 0.930, 0.694, and 0.422 of the expected cash flows. Since certaintyequivalents are used, we discount at the risk-free rate,3 which is assumed to be 8 percent. The
1Mutually exclusive may be defined in terms of the conditional probabilities as P(A|B) = 0 and P(B|A) = 0.2Some authors have suggested that the equal market valuation curve be employed in calculating certainty equivalents. Although suchan approach may be useful in securities investment analysis, it is questionable within the context of capital investment analysis for anindividual firm.3Here risk free means free of default risk. The rate is generally taken to be the rate on US government securities of the same futurematurity as the cash flow’s receipt.
Single Project Analysis under Risk 193
Figure 18.1 Risk–return trade-off curve for the enterprise
NPV of the mixer is
NPV = $ − 25,000 + (0.982)($10,000)
1.08+ (0.930)($15,000)
(1.08)2
+ (0.694)($20,000)
(1.08)3+ (0.422)($20,000)
(1.08)4
= $ − 25,000 + $9092.59 + $11,599.88 + $11,018.39 + $6203.65
= $13,274.51
which is positive; thus the project is acceptable under the NPV criterion.
CERTAINTY EQUIVALENTS: METHOD II
The previous section shows one way in which certainty equivalents may be found and usedto calculate a project’s certainty equivalent NPV. Another method that is sometimes used isfirst to calculate the project’s NPV at the risk-free interest rate. The next step is to convertto a “certainty” equivalent by subtracting an allowance for risk, usually taken to be V [NPV],the variance of NPV. To illustrate, assume the variance of the NPVs of the cash flows inExample 18.1 is V [NPV] = $14,422.07. First calculate the NPV of the expected cash flows atthe risk-free rate (again assumed to be 8 percent):
NPV = $ − 25,000 + $10,000
(1.08)1+ $15,000
(1.08)2+ $20,000
(1.08)3+ $20,000
(1.08)4
= $ − 25,000 + $9259.26 + $12,860.08 + $15,876.64 + $14,700.60
= $27,696.58
Next, subtract the variance to obtain the certainty equivalent:
NPV = $27,696.58
−V (NPV) = −14,422.07
$ 13,274.51 certainty equivalent
194 Capital Asset Investment: Strategy, Tactics & Tools
In this case the certainty equivalent is identical to that obtained with method I; this is becausethe variance was chosen to make the results identical.
In general, it is unlikely that this method will yield results close to those obtained withmethod I. Although this is empirically questionable, we can say a priori that for this methodto yield the same results as method I, it is necessary that the variance measure risk in thesame way that calculation of certainty equivalents for the individual period cash flows does.Considering that the certainty equivalents of the individual cash flows are based on utility,and the variance is based on dollars alone, the two methods will not yield comparable resultsexcept for investment curves of a most particular type.4
Despite the conceptual problems imbedded in method II, it is sometimes used, and it isthen necessary to have equations for calculating the expected value of NPV, E[NPV], and thevariance of NPV, V [NPV]. The expected value of NPV is found from
E[NPV] =N∑
t=0
E[Rt ]
(1 + i)t(18.2)
where R0 is the required investment outlay. The variance is found from the relationship
V [NPV] = σ 20 + σ 2
1
(1 + i)2+ σ 2
2
(1 + i)4+ · · · + σ 2
N
(1 + i)2N(18.3)
if the cash flows are independent of one another (uncorrelated). If the cash flows are perfectlycorrelated, the variance is defined by
V [NPV] = σ 20 + σ 2
1
(1 + i)2+ σ 2
2
(1 + i)4+ 2
{Cov(0, 1)
(1 + i)1+ Cov(0, 2)
(1 + i)2+ Cov(1, 2)
(1 + i)3· · ·
}(18.4)
For the general case with three cash flows that are random variables, X0, X1, X2 with weightsa, b, c, the variance is given by
V [NPV] = V [aX0 + bX1 + cX2]
= a2V [X0] + b2V [X1] + c2V [X2](18.5)+ 2ab Cov[X0, X1] + 2ac Cov[X0, X2]
+ 2bc Cov[X1, X2]
where a = (1 + i)–0 = 1, b = (1 + i)–1, c = (1 + i)–2. The complexity of this expressionsuggests why we consider the special cases of zero correlation and perfect correlation, forwhich the considerably simpler equations apply.
RISK-ADJUSTED DISCOUNT RATE
Another method for dealing with risky cash flows is that of the risk-adjusted discount rate.Although the certainty equivalent method essentially adjusts the cash flows or the NPV of the
4The assumption that the distribution of returns is normally distributed means that the distribution is fully described by two parameters:mean and variance. This implies that utility may be maximized by appropriate selection of assets according to a function of meanand variance, provided we know the equation for the investor’s utility in terms of these parameters. Certainty equivalent method II,by subtracting the unweighted variance of returns, assumes a very special type of utility function in addition to normally distributedreturns.
Single Project Analysis under Risk 195
unadjusted flows, the risk-adjusted discount rate makes the adjustment to the discount rate foreach cash flow period prior to calculating the NPV.
In Chapter 6 the NPV method was discussed in the context of a certainty environment.There it was assumed that the enterprise’s cost of capital was the appropriate rate of discount,5
and that this was constant from period to period over the asset’s economic lifetime. Now,assuming a risky environment in which the cash flows differ both in futurity and in uncertainty,it is no longer reasonable to assume that a constant discount rate is appropriate. Ceteris paribus,the riskier an investment return occurring sometime in the future, the greater the required rateof return — discount rate — that investors will apply to it.
Example 18.2 To illustrate the method of risk-adjusted discount rate let us again considerthe Ajax Company mixer from Example 18.1. The cost of the mixer is $25,000 and the cashflows expected at the end of each year of its four-year economic life are $10,000, $15,000,$20,000, and $20,000.
The NPV, using the risk-adjusted discount rate approach with rates k = 9.98 percent, k2 =11.99 percent, k3 = 21.98 percent, and k4 = 34.00 percent, is
NPV = $ − 25,000 + $10,000
(1.0998)1+ $15,000
(1.1199)2+ $20,000
(1.2198)3+ $20,000
(1.3400)4
= $ − 25,000 + $9092.56 + $11,960.04 + $11,019.56 + $6203.68
= $13,275.84
which is within 0.01 percent of the certainty equivalent approach result. The difference is dueto rounding errors in this case because the risk-adjusted discount rates were chosen to yieldidentical results.
If α is the constant that relates the certainty equivalent to its expected value of cash flow,and i the risk-free rate, then the relationship between these variables, as Mao has shown, is
Certainty equivalentαE[Rt ]
(1 + i)t
Risk-adjusted discount rateE[Rt ]
(1 + kt )t
(18.6)
which, after rearranging terms, yields
kt = 1 + i
α1/t− 1 (18.7)
An investor who is perfectly consistent in applying this method and the certainty equivalentmethod should obtain identical results (except for rounding errors). Whether or not decision-makers are consistent in applying the methods is an empirical question, as is the question asto which method is preferred (if either is) by the same decision-makers.
COMPUTER SIMULATION
Although the foregoing methods for dealing with risk can be useful, they have some short-comings. Neither the certainty equivalent nor the risk-adjusted discount rate provides more
5Some would argue that the cost of common equity is the appropriate rate since the net, after-tax cash flows accrue to the commonshareholders. See, for example, J. C. T. Mao [100, p. 138].
196 Capital Asset Investment: Strategy, Tactics & Tools
than a single, point estimate of the returns on a capital investment. Neither is suited to riskyinvestments for which characteristics other than the cash flows themselves may be randomvariables. For example, the project life itself may be a random variable, or at least we maywish to examine the implications if it is treated as one. Circumstances may exist that compelus to treat the discount rate itself as a random variable, and we may have reason to want totreat the related cash flows according to a more complex relationship than simple correlationallows. And, we may want to examine the effect of having a particular policy that requires theasset to be abandoned if certain conditions are met, or we may wish to examine the effects ofrandom shocks, such as a sudden increase in petroleum price.
Computer simulation (also referred to as Monte Carlo experimentation) enables us to an-alyze complex investments for which direct mathematical methods either do not exist or areinadequate for our purpose. Because of the speed with which modern digital computers canperform complex, repetitive calculations, we can examine the simulated outcome of thousandsof iterations that, taken together, provide a profile of the investment. Through analysis of thesimulated profile, within the context of management’s investment curve (a notion developedin the previous chapter), a rational decision may be reached to accept or reject the project.
The topic of computer simulation is covered thoroughly in texts devoted exclusively tothe subject. For those who desire a detailed development we recommend those works beconsulted. It is not our purpose here, however, to become sidetracked into a detailed discussionof computer simulation. For our purposes it should suffice to state that three basic things arerequired: (1) a computer; (2) a (pseudo-) random number generator; and (3) a mathematicalmodel of the thing we wish to simulate. Since serviceable random number generators are anintegral part of much computer software nowadays, to perform a simulation analysis we needbe mainly concerned with the task of specifying our model carefully, then translating it intosuitable form for the computer.
It is often recommended that one should use the risk-free rate (i.e. the yield on 90-day USTreasury bills) in computer simulations because the simulated distribution itself reveals theproject’s risk. However, the NPV profile obtained with the risk-free rate is difficult to interpretand relate to the NPVs of other projects. Therefore, simulations are in practice often performedusing the organization’s cost of capital. The following example was analyzed with KAPSIM,a program written by this author to simulate single capital investment projects. It can be donewith other simulation packages and add-ins for popular spreadsheet packages or even withina spreadsheet itself with contemporary spreadsheet programs with a little programming work.
Example 18.3 An investment costs $2.5 million and is expected to last nine years. The projectlife is thought to be Poisson distributed, so the variance is also nine. Management policy is todispose of such assets after 20 years; thus if the project should last that long, it would then beabandoned.
The variance of the net, after-tax cash flows is estimated to be ($2 million)2 every year. Thefirst-year cash flow has expected value of −$5 million; that of the second year is +$5 million;from the third through the ninth year the returns are expected to grow by $1 million each year,starting at $6 million at the end of the third year. The returns in years 10 and beyond are zerobecause the project is expected to last only nine years.
A further complication is that, because of structural changes in the economy, the discountrate is expected to increase over time. The expected rate is 10.1 percent for the first year’s cashflow, with variance (1.01 percent)2. Each subsequent year is expected to be 0.1 percent higher,and the standard deviation to be 10 percent of expected value.
Single Project Analysis under Risk 197
Table 18.1 Example 18.3 problem description (Part 1)
Item Distribution Parameter 1 Parameter 2 Parameter 3
Initial outlay Constant 2.50000E + 06 0 −0.0Period 1 return Normal −5.00000E + 06 2.00000E + 06 −0.0Period 2 return Normal 5.00000E + 06 2.00000E + 06 −0.0Period 3 return Normal 6.00000E + 06 2.00000E + 06 −0.0Period 4 return Normal 7.00000E + 06 2.00000E + 06 −0.0Period 5 return Normal 8.00000E + 06 2.00000E + 06 −0.0Period 6 return Normal 9.00000E + 06 2.00000E + 06 −0.0Period 7 return Normal 1.00000E + 07 2.00000E + 06 −0.0Period 8 return Normal 1.10000E + 07 2.00000E + 06 −0.0Period 9 return Normal 1.20000E + 07 2.00000E + 06 −0.0Period 10 return Normal 0 2.00000E + 06 −0.0Period 11 return Normal 0 2.00000E + 06 −0.0Period 12 return Normal 0 2.00000E + 06 −0.0Period 13 return Normal 0 2.00000E + 06 −0.0Period 14 return Normal 0 2.00000E + 06 −0.0Period 15 return Normal 0 2.00000E + 06 −0.0Period 16 return Normal 0 2.00000E + 06 −0.0Period 17 return Normal 0 2.00000E + 06 −0.0Period 18 return Normal 0 2.00000E + 06 −0.0Period 19 return Normal 0 2.00000E + 06 −0.0
Note: Number of iterations equals 1000; expected value of project life is 9 with variance of 9;distribution Poisson; maximum project life is 20.
Finally, because the project may be abandoned prior to the end of its economic lifetime, itis necessary to estimate salvage at the end of each year. Salvage value is assumed to declinefrom the original investment by 20 percent each year. However, the rate of decline is estimatedto be Poisson distributed.
This information is summarized in Tables 18.1 and 18.2, which contain the provided infor-mation as part of the computer-printed simulation results. One thousand iterations are run toobtain a sufficient sample of results, which are measured by the NPV. If too few iterations areused, we cannot draw conclusions from the scattered results, whereas too many iterations wouldoffer little or no additional useful information. The cost of computation has come down greatly(thanks to constantly improving technology). So, it may be best to err somewhat on the side oftoo many iterations rather than too few if there is any doubt about how many should be used.
Tables 18.1 and 18.2 provide a record of the information we put into the computer simulationof the investment project. All simulations should be accompanied by such “echo” output for tworeasons: (1) in order to find mistakes that may have been made in specification of the problemsor preparation of the data and (2) in order to provide an organized record to accompanythe simulated results. Simulation output for one case may look about the same as output foranother. If the input data are attached to the computer printout, one can avoid the frustrationand difficulty of trying to reconstruct the assumptions that produced a particular computerrun’s output.
Table 18.3 contains the frequency interval bounds, and frequency distribution of the simu-lated NPV results for Example 18.3. The frequency count for an interval is determined by thenumber of results that are greater than or equal to the lower bound and strictly less than theupper bound. Figure 18.2 contains a graph corresponding to the values of Table 18.3.
198 Capital Asset Investment: Strategy, Tactics & Tools
Table 18.2 Example 18.3 problem description (continued)
Item Distribution Parm 1 Parm 2 Parm 3
Period 1 capital cost Normal 0.101000 0.010100 000000Period 2 capital cost Normal 0.102000 0.010200 000000Period 3 capital cost Normal 0.103000 0.010300 000000Period 4 capital cost Normal 0.104000 0.010400 000000Period 5 capital cost Normal 0.105000 0.010500 000000Period 6 capital cost Normal 0.106000 0.010600 000000Period 7 capital cost Normal 0.107000 0.010700 000000Period 8 capital cost Normal 0.108000 0.010800 000000Period 9 capital cost Normal 0.109000 0.010900 000000Period 10 capital cost Normal 0.110000 0.011000 000000Period 11 capital cost Normal 0.111000 0.011100 000000Period 12 capital cost Normal 0.112000 0.011200 000000Period 13 capital cost Normal 0.113000 0.011300 000000Period 14 capital cost Normal 0.114000 0.011400 000000Period 15 capital cost Normal 0.115000 0.011500 000000Period 16 capital cost Normal 0.116000 0.011600 000000Period 17 capital cost Normal 0.117000 0.011700 000000Period 18 capital cost Normal 0.118000 0.011800 000000Period 19 capital cost Normal 0.119000 0.011900 000000Period 20 capital cost Normal 0.120000 0.012000 000000
Note: Salvage assumed to decline from initial outlay by percent per year of: Poisson2.00000E − 01, 2.00000E − 01, 0.
Table 18.3 Example 18.3 NPV frequency distribution
FrequencyInterval Lower bound Upper bound ≥ = LB, < UB
1 −0.308229000E + 08 −0.284193447E + 08 12 −0.284193447E + 08 −0.260157834E + 08 43 −0.260157834E + 08 −0.236122341E + 08 24 −0.236122341E + 08 −0.212086788E + 08 65 −0.212086788E + 08 −0.188051235E + 08 76 −0.188051235E + 08 −0.164015682E + 08 207 −0.164015682E + 08 −0.139980129E + 08 278 −0.139980129E + 08 −0.115944576E + 08 209 −0.115344576E + 08 −0.919090230E + 07 35
10 −0.919090230E + 07 −0.678734700E + 07 4311 −0.678734700E + 07 −0.438379170E + 07 3712 −0.438379170E + 07 −0.198023640E + 07 7413 −0.198023640E + 07 0.423318900E + 06 10514 0.423318900E + 06 0.282687420E + 07 15615 0.282887420E + 07 0.523042950E + 07 17616 0.523042350E + 07 0.763398480E + 07 14017 0.763398480E + 07 0.100375401E + 08 10118 0.100375401E + 08 0.124410354E + 08 3519 0.124410954E + 08 0.148446507E + 08 920 0.148446507E + 08 0.172482060E + 08 1
Note: Avg. NPV = 0.631806495E + 07; std. dev. of NPV = 0.762450328E + 07; Low =−0.308229000E + 08; high = 0.172482060E + 08; range = 0.480711060E + 08.
Single Project Analysis under Risk 199
Figure 18.2 Frequency histogram for Example 18.3 computer simulation
The decision to accept or reject the investment may be made from the frequency distribution,which is clearly not normally distributed. The first 12 intervals contain only negative NPVresults, whereas the thirteenth contains mixed results. If we may assume the distribution withinthe thirteenth interval to be uniformly distribu ed over that interval −0.198023640E + 07 to+0.423318900E + 06, a range of 0.2403553E + 07, then (198/240) × 105, or 87 of the 105simulated observations in that interval are negative, and 18 are positive. Therefore, there isprobability of (276 + 87)/1000 or 363/1000 = 0.363 that this project, if accepted, wouldyield a negative NPV, or better than one chance in three.
Let us say that management will not accept any project for which the odds of losing morethan the initial investment (in this case $2.5 million) exceed 1/10. If we may again assumeresults to be proportionately split over the twelfth interval, 58 of them in the interval are −$2.5million or less. This means the odds are 260/1000, or 0.260 of losing more than the initial
200 Capital Asset Investment: Strategy, Tactics & Tools
investment (a little less if we were to consider the negative cash flow at the end of the first yearas part of the initial investment). This project would be unacceptable to management underthis policy, even though the mean NPV is $6,318,065.
A virtually limitless number of management policies may be applied to the frequency dis-tribution. The particular ones applied will depend on the enterprise’s investment curve, whichreflects the risk attitudes of its management. For instance, management might consider theproject acceptable if the odds for an NPV exceeding $10 million were greater than the oddsof NPV loss of more than $20 million (on the assumption that management could, and would,intervene to prevent such loss actually accumulating; that is, changing the odds during thegame). Alternatives are limited only by one’s imagination.
This example was not designed to represent necessarily a realistic project, but rather toillustrate how simulation analysis can facilitate the decision-making process for a complexproject. No doubt the example may be made more realistic simply by changing some of theparameters. As it stands, this hypothetical project would probably not be very attractive toa risk-averse, capital preservation-oriented management because there are fairly large oddsthat losses with NPV as large as −$30 million could result from adopting this project. Unlessmanagement believes that the odds can be significantly improved by intervention as time passes,this project will be passed over by most executives if we may extrapolate from Mao’s surveyfindings cited in the previous chapter.
LEWELLEN–LONG CRITICISM
Wilbur G. Lewellen and Michael S. Long, in an insightful comparison of simulation resultsto those obtained from point estimates, draw some thought-provoking conclusions. Theseshould be considered carefully, because of implications that may limit the classes of invest-ment projects that are suitable for Monte Carlo treatment. In summary, the Lewellen–Longconclusions are:
1. If the IRR is used in simulating a capital investment, the relationship between the cash flowsand the IRR will cause the mean IRR to be lower than the IRR obtained by discounting themeans of the respective cash flow distributions. This holds even if the cash flow distributionsare symmetrical about their respective means.
2. By using NPV rather than IRR as the measure of a simulated investment, the bias is elimi-nated because the present value of a future cash flow is a linear function of the size of thecash flow, whereas the IRR is not.
3. The discount rate that should be used in present value simulations is the risk-free rate (freeof default risk, such as federal government securities).
4. Even with the NPV, problems arise when the project lifetime is uncertain. Variable projectlife will, ceteris paribus, cause the mean of the resulting simulated NPV distribution to beslightly less than the NPV obtained by discounting the cash flows over the mean project life.
5. Further disparities can be caused by nonsymmetrical cash flow distributions. This problemis likely to arise when a “most likely,” or modal value, is used in place of the mean inspecifying the simulation.
6. Since, because of the cost and time required to properly perform a computer simulation,it will be limited to the larger and more important projects, most will be evaluated by othermethods, using a point estimate of each cash flow. It is therefore important that expectedvalues rather than modal values be employed.
Single Project Analysis under Risk 201
7. If any elements of the cash flows depend on multiplicative combinations of stochasticvariables, it is important to use the cash flows directly in obtaining the expected values.Only if the multiplicative components are completely independent can E[p · q] be treatedas equal to E[p] · E[q]; otherwise E[p · q] = E[p] · E[q] + σpq where σpq is the co-variance between p and q.
For example, if p represents price and q quantity sold, we should recognize that in allbut perfectly elastic or inelastic demand, price and quantity are correlated. Since price andquantity demanded are inversely related, indirect estimation obtained by multiplying E[p]by E[q] will overstate E[p · q]. Because the correlation coefficient ρ is defined by
ρ = σpq
σpσq
the relationships are
E[p · q] > E[p] · E[q] if ρ > 0
E[p · q] < E[p] · E[q] if ρ < 0
8. All the above problems are either artificial or avoidable, with the exception of the variableproject life. Therefore, as an index of investment project worth the point-estimate analysisshould serve quite well. While not incorrect in determining a proposal’s expected return,simulation is unnecessary.
9. The claim that a major benefit from simulation analysis is that it provides an improvedappreciation of the risk the enterprise would assume may be misleading and evendangerous. It is not the riskiness of the individual investment (its “own” risk) which isimportant, but rather its risk within the context of a collection of assets to which its returnsmay be related. To quote Lewellen and Long [89]:
. . . the one irrelevant feature of an asset’s prospective returns is its “own risk” — the outcome uncer-tainties unique to the asset itself. These can, and will, be diversified away by individual investors andby institutions in their securities portfolios, leaving only the degree of correlation between the asset’sreturns and those of the so-called “market portfolio” as relevant to value, since this connection andits implied risks cannot be extinguished via diversification.
And they conclude by stating that [89, p. 31]:
Simulated profiles of capital expenditure proposal outcomes are, perforce, chiefly descriptions of “ownrisk.” They do not reveal a project’s addition to . . . total risk. Having simulated, therefore, the analyststill cannot legitimately compare and choose among alternatives, because nowhere in the informationhe obtains is the single most important criterion of investment worth — portfolio impact — addressed.However imperfectly, the cost-of-capital/single-point-expected value cash flow forecast approach doesget at the issue of portfolio context. For that reason, . . . not only is such an approach less onerous inexecution but, paradoxically, also has a higher potential for relevant risk recognition. . . .
Although portfolio effects and diversification are topics that are addressed in the followingchapters, the Lewellen–Long criticism of simulation should cause us to reflect carefully onthe facts and circumstances surrounding an investment proposal before boldly embarking ona simulation analysis. And there are circumstances under which a simulation analysis may bepreferable to using the simplified analysis based on point estimates. These might include thefollowing:
202 Capital Asset Investment: Strategy, Tactics & Tools
1. The proposal is under consideration by an entrepreneur who will be putting all his or herassets into the project if it is accepted.
2. It is not possible to obtain meaningful estimates of the proposal’s correlation with theexisting assets of the firm, or the project being considered is estimated to be uncorrelatedwith them.
3. The project is large in relation to the firm and has a potential for large loss that could threatenthe firm’s survival.
4. Possible pressure on executives to avoid investments in risky assets that would reduce orleave unchanged the organization’s overall risk. If such pressures have been responsiblefor portfolio managers generally avoiding “second-tier” securities, gold, and commodityfutures then we may surmise that similar peer group or external pressures may act toconstrain decisions to “own risk” more than portfolio considerations.
5. The cash flows may be governed by complex relationships which preclude analysis ofpoint estimates. (The Lewellen–Long article did not deal with the problem of simultaneousvariability in cash flows and project life or contingent cash flows, for example.) Althoughsome fairly complex proposals may be analyzed with point estimates, considerable timemay pass before the analyst develops the insight and inspiration that may be necessary.Contingent asset relationships of the type treated in the following chapters, for example,may be impossible to deal with on a point-estimate basis.
6. The matter of risk in the public sector may require a different approach even if the CAPMwere unequivocally and universally accepted for firms in the private sector.
7. The CAPM rests upon a foundation of restrictive assumptions that may not always apply,particularly for smaller firms. For additional insight into the pros and cons of simulationfor the analysis of risky investments, an article by Stewart Myers [113] is recommended.The matter remains controversial and is likely to remain so, at least until some additionalquestions about the CAPM can be unequivocally answered.
With the conclusion of this chapter we shall leave the domain of risk without diversificationand enter that of capital investment in which project proposals are not independent and portfolioaspects must be considered.
19Multiple Project Selection under Risk:
Computer Simulation and Other Approaches
Previously we primarily considered capital investment selection under risk for individual can-didate projects. To simplify exposition we assumed they and existing assets were independent.But projects often are interrelated, and therefore cannot be evaluated in isolation. Up to now itwas useful to assume that all relevant asset characteristics were contained in expected returns,variances and covariances. That enabled selecting efficient combinations of investments: thosewith maximum expected return for a given level of risk or, alternatively, those with minimumrisk for a given level of return.
In this chapter we examine further the problem of capital investment selection. Now, however,we shall not require that returns be normally distributed or that the relationship between projectreturns and existing assets be constant intertemporally. Additionally, we shall consider theproblem introduced by sequential, event-contingent decisions over time. Specifically, mattersrelating to parameters other than the mean and the variance–covariance are considered, as aresimultaneous variability of project characteristics.
DECISION TREES
The decision tree approach is a widely employed analytical device for addressing investmentprojects that contain sequential, event-contingent decisions. Such projects, in other words, havecertain chance-determined outcomes during their lifetimes and these outcomes may influencedecisions made during the project lifetime and interact with them.
Magee’s paper [95] is now a classic on the topic of decision trees, and was responsiblefor focusing attention on this important technique, which others subsequently built upon.Two years prior to Magee, Masse [102] criticized the view of investments as single-perioddecisions isolated from future events. He noted that future alternatives are conditioned bypresent choices. Because of this, investment evaluation cannot be reduced to a single decision,but must be viewed as a sequence of decisions over time.
Decision trees are similar in some respects to Markov chains. They differ in that they includeimbedded decisions and not solely probabilistic event nodes. A Markov chain has the propertiesthat (1) the possible outcome set is finite; (2) the probability of any outcome depends only onthe immediately preceding outcome,1 and (3) the outcome probabilities are constant over time.Markov chains are suitable for many types of problems and the mathematics for their analysisis well developed.
Figure 19.1 contains the structure of a generalized decision tree. The letter D denotes adecision and C a chance event. At D1,1 the decision-maker is faced with two choices.2 The
1This is true only for first-order Markov chains. If the chain is second-order, then an outcome may depend on the two prior events, andso on.2In general, there will be n decision choices, where n is a finite number. Our space limitations dictate that for illustrative purposes onlytwo choices be used.
204 Capital Asset Investment: Strategy, Tactics & Tools
Figure 19.1 General structure of a decision tree
first subscript number refers to the point in time at which the decision or chance event occurs.Time need not be cardinal. In other words, the actual time between t = n and t = n + 1 maybe either more or less than the time from t = n − 1 to t = n. The second subscript refers tothe position of the event or decision from top to bottom in the diagram.
The problem illustrated in Figure 19.1 requires two decisions, one at the outset and one aftera chance event has occurred. Although they are shown as possibly different events, C2,1 andC2,2 may refer to the same chance event. The effect on the enterprise, however, is differentbecause of the decision made at D1,1. Similarly, D3,1, D3,2, and so on, could refer to thesame decision, tempered by different precedents. Alternatively, the decisions could be quitedifferent.
To find the optimal decisions we examine the payoff utilities associated with a particularsequence of decisions and chance events. The final result, denoted by x and y, is determinedby conditional probability of preceding events. The optimal decisions are found by backward
Multiple Project Selection under Risk: Computer Simulation 205
induction, by looking at the most distant result, and tracing back from there to the very beginningat D1,1. As an alternative to working with the utilities of each separate terminal point, itis possible to find the final monetary outcomes and their associated probabilities for eachbranch from the final decision nodes, then take the certainty equivalents of the uncertainoutcomes.
An example will help clarify how decision trees may be applied to capital investmentdecisions.
Example 19.1 The Northern Lites Manufacturing Company has successfully produced andmarketed home heating units for 30 years. The units are designed to use either gas or fuel oil, butnot both. The engineers of the firm have now designed a new unit that can use interchangeablyeither gas, fuel oil, coal, or cord wood and, with an optional adapter, peanut shells andother waste products. The marketing staff is enthusiastic about the sales prospects for the newmultiple-fuel furnace that they are calling the “Hades Hearth Home Heater.”
The new heating unit could be produced in the existing factory, but changeover would takeat least a year, perhaps longer, and an inventory of replacement parts would have to be builtup to meet the current owners’ needs for repairs of the present models. A new plant could beconstructed to be in operation within a year, and a nearby facility could be leased and readiedfor production in the same time span.
Because of present shortages with rising fuel oil and natural gas prices, sales are expectedto be brisk. However, there is a 30 percent chance of an economic depression in a year thatcould be so severe that even though they might like to buy the new furnace, consumers willnot have the money. If there is such a depression, the firm will likely lose $1000 with a newplant facility, $300 with the leased facility, and $100 if it delays its production by deciding touse the existing plant. A deep depression would mean a halt to production until the economyimproved.
A new plant will allow maximum production. Delaying production to use the existing plantwill allow competitors to take the lead and sales will therefore be less. In two years theengineering staff feels it will have a much improved design that may be required to keep salesup. There is some risk, however, that the improved design may not be as popular as anticipatedbecause of competitive developments and the possibility of meaningful government incentivesfor fuel conservation.
Figure 19.2 illustrates the tree diagram for this problem. If a new plant is built, if there isno depression, and the design is changed and successfully received in the market, the presentvalue to the firm is $900. If not well received, the present value is $360. The remainingresults are similarly interpreted. Note that the expected values shown in Figure 19.2 areexpected monetary values, not expected utilities or certainty equivalents. Table 19.1 showsthe calculations corresponding to the expected monetary values of Figure 19.2 for finding theexpected utilities.
Implicit in the approach illustrated in Table 19.1 is the assumption that the expected utilityis the proper measure by which the optimum course may be determined. However, we shouldnote that this approach does not take into account the distributions of the outcome utilities butonly their expected values. It may well be the case that certainty equivalents correspondingto the monetary outcomes would yield a different ranking. Therefore, it may be important inpractice to have management decide on the basis of the certainty equivalents that it assigns tothe various choices.
206 Capital Asset Investment: Strategy, Tactics & Tools
Figure 19.2 Decision tree for Example 19.1
As matters stand, the expected utilities in Table 19.1 favor leasing a plant for productionof the multiple fuel furnace instead of either building a new plant or delaying and using theexisting plant. Note that the spread of both dollar and utility outcomes is greater for leasingthan for delaying until the current plant can be used, thereby making the lease arrangementmore risky. By the same reasoning, the building of a new plant would be both more risky andoffer a lower expected utility than leasing; it is thus dominated by the leasing alternative.
An alternative to either choosing the path offering the highest expected utility or choosingthe path offering the highest certainty equivalent is found in the next chapter. There the capitalasset pricing model (CAPM) is discussed and it is shown that the optimal choice might bemade by employing the risk/return trade-offs prevailing in the market.
Multiple Project Selection under Risk: Computer Simulation 207
Table 19.1 Calculation of payoff utilities forExample 19.1
Build new plant0.7 × 0.8 ×U ( $900) = 0.56 × 0.80 = 0.4480.7 × 0.2 ×U ( 360) = 0.14 × 0.50 = 0.0700.7 × 0.6 ×U ( 720) = 0.42 × 0.73 = 0.3070.7 × 0.4 ×U ( 180) = 0.28 × 0.30 = 0.0840.3 × 1.0 ×U (−1000) = 0.30 × −0.90 = −0.270
0.639Lease plant0.7 × 0.8 ×U ( $630) = 0.56 × 0.68 = 0.3810.7 × 0.2 ×U ( 270) = 0.14 × 0.40 = 0.0560.7 × 0.6 ×U ( 540) = 0.42 × 0.63 = 0.2650.7 × 0.4 ×U ( 225) = 0.28 × 0.35 = 0.0980.3 × 1.0 ×U ( −300) = 0.30 × −0.32 = −0.096
0.703Delay production0.7 × 0.7 ×U ( $450) = 0.49 × 0.57 = 0.2790.7 × 0.3 ×U ( 135) = 0.21 × 0.24 = 0.0480.7 × 0.5 ×U ( 360) = 0.35 × 0.49 = 0.1720.7 × 0.5 ×U ( 90) = 0.35 × 0.18 = 0.0630.3 × 1.0 ×U ( −100) = 0.30 × −0.14 = −0.042
0.520
OTHER RISK CONSIDERATIONS
In addition to the types of risk considerations introduced by sequential, event-contingent deci-sions and intervening stochastic events, there are other situations in which portfolio approachesmay not be employed to best advantage. Since portfolio approaches developed by Markowitzand Sharpe assume stabile (or at least not suddenly changing) relationships between projectsand the enterprise’s existing assets, problems in which the relationships may change in re-sponse to some future chance event or events do not lend themselves to portfolio selectionapproaches as they currently exist.
Recognizing the shortcomings pointed out in Chapter 18 for computer simulation of singleinvestment proposals, computer simulation may nevertheless offer the most suitable approachto the analysis of certain multiple-project problems. One such application arises in the caseof situations involving assets with useful lives governed not only by their own unique char-acteristics but also by the life of the aggregate. An example will serve to illustrate this typeof problem. The following example is rather lengthy because of the nature of the problemand because it illustrates a way of analyzing multiple-project problems by means of computersimulation.
Example 19.2 A comprehensive simulation example3 In oilfield primary recovery oper-ations, equipment deterioration and breakdown create problems over the lifetime of the field.These problems are of a reinvestment/abandonment type for which little theoretical guidanceis to be found in texts on capital budgeting, engineering economy, or elsewhere. For instance:
3Extracted from a paper by Anthony F. Herbst and Sayeed Hasan [64]. By permission of the American Institute for Decision Sciences.
208 Capital Asset Investment: Strategy, Tactics & Tools
(1) When should a particular well be rehabilitated rather than abandoned? And (2) whenshould the entire field be abandoned for continuing primary recovery and secondary recoveryoperations begun?
Discussion is limited in scope to consideration of decision algorithms for determining:
1. Whether a particular well should be repaired after equipment breakdown, or should insteadbe shut down.
2. Under what circumstances the entire field should be abandoned as a primary recovery field.
We consider the related problem of replacement of primary recovery with secondary recovery(by means of gas or water injection) before primary recovery in itself becomes economicallynonviable, though our emphasis is on the above two problems.
For purposes of classification we distinguish between two cases of primary recovery fieldabandonment. First, the weak case, in which the primary recovery operation is in itself stilleconomically viable, but inferior to and economically dominated by a change in technology tosecondary recovery methods. Second, the strong case, in which primary recovery by itself iseconomically nonviable regardless of secondary recovery operations.
A difficult but interesting feature of the problems considered in this section is interdepen-dence between the maximum expected economic life of each well and that of the entire field.Abandonment of any given well or set of wells may result in the entire field becoming unprof-itable with primary recovery. Adoption of the decision to abandon the entire field, of course,implies abandonment of all individual wells whether or not they are operating profitably at thetime.
The decision to abandon the entire field for primary recovery (strong case) is relativelystraightforward. The field should be shut down when the variable cash revenue contributionof the field no longer exceeds variable cash cost. In other words, in the strong case the fieldshould be abandoned when net cash contribution to fixed cost coverage and profits becomeszero.
The weak case decision to abandon primary recovery and replace it with secondary recov-ery technology should be made when the expected net present value (NPV) of investment insecondary recovery by gas or water injection exceeds the expected NPV of the remaining cashflows with primary recovery. We will not at this time get into the problem of determining theNPV of secondary recovery investment, as this appears to be amenable to the more standardcapital budgeting–engineering economy analysis. We do, however, address the problem ofestimating the NPV of the remaining cash flows with primary recovery.
The more difficult question than field abandonment for primary recovery is that of whenany particular stripper well should be abandoned. At least two circumstances must be consid-ered. First, under what circumstances should a given well be abandoned prior to equipmentfailure? Second, given an equipment failure of a particular type, when should repair or re-placement be undertaken, and when should the well be shut down and abandoned for primaryrecovery?
Characteristics of the Problem
The Oilfield
Type of Recovery Process We adopt the terminology that primary oil and gas recovery isthat undertaken by use of pumps and secondary recovery is that undertaken by use of water or
Multiple Project Selection under Risk: Computer Simulation 209
gas injection.4 When pumps are employed to raise the oil to the surface, the wells are calledstripper wells. Each stripper well will have its own pump, and therefore there will be a closecorrespondence between the number of wells and the amount of capital equipment required.
Secondary recovery with water or gas injection is accomplished by drilling wells aroundthe field through which water or gas may be injected to drive out the oil ahead of it. Unlikeprimary recovery, the number and location of injection wells are not strictly determined by thenumber of oil wells and there is no one-to-one correspondence between injection sites and oilwells as there is between pumps and stripper wells.
Breakdown of Stripper Wells Stripper wells are subject to a variety of breakdowns whichrequire the decision be made to either repair the problem or else to abandon that well duringprimary recovery. Major types of breakdown include pump overhaul, need for condensatewash, broken pump rod, and rupture in production tubing. In addition to the normal costs ofrepair, a condensate wash involves several days’ lost production while it is being carried out.Periodic, scheduled maintenance may entail pump overhaul and condensate wash to preventunscheduled breakdowns which may be more costly than planned preventive maintenance.
The probabilities associated over time with different major breakdowns may not be inde-pendent. For instance, a broken pump rod may do other damage to other well equipment, anda condensate wash may sometimes be required after another type of breakdown.
It is not our purpose to become involved in the technical details of the recovery process, butrather to deal with the methodology for determining the optimal reinvestment policy for the oilcompany. Therefore, after this brief introduction to the technology of recovery, we will moveon to the financial, economic, and methodological consideration of our work.
Decision Required
There are three basic decisions which must be made continually over the life of the field:
1. Given a well breakdown, whether to repair or shut down (abandon) the well.2. Whether to replace all well equipment with new equipment of the same type or switch to
another extraction technology and equipment.3. When to shut down the entire field to recovery with the currently employed technology.
The second decision, we propose, is amenable to more or less familiar approaches, such asthe MAPI (Machinery and Allied Products Institute) method of replacement analysis.
Financial Measures of Investment Return
Modern financial management heavily favors the use of discounted cash flow (DCF) methodsof evaluation, and favors the NPV approach over IRR. While the DCF approach has beenfavored, the payback criterion has simultaneously been attacked for use in investment decisions.However, DCF methods require certain information and conditions which may not exist.
In the problem we consider, the economic lives of the oilfield and the individual wells areinterdependent. We do not know the useful life of any individual well, as that life depends on the
4Another classification scheme would relegate to primary recovery the removal of oil by employing the natural subsurface pressure thatmay exist, secondary recovery to the use of pumps, and tertiary recovery to the water or gas injection methods. Under this alternativeclassification a field may be taken from primary to tertiary recovery by omitting the use of pumps entirely.
210 Capital Asset Investment: Strategy, Tactics & Tools
nature and frequency of required repairs and especially on the life of the entire field. Similarly,the life of the field depends on the lives of the individual wells. When all wells are shut down,obviously the field life, with the particular technology, has ended. When it has been decided toclose the field all wells in it will be closed. It is not known a priori how long the useful lives ofeither the individual wells or the field will be.
Additionally, the DCF methods require that the cash flows for each period be known orestimated. However, in the case of well breakdown the cash flows will be affected by thepattern and types of breakdown experienced, which are stochastically determined. The DCFmeasures are known to be affected by the pattern of cash flows as well as their magnitudes,and although the expected value of the cash flows could be determined the pattern cannot beon an a priori basis.
The absence of a priori knowledge of useful lives and cash flow patterns, coupled with theinterdependence of the well and field characteristics, negates the usefulness of ordinary DCFmethods to the problem we consider.
Methodology
We are interested in determining a policy for repairs that will yield the maximum expected netdiscounted return, and in the absence of a priori information, use the approach of simulationexperience with the field under alternative policies. The results of the simulation then provideinformation that can be converted to the DCF present value measure. In other words, weapproach the problem by simulating the cash flows of the field under different repair policies.The cash flows are carried forward with compounding at the firm’s cost of capital as futurevalues. At abandonment or shutdown of the field the future values are then converted to presentvalue equivalents. Choice of the proper policy will yield results that would be obtainable inordinary circumstances by a direct DCF approach.
Since the payback criterion is already familiar and widely used in industry we adopt it asour policy variable for repair decisions.5 If the net cash flow over n periods, where n is thepayback policy, is insufficient to recover the repair cost, we shut down the well. If the cash flowis greater than the repair cost, we repair and continue to operate the well. After a representativenumber of simulations, employing different seed values for the generation of (pseudo-) randomnumbers, we propose adopting that payback policy that promises the highest NPV for the field.
We also consider as a secondary policy the shutdown of the field when the rate of change inthe cash revenues becomes less than the rate of change in cash costs (including repair costs)over m periods. This provides a means for determining when to shut down the entire fieldbefore all wells are shut down and before net cash flow becomes negative. We define shutdownunder this secondary policy variable as “weak case” shutdown.
The field will be shut down to primary recovery whenever the net cash flow becomes negative.That is, when cash revenues are inadequate to cover cash costs, we will shut down the field.We define shutdown of the field under this condition as “strong case” shutdown.
The Model
I. Net per-period cash revenue for the field in period j:
NR = Total cash revenue − Variable cash cost − Fixed Cash Cost:6
5We do, however, adjust the payback criterion to include the time-value of funds.6Cash costs that would not exist were the field to be closed.
Multiple Project Selection under Risk: Computer Simulation 211
NR = PCFOILN W E L L S∑
i=1
OILBBLi, j
+ PCFGASN W E L L S∑
i=1
GASMCFi, j
− (ROALTO + PROCO)N W E L L S∑
i=1
OILBBLi, j
− (ROALTG + PROCG)N W E L L S∑
i=1
GASMCFi, j
− CASHFC −N W E L L S∑
i=1
REPRCi, j
Strong case decision:
If NR ≤ 0, shut down the entire field where i denotes individual wells.
PCFOIL and PCFGAS = the price per barrel of oil and MCF7 of gas, respectivelyOILBBL and GASMCF = the barrels of oil output and MCF gas output for the ith well,
j th periodROALTO and ROALTG = the royalty charges levied per barrel of oil and MCF of gasPROCO and PROCG = processing costs per barrel of oil and MCF of gasCASHFC = fixed cash expense per period while the field is openREPRC = repair cost for the i th well and jth period if a breakdown
occurs and is repaired
II. For a given well i :Let bk, j be the number of breakdowns of type k in period j . Then the reinvestment costfor repairing the breakdown is Ci, j . Let Pk be the cost of repair for type k breakdown. Werequire recovery of the cost with a payback period of n time periods.
If Ci, j <
j+n∑l= j+1
OILBBLi,l(PCFOIL − ROALTO − PROCO)/(1 + RATE)l− j
+j+n∑
l= j+1
GASMCFi,l(PCFGAS − ROALTG − PROCG)/(1 + RATE)l− j
where RATE is the firm’s per period cost of capital, we repair the well and continueoperation. Otherwise we shut down the well.
III. Breakdowns are assumed to be:
1. Poisson distributed over time, and be2. Independent of previous breakdown history of the well.
Figures 19.3(a) and (b) contain a flow chart of the model.8
7MCF is defined as 1000 cubic feet.8The paper published in the Proceedings did not contain this flow chart because of space limitations.
212 Capital Asset Investment: Strategy, Tactics & Tools
Figure 19.3(a) Flow chart of the model
Estimating NPV of Remaining Cash Flows with Primary Recovery
Empirical experience with our model, which provides for application of production deterio-ration gradients and changes in price, processing costs, and so on, suggests how remainingfield NPV may be estimated to any arbitrary point in time. Since the accumulated net futurevalue is calculated for each time period prior to shutdown, the NPV of cash flows over anytime span prior to shutdown may be calculated. Thus, if shutdown should occur in period 85,
Multiple Project Selection under Risk: Computer Simulation 213
Figure 19.3(b) Continuation of flow chart of model
and we are interested in the NPV of flows between period 40 and 85, we can determine thiseasily with customary financial mathematics.
Analysis of the remaining value of net cash revenues over time will provide informationuseful in the decision of whether, and when, to switch to other technology for recovery.
Empirical Results
The model was tested with empirical data provided to one of the authors, for a stripper fieldin western Canada. Results of a typical simulation run are summarized in Table 19.2. Thecondition of this field is such that the maximum NPV is obtained with a payback policy of fourperiods (one week equals one period in this run). Such a short payback meant that after amajor breakdown of any type the well involved would be abandoned. This may be attributedto the low oil and gas yields of the wells in a declining old field in relation to the cost ofrepairs.
Breakdown probabilities were assumed to be independent of prior breakdown history ofany well. This assumption may not be warranted, but the authors had no information tothe contrary which would enable them to modify the breakdown probabilities for furtherbreakdowns once a well had been repaired. Because of this, breakdowns of the same type couldoccur within the payback period. This meant that a repair could be carried out only to havethe same type of breakdown result in abandonment before the cost of the first repair had beenrecovered.
214 Capital Asset Investment: Strategy, Tactics & Tools
Table 19.2 Simulation results
All wells NPV by Strong caseshut down waiting until shutdown NPV within period all wells are in period strong case
NPAOUT No. shut down no. shutdown
4 15 $10,888.50 15 $10,888.505 18 −1,756.10 9 8,312.776 18 −1,756.10 9 8,312.777 18 −2,592.75 5 5,263.988 18 −2,592.75 5 5,263.989 18 −2,592.75 5 5,263.98
10 26 −18,184.30 5 5,263.9811 26 −18,184.30 5 5,263.9812 26 −18,184.30 5 5,263.9813 26 −18,184.30 5 5,263.98
CONCLUSION
The aim of this chapter was to illustrate two approaches to capital investment decision-makingunder risk that may provide useful results when portfolio approaches are not applicable. Thefirst approach shown was that of decision tree analysis. The second was computer simulationof multiple project problems.
In decision tree analysis the problem is characterized by a sequential decision–event–decision–event . . . chain over time. Because decisions are contingent on chance events, it maynot be possible to specify a priori which decision beyond the first will be made. The goalof decision tree analysis is to identify the mixed decision–event sequence that promises themaximum expected utility. By doing so it is possible to identify the superior initial decisionbecause it is determined by all that follows it. Decisions reached through application of thedecision tree approach will often be different from those reached through viewing the problemas an isolated, period decision because this latter approach ignores future choices.
Computer simulation of multiple project capital investment problems can be used to reachoptimal decisions where the interrelationships between projects are complex. It is difficult togeneralize about the type of problem warranting the time and expense of computer simulation.However, problems involving many projects, where their relationships cannot be describedby variance–covariance or correlation, will be candidates for simulation, especially if theparticular problem represents large resource value to the enterprise. Computer simulationshould be viewed not as a replacement for other methods of analysis in normal situations, butrather as a last line of defense when other approaches fail, and a powerful last defense it is. Innormal circumstances the extra time, effort, and expense of computer simulation are difficultto rationalize when other, less arcane methods yield correct decisions.
20Multiple Project Selection under Risk:
Portfolio Approaches
In Chapter 16 we discussed the matter of investment selection from an array of individu-ally acceptable independent1 projects. The selection was constrained by conditions of capitalrationing (that is, budget constraints), management policy, and technical considerations.Chapter 18 considered the analysis of individual risky projects within a framework of completeproject independence. This chapter takes up capital investment analysis when we recognizestochastic dependence between project proposals themselves and between project proposalsand the existing assets of the firm.
INTRODUCTION
When investment projects are statistically related to one another and/or to the currently existingassets of the firm, it is no longer appropriate to treat them as independent for purposes ofanalysis. When there is dependence, it is to be expected that we may find that project A ispreferable to project B, even though A has a greater variance and coefficient of variation inits returns than B, and it is therefore more risky as an individual project. The reason for suchresults as these is that project A’s covariance makes it more favorable for selection than doesproject B’s covariance. Up to this point we assumed (or pretended) that all project covariancesto other projects and to the enterprise’s existing assets to be zero, i.e. complete independence.We shall now drop the assumption of zero covariances. And, because the covariance termsare no longer zero, we shall have to take them explicitly into account; we can no longer basedecisions solely on a project’s individual characteristics of risk and return as isolated factors.
To illustrate the effect of nonindependence between a project proposal and the firm’s existingassets, let us examine the following example.
Example 20.1 The owners of Al’s Appliance Store are considering a major expansion that,if undertaken, would mean construction of an attached laundromat and an equipment rentalshop. In that case, the existing building would then become one part of a triplex. The ownershave carefully considered the disparate natures of the businesses and have determined the datain Table 20.1 to be representative.
Although the return on investment is higher, the riskiness of the proposed expansion isgreater in both absolute and relative terms than those of the firm’s existing assets. If we wereto treat the proposal as independent of the appliance store, it might well appear to be more riskythan the business to which the owners are accustomed. Analysis of the proposed expansion inisolation could be carried out with techniques suggested in previous chapters. Here we are
1We assume here that contingent project relationships do not necessarily reflect stochastic dependence. We shall reserve the termstochastic dependence for dependence between the cash flows of projects and not dependencies seen only at the initial accept–rejectdecision. Contingent relationships are often asymmetrical (“accept B only if A has been accepted,” not vice versa), whereas stochasticdependence is generally taken to be symmetrical.
216 Capital Asset Investment: Strategy, Tactics & Tools
Table 20.1 Data for Example 20.1
Proposed attachedAppliance store laundromat and(existing assets) equipment rental shop
Market value/cost $1,000,000 $500,000Expected annual return E[R0] = $100,000 E[R1] = $75,000Variance of annual return σ 2
0 = ($10,000)2 σ 21 = ($15,000)2
Coefficient of variation σ0/E[R0] = 0.100 σ1/E[R1] = 0.200Covariance σ 2
0,1 = −($12,000)2
Table 20.2 Combined asset characteristics,Example 20.1
Value of combined assets $1,500,000Expected combined annual returns E[Rc] = $175,000Variance of combined returns σ 2
c = ($2,333)2
Coefficient of variation σc/E[Rc] = 0.013
not interested in the proposal as an entire, stand-alone investment, but as an addition to aninvestment portfolio that already contains assets. Therefore, it is not appropriate to analyzethe expansion without reference to the combined, portfolio effect that its acceptance wouldbring about.
Individually, the proposal is relatively twice as risky as the appliance store. However, thecovariance between them is negative because it is expected that the proposal will responddifferently to changing status of the economy than will the existing business; in fact, it willrespond oppositely. Table 20.2 contains data for the firm assuming the expansion has beenundertaken.
Note that the relative riskiness as measured by the coefficient of variation has dropped farbelow what it was for the appliance store alone, which individually was less risky than theproposal. The variance of returns on the combined assets2 is less than on the proposal alone oron the existing assets, so the absolute risk is also less than it would be without the expansion.This highlights the benefits to be gained by diversification among assets whose returns tend tomove in opposite directions. The correlation in this case between the appliance store and thelaundromat-cum-rental shop is ρ = −0.96, an almost perfect negative correlation. So perfectis the relationship that 92 percent of the variation in the return on one may be statisticallyexplained by variation in the other (ρ2 = 0.92).
Generalizations
In this example the high negative correlation yields dramatic results. But what if the correlationis not so strongly negative? And, what if, instead of negative correlation, there is positivecorrelation? Table 20.3 contains comparative figures.
2The variance of a linear combination of variables, with weights a, b, c, and so on, is given by
Var (ax + by + cz + · · ·) = a2Var (x) + b2Var (y) + c2Var (z) + · · · + 2ab Cov (x, y) + 2acCov (x, z) + · · · + 2bc Cov (y, z) + · · ·
Multiple Project Selection under Risk: Portfolio Approaches 217
Table 20.3 Risk–return relationships fordifferent return correlations between existingfirm and proposed expansion
ρ Cov(0, 1) σ 2c σc/E[Rc]
−1.0 −($12,247)2 (1,667)2 0.010−0.5 −($8,660)2 (6,009)2 0.034
0.0 0 (8,333)2 0.048+0.5 ($8,660)2 (10,138)2 0.058+1.0 ($12,247)2 (11,667)2 0.067
Comparison of the coefficients of variation for the combined assets with the existing assetsreveals the following:
1. As the correlation coefficient changes from perfect negative correlation to perfect positive,the riskiness of the combined enterprise increases.
2. For all correlations, acceptance of the proposal will reduce relative risk for the combinedenterprise to a lower level than that of the original assets (in this example, the appliancestore).
3. The case of ρ = 0.0 corresponds to a covariance of zero and independence of the proposaland the existing firm.
4. Even with perfect positive correlation (ρ = +1.0) the relative risk for the combined assetsis less than that of the proposed expansion alone.
Project Independence
An important implication of point (3) above is: even if the proposed business expansion wereindependent of the existing assets of the firm, proper analysis cannot be done without referenceto them. The coefficient of variation for the proposal is 0.200. Yet, the coefficient for the firmafter undertaking the investment is 0.013,when ρ = 0.0. Thus, although the expansion itselfis per se more risky than the existing firm, its acceptance would actually decrease the riskof the firm. The point is that if the expansion is viewed separately, it may be rejected becauseit is relatively twice as risky as the firm as it exists, without the investment. If, on the otherhand, the accept/reject decision is based on the riskiness of the firm before the investment tothe firm after the investment it is apparent that relative risk has decreased measurably. Therelevant comparison is not in the terms of the “own” risk of the proposal vis-a-vis the firmwithout the proposal, but rather the firm without the proposal to the firm with the proposal. Thisis not to say that techniques of analysis such as computer simulation of individual projectsshould be rejected, but that their use must be appropriately modified to take the foregoinginto account even when the project is statistically independent of the existing assets of theenterprise.
Project Indivisibility
Portfolio selection theory as it is applied to securities, generally assumes that the investmentin any particular asset is finely divisible. This means that the percentage of the portfolio
218 Capital Asset Investment: Strategy, Tactics & Tools
committed to a particular asset can be any amount between 0 and 100 percent. In the caseof securities such an assumption is reasonable because discontinuities are relatively small,amounting to no more than the smallest investment unit. With capital investments, however,the assumption of finely divisible investments is seldom, if ever, justified. In the foregoingexample the question was not what proportion of the proposal should be invested in between0 and 100 percent inclusive, but whether the investment would be rejected (0 percent) oraccepted (100 percent). Only the extremes were considered because the project was consideredindivisible.
Project indivisibilities are much more troublesome in capital budgeting than in (stock andbond) portfolio selection. Therefore, techniques that were originally developed for securitiesinvestment portfolio selection must be judiciously modified if correct decisions are to be made.Choices of percentage other than 0 percent (rejection) and 100 percent (acceptance) aregenerally not possible.
MULTIPLE PROJECT SELECTION
The formal model for optimal choice of risky assets for an investment portfolio is generallycredited to Markowitz [101]. What is optimal for one investor will generally be different fromwhat is optimal for another because of different attitudes toward risk-to-return relationships.The basic nature of risk and return within the framework of utility was covered in Chapter 17for individual asset choice decisions. Here we extend the treatment of risk–return trade-off tothe multiple asset case.
Let us denote the overall return from a portfolio, R, as a weighted average of the returns,Ri , with weights xi , as
R = x1 R1 + x2 R2 + · · · xn Rn (20.1)
Each xi is the proportion of the total invested in asset i , and Ri is the expected return fromthat asset. The Ri are random variables and therefore R, the portfolio return, is also a randomvariable. If we denote the expected values of the Ri by µi , the variance of asset i by σi i , andthe covariance of asset i and asset j as σi j , we obtain for the portfolio
E[R] = x1µ1 + x2µ2 + · · · + xnµn (20.2)
Var[R] = x21σ11 + x2
2σ22 + · · · + x2nσnn
+ 2x1x2σ12 + 2x1x3σ13 + · · · + 2x1xnσ1n(20.3)
+ 2x2x3σ23 + · · · + 2x2xnσ2n, · · ·=
n∑i=1
n∑j=1
xi x jσi j
Although the optimum portfolio for any particular individual depends on one’s utility func-tion, we can nevertheless narrow considerably the field of choice. For instance, we can eliminatefrom further consideration those portfolios that should not be selected by any rational personregardless of utility function. For this we use the concept of efficient portfolio.
An efficient portfolio is one for which a higher return cannot be had without incurring higherrisk or, equivalently, a portfolio for which one cannot reduce risk without a corresponding lossof return. A portfolio is inefficient if, by changing the proportions of the assets held, wecan obtain a higher return with no more risk, or reduce risk without sacrificing some return.
Multiple Project Selection under Risk: Portfolio Approaches 219
Figure 20.1 Selection of optimal portfolio of risky assets
Alternatively, an efficient portfolio is the minimum variance portfolio among all portfolioswith the same expected return and it is the portfolio with the maximum expected return amongall with the same variance of return.
Markowitz suggested two steps in finding an investor’s optimum portfolio. First, identify theset of efficient portfolios. Second, select that efficient portfolio that best matches the investor’srisk–return attitude, which maximizes the investor’s utility. The exact portfolio will vary frominvestor to investor, but all of them must be efficient portfolios. Figure 20.1 summarizes theprocedure.
Line AB is the efficient set of portfolios. The Ui represents a particular investor’s set of utilitycurves relating risk to returns.3 The shaded area represents the set of all other portfolios thatcould be formed from the same assets (assuming infinite divisibility). The point of tangencybetween the highest indifference curve (U0) and the efficient set corresponds to the optimumportfolio for this particular investor.
Markowitz considered several different possible forms of utility functions. It was thequadratic form that Farrar [46] employed to determine the coefficients of risk aversion formutual funds; also the quadratic form of utility functions has been widely employed by manyother writers, in no small measure due to its consistency with diminishing marginal utility andother generally accepted aspects of preference theory. The discontinuous utility function usedby Roy [135] provides a linkage to the notion that survival of the enterprise is an importantinvestment objective.
Roy’s utility function was of the form
U [R] = 1 for R ≥ a
= 0 for R < a
which yields expected utility of
E[U ] = 1 · Pr(R ≥ a) + 0 · Pr(R < a)
= 1 − Pr(R < a)
3For investments of the size represented by this portfolio. Chapter 17 suggests that the utility function is related to the size of therequired investment and to risk of ruin as well.
220 Capital Asset Investment: Strategy, Tactics & Tools
for which maximization of utility means minimization of the probability that the portfolio returnwill fall below amount a. Graphically, this corresponds to selecting the portfolio determinedby the tangency of a line through point a with the efficient set in Figure 20.1. Roy’s modelis consistent with the responses elicited in Mao’s survey of executives cited in Chapter 17.Together, they suggest that where the survival of the enterprise itself is a factor in the decisionprocess, the decision-maker may apply a utility function distinct from that which is used incircumstances in which the enterprise’s survival is not in question.
Finding the Efficient Set
Since the first step in the procedure for finding the optimal investment portfolio is to findthe set of efficient portfolios, we require a systematic method. This may be formulated as amathematical programming problem:
Maximize � = E[R] − λ · Var[R] (20.4)
= xµ − λσ 2 (in vector notation)
subject to∑
i
xi = 1.0
or xi ≥ 0 (20.5)
xi = 0 or 1
Lambda (λ) is the coefficient of risk aversion, which we restrict to the range 0 to + ∞.To obtain the efficient set we must solve the above problem in equation (20.4) for differentvalues of λ until we have enough points on the efficient set to define its curve. It is importantto recognize that this model assumes that the investor’s risk attitude is adequately reflectedby the variance–covariance matrix of all candidate investments. The final constraint restrictsus to either accepting a project entirely (x = 1) or else rejecting it entirely. This constraintis not necessary (under most circumstances) when we are considering securities portfolioinvestments. However, most capital investment projects are indivisible; we either accept orreject a particular project; we do not have the option of owning a partial share of it. In securitiesportfolio selection this constraint is replaced by the restriction that the xi (the weights) sum toa value of 1.0 (100 percent).
If we measure the return on an investment by its net present value (NPV), then we canformulate the problem of finding a point in the efficient set as
Maximize F = p1x1 + p2x2 + · · · + pn xn
−λ(x2
1σ11 + x1x2σ12 + · · · + x1xnσ1n
x2x1σ21 + x22σ22 + · · · + x2xnσ2n
...xn x1σn1 + xn x2σn2 + · · · + x2
nσnn)
(20.6)
Multiple Project Selection under Risk: Portfolio Approaches 221
subject to a11x1 + a12x2 + · · · + a1n xn ≤ b1 (20.7)
a21x1 + a22x2 + · · · + a2n xn ≤ b2
...
am1x1 + am2x2 + · · · + amn xn ≤ bm
and for all i, xi = 0 or xi = 1
which, in matrix algebra notation, becomes
Maximize F = p · x − λ(x · V · x) (20.8)
subject to A · x ≤ bxi = 0 or 1
(20.9)
where V is the variance–covariance matrix of returns.A point regarding problem setup that seems to be often ignored in attempts to transfer this
model to capital investments is that existing assets of the enterprise need to be included inthe formulation. In other words, we cannot look solely at the proposals under considerationand their interrelationships, but need to include their relationships to the firm as it is now. Inthe programming framework above, this may be done by treating the existing enterprise as a“project” with zero required investment outlay. If divestiture is not to be considered, it willalso be necessary to include a constraint that requires the existing enterprise to be included inthe optimal solution. It may be interesting, however, to solve again without the constraint todetermine if divestiture would be worthwhile.
This formulation may be compared to that of the problem of project selection under certaintyin Chapter 16. In fact, certainty means that the elements of the variance–covariance matrixare all zero, and the problem specified in equations (20.6) through (20.9) reduces to equations(16.1) and (16.2).
Mao [98] shows how the Lawler and Bell zero–one integer programming algorithm (dis-cussed in Chapter 16) may be modified to solve problems with objective functions of theform in equations (20.6) and (20.8). However, Baum, Carlson, and Jucker showed that theMarkowitz approach, when applied to problems with indivisible projects, “may result in either(1) a solution set that does not contain all solutions of interest to the decision maker or (2) arequirement that the (implicit) utility function describing the decision maker’s preferences bea linear function of the mean and variance of return” [7]. By examining all feasible solutions toa problem solved by Mao (pp. 295–296), Baum, Carlson, and Jucker show that no matter whatthe value of λ, the coefficient of risk aversion, an efficient point is missed by the Markowitz ap-proach. Furthermore, the Markowitz approach to Mao’s problem, although missing an efficientpoint, produced a dominated point:4 “More generally, any efficient point which is not locatedat a corner point of the upper boundary of the convex hull of the complete set of efficient pointswould also be missed” [7, p. 338].
The Baum, Carlson, and Jucker article is significant in that it demonstrates that incorrectresults may be obtained when the Markowitz approach is employed with problems involvingselection of indivisible assets. This is a serious problem requiring suitable modifications to
4A dominated point is one that is inferior to other asset combinations for all values of X , the coefficient of risk aversion.
222 Capital Asset Investment: Strategy, Tactics & Tools
the solution algorithm or alternative methods of solution that find only dominant points, forexample.
Although problems involving complex constraints of the type illustrated in Chapter 16 donot lend themselves to solution by complete enumerations, an enterprise which has only a fewfairly large investment proposals may be able to analyze them quite successfully without relyingon the Markowitz approach per se. Baum et al. found the error in the Mao problem solution byevaluating all 32 possible selections of the 5 candidate projects.5 Full evaluation can get out ofhand quickly, however, as the number of projects considered increases. Nevertheless, with upto 10 projects or so (210 = 1024) evaluation of all combinations of proposals by modern digitalcomputers is not only possible but not particularly costly. After about 10 projects, however,the number of combinations becomes rapidly so great that full enumeration is not possible.In all but the largest enterprises this should not be an insurmountable problem, particularly ifminor projects are aggregated together.
Another possibly rewarding substitute approach might be found in using a modified zero–oneprogramming algorithm, such as that discussed in Chapter 16 to obtain the partial enumerationcombinations that will need to be evaluated in terms of risk in a second step. In other words, if wecan eliminate infeasible combinations before looking at risk, then the number of combinationsto be examined will be reduced, often substantially, particularly where several proposals aremutually exclusive.
The Sharpe Modification
William Sharpe is credited with a modified version of the Markowitz model that has far-reachingimplications for the valuation of assets [142]. Sharpe’s model provides the foundation of thecapital asset pricing model (CAPM) discussed in Chapter 21. The Sharpe model is oftenreferred to as the “diagonal model” of portfolio selection. His model assumes that returns onassets are related only through their correlations with some index. Returns are defined as
Ri = Ai + bi I + µi (20.10)
where I is a random variable denoting an index, and Ai and bi are constants for asset i . Thevariable I is assumed to have a finite mean and finite variance. The µi are random errors due toindependent, external causes, and satisfy the usual least-squares assumptions of zero expectedvalue, finite variance, and independence (zero covariances).
With these specifications any investment can be split into two parts: (1) an investment inAi and µi , the asset’s basic, or unique characteristics and (2) an investment in I , the externalindex. This means that the variance–covariance matrix for n securities plus the external index(n + 1 in total) has zero elements except on the diagonal containing the covariances betweeneach of the n securities and the external index.
Elimination of the covariances between individual assets makes the Sharpe model muchmore amenable to computer solution than the original Markowitz formulation, because thecomputational burden is substantially lessened. It has yet to be determined whether or not theSharpe model suffers from similar shortcomings to those Baum et al. found for the Markowitzformulation when assets are indivisible.
5The number of possible selections is given by 2n , where there are n candidate proposals.
Multiple Project Selection under Risk: Portfolio Approaches 223
Figure 20.2 Selection of optimal portfolio of risky indivisible assets
RELATING TO INVESTOR UTILITY
After determining the set of efficient assets for various6 coefficients of risk aversion (λ), theactual combination of projects selected will depend on the decision-maker’s utility function.Graphically, the optimum project selection is that which just touches the indifference curve(that is, constant utility curve) corresponding to the highest utility.
In the case of infinitely divisible assets as depicted in Figure 20.1, the point of tangencyequates the marginal rates of substitution of the efficient set curve7 and the tangent indiffer-ence curve. For indivisible assets, the optimum efficient point is the one lying on the highestindifference curve. Figure 20.2 illustrates this.
In practice, it may not be necessary to estimate the family of indifference curves; in fact,most managements will insist on making the final decision on which efficient set of capitalinvestments will be undertaken. There is a marked tendency for executives to resist whatthey interpret as arrogation of their authority and responsibility by technicians armed withcomputers. It may suffice to submit to those who have the decision authority only the fewefficient alternative combinations that are not dominated by others. This will generally meanconsidering which one, two, or few projects will be either removed or entered into the finalselection, because alternative efficient collections are often similar over a rather wide range ofλ values, differing by only few individual included or excluded projects.
EPILOGUE
Subject to qualifications resulting from the article by Baum, Carlson, and Jucker, the quadraticprogramming approaches to asset selection used by Markowitz and by Sharpe yield usefulinsights into the process by which risk-averse investors select investment portfolios. Extensionfrom the domain of securities analysis, where individual investments may be considered tobe finely, if not infinitely, divisible to the domain of indivisible capital project investments,can yield incorrect results. This appears, however, to be more a mechanical problem withimplementation of the procedure than it is a problem with the theoretical concepts. As such itis a flaw that should be amenable to correction by modification of the algorithm.
6If we knew λ in advance, we could solve directly for the optimal portfolio.7The curve defined by all efficient points is called the “efficient frontier.”
224 Capital Asset Investment: Strategy, Tactics & Tools
Of a more fundamental nature are questions arising from concerns relating to the choiceof measure of return that is used as well as questions about the adequacy of the variance–covariance relationship to reflect all relevant risk–return attitudes, although numerous authorshave defended this latter factor. Still further unresolved concerns stem from the nature of thetrade-off function relating uncertainty and futurity, from considerations of abandonment priorto final project life maturity, and from complex intertemporal contingencies between today’sprojects and future candidate projects.
Estimation of project variances and covariances is a difficult problem. And the question ofsolution sensitivity to misspecification or misestimation of the variance–covariance structureand the stability of these parameters over time has not been resolved to everyone’s satisfactionfor securities investments, not to mention capital expenditure proposals.
These concerns and cautions have not been raised to disparage application of portfolioselection techniques to capital investments, but rather to alert the reader that there are unresolvedand controversial matters that tend to be ignored by the more zealous advocates of portfolioselection techniques to capital investment expenditure analysis.
21
The Capital Asset Pricing Model
Early in this book, the capital asset pricing model (CAPM) was mentioned in relation to thefirm’s cost of equity capital. In Chapter 18 it was mentioned again in discussion of the criticismof computer simulation of individual capital investment proposals. In this chapter we examinethe CAPM more closely, to see how it may enable the firm to rationalize the risk–return trade-off and how it may assist in estimating the firm’s cost of equity capital and in assessing theinteraction between capital investment and cost of capital.
The purpose of this chapter is to present the CAPM as a means for dealing with risk incapital investment. In keeping with this purpose, the model is examined critically in terms ofhow well it facilitates making better decisions, as well as problems that affect application ofthe model to capital investment projects.
ASSUMPTIONS OF THE CAPM
The CAPM is based on several of the following assumptions, some of which do not conformwell to the reality of capital investment projects:
1. Investors are risk-averse maximizers of the expected utility based on wealth at end-of-holding period.
2. Expectations about asset returns are homogeneous: everyone agrees on the probabilitydistributions governing returns.
3. Assets are fixed in number, marketable, and infinitely divisible.4. A risk-free asset exists and investors may borrow or lend without limit at the rate of return
paid by the risk-free asset.5. The market is perfect: no taxes, no transactions costs, no restrictions.6. All investors have the same planning horizon and holding period.7. Investors do not distinguish between sources of returns. They are indifferent between divi-
dends and interest of equal dollar amount.
THE EFFICIENT SET OF PORTFOLIOS
In Chapter 20 portfolio selection techniques were examined. They require that we first findthe set of efficient portfolios, those that cannot be improved upon in this sense: an efficientportfolio is one that has maximum expected return for a given level of risk or, alternatively,has minimum risk for a given expected rate of return. In other words, if a portfolio is efficient,we cannot improve upon it by finding another portfolio that has the same (or greater) returnand lower risk or the same (or less) risk, and greater return.
Since the investor is assumed to be risk averse and utility maximizing, and since greaterutility is associated with greater returns, the only portfolios he or she is interested in are efficientportfolios. In Figure 21.1 the section of curve II′ above point 0 is the efficient frontier. Points
226 Capital Asset Investment: Strategy, Tactics & Tools
Figure 21.1 The efficient set and investor portfolio choices
on II′ below point 0 represent nonefficient portfolios because for the same risk each has analternate providing higher expected return on the segment above.
Portfolio Choices
The risk-averse, utility-maximizing investor that we have assumed can be interested onlyin efficient portfolios. It remains for us to determine exactly which efficient portfolio isoptimum for that investor. In Chapter 20 this was found to be the point of tangency betweenthe efficiency frontier and the investor’s highest indifference curve. These indifference curvescontain risk–return combinations which, to the particular investor, offer constant utilityalong a given curve. In Figure 21.1 the portfolio choices of three investors (A, B, and C) aredepicted by points 1, 2, and 3. Investor C may be considered the most aggressive of the threebecause point 3, although corresponding to higher returns than the others, is also associatedwith proportionately greater risk. By similar reasoning, investor A may be considered themost conservative of the three. Minimum risk is at point 0, but the return of the portfoliocorresponding to that point is also at a minimum.
So far, we have not admitted the existence of a risk-free asset into our portfolio considerations.At this point we shall assume such an investment exists, and examine implications.
Enter a Risk-free Investment
The foregoing assumed that the only assets that existed were risky. Now we assume thereis a risk-free asset, paying a certain rate of return that is denoted by R f . Because federalgovernment securities are risk free in this sense, and they are financial instruments, we mightalternatively use i to denote the fixed, certain rate of return. Those who are totally averse torisk would be expected to purchase only the risk-free asset for its certain return. Others wouldpurchase combinations of the risk-free asset and the market portfolio.
The risk-free assumption, it should be noted, requires several subsidiary assumptions.Although the payment of a fixed, periodic money interest on federal government bondsis reasonable (except possibly in time of war or revolution that may bring to power newgovernment that will repudiate the debt), it raises questions. Is the rate of return in nominalterms or in real terms? What about capital gains or losses brought about by changes in the
The Capital Asset Pricing Model 227
Figure 21.2 Risk-free asset, market portfolio, and capital market line (CML)
market rate of interest? If there can be capital gains, then the rate of return is not fixed andcertain.
In order for the risk-free return to be in fact risk free, or certain, we need to require either that(1) we measure nominal returns, unadjusted for inflation or (2) we have constant inflation orknow the rate of inflation over the holding period, or (3) the risk-free security is a floating ratebond that pays a certain real return, or (4) the holding period is short enough that the effects ofinflation can be ignored. If both the risk-free return and the market portfolio return are equallyaffected by inflation, the problem diminishes and it is not necessary to do more than insurethat returns are measured on the same basis in either nominal or real terms.
The problem posed by capital gains may be resolved in either of two ways. We can addthe assumption of costless information, which is sometimes listed as one of the requiredassumptions of the CAPM, or we may instead assume a constant interest rate over the holdingperiod or else a short holding period over which interest-induced capital gains and losses wouldbe minimal, such as with three-month Treasury bills, for example.
For our purposes we need not be overly concerned about these questions about the risk-freeasset. However, because such questions tend to arise from students of the CAPM, this seemsan appropriate place to bring them into the open. We shall not dwell on this matter further, butcontinue, assuming that the risk-free rate is indeed risk free.
A risk-free asset in addition to the efficient set means that now every investor can have aportfolio composed of the risk-free asset plus an efficient portfolio of risky assets. Figure 21.2illustrates this. Any point on the straight line may be obtained by the appropriate linear com-bination of the risk-free asset and the efficient portfolio at the point of tangency. If q is theproportion of the investor’s wealth invested in the risk-free asset, then 1 − q is the proportioninvested in the tangent portfolio, and 0 ≤ q ≤ 1.
The line from R f tangent to the efficient set is called the capital market line (CML). Nolonger are investors content to hold only risky assets in efficient portfolios. Now they want tohold some amount of the risk-free asset in combination with an efficient portfolio. And theone efficient portfolio they will hold in such combination is the market portfolio. Every otherportfolio is inferior, because now investors can achieve higher returns for a given level of risk.In terms of indifference curves this means that such curves will not be tangent to the efficientset, but rather will be tangent to line R f R′
f that lies above the efficient set at every point exceptM, the point of tangency corresponding to the market portfolio.
228 Capital Asset Investment: Strategy, Tactics & Tools
The market portfolio must include every risky asset; otherwise the prices of some wouldrise and others fall until all were included, and all assets must be owned by someone. A changein the risk-free return vis-a-vis the return on the market portfolio means that a new pointof tangency will exist and risky asset prices will adjust until a new equilibrium is reached,corresponding to a new market portfolio.
The line segment of R f R′f above point M corresponds to margin purchases of the market
portfolio. In other words, investors who seek a higher return, albeit with higher risk, can borrowat the risk-free rate (by assumption) and invest the proceeds in the market portfolio. Therefore,linear combinations of the risk-free asset and the market portfolio are all that investors willwant to hold above point M as well as below it. Again, above point M as well as below it,investors will want to stay on line R f R′
f because it offers greater utility.The equation of the capital market line R f R′
f is
E[RM ] = R f + λσM (21.1)
or if we can denote E[R] byR:
RM = R f + λσM (21.2)
The line slope, λ, is the “price of risk.” The risk-free rate and market expectations may changeover time and, accordingly, the price of risk may change as a new equilibrium is reached.Equilibrium is assumed to exist now and to reflect expectations over one period into the future.Lambda (λ) is defined by
λ = RM − R f
σM(21.3)
Thus the price of risk is not simply the difference between the price of the risk-free asset andthe expected return on the market portfolio, but must be adjusted for the standard deviation ofthe portfolio return.
THE SECURITY MARKET LINE AND BETA
The equilibrium conditions for efficient holdings of a risk-free asset and the market portfolioare given by the capital market line. But we are also interested in the return on inefficientholdings, whether portfolios or individual assets. Every asset is held in the market portfolio, aswe have previously stated; otherwise its price would change until it was included. The capitalmarket line provides a measure of the price of risk for the overall market portfolio. But whatis the risk–return trade-off for a given asset or inefficient asset portfolio?
By equating the “price of risk” with the slope of the efficient frontier at point of tangencyM (Figure 21.2), we obtain
RM − R f
σm= E[R j ] − RM(
σ j M − σ 2M
)/σM
(21.4)
from which can be obtained the relationship usually called the capital asset pricing model(CAPM):
E[R j ] = R f + σ j M
σ 2M
[RM − R f ] (21.5)
The Capital Asset Pricing Model 229
which is usually written in the form
R j = R f + β j [RM − R f ] (21.6)
where beta (β) is defined as
β j = Cov(R j , RM )
Var(RM )= σ j M
σ 2M
(21.7)
Beta can be thought of as the relative volatility of the jth asset and may be estimated (if dataare available) by fitting an ordinary least-squares regression to the equation:
R j = α j + β j RM (21.8)
Several investment services publish calculated betas for US common stocks. However, becausetrue market returns would have to be based on a market portfolio including every asset weightedaccording to its importance, and because this would be impossible to achieve and prohibitivelycostly to try to approach, the “market” portfolio is in fact a portfolio of New York Stock Ex-change listed common stocks or some more restricted list such as the Standard and Poor’s 500.
If we define total risk of any asset as systematic risk plus unsystematic risk, then betarepresents the systematic risk. The actual return on any asset j is given by
R j = α j + β j RM + µ j (21.9)
with variance
σ 2j = β2
j σ2M + σ 2
µ (21.10)
where µ represents an independent random error term, and R j and RM are random variables.Since in (21.10) the variance is the total risk, the right-hand side contains the two component
risks. The unsystematic risk is σ 2µ, and this can be eliminated by diversification. Because un-
systematic risk can thus be eliminated, the market will not pay to avoid it. However, systematicrisk is another matter entirely. Investors will pay to avoid systematic risk, and in equilibriumevery asset must fall on the security market line. Figure 21.3 illustrates the security marketline. The market portfolio itself has β = 1, as indicated.
Figure 21.3 The security market line (SML)
230 Capital Asset Investment: Strategy, Tactics & Tools
Figure 21.4 Empirical security market line
Although the capital market line relates a risk-free asset to the market portfolio, the securitymarket line relates risk and return for individual securities. These same individual assets areincluded in the market portfolio. Each has its own systematic, nondiversifiable risk associatedwith it and therefore its own rate of return that the marketplace will require of it. If the returnis higher than necessary, investors will want to add more of it to their holdings, bidding up theprice until, in equilibrium, it lies on the security market line. On the other hand, if the returnis lower than necessary, holders will want to sell and the price will decline until it rests on thesecurity market line.
If we plot data points for risk (that is, beta) and return on securities, we will obtain ascattergram such as that shown in Figure 21.4, to which an empirical security market linemay be fitted by least-squares regression. We shall not address here the question of how wellhistorical data reflect expectations about future performance, the stability of β, and so on.These significant questions have been considered by many researchers, and discussion herewould put us on a lengthy detour from the main track.
An important property of the CAPM stated here without proof is: the β of a portfolio is alinearly weighted average of the individual constituent assets. If a portfolio is composed of xpercent asset A, y percent asset B, and z percent asset C, then the portfolio β is
β = xβA + yβB + zβC (21.11)
This is a very important result because it means that the risk of a portfolio may be foundwithout resort to quadratic programming to determine the efficient set.
THE CAPM AND VALUATION
The CAPM (through the security market line) provides the relationship between risk andrequired return imposed by the interactions of all investors in the market. Expected returnis determined by the price paid for an investment. The return is determined as the ratio ofearnings return (dividend or interest) plus (or minus) the price change in the asset over theholding period to the price paid for the asset. If we assume a single-period holding, the return
The Capital Asset Pricing Model 231
is then
R1 = D + (P1 − P0)
P0(21.12)
where D is the income return component and (P1 − P0) the capital gain or loss. Combining Dwith P1 and calling it P ′
1, this becomes
R1 = P ′1 − P0
P0(21.13)
Equating this to the CAPM of equation (21.5) and taking expectations of both sides, we get
P ′1 − P0
P0= R f + σ j M
σ 2M
[RM − R f ] (21.14)
This may be rearranged to obtain P0:
P0 = P ′1
1 + R f + (σ j M/σ 2
M
)[RM − R f ]
(21.15)
or
P0 = P ′1
1 + R f + (λ/σM )σ j M(21.16)
where λ was defined in equation (21.3) as the price of risk. Note that this corresponds to therisk-adjusted discount rate discussed in Chapter 18. The certainty equivalent formulation maybe obtained from (21.16) by substituting for σ j M the equivalent (1/P0)σP1 RM :
P0 = P ′1 − (λ/σM )σP1 RM
1 + R f(21.17)
THE CAPM AND COST OF CAPITAL
Early on we stated that the cost of capital to the firm is the required rate of return. The costof equity is the rate required by common shareholders, the cost of debt is the rate required bycreditors, and so on. Therefore, we can write k for E[R j ], where k is the required rate of return.Then equation (21.6) becomes
ke = R f + βe[RM − R f ] (21.18)
for the cost of equity capital, ke, where βe is the systematic risk of equity in the particular firm.The weighted average cost of capital may be found as illustrated in Chapter 4 once the
component costs have been obtained. Dropping the assumptions used to develop the CAPM,which mean that debt is debt, and therefore risk free, how can the cost of debt be found?In principle the CAPM could be used, but how does one estimate beta for a firm’s bonds,particularly for a firm that has never defaulted? The cost of debt may be observed directly inthe market for publicly traded bonds or estimated from market data on those that are. Companydebt will always be required to yield more than federal government debt because the formeris not risk free, whereas federal government debt is as close as we can come to approximatinga risk-free security.
232 Capital Asset Investment: Strategy, Tactics & Tools
Figure 21.5 The SML and capital investment
THE CAPM AND CAPITAL BUDGETING
One of the results yielded by the CAPM is that once we determine an asset’s risk we willknow the required rate of return it must yield. This being the case, if we can estimate thesystematic risk of the capital investment proposal, we can apply the CAPM. Estimation of betafor a capital investment is a formidable task because we normally do not have a foundationof historical data on this or similar projects from which we might comfortably estimate theproject’s beta. Assuming we have obtained a beta, we could apply the risk-adjusted discountrate or the certainty equivalent method to find the NPV. Or, as illustrated in Figure 21.5, wemay employ the security market line to determine whether or not individual capital investmentprojects should be undertaken.
When we evaluate a capital investment proposal that has the same risk as the firm as a whole,we can correctly use the firm’s weighted average cost of capital. However, for projects whoserisk is either greater or less than that of the firm, it is not appropriate to use the weightedaverage marginal cost of capital.
In Figure 21.5 proposal B has a higher expected return than the overall firm. Should theproject be accepted? No. The return is greater but the risk is greater yet. In fact, because pointB lies below the security market line, we can say that its risk is too great in relation to itsexpected return, or equivalently that its return is too low in relation to its risk. Proposal A hasa expected return less than the firm. But its risk is also less. In fact, because point A lies abovethe security market line, its return is greater than it need be for its level of risk or its risk lessthan it need be for the expected return. For projects having different systematic risk than thefirm as a whole, the CAPM provides a rationale for accepting or rejecting them.
COMPARISON WITH PORTFOLIO APPROACHES
Two major theoretical approaches to evaluation of risky, interrelated projects are those ofportfolio diversification along the lines pioneered by Markowitz (1962) and the approachimplied by the CAPM.
The Capital Asset Pricing Model 233
The Markowitz approach requires that we solve a programming problem with a quadraticobjective function. To do so we need the matrix of the variances and covariances of all assetsthat may be included in the portfolio. Then, once the quadratic programming problem has beensolved, we must either solve again repeatedly until we have the efficient set or efficient frontierand then find a point of tangency with the investor’s utility function or, alternatively, we mustfind the investor’s coefficient of risk aversion at the outset, using this in the quadratic programto solve once for the optimum portfolio.
With the Markowitz portfolio approach we must do considerable computation after obtaininga burdensome amount of data. There may be serious questions about the accuracy of variance-covariance data and utility function estimates as well. Assuming that we have obtained accuratedata and solved the quadratic programming problem, the result is a selection of proposals thatmay be unique to the utility function used or the coefficient of risk aversion. Different utilityfunctions may be expected to yield different optimal portfolios.
In contrast, the CAPM requires only that we know the security market line and the ex-pected return and the beta of any given investment. Since beta measures the systematic ornondiversifiable risk of an asset, other risk is assumed to be unimportant because (in theoryat least) it can be eliminated by suitable diversification. In the case of the CAPM, the marketdetermines, through the security market line, what is an acceptable investment and what is not.In the Markowitz approach, the question of whether or not a particular project is acceptableis internal to the enterprise. It depends on the existing firm, the array of candidate projects,and a utility or investment function. The Markowitz approach assumes there are benefits to begained through active diversification. The CAPM suggests, however, there is no advantage todiversification by the firm, although it implicitly requires that diversifiable risk is eliminatedthrough appropriate portfolio construction.
SOME CRITICISMS OF THE CAPM
The question we need to be concerned with in a capital investment text is how appropriatea model or technique is to capital investments. The CAPM was developed, and its validitytested, for common stocks. Can it be transferred to capital-budgeting applications without lossof validity?
There are some major differences between publicly traded securities and capital investments.Such securities are highly liquid, whereas capital investment projects often do not have asecondary market other than for scrap. Securities approach the CAPM assumption of infinitedivisibility much more closely than most capital investment proposals. There is usually littleinformation available to the public about capital investment projects, much less a consensusabout expected return and risk.
Commodity markets meet the conditions assumed by the CAPM much more closely thancapital investments. Yet, in an empirical test of the CAPM, Holthausen and Hughes concludedthat the CAPM may not fit commodity markets as well as it does security markets. Theyalso observed that returns on commodities do not seem closely related to the measures ofnondiversifiable risk they employed (that is, the betas) [70].
Rendleman points out that although the CAPM provides a basis for valuing securities ina perfect and efficient capital market, there are problems in attempting to use it in capitalbudgeting. He examines the application of the CAPM under capital rationing, concluding thatit is not appropriate “for a firm to use its own beta when computing the expected excess returnof a project with risk characteristics identical to the firm itself” [129, p. 42].
234 Capital Asset Investment: Strategy, Tactics & Tools
Myers and Turnbull showed that under certain circumstances a capital project’s beta willbe a function of the growth rate of the cash flows and the project life. They suggest it maynot be possible to obtain betas that truly represent the systematic risk for the firm’s cash flows[115].
Levy was concerned with the use of the CAPM in public utility regulation. Although hisremarks are addressed to utility company securities, they apply also, perhaps more strongly,to use of the CAPM for capital budgeting. His comments are:
1. If a public market for the utility’s stock does not exist, the beta cannot be computed.2. Similarly, if a list of diversified companies were determined (on qualitative grounds) to be
equivalent in risk to a particular nontraded utility, the average standard deviation of earningsof firms in the list could probably be used as an estimate of the standard deviation for theutility; but the same inference might not be appropriate regarding the beta.
3. At least two studies . . . find that investors receive some incremental return for incurringdiversifiable risk.
4. Unless the independent variable in the regression equation is fully diversified to includebonds, real estate and other investments, the beta coefficient will not properly distinguishbetween diversifiable and nondiversifiable risk.
5. Large disparities can exist in the beta coefficients for individual stocks when differentcomputational methods are employed [such as time span].
6. Independent studies reveal marked instability of [calculated] betas over time.7. [Studies] indicate that returns on high beta stocks were lower than would be expected and
returns on low beta stocks higher . . . the expected relationship between betas and individualstock returns prevailed less than half the time.
8. . . . tests will show, almost invariably, that the beta coefficients for individual stocks are notsignificantly different in a statistical sense from the general market beta of 1.00 [88].
Fama demonstrated that the future riskless rate, the market price of risk, or the elasticityof a capital project’s expected cash flows with respect to the market return must be certain.Otherwise the CAPM cannot be properly employed. The only parameter that can be uncertainthrough time is that of the cash flows themselves [44].
THE ARBITRAGE PRICING THEORY (APT)
In recent years the CAPM has yielded to the notion of the arbitrage pricing theory. The CAPMprovides a theoretically simple model for measuring and matching risks to expected returns.It affords a way of determining the return that in the view of the overall market is appropriateto the level of risk, and vice versa.
The CAPM postulates that the relevant, or nondiversifiable risk of an asset is fully measuredby its sensitivity to the risk premium on the market portfolio; that is, to (RM − R f ). Thatsensitivity is measured by the asset’s beta (β). While the CAPM is useful, it fails to explainwhy there appear to be persistent differences in common stock returns from influences arisingfrom the industry a firm is in, its size, the term structure of interest rates, and so on. And, ithas been suggested by Roll [133] that the CAPM cannot be adequately tested empirically.
The fundamental idea behind the APT is that there are several influences that togetherdetermine the return on a security or other investment, and also the risk. These influences arecalled factors and have a particular statistical meaning and interpretation.
The Capital Asset Pricing Model 235
The general awakening of interest in a multiple factor model by financial economistsfollowed Ross [134], who provided a theoretical foundation that earlier multiple factor modelslacked.
Factors — What Are They?
In multivariate statistics there are two closely related techniques called principal componentsanalysis and factor analysis. Both are based on the notion that observations on real, observable,economic, or other variables can be viewed as being a weighted sum of unobservable variablescalled principal components, or factors. Alternatively, the principal components or factors maybe viewed as being a weighted sum of the observable variables.
In principal components analysis, if there are j variables then it will require j principalcomponents to account for the total variance [9]. The jth principal component accounts forthe jth greatest variance. A small number of principal components may account for most ofthe original variance, but all j of the principal components are required to fully account forthe variance. The principal components are linear combinations of the original, observablevariables.
In common factor analysis the model is based on the premise that there are fewer factors thanthe original j variables. However, the factors may not account for as much of the variance asthe same number of principal components, since the common factors are linear combinationssolely of the common parts of the variables.
Both techniques have been applied to the APT. However, factor analysis is preferred. Factoranalysis is directed toward the correlation, or covariance between the original variables. Prin-cipal components analysis is directed toward the variance of the variables, not the commoninfluence between variables. Since the APT postulates influences common to many assets in themarket, researchers thus normally should use the technique that better probes for such commoninfluences in preference to one that looks for something that might be unique to each asset.
The idea behind the techniques is perhaps best grasped by seeing how they apply graphically.Consider Figure 21.6, which shows a scatter of points in an axis system with returns to assetA on the vertical and returns to asset B on the horizontal. The two assets have returns that
Figure 21.6 Two assets, two factors
236 Capital Asset Investment: Strategy, Tactics & Tools
are related, and that is why the scatter of points is elliptical and not circular. Perfect positivecorrelation would, of course, appear as a straight line rising at a 45-degree angle from theorigin; perfect negative correlation would appear as a straight line falling from left to right ata 45-degree angle.
The longer axis fitted to the ellipse containing the returns accounts for the greater proportionof the common variance in returns between the two assets. The shorter axis accounts for the restof the common variance since there are only two assets. Points on the two axes that correspondto the original pairs of returns, such as pair x, coincide with an “observation” on a factor orprincipal component. Point q is the value of factor 1, point r the value of factor 2. Because theprojection of the first (i.e. principal) axis on the RB axis is longer than that on the RA axis, weobserve that it is more closely related to asset B’s returns. Factor 1 is said to load more heavilyon asset B.
This graphical model may be extended to three dimensions. Beyond three dimensions themathematical techniques can deal with the calculations even though there may be no meaningfulgraphical exposition of the hyperellipsoids to which an axis system is being fitted.
Multiple correlations between the original observations and the points on the axes providewhat are termed factor loadings. The axis system may be rotated to obtain factor loadings withthe original variables that may facilitate interpretation of the factors. For example, if one wereusing data for interest rates, a high loading of factor n with the yield on US Treasury bonds,corporate AAA-rated bonds, and mortgages, a promising interpretation would be that factor nis a long-term interest rate factor. When a factor loading is very high on an observable variable,we then may use the observable variable in place of the factor, as a surrogate. In searchingfor macroeconomic variables that may be called “factors” such high loadings are what onelooks for.
Rotation may be orthogonal or nonorthogonal. That is, rotation may be done with the axesperpendicular to one another, or nonperpendicular. Principal components are always donewith orthogonal axes. Factor analysis may be done either way, but the nonorthogonal rotationschemes, despite some subjectivity necessary in their use, provide a richer domain of possibleinterpretations.
In comparison to other statistical techniques, factor analysis has not had a large variety oftests of significance developed. A traditional rule is that factors (or principal components)having eigenvalues (that is, factor or component variances) greater than 1.0 are likely to besignificant. The intuitive interpretation of this is that those factors account for more of thecovariance or correlation than any single one of the original variables. The lack of powerfulstatistical tests for determining which, or how many, factors are significant (that is, “priced” inthe market) has probably been central to the controversy that continues over how many factorsaffect the returns of assets, and what interpretations may reasonably be placed upon them.In examining the history of the APT this knowledge will provide a necessary perspective forunderstanding its evolution.
APT and CAPM
In applying the CAPM we fit a regression equation with return on the jth asset as the dependentvariable to one independent variable, return on the market portfolio or index. The APT takesthe process a step further, by adding to the market return the macroeconomic variables that areconsidered to be important in determining the returns on assets. Thus, APT is a multivariateanalog to the CAPM and includes the CAPM as a special case.
The Capital Asset Pricing Model 237
Development of the APT
APT is considered to offer a testable alternative to the CAPM introduced by Sharpe [143].Ross argues that the APT is an appropriate alternative because it agrees with the intuitionbehind the CAPM since it is based on a linear return-generating process in which risk maybe separated into diversifiable and nondiversifiable portions. Unlike the CAPM, however, theAPT is claimed to hold in both multi-period and single period instances, and the “market”portfolio plays no special role in APT.
Major differences between APT and Sharpe’s model are that (1) APT allows for more thanone return-generating factor and (2) because no arbitrage profits are possible in a market atequilibrium, every equilibrium may be characterized by a linear relationship between eachasset’s factors. The easily accepted assumption that no riskless arbitrage profits can exist inequilibrium, given the factor-generating model, leads directly to APT.
Bower, Bower, and Logue (BBL) provide an example application of the APT to a sampleof companies [16]. Though they had worked with a larger sample, the one for which resultswere presented in the article cited here was very heavily composed of firms in the broadcastingindustry. Nevertheless, the results obtained are worth looking at for the general lesson theycontain. The authors used four alleged factors (industrial production, inflation, interest rateterm structure, and the spread between low and high grade bonds) in regressions, along withthe market index return as a fifth macroeconomic variable. They compared the results withthose from regressing the stocks’ returns on the market index return, the CAPM model.
BBL found that the R2 for the APT was 0.36 compared to 0.32 for the CAPM. Taking theregression coefficients they had obtained, they applied them to estimate the required rate ofreturn on each firm’s stock. For the sample they found the APT yielded a lower required rate ofreturn than the CAPM, an average of 18.8 percent versus 23.0 percent. However, the standarddeviation of the required returns was 6.05 percent with the APT, compared to 4.43 percent forthe CAPM.
Since BBL do not state whether or not they used the R2 adjusted for degrees of freedom, it isnot possible to determine from the article if the four additional macroeconomic variables madea genuine contribution or not. For each additional independent variable used in a regressionthere is generally an increase in the R2 due to loss of degrees of freedom. Use of R2 (called“r-bar squared”) takes this into account.
Close examination of BBL’s regression coefficients reveals that with the consistent exceptionof the return on the market index, most of them are very close to zero, even though some of themare statistically significant. The reason for mentioning this is to focus attention on the differencebetween statistical significance and economic, or practical significance. For example, for CookInternational, BBL calculate a coefficient of 0.05 (one of the larger such coefficients), witha t-value of 3.32 indicating high significance, for the third macroeconomic variable (interestrate term structure). The coefficient for the market return in contrast is 0.90 with a t-value of3.12. Thus, the statistical significance of interest rate term structure is greater than that of themarket, but its impact on the required return only 1/18th as great!
If indeed there are only four or five factors, one might be forgiven if he or she looks forsomewhat greater economic significance from each.
22
Multiple Project Selection under Risk
Previously we discussed how to select individual risky investments when they are independentof the existing assets of the firm or were very large relative to the firm’s existing assets. Thecapital asset pricing model (CAPM) considered how to select individual risky assets when theycould be considered related to the existing assets of the firm only through an index of returnson a “market” portfolio. This chapter focuses on how to select risky projects that are relateddirectly to the existing assets of the firm, and not solely through a market index.
Thus, we are going to consider an alternative to the CAPM approach for selecting risky assets.This is appropriate for many capital investments when we do not have a reliable estimate of theirbetas, or have reason to question the appropriateness of the CAPM for them. The CAPM worksbest for large projects, those on the scale of acquisition of other, publicly traded companies.The approach illustrated in this chapter can offer advantages for other projects.
Recognition that risky investment projects are statistically related to one another or to thecurrently existing assets of the firm can facilitate better investment decisions. When risky assetshave nonzero correlation between their cash flows we say the project proposals are (statistically)dependent. When different project proposals have different degrees of dependence in relationto the existing assets of the firm we often find that a project that is individually more risky ispreferable to one that is individually less risky.
The correlation coefficient (ρ, the Greek letter rho) between two sets of cash flows (forprojects A and B) is defined as the ratio of the covariance between the two divided by theproduct of their standard deviations:
ρ = σA,B/(σAσB)
The covariance between the cash flows of A and B is analogous to the variance of the cashflows of A, or of B. If we denote an individual cash flow of project A by X, and the average,or expected cash flow by µX (Greek letter mu), then the variance is calculated as the expectedvalue of (X − µX )2; that is, the average or expected value of the quantity X less its expectedvalue squared. If we denote an individual cash flow of project B by Y, then the covariance isdefined as the expected value of (X − µX )(Y − µY ). The value of ρ2 (rho-square) measuresthe proportion of the variability in A that is statistically accounted for by the variability in B.And, since we measure risk as variability, a ρ2 of 1.00 tells us that the cash flows are perfectlyrelated, while a ρ2 of −1.00 tells they are perfectly but oppositely related.
Up to this point we have assumed that covariances, and thus the correlations, betweenprojects and the enterprise’s existing assets were zero, that is, they were completely indepen-dent. Now we will drop the assumption of independence and take explicitly into account thecovariance terms. We will no longer look at decisions based solely on a project’s individualcharacteristics of risk and return as isolated factors, or solely on its relationship with a marketindex.
To illustrate the effect of nonindependence between a project proposal and the firm’s existingassets the following example will be useful.
240 Capital Asset Investment: Strategy, Tactics & Tools
Table 22.1 Al’s appliance store, etc.
Existing appliance store Proposed laundromat, etc.
Market value (or cost) $1,000,000 $500,000Expected annual return 100,000 75,000Variance of annual return (10,000)2 (15,000)2
Coefficient of variation 0.100 0.200Covariance –($12,000)2
Table 22.2 Combined asset characteristics, Example 22.1
Value of combined assets $1,500,000Expected combined annual returns $175,000 = E[Rc]Variance of combined returns ($2,333)2 = 5.443E + 06 ($2)Coefficient of variation 0.013Correlation coefficient −0.960
Example 22.1: Al’s Appliance Shop, Revisited In Chapter 21 we considered a problemfaced by the managers of Al’s Appliance Arcades, Inc. That chapter used the CAPM framework.Here we will look at it again, but this time in light of the variance, covariance characteristicsof the projects and the firm. As we return to the scene, the owners of Al’s Appliance Arcades,Inc., are again considering a major expansion that would mean construction of an attachedlaundromat and an equipment rental shop. The existing building would become one part ofa triplex. Management has investigated the natures of the businesses and has determined thedata in the following table to be representative. The statistical measures were calculated fromthe historical cash flows of the existing business and data for a laundromat similar to the onethat is being considered for investment. Table 22.1 summarizes the data.
Although the return on investment is higher, the riskiness of the proposed expansion isgreater in both absolute and relative terms than those of the firm’s existing assets.
Analysis of the proposed expansion in isolation could be carried out with techniques sug-gested in previous chapters. That would be appropriate for an investor who did not have otherbusinesses besides the proposal. But here we are not interested in the proposal as a solitaryinvestment holding. Rather, it would be an addition to an investment portfolio that alreadycontains assets.
Individually, the proposal is relatively twice as risky as the appliance store, as indicated bythe respective coefficients of variation, which are defined as the standard deviations dividedby the respective expected returns. However, the covariance between them is negative. Thisindicates that the proposal will tend to respond not only differently, but oppositely, to changesin the economy than the existing business. Table 22.2 contains data for the firm assuming theexpansion has been undertaken.
Note that the relative riskiness as measured by the coefficient of variation has dropped farbelow what it was for the appliance store alone, which was less risky than the proposal. Thevariance of returns on the combined assets1 is less than that of either the proposal alone, or
1When we measure returns and variances in dollars, the variance of a combination of assets is given by
Var(x + y + z + . . .) = Var(x) + Var(y) + Var(z) + . . . + 2 Cov(x, y) + 2 Cov(x, z) + . . . 2 Cov(y, z) + . . .
Multiple Project Selection under Risk 241
Table 22.3 Risk–return relationships for different return correlationsa
between existing firm and proposed expansion
Correlation Cov(0,1) Var(combination) Coefficient of variation
−1.00 −1.5000E + 08 2.5000E + 07 0.029−0.90 −1.3500E + 08 5.5000E + 07 0.042−0.80 −1.2000E + 08 8.5000E + 07 0.053−0.70 −1.0500E + 08 1.1500E + 08 0.061−0.60 −9.0000E + 07 1.4500E + 08 0.069−0.50 −7.5000E + 07 1.7500E + 08 0.076−0.40 −6.0000E + 07 2.0500E + 08 0.082−0.30 −4.5000E + 07 2.3500E + 08 0.088−0.20 −3.0000E + 07 2.6500E + 08 0.093−0.10 −1.5000E + 07 2.9500E + 08 0.098
0.00 0.0000E + 00 3.2500E + 08 0.1030.10 1.5000E + 07 3.5500E + 08 0.1080.20 3.0000E + 07 3.8500E + 08 0.1120.30 4.5000E + 07 4.1500E + 08 0.1160.40 6.0000E + 07 4.4500E + 08 0.1210.50 7.5000E + 07 4.7500E + 08 0.1250.60 9.0000E + 07 5.0500E + 08 0.1280.70 1.0500E + 08 5.3500E + 08 0.1320.80 1.2000E + 08 5.6500E + 08 0.1360.90 1.3500E + 08 5.9500E + 08 0.1391.00 1.5000E + 08 6.2500E + 08 0.143
aNote that the covariance between variable 0 and variable 1 may be found from thecorrelation (R) and the standard deviations as Cov(0, 1) = ρσ0σ1.
that of the existing assets. Therefore, the absolute risk is also less than it would be withoutthe expansion. This highlights the benefits to be gained from diversification. The results ofthe combination are particularly attractive for assets whose returns tend to move in oppositedirections; the correlation in this case between the appliance store and the laundromat-cum-rental shop is ρ = −0.96, an almost perfect negative correlation. Thus, ρ2 = 0.92 and thismeans that 92 percent of the variation in the return on one is statistically accounted for byvariation in the other. In this example, the correlation is computed from
(−12,000)2/(10,000∗15,000) = −(144,000,000)/(150,000,000) = −0.96
Generalizations
In this example the high negative correlation yields dramatic results. But what if the cor-relation were not strongly negative? What if, instead of negative correlation, there is posi-tive correlation? Table 22.3 contains a range of comparative figures. Figure 22.1 illustratesthe relationship between the correlation coefficient and the coefficient of variation betweenthe existing appliance store and the proposed laundromat. From this we can make severalgeneralizations.
Note: If we were working with percentage returns we would have to use a formula that includes the weights (relative proportions) ofthe assets in the portfolio. Thus, the variance of a combination of assets, with weights a, b, c, and so on, is given by
Var(ax + by + cz + . . .) = a2Var(x) + b2Var(y) + c2Var(z) + . . . + 2ab Cov(x, y) + 2ac Cov(x, z) + . . . + 2bc Cov(y, z) + . . .
242 Capital Asset Investment: Strategy, Tactics & Tools
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Coe
ffic
ient
of
Var
iatio
n
−1 −0.5 0 0.5 1Correlation
Figure 22.1 Al’s Appliance Store correlation vs coefficient of variation
1. As the correlation coefficient changes from perfect negative correlation to perfect positive,the riskiness of the combined enterprise increases. However, it is less than that for eitherthe existing business or the proposal alone for all negative correlations.
2. The greatest reduction in relative risk occurs from investing in the proposed expansion ifthere is perfect negative correlation between the existing assets and the proposed expansion.
3. The case of ρ = 0.0 corresponds to covariance of zero and consequently statistical inde-pendence of returns from the proposal and those from the existing firm.
The combination of existing assets with the proposed expansion project reduces the risk butnot the expected return. The returns are linearly additive (no terms are raised to an exponentother than 1.0). The risk, as measured by the variances or standard deviations, is not linearlyadditive.
PROJECT INDEPENDENCE — DOES IT REALLY EXIST?
Unless the proposed business expansion is independent of the existing assets of the firm, properanalysis cannot be done without reference to them.2 The coefficient of variation for the proposalis 0.200. A ρ = 0.0 means that the projects are statistically independent. Yet, when ρ = 0.0,the coefficient for the firm after undertaking the investment is 0.103, which is greater thanthe 0.100 it would be without the project. Thus, although the expansion itself is per se twiceas risky as the existing firm, and its returns statistically independent, its acceptance wouldactually increase the relative risk of the combined enterprise, the overall firm, very little. Thepoint is that if the expansion is viewed separately, it would probably be rejected because it isrelatively twice as risky as the firm as it exists now, without the investment. If, on the otherhand, the accept/reject decision is based on the riskiness of the firm before the investment tothe firm after the investment it is apparent that relative risk has increased insignificantly. Therelevant comparison is thus not in the terms of the “own” risk of the proposal vis-a-vis the firmwithout the proposal, but rather the firm without the proposal to the firm with the proposal.
2Therefore, certainty equivalents and risk-adjusted discount rates should be adjusted to reflect this fact.
Multiple Project Selection under Risk 243
PROJECT INDIVISIBILITY: A CAPITAL INVESTMENTIS NOT A SECURITY
In portfolio selection as it is applied to securities, it is generally assumed that the investment inany particular asset is infinitely divisible and, excluding short sales, that the amount committedto any asset lies between 0 and 100 percent. In the case of securities such an assumption isreasonable because discontinuities are relatively small, amounting to no more than the small-est investment unit. With capital investments, however, the assumption of infinitely divisibleinvestment units is seldom justified. In the foregoing example the question was not what pro-portion of the proposal should be invested, between 0 and 100 percent inclusive, but whetherthe investment would be rejected (0 percent) or accepted (100 percent). Only the extremeswere considered because the project was considered indivisible.
Project indivisibilities are much more troublesome in capital budgeting than in (stock andbond) portfolio selection. Therefore, techniques that were originally developed for securitiesinvestment portfolio selection must be judiciously modified if correct decisions are to result.Choices of percentage other than 0 percent (rejection) and 100 percent (acceptance) are gener-ally not possible. Thus, in adapting portfolio theory, which was developed for common stockinvestments, we cannot assume that for capital investments there is the same (relatively) smoothset of risk–return combinations. With two assets, A and B, we either have 0 percent of each,100 percent of each, or 0 percent of one and 100 percent of the other. We do not have the optionof considering investment of x percent of the firm’s assets in A and (100 −x) percent of thefirm’s assets in B.
Figure 22.2 graphs the risk–return combinations available for the two assets we have beenexamining. Clearly the proposed investment, in combination with the existing firm, can offeran attractive result for some possible correlations with the existing firm. The expected returnequals the sum of those for the individual projects. But the risk, as measured by the standarddeviation, would be greatly reduced with negative correlation, not only from the proposedinvestment’s own rather high level, but from that of the firm without the proposal.
In the case of a corporation with widely held ownership, the shareholders may prefer that theCAPM criterion be used. If only a small percentage of their wealth is held in the form of shares
6.00E+04
8.00E+04
1.00E+05
1.20E+05
1.40E+05
1.60E+05
1.80E+05
Mar
ket V
alue
0.00E+00 5.00E+07 1.00E+08 1.50E+08 2.00E+08 2.50E+08
Variance
Combination
Existing
Proposed
Figure 22.2 Risk–return set Al’s Appliance Store, etc.
244 Capital Asset Investment: Strategy, Tactics & Tools
Table 22.4 Variances and correlations
Project 1 Project 2 Noah Zark
Variance–covariance matrixProject 1 0.00004119 0.00000702 0.00000595Project 2 0.00000702 0.00001155 0.00000999Noah Zark 0.00000595 0.00000999 0.00000918
Correlation matrixProject 1 1.0000000 0.3218166 0.3059307Project 2 0.3218166 1.0000000 0.9699338Noah Zark 0.3059307 0.9699338 1.0000000
in the original business, risk reduction for this one firm is not likely to figure as prominentlyin their concerns. They can achieve their own, individual, preferred risk–return combinationsimply by buying or selling stock shares in this and other businesses. But, if the firm is ownedby one individual (or a few persons) the proposal could be accepted based on the risk–returncombination of the merged capital assets.
Example 22.2 Noah Zark In Chapter 21 we illustrated the use of betas to determinewhether an investment should be undertaken. This information was then used to decide whetheror not to accept either or both of two proposed investments.
Noah Zark, Inc. is a major manufacturer of pet foods whose management feels it is timeto significantly expand its operations. Two capital investment proposals are being considered,each with very different characteristics. One project is a fish farm that has been operatingfor several years. The other is a factory that can synthesize protein from the carbohydratesin corn. Management requires a recommendation separate from that based on betas aboutwhether either or both of the projects should be accepted for investment. Table 22.4 con-tains the variance–covariance matrix and correlation matrix for Noah Zark and the proposedprojects.
Note that in Table 22.5 the variance is calculated from the relationship
(9.18E − 6)(10/11)2 + (4.119E − 5)(1/11)2 + 2(5.95E − 6)(10/11)(1/11) = 8.911E − 6
But this is not in dollars. To convert it to dollars we need to multiply by the combined assetvalue squared ($1100)2, to obtain ($$)10.78. Then the standard deviation, the square root ofthe variance, is 0.2985073 percent, or $3.28. This dollar measure is obtainable either as thesquare root of ($$)10.78 or from 0.002985073 times $1100. The expected return is
(4.353%)($1000) + (4.500%)($100) = $48.03
The expected returns and variances for the other combinations are obtained in like manner.The results in Table 22.5 can be used to decide whether or not projects 1 or 2 should
be accepted. Figure 22.3 graphs the expected returns against the coefficients of variationfor combinations of Noah Zark with the proposed capital investments. Comparison of thecoefficient of variation for the Noah Zark firm alone with (1) the portfolios composed of theexisting Noah Zark firm with project 1, (2) the Noah Zark firm with project 2, and (3) the NoahZark firm with both projects 1 and 2, reveals several facts.
Multiple Project Selection under Risk 245
Table 22.5 Noah Zark with projects 1 and 2—risk : return characteristics
Costs Market value
Project 1 Project 2 Noah Zark
Value weights $100 $200 $1,000Dollar returns $4.50 $6.60 $43.53Dollar (squared) $$0.42 $$0.46 $$9.18
variances
Firm with Firm with Firm withCombinations Firm alone project 1 project 2 projects 1 and 2
Variance ($$) $$9.18 $$10.78 $$13.64 $$15.52Standard deviation $3.03 $3.28 $3.69 $3.94
E[$R] $43.53 $48.03 $50.13 $54.63Coefficient of 0.0696 0.0684 0.0737 0.0721
variation
Weights Combinations Values Weights Squared weights
Firm $1,000 0.90909 0.82645Project 1 100 0.09091 0.00826Total $1,100 1.00000Firm $1,000 0.83333 0.69444Project 2 200 0.16667 0.02778Total $1,200 1.00000Firm $1,000 0.76923 0.59172Project 1 100 0.07692 0.00592Project 2 200 0.15385 0.02367Total $1,300 1.00000
4.1500%
4.2000%
4.2500%
4.3000%
4.3500%
4.4000%
3.00 3.20 3.40 3.60 3.80 4.00
Standard Deviation
Exp
ecte
d R
etur
n
Firm with Project 2
Firm with Both 1 and 2
Firm with Project 1
Firm Alone
Figure 22.3 Risk–return combinations: Noah Zark and its investments
First, acceptance of project 1 increases the total return on investment to 4.37 percent perquarter ($48.03/$1100), or a compound annual rate of 18.64 percent (1.04374 − 1) from4.35 percent per quarter and 18.57 percent per year. Second, along with the increased re-turn there is a decline in the relative risk. The coefficient of variation drops from 0.0696
246 Capital Asset Investment: Strategy, Tactics & Tools
for the firm alone to 0.0683 for the firm combined with project 1. Thus, project 1 should beaccepted.
In comparison, project 2 if accepted would decrease the expected return on investment to4.18 percent per quarter ($50.13/$1200) or 17.8 percent per year, and at the same time raisethe relative riskiness, as measured by the coefficient of variation, to 0.0737. There would haveto be some compelling intangibles associated with project 2 for management to consider itfurther. It should not be accepted.
If both projects 1 and 2 were to be accepted, the result would be an expected return of 4.20percent per quarter ($54.63/$1300), or 17.7 percent per year. And the coefficient of variationwould be 0.0721. Since this is inferior to the firm alone, or the firm combined with project 1,project 2 will not be accepted.
The acceptance of both projects was indicated when the CAPM was applied to these sameassets in Chapter 21. However, here we look at the firm and the proposed projects, and all theirinterrelationships, but not at the market. With the CAPM we considered only their relationshipsas manifested through the market, and ignored their covariances with one another. The CAPMis appropriate for companies whose shareholders have well-diversified portfolios, such thatthe market for securities governs their risk–return preferences. However, for companies thathave concentrated ownership, and whose owners are not (and perhaps cannot be) so welldiversified, the approach illustrated in this chapter may be more useful.
This example illustrates that accept/reject decisions for capital budgeting proposals canbe obtained which take risk into account even when suitable estimates of their return char-acteristics vis-a-vis the “market” are not available. The decision may be based on the vari-ance/covariance relationships between the firm and the projects calculated from the returnson each of them alone. Thus we have a useful alternative to the CAPM in situations whereeither the CAPM is considered inappropriate or else a second measure is desired to augmentthe CAPM-based recommendation.
It must be mentioned that a project proposal may have both higher expected return andhigher risk than the existing firm. If the correlation is not such that the combination of theproject with the firm produces a coefficient of variation that is not greater than that of the firm,that will create some indeterminacy. Should the project be accepted if its combination with thefirm produces both a higher return and higher relative risk? This is a question that managementmust answer. Without reference to what the market judges to be the appropriate relationship ofrisk to return, management must apply its collective utility function to the question and decideif the return is sufficiently high to justify the greater associated risk. In the case of the CAPMthe decision could be made without reference to management judgment about the risk–returntrade-off. With the present method management must decide, it cannot delegate the decisionto the market.
GENERALIZATION ON MULTIPLE PROJECT SELECTION
The formal model for optimal choice of risky assets for an investment portfolio (generally calledthe Markowitz (1962) model) suggests that what is optimal for one investor will generally bedifferent from what is optimal for another because of different attitudes toward risk–returnrelationships. The basic nature of risk and return within the framework of utility is coveredin Chapter 14. Here we shall adapt the treatment of risk–return trade-off to the multiple assetcase.
Multiple Project Selection under Risk 247
The overall return from a portfolio,3 R, is a weighted average of the returns, Ri , with weightsxi , as
R = x1 R1 + x2 R2 + · · · + xn Rn (22.1)
Each xi is the proportion of the total invested in asset i, and Ri is the expected return fromthat asset. The Ri are random variables and therefore R, the portfolio return, is also a randomvariable. If we denote the expected values of the Ri by µi , the variance of asset i by σi i , andthe covariance of asset i and asset j as σi j , we obtain for the portfolio:
E[R] = x1µ1 + x2µ2 + · · · + xnµn (22.2)
Var[R] = x21σ11 + x2
2σ22 + · · · + x2nσnn (22.3)
+2x1x2σ12 + 2x1x3σ13 + · · · + 2x1xnσ1n
+2x2x3σ23 + · · · + 2x2xnσ2n, . . .
=n∑
i=1
n∑j=1
xi x jσi j
Securities
When working with securities — which were assumed to be infinitely divisible — varyingthe weights of the individual assets allows us to obtain an unlimited number of portfolios.However, we want to ignore all portfolios other than those that have expected returns superiorto all others with the same risk4. In other words, we want to consider only those risk–returnpoints which constitute the efficient set, that when plotted on a graph define the efficient frontier.For each point on the efficient set there is no other point having lower risk for that expectedreturn or higher expected return for that same risk.
Capital Investments
Unlike securities, capital investment projects are not usually divisible, and often they are one-of-a-kind investments. What this means is that the xi weights above must take on values of 0 or1 only5. The practical consequence of this is to narrow immensely the possible combinationsof projects with the existing firm.
In the case of a firm that is considering only one large capital investment, the expected returnand variance of the combination reduces to
E[R] = x1µ1 + x2µ2
Var[R] = x21σ11 + x2
2σ22 + 2x1x2σ12
where x1 and x2 are either 0 or 1.
3A portfolio is nothing more nor less than a set, or collection, of assets. Thus it may be the stocks and bonds owned by an individualor a pension fund or insurance company. But it may just as correctly be considered as the individual building, fixtures, and so forthof one retailing store in a national retailing chain, or the collection of all the individual stores owned by the chain. Thus it may be atwhatever level of aggregation we wish to consider.4An equivalent way of stating this is that only portfolios having the lowest risk among those having the same expected return are found.5If there were two projects that were virtually identical then they could either be combined and treated as one project, or else treatedas separate projects that would either be accepted or rejected on their own merits.
248 Capital Asset Investment: Strategy, Tactics & Tools
Clearly, if asset 1 is the existing firm, then its weight must be 1 unless we are consideringselling the business. Thus, as a practical matter we are considering the result of giving a weightof 1 to the proposed project (accepting it) versus continuing the existing business without it.We are interested in what would happen to the risk–return characteristics of the firm if theproject were to be accepted.
Extension to two proposed projects plus the firm is straightforward:
E[R] = x1µ1 + x2µ2 + x3µ3
Var[R] = x21σ11 + x2
2σ22 + x23σ33 + 2x1x2σ12 + 2x1x3σ13 + 2x2x3σ23
where x1, x2, and x3 are either 0 or 1.A company that has only a few fairly large investment proposals may be able to analyze them
without relying on sophisticated and technical computer programs. For example, Baum et al.[7] evaluated all 32 possible selections of 5 candidate projects. However, full evaluation canget out of hand quickly as the number of projects considered increases; with N projects thereare 2N unique combinations. After about five projects the number of combinations rapidlybecomes so great that full enumeration is unduly burdensome if not impossible. However, inall but the very largest enterprises this should not be a significant problem, especially if minorprojects are aggregated together and treated as a single project.
Most managers are wise to insist on making the final decision on which set of capital in-vestments will be undertaken. In the author’s experience executives tend to resist arrogationof their authority and responsibility by technicians armed with computers. And they are gen-erally correct in this view, because there is as yet no adequate substitute for the judgment ofexperienced managers. And when all is said and done, it is the executive who decides thatis responsible for the consequences. Those performing the detailed project analysis shouldnormally present the executives who have the decision responsibility with a summary of thecharacteristics of the efficient alternative combinations. In the best capital investment analysisthere will be dialogue between those responsible for the final decision and those doing thedetailed analysis as the analysis proceeds. Few things could be expected to result in worsecapital-budgeting decisions that to have an assignment given to the analyst who will have nofurther contact with the decision-makers until he or she presents a recommendation.
SUMMARY AND CONCLUSION
This chapter presented an alternative to the CAPM for determining the acceptability of proposedcapital investment projects. It was shown that the combination of a proposed project with thefirm can lower or increase the enterprise’s risk depending upon the correlation of its cash flowswith those of the existing firm.
Projects that decrease the firm’s risk while not decreasing its expected return on investmentare accepted. Projects that increase the risk may be acceptable if management judges that theyincrease the expected return enough to compensate for the greater risk. Thus this method,in contrast to that of the CAPM, requires that management decide whether the risk-to-returntrade-off is appropriate. This is both a strength and a weakness of the method. It is a weaknessif judged by the extent to which decisions may be automated or delegated to lower levels ofmanagement. It is a strength if judged on the basis of management having to commit to adecision based on its collective judgment.
Multiple Project Selection under Risk 249
The method illustrated in this chapter provides an alternative to the CAPM. In cases wherethe CAPM is judged to be inappropriate or inapplicable it will allow rational decisions to bereached on the basis of risk-to-return trade-offs. In cases where the CAPM is applicable it canprovide another angle of view by which a proposed project may be examined.
A BETTER WAY TO CALCULATE RISK-ADJUSTED DISCOUNT RATESRecall that the capital asset pricing model (CAPM) can be written as the equation
k = Rf + β(RM − Rf)
Thus if we have the beta for an asset we can determine the appropriate discount rate touse in calculating its net present value (NPV). This suggests that there is a superior wayby which we can calculate risk-adjusted discount rates. To apply the method we need theproject proposal’s beta (β), the risk-free rate of interest (Rf), and the expected return on themarket (RM). For the risk-free rate we normally would use the yield on three-month USTreasury bills. The expected return on the market is the most troublesome item to estimate.Normally we would use the historical rate of return as the estimate of the expected rate.
For the Noah Zark company the risk-adjusted rates of discount we would use for calcu-lating the NPVs of projects 1 and 2 would be calculated as follows, using the data from theexample:
Project 1:k1 = 1.750% + 0.689(3.058% − 1.750%) = 2.651% per quarter which on a compound
annual basis is (100(1.02651)4 − 1) = 11.034%.
Project 2:k2 = 1.750% + 1.193(3.058% − 1.750%) = 3.310% per quarter which on a compound
annual basis is (100(1.0331)4 − 1) = 13.914%.
Thus we see that the more risky project is discounted at a higher rate than the less riskyproject. Not only that, but the discount rate reflects the market’s consensus judgment aboutthe “price of risk” and is therefore an objective alternative to other ways of calculatingrisk-adjusted discount rates.
23
Real Options∗
The term “real options” might better have been called “options on tangible or physical assets”but the term is solidly entrenched now. Real options in the realm of capital investment are thosecontaining choices about capital assets. These choices include whether to invest now, or investlater, whether to divest now or divest later, whether to expand or contract scale of operations,whether to switch use (input, output, or the application of a real asset) and combinations thatinvolve choices of actions over time about capital assets. Stewart Myers coined the term “realoptions” in a 1984 article to contrast them with financial options.
Why should we use real options? The main reason is that the traditional methods of capitalinvestment analysis fail to recognize that effective managers do not stand by passively, but areactive, and adapt to changes in markets and to opportunities for capturing strategic value. “Anoptions approach to capital budgeting has the potential to conceptualize and quantify the valueof options from active management and strategic interactions” [161, p. 4].
ACQUIRING AND DISPOSING: THE CALL AND PUT OF IT
Call options provide their owners with the right to buy something at a specific price, termedthe strike price or exercise price, until (American-style call options) or at (European-style calloption) some specified expiration date. Put options endow their owners with the right to sellsomething at a strike price at, or until, expiration. As a practical matter, there is little differencebetween the theoretical prices of an option as American or European. This is especially sowhen one considers that in the case of capital assets one usually is working with forecasts,approximation, insights, and managerial intuition about the relevant parameters. Other factorsare typically more important in determining the value of a real option on a capital asset.
The theoretical value of an option, which we assume is the market price if it is traded publicly,is determined by the difference between the exercise price and market price of the underlyingasset, the time to expiration, the price volatility of the underlying asset, the risk-free interestrate, and the cash flow from the underlying asset (interest or dividend or cost of storage if aphysical commodity). Most finance texts do not treat options on physical commodities, insteadconcentrating on options on common stock, and sometimes including those on futures andcurrencies. However, there are important differences between options on stocks, futures con-tracts, and physical commodities.1 In the case of real options the valuation models for financialinstruments and commodities may sometimes work, but often other approaches must be used.
The fundamental concepts of put and call options can be viewed in the real option schemeof things as abandon or invest, stop or proceed. The time aspect of a real option is readily
∗The author is indebted to J. Barry Lin in for his advice and co-authorship on this chapter. We both appreciate the critical insights and
valuable advice on this chapter provided by Marco Antonio G. Dias.1Users of this book may download a free ExcelTMspreadsheet, developed by the author, that is designed to calculate option prices andother parameters (the “Greeks”) and provide sensitivity analysis graphics. The spreadsheet is based on an article by James F. Meisnerand John W. Labuszewski [103].
252 Capital Asset Investment: Strategy, Tactics & Tools
seen to correspond to decide now, or decide up to some future date. Transfer of option pricingmethods from financial instruments and commodities to real options presents difficulties inmost cases because valid estimates of the critical parameters are not available. One needs tohave a sense of the market price of the underlying asset, for instance, and that might not beavailable. Similarly, the volatility of the asset price may present estimation obstacles. On topof that, the time to expiration — the useful life of the real option — may depend on events thatcannot be gauged in advance.
MORE THAN ONE WAY TO GET THERE
In previous chapters we examined capital investment decisions using decision trees, dynamicprogramming, and computer simulation. These offer both competing and complementary meth-ods for dealing with projects that have real options embedded in them. Which method is bestsuited to a particular situation is something that the decision-maker must choose, somethingthat cannot be specified in advance for all cases. Where it is possible, analysis of a project bythe real options approach and one or more other approaches can provide insights that raise theodds for success — for sound decisions — those that foster success of the organization.
Success with the real options approach, as with any analytical technique, rests heavily uponcareful, accurate specification of the problem situation and all the relevant factors. A techniquethat is based on ad hoc, seat-of-the-pants “guesstimates” risks becoming little more than aburlesque that consumes resources and misleads management into costly mistakes. This is nomore a problem with the real options approach than with any of the others. However, whenevera new approach becomes so suddenly popular as the real options way of looking at things,one must resist the temptation to throw data into it haphazardly in the belief or hope that themethod itself will make things right.2
WHERE ARE REAL OPTIONS FOUND?
Real options are found wherever there is potential for management action beyond the presenttime, t = 0. They are found in such diverse decisions as mineral and petroleum exploration;valuation of and investment in a start-up business; research, development and testing of a newmedicine; valuation of raw land; licensing a technology; investment in infrastructure/ancillarysupport operations; and contingency planning for flexibility in responding to competitors’actions. These are often referred to as strategic investments.
Besides finding real options in projects already planned or extant, they may also be deliber-ately engineered into a project from the beginning. In other words, they arise due to the naturalattributes of the project (abandonment, for instance), or they may be created through the designof the project. For example, a firm can make a product that can be retrofitted later in the eventa technology becomes developed sufficiently. This is what happened with the now ubiquitousglobal positioning system (GPS). The cost of engineering the option into the project at theplanning stage essentially makes the initial outlay endogenous.
2The popularity of real options as a tool is attested to by the seminar-workshops on the topic as this chapter is written. One need onlylook to the zeal with which some promoted value at risk (VAR), and before that to value-added accounting and the capital asset pricingmodel (CAPM), to see why some sober caution is in order before leaping onto the bandwagon. That said, the real options approachcan offer genuine value if applied properly and to appropriate problem situations. At the very least it offers a different angle of viewthat can be compared to other measures, such as discounted cash flow.
Real Options 253
Table 23.1 Comparison of Real to Financial Options
Parameter Capital asset project Financial Option
X Initial investment cost Exercise premium (option price)S Value of capital assets Price of underlying securityt Time of viability for option Time to expiration (i.e. days/365)σ 2 Volatility of project cash flows Volatility of security returnsi Risk-free discount rate Risk-free discount rated Annual cash flow return Annual dividend or interest return
In contrast to real options, discounted cash flow (DCF) analysis is like a “buy-and-hold”strategy over the span t = 0 to t = n. DCF analysis implicitly assumes that a decision is takenimmediately, then not revisited before the end of the project’s useful life.
One can find real options embedded in such management decisions as:
1. Selecting a production facility design (plant layout design) and choosing the technologiesthat will be built into it.
2. Building a parking garage or high-rise office or apartment building on a parking lot.3. Investing in new cable TV lines and nodes when a superior technology appears to be
imminent.4. Entering a foreign market by licensing, joint venture, or other means.5. Leasing versus purchasing land, buildings and equipment.6. Investing in a new computer system today or waiting for a likely better one next year.7. Selling a business now or delaying until the economy improves.8. Investing in corporate “capabilities” (training, R&D, distribution channels, customer and
supplier relationships, etc.), allowing the corporation to be better prepared for the businessgame, by creating new real options that can be quickly exercised in favorable scenarios [seee.g. 81].
A project such as a research and development (R&D) program may be required at stage 1 inorder for the organization to be able to undertake a more advanced R&D program later on atstage 2 or 3. In other words, R&D may offer no short cuts; one must ante up early on to be inthe game later. However, each stage of R&D may produce some cash generating opportunities.These might be foreseen or be the result of serendipity, like 3M’s ubiquitous Post-ItTM notes,discovered when an adhesive “failed” to hold tightly.
COMPARISON TO FINANCIAL OPTIONS
It is useful for those who are familiar with financial options to see how real options comparewith them. Table 23.1 contains a comparison of the corresponding terms. Some authoritiesomit the term d, for cash flow, but sometimes this can be important in the analysis of bothfinancial and real options.
FLASHBACK TO PI RATIO — A ROSE BY ANOTHER NAME . . .
Chapter 6 examined the profitability index (PI) as a means for comparing two or more competingcapital assets for which the initial investments were different. The PI is usually defined as the
254 Capital Asset Investment: Strategy, Tactics & Tools
ratio of the present value of returns to present value of initial project investment cost. In theliterature on real options the PI is used under the label quotient, or NPVq [see e.g. 92]. In optionterminology it is defined as NPVq ≡ S/PV[X ]. The PV of the exercise or initial investmentcost indicates that the exercise may occur some time into the future.
The value of an option depends most heavily on the relationship between X and S, t and σ . Thecumulative variance is given by the quantity σ 2t, and the corresponding standard deviationby σ
√t. Cumulative variance is very important in valuing real options. A low cumulative
variance means that there is a low probability of the option ever achieving a major changefrom its current value over its remaining useful life. A high cumulative variance means thatthere exists a good chance that the option will experience a significant change in value over itsremaining life.
TYPES OF REAL OPTIONS
Real options have been placed into as many as half a dozen categories, based upon the type offlexibility they offer. The categories may conveniently be reduced to the following:
1. The option to delay, to put off a decision until some date in the future. The future datemay be determined by some event or by management calendar. The event might be anaction from a competitor or change in some government regulation or tax, for example. Thedelay is valuable in general because new information may arrive to enhance managerialdecision-making.
2. The option to stage development with a series of investment outlays. The project can beterminated at any stage. Thus the option to abandon is a type of staged development option.So is the option to grow, in which initial R&D and prototyping are required to provide infor-mation for further development, abandonment, or delay until market or technology develop.
3. The option to change scale. This is basically a staged development option, though it is usefulto consider separately. An initial investment decision might be for production equipment thatis less costly, but also incapable of high-volume output. This contrasts with an initial invest-ment decision to adopt higher-cost, high-volume production equipment. Both decisions haveembedded options to change scale, with their own favorable and unfavorable consequences.
4. The option to switch production function and thus either input or output mix, depending onmarket conditions. This is similar to, and can be a special case of, the option to change scale,but is usefully treated separately. For example, a petroleum refinery may be optimized fora particular type of crude oil and desired mix of refined products, or for flexibility in both.
In addition, the real option can be naturally embedded into a project (for example the optionto shut down a plant) or can be created by an investment to embed flexibility into the project(e.g. buying vacant land neighboring the plant, in order to expand operations at low cost in afavorable scenario). In the first case, managers have to identify the options and plan how totake advantage of them. In the latter, managers need to value the real option and compare thatvalue to the costs.
A paper by Jerry Flatto3, presents the results of a questionnaire he sent to the chief infor-mation officers at Life Office Management Association (LOMA) member companies in theUnited States and Canada. He found that very few had ever heard of the term “real option”,
3See http://www.puc-rio.br/marco.ind/LOMA96.html. The article was written for Resource — The Magazine for Life Insurance Man-agement Resource, published by the Life Office Management Association (LOMA), the education arm of the insurance industry.
Real Options 255
but after having it explained to them most said they do include some aspects in their analysisand that in about 60 percent of the cases this made a difference in the approval process.
Remarkably, he found that no companies were using formal models designed to value realoptions. Instead, those who took real options into account said they incorporated real optionconsiderations only qualitatively. It is a tribute to the importance of real options that he foundthat 75 percent of the companies accept projects despite quantitative analysis to the contrary.4
However, the lack of formal models casts a shadow over the procedure under which quantitativemodels are overruled.
REAL OPTION SOLUTION STEPS
There are four steps for formulating and solving a real option problem:
1. Frame the problem.2. Apply the option valuation model.3. Examine the results.4. Reformulate and return to step (1).
The framing step is the most critical for successful application of real options methodologyto a problem situation for which a real options approach is appropriate. If it is done well theprospect for valid analytical results will be greatly improved. But if it is done poorly the processbecomes a burlesque or parody that it would be better to have avoided.
In framing the problem it is necessary to answer a number of questions:
1. What are the critical decision variables?2. How reliable are the estimates, or forecasts of those numbers?3. What are the uncertain elements that could cause the decision(s) to change?4. What is the risk-to-return profile of the investment? How will it affect the firm’s risk
exposure, both market and private?5. Can the option, or a close surrogate, be found at a better price in the market?6. Does the valuation of the option pass a commonsense filter?7. Who are those who have the authority to exercise the option?8. What changes, if any, are required in the organization to adapt real options and use them
profitably?9. What similar projects, if any, has the organization dealt with in the past?
Each source of uncertainty should have its own payoff diagram. And, the decision rule foroption exercise must be specific. Ambiguity does not make for clear management action. Thedecision rule may be modified as time passes and new information is obtained. But at eachstage it must be clearly understood by those who are authorized to act, to exercise the option.It is possible that managerial inertia may lead to systematic overvaluation of real options. Ifso, then specific decision rules could reduce that bias. For example, managers often delayabandonment until a project has gone so far down in value as to be indubitably a lost cause,when they should have recognized the problem earlier and acted then.
4The author recalls a case in which a major US manufacturer leaned toward accepting a project to manufacture a product for com-mercial aircraft that was marginal in terms of the conventional NPV, IRR, and payback criteria. The reason was that by producingand selling the product their sales staff would have access to customers that would facilitate sales of other products, and also that theproduct in question could be further refined as information was obtained from customers based on their experience with it.
256 Capital Asset Investment: Strategy, Tactics & Tools
The option valuation step may be carried out in a number of ways, depending on the exactproblem specification. Among these are application of the Black–Scholes model for valuingAmerican-style options; use of the binomial option valuation model, and computer simulation.The choice among these is less critical than the careful specification of answers to the framingquestions above.
The difference between the valuation of an American-style and European-style option is notcritical in most instances. If one uses a model for valuing a European-style option in a situationthat really offers an American-style exercise choice, then one should view the valuation as alower bound on the option’s value.
OPTION PHASE DIAGRAMS
Real options may usefully be viewed with a phase diagram, in which cumulative variance isplotted against how far in-the-money or out-of-the-money the option is at the present. ProfessorLuehrman offers a very nice example with his “tomato garden” which has tomatoes at variousstages of ripeness [92, p. 8]. Figure 23.1 contains a similar phase diagram. The density ofthe shading indicates the option value at present; the darker the shading the higher the optionvalue. The line labeled NPVq indicates the project’s value according to the DCF quotientmeasure. This measure and the conventional NPV must converge at expiration to an agreement
HighNPVq
IV. High Cumulative Variance
V. Unlikely Future II. Wait, if possible
III. Maybe, but NPV < 0
VI. Never Exercise the Option
Out of the Money In the Money
I. Exercise the Option NowLow
Cum
ulative Variance
Figure 23.1 Real option value
Real Options 257
on whether or not to undertake the project. But prior to expiration NPVq may be greater thanone, while conventional NPV is negative [92, p. 7].
To see how the phase diagram may be useful consider Alluvial Inc., which owns and operatesa profitable sand and gravel pit in the path of a major city’s expansion. The company can sellnow to the city, which would use the land for a major park and recreation area. The price isconsidered fair in light of current use of the land, but not much better. The company has tofactor in some other considerations.
The company can continue to operate the sand and gravel business until the deposits aredepleted. During this time span the value of the land is likely to increase, but so are taxes,especially since some in the city government want to encourage a sale to the city. Alluvial’smanagement is thinking of converting the gravel pit, now filled with pure fresh water, into acenterpiece lake around which a housing development will be constructed. The pit could alsobe used for a landfill, but this would create problems with the groundwater supply, and thus runafoul of environmental protection laws. A suitable alternative is conversion into a golf courseand private recreation area (boating sports, fishing). This would allow Alluvial to convert lateron to housing or other use with minimal cost.
Under the above description, Alluvial has a project that has a positive NPV and also containsuncertainty. The uncertainty as specified puts the sand and gravel operation into the “wait” cat-egory. It is likely to become worth much more, and in the meantime can continue to be operatedprofitably. This project falls in the middle right-hand portion of the phase diagram, phase II.
Now consider OO Corporation, which for the past five years has leased land it owns in NewJersey to LCN Inc. LCN has broken the terms of the lease, failed to make the last three monthlypayments, and its nominee directors and managers have disappeared. The land is determinedto have a great deal of toxic waste stored on and under it (and possibly some other things). Thesurrounding municipality is concerned about groundwater contamination and noxious fumesemanating from the land, and fears that going to court may delay action that could clean it upunder a federal superfund grant. The municipality has made an offer that its lawyers believeOO Corp. cannot refuse. The offer is to buy the land for the amount owed by LCN Inc. andwithout recourse for the costs of cleanup. The cost of cleanup would be prohibitive to OOCorp. to undertake on its own.
In this case immediate exercise by the company of this put option is indicated. To delay wouldlikely mean costly legal battles with little prospect of winning. And the eventual judgmentagainst OO Corp. could mean its demise. This is a project that will never have a positive NPVand the option to sell should be exercised immediately. It falls in the lower right portion of thephase diagram, phase I.
COMPLEX PROJECTS
A project may have the characteristics of both “assets-in-place” and “growth options.” Suchprojects should be analyzed by separating the parts and analyzing them separately. The assets-in-place may be properly handled with DCF techniques, but the growth components shouldbe handled with an option approach. The process of separation may, however, be difficult toaccomplish. For instance, assets-in-place may well contain embedded options, such as that ofabandonment. Nevertheless, for such projects DCF alone should be avoided because it yieldsincorrect and misleading values.
In framing a real options problem one must be careful to exclude sources of uncertaintythat are not truly relevant and significant. Options with more than two or three sources of
258 Capital Asset Investment: Strategy, Tactics & Tools
uncertainty are computationally difficult to value, and most numerical solutions cannot handlemore than two, or in some cases three. A new approach to solve several sources of uncertainty,Monte Carlo simulation for American (real) options, offers great practical promise, but is stillunder research development [30].
If the problem frame cannot be explained to experienced, senior management it is probablytoo complex, and should be reformulated. The frame should be capable of explanation toexperienced decision-makers in the industry or area of business affected. And it should bealigned with their experience and understanding. Managers who are too busy to peruse complexdetails should quickly grasp it. Decisions are seldom profitable or unprofitable because of detailbut rather because of the major factors involved.
One of the crucial questions that those framing the problem as one in real options mustask is how the relevant uncertainty affects the payoff function. Under some circumstancesthe problem may lend itself to the Black–Scholes model; in others the binomial model orsimulation may be indicated. Besides the sources of uncertainty, the option model inputs mustbe defined carefully.
In many cases a DCF analysis will have been performed already. If it has, then most ofthe input data for a real options analysis may already be at hand.5 Other than reviewing it foraccuracy and updating to incorporate new information, one would not develop the data overagain from scratch.
The number of options identified in a particular project depends on the number of decisionstages it may be broken into. The greater the number of stages the greater the number ofidentifiable options in most cases.
ESTIMATING THE UNDERLYING VALUE
Estimating the value of the underlying asset for a real option presents a problem not encounteredwith financial options for which the underlying asset is traded actively on the stock or futuresexchange. Sometimes one can estimate the value of the underlying real asset from market data.For example, if the real asset is thought to be worth 15 percent of the value of a competitor’sbusiness that is publicly traded, then one may use that amount as a proxy for the underlyingasset value.
One might argue that the exercise price seldom presents a problem because the cost ofexercise can be determined with relative precision compared to the market value of real assets.However, salvage value of an assembly plant five years into the future is not likely to be knownwith anything close to certainty today. And, in R&D, the exercise prices (cost of developmentfollowed by cost of commercial-scale production) are just wild guesses too.
WHAT TO EXPECT FROM REAL OPTION ANALYSIS
Before committing time and resources to real options analysis one should ask what is reasonableto expect from it. One usually seeks numerical output from the exercise of valuing an investmentproject; that is, a dollar amount that management can gauge for how well it measures up to
5Real options analysis can be considered a sensible generalization on static DCF analysis. After one basic estimation has been donefor one DCF analysis, management can then ask whether any real option is present, and what the parameters of these real optionsare. The value of the real options can then be estimated, and value of the project can be critical when the traditional capital-budgetingcriteria such as NPV and IRR are marginal for a project. Static NPV is a special case of “post modern” NPV where the options havenegligible value.
As parameter estimation is much more problematic with real options, in comparison to financial options, scenario analysis andsensitivity analysis using a range of likely parameters are often helpful.
Real Options 259
benchmarks for such projects. But “ . . . the most important output is not the dollar value ofthe strategy but the three types of decision-making tools shown. These can be used to manageand design investments and to build consensus around the investment strategy” [3, p. 104].The three decision-making tools mentioned are critical values of the asset over time, strategyspace (i.e. abandon, continue, modify), and investment risk profile, the last of which can berelated to probabilities of abandoning, continuing, modifying the investment. In other words,dollar values can be useful, but other factors that management can weave into the fabric of thedecision may be even more important.
Dollar value provides a point estimate, such as the NPV of a project. Knowledge of theinvestment’s risk profile can indicate to management the probabilities associated with rangesof dollar outcomes. And the decisions to abandon, continue, or expand the project are tied tothe probabilities revealed by the risk profile of the project. (For further insight on this, reviewthe discussion of risk profile in Chapter 19, in which computer simulation is covered.)
IDENTIFYING REAL OPTIONS — SOME EXAMPLES
Real options are everywhere, it merely takes practice to develop an eye for seeing them. Forexample:
1. Corel Corporation’s decision to port WordPerfect Suite 2000 to the Linux operating systemhad real options characteristics. Corel in essence decided for early exercise of the option toadapt its sound, but declining-in-popularity office suite to the relatively new, rapidly grow-ing, Unix-like open operating system, Linux. In contrast, Microsoft Corporation decidednot to exercise the same type option. If Corel could have captured a significant share ofthe Linux office suite market before Microsoft moved into it, the early exercise would havepaid off. Microsoft may have difficulty adapting its software to Linux because the Linuxoperating system is open software, basically free to those who wish to use it. Porting itsoffice suite to Linux would assist in making a significant rival to its core operating systembusiness even more popular. However, by delaying what may be the inevitable, Microsoftwas risking widespread adoption of Corel’s rival office suite, which might be difficult torecapture in the Linux market later on.6
2. Rain forest/jungle versus cattle ranches/coffee–tea–cocoa plantations/etc. suggests severalreal options for tropical nations. On the one hand, the wait and see, or wait to exerciseoption for developing jungle or rain forest into use for cattle ranching, or into plantationsfor coffee, bananas, teas, cocoa, or hundreds of other crops, may mean opportunity lossesof export sales, employment growth, and foreign exchange earnings. On the other hand,nonconsumptive development for tourism, biomedical plant research and sustainable har-vesting of timber and medicinal and food plants offers an option that is similar in respect togenerating revenues, yet different in terms of being far more reversible later on if desired.(One cannot turn a cattle ranch or coffee plantation back to the original state of the land, atleast not quickly and at low cost.) And the flexibility to change from one type of reversibledevelopment to another is clearly different than an irreversible conversion of land to a usethat may or may not remain viable in a few years.
3. Farmland in path of city expansion can be — depending upon zoning, which can often bechanged if necessary — converted to a golf course, cemetery, housing or industrial devel-opment. If there are subsurface minerals, or even sand or gravel of good quality that may be
6In 2001 Corel got out of Linux altogether. A visit to its website at http://www.corel.com reveals Corel no longer offers or supportsCorel Linux and is no longer offering an office suite to run under Linux or support for the one it did offer.
260 Capital Asset Investment: Strategy, Tactics & Tools
extracted, some of the land may be converted to a landfill upon which the golf course can bebuilt later, or as a lake at the center of a luxury housing or high-rise building development.This type of project offers many option possibilities and opportunities for sequencing thatmay improve the overall project value. For instance, conversion to a golf course leaves openthe possibility for later conversion to construction of luxury housing or high-rise buildingseither upon or around the periphery of the golf course. In contrast, an early decision to exer-cise the option of building an industrial park or a cemetery would preclude or at least hinderother opportunities afterward. And, until a decision to do otherwise, the farm might continueto be operated as a farm, earning cash flows until an alternative use decision is exercised.
CONTINGENT CLAIM ANALYSIS
Contingent claim analysis (CCA) is based on two simple, yet powerful ideas of modern financetheory. First, the present value of a risky cash flow stream can be found by discounting thecertainty equivalent of the risky cash flows at the risk-free interest rate. In contrast, conventionalmethods discount the risky cash flow at the required rate of return (risk-free rate plus a riskpremium) of the cash flow stream. Second, the value of a contingent claim on an asset can bederived by finding the value of an equivalent tracking portfolio of similar traded or observedassets. The binomial option pricing model is the best known application of CCA in a binomialtree-based structure.
THE BINOMIAL OPTION PRICING MODEL
The principles behind the binomial option pricing model are elegantly simple. It is based onthe assumption that from one time period to another the asset can take on only one of twovalues. If we represent the value of an asset at time t = 0 as A, then at the end of the first timeperiod the asset will have either value Au for an upward change, or Ad for a downward change.At the end of period 2 the asset will have either value Au2 or Ad2. And the process continuesover all the time periods in the uncertain life of the asset.
This model assumes a risk-neutral or hedged asset. That is, it assumes the asset’s risk ishedged with a tracking portfolio, or tracking instrument. This resolves the question of whatdiscount rate is appropriate to the asset values over time — it is the risk-free rate, r. The reasonit is only this much is that we assume risk has been eliminated by the hedge. For convenience,continuous compounding is used.7 The expected return on the asset is r, the risk-free rate, thevariability is given by the variance, σ 2.
Figure 23.2 shows the process of option evolution. The risk-neutral probability parametersfor upward ( p ) and downward ( 1 − p ) are calculated so that the distribution of final outcomesmatches the conditions of the situation. From the definition of return at the risk-free rate weobtain the equation
p Au + (1 − p)Ad
A= er (23.1)
Next, equating the variance of return to the estimated distribution8 yields
pu2 + (1 − p) d2 − (pu + (1 − p) d)2 = σ 2 (23.2)
7As the number of time periods in a year increases the continuous compounding result is rapidly approached in any case.8It is generally assumed that the normal distribution applies.
Real Options 261
0
12
34
56
7
Binomial Lattice, or Tree
Outcome Distribution Histogram
Figure 23.2 A binomial model of uncertainty
so that
p = Aer − Ad
Au − Ad(23.3)
If we assume symmetry of the upward and downward movements we can obtain one solutionto p from the equations.9 Then
d = 1
eσ= e−σ and u = eσ (23.4)
and
p = er − d
u − d(23.5)
9The point of such an assumption, of course, is to reduce the number of unknowns to the number of unique equations so that a uniquesolution exists.
262 Capital Asset Investment: Strategy, Tactics & Tools
An advantage of the binomial option pricing model is that it is adaptable to modification to suita particular situation where standard assumptions are inappropriate. It is adaptable to optionpricing situations for which the Black–Scholes model is not.
We will not go into the assumptions of the Black-Scholes model or closed-form, partialdifferential equation (PDE) option pricing here. There are many good references to the former,and the latter is best left to specialists in financial engineering.10 Furthermore, most personswho are using this book already have some acquaintance with the Black–Scholes model.
Application of option pricing models to real options is best shown by some examples.
Example 23.1 Value of Strategic Flexibility in a Ranch/Farm Consider the case of theKaiser Ranch that raises both native and “exotic” game animal species, and whose sharesare publicly traded, and thus its market value is observable. The ranch also operates privatehunting activities that attract a worldwide clientele of wealthy sportsmen.11 Figure 23.3 depictsthe (uncertain) growth path of the value12 of the business (not the cash flows). The value of theranch is driven by uncertainty about the state of the economy, the amount of hunter spendingat the ranch, and the market price of the wild game meat. Further, suppose that the risk of thebusiness requires a k = 17.5 percent required rate of return, while the risk-free rate is r =
S0 � 100
Su1 � 175
Sd1 � 60
Sd2 � 36
Sud2 � 105
Su2 � 306.25
Figure 23.3 The growth path of ranch
10Fisher Black, cocreator of the Black–Scholes model, speaking at an annual meeting in New York of the International Associationof Financial Engineers, where he was honored as 1994 IAFE/Sunguard Financial Engineer of the Year, made some emphatic pointsin his luncheon speech. The gist of it was that while closed-form solutions are elegant, they often do not exist for an option. And,with computers so powerful and available, one should not waste time and effort searching for closed-form solutions when computerapproximations are perfectly adequate.11Those who object to hunting for sport, or to it being done on farms like this, may substitute — without loss of generality — a duderanch cum photo safari operation combined with a ranching operation that raises endangered species for reintroduction in their nativehabitats and for wildlife parks.12The assumption here is that the firm value each period/node has been found by DCF analysis for the given scenario to derive thevalue tree.
Real Options 263
A0 � 300
Au1 � 525
Au2 � 918.75
Aud2 � 315
Ad2 � 108
Ad1 � 180
Figure 23.4 Growth path for new project
5 percent. There is a 50 percent probability that the price will go up by 75 percent each period,and a 50 percent probability that it will go down by 40 percent each period. Kaiser is traded,its asset value is observable. We use Kaiser and the risk-free asset to derive the risk-neutralprobabilities for the contingent claim valuation of a new project with embedded real options.
Suppose there is a similar ranching project available for adoption. Figure 23.4 depictsthe (uncertain) growth path of the asset value of the new project. The new project is exactlythree times the scale of the existing ranch. Using conventional DCF technique, the NPV of theproject is
PV (future asset value, not cash flow13) = (0.5 × 525 + 0.5×180)/(1 + 17.5%) = 300
This is exactly the current value given for the project. If the current owner asks for a price of310, then the NPV of this project is
NPV = 300 − 310 = −10
The CCA approach uses a backward risk-neutral valuation process.14 As derived above, therisk-neutral probability p can be found as15
p = [(1 + r )S0 − Sd ]/(Su − Sd )
We do not here include the complicating factor of dividend yield. However, in the case ofdividend yield (or some other cash flow rate from the underlying asset S, in percent perannum) δ, one may use the preceding equation but include (by summing) the dividend in the
13It is important to recognize that if we were to use cash flows instead of asset values it would be necessary to include each and everycash flow. By using asset values we need only work with the nearest values.14Varian [165] contains a detailed discussion of the theoretical foundation and derivation of CCA in general, and compares risk-neutralprobabilities to decision-tree probabilities.15Note that we use discrete compounding here instead of the er term of continuous compounding introduced earlier.
264 Capital Asset Investment: Strategy, Tactics & Tools
values of Su and Sd, or alternatively one may use the equation
p = ([(1 + r )/(1 + δ)]S0 − Sd)/(Su − Sd)
Given the risk-neutral probability, the certainty equivalent value is computed by taking inte-gration of the risky asset values over the risk-neutral probabilities, or
AVCE = pAu1 + (1 − p)Ad1
The present value of project cash flows is then found by discounting the certainty equivalentcash flow at the risk-free rate:
A0 = AVCE/(1 + r ) = [pAu1 + (1 − p)Ad1]/(1 + r )
We have:
p = ((1.05) × 100 − 60)/(175 − 60) = 45/115 = 0.3913
(1 − p) = 0.6087
and
A0 = (0.3913 × 525 + 0.6087 × 180)/(1 + 0.05) = 315/1.05 = 300
Again
NPV = 300 − 310 = −10
The CCA result confirms the valuation derived from conventional DCF valuation. In thisbasic scenario, no real option was considered, and the two approaches yield identical results.In situations involving complex options, conventional DCF would generate erroneous value(ignoring the value of the real option), whereas CCA is a straightforward and elegant tool thatcan be used to properly account for all the embedded options in project valuation, as will beillustrated below.
Although our simple two-period tree ends with only two possible outcomes, in practicalapplications it is quite easy to model more subperiods, each of shorter span, and thus gen-erate as many outcomes as required for the analysis. It is important to note that the risk-neutral probability is distinct from the probabilities used in a conventional decision-tree typeof analysis.
For the project, an NPV of −10 would seem to make it unprofitable to undertake. However,considerable operational flexibility is present in this project. This operational flexibility canbe analyzed and valued as real options using the CCA approach. These embedded optionsenhance the value of the project. In using conventional DCF analysis, these valuable optionsare ignored, and consequently misallocation in the form of under-investment results. We define:
Strategic NPV = Passive NPV + Value of operational flexibility (real options)
The correct decision rule for investment then is to accept projects with positive StrategicNPV.
A GAME FARM, RECREATIONAL PROJECT
We now examine a ranch project in more detail. Using the payoff trees given above, assumethat we are hired as consultants to work out for Eric Brand, who is considering conversion of
Real Options 265
his large cattle and alfalfa ranch to a ranch that raises deer and bison for sale of the meat tohealth-conscious consumers. The ranch is surrounded by large tracts of national forest land,bureau of land management lands, and Indian reservation lands. The first payoff tree above(S’s) illustrates the future path of the value of a similar, existing ranch. The Brand project(A’s) requires initial investment of 310 in lease of some adjoining land, purchase of breedingstock, equipment, and infrastructure development costs — the same as the asking price for theKaiser Ranch.16 As illustrated above, the conventional (i.e. passive) NPV is −10, and wouldrender the project unacceptable. There are several interesting, different embedded strategicreal options in this project that Mr Brand is considering.
Option to Switch Operation
It is especially true of farms and ranches, but also true for many other businesses, that anoperation can be switched into a different process/technology, different input combination, ordifferent output mix. In times of economic downturn, managers have the option of running agame ranch as a conventional ranching operation, and selling the meat in more than just thespecialty market. Even in times of high sport hunting spending, the state of the economy couldbe such that meat prices are high enough to justify a switch from the sport hunting operation toconventional game animal farming, or some mix of the two. We discuss two possible scenarios.
Case 1: 100 percent Switch
Assume that the ranch has to be run completely (100 percent) either in sport hunting or inconventional game animal farming. Further, assume the following asset valuations (A′) fromthe conventional game farming operation. Figure 23.5 Illustrates the evolution of asset values.
Under the switched operation, cash flows are lower and less volatile than the sport huntingincomes. The income fluctuation is driven by uncertainty in game animal meat prices and gameanimal farming costs. Current value of the switched operation (A′
o) is lower than that of thesport hunting project (A0). Otherwise management will immediately switch.
The strategic flexibility embedded in this case is that the management has the option ofswitching. The switching decision will be based on year 1 realized state of economy. Themanagement compares the asset values of current use (A) to that of the alternative operation(A′). Year 1 asset value will be the maximum of the two, max (A, A′).
Vu1 = max (Au1, A′u1) = max (525, 412.5) = 525
when sport hunting is chosen, and
Vd1 = max (Ad1, A′d1) = max (180, 233.75) = 233.75,
when operation is switched. Applying CCA, the strategic NPV (value of the project plus theoption to switch operation) is
Strategic NPV = [pVu1 + (1 − p)Vd1]/(1 + r ) − I0
= (0.3913 × 525 + 0.6087 × 233.75)/(1.05) − 310
= (347.72/1.05) − 310 = 21.16
16Results may be scaled to make the analysis easier, and adjusted accordingly afterward.
266 Capital Asset Investment: Strategy, Tactics & Tools
A'0 � 275
A'u1 � 412.5
A'd1 � 233.75
A'u2 � 618.75
A'd2 � 198.688
A'ud2 �350.625
Figure 23.5 Evolution of asset value, switch operation
The value of the strategic option to switch can be calculated as
Value of option to switch = Strategic NPV − Passive NPV
= 21.16 − (−10) = 31.16
The value of the option to switch accounts for a full 10 percent of the project’s gross value(310). The presence of this strategic option makes the project viable (strategic NPV =21.16 > 0), given a negative conventional NPV.
It is important to note that, although incomes from conventional game animal farming aresubstantially lower than the sport hunting operation in the good states, the lower volatility, andthus higher incomes in the bad states, creates a valuable strategic option for management.
Case 2: Mix Operation
In most manufacturing operations, assets are flexible only on an exclusive basis, capable ofa complete switch in term of inputs or outputs, a situation similar to case 1 analyzed above.In a sport hunting operation, however, a mixed operation is not only possible, but very likelyoptimal in many situations. We next consider a scenario of mixed operation.
Assume that management can reconfigure the operation by separating one part or the ranchfacility, say one or two sections, for conventional game farming and still maintain a certainlevel of sport hunting activity. Specifically, suppose the optimal mix is to run 40 percent ofplanned sport hunting activity and at the same time raise game animals to generate 75 percentof the cash flow from complete (100 percent) conventional game farming. The manager will
Real Options 267
compare a 100 percent sport hunting outcome (A) to the outcomes from the optimal mix(0.4 × A + 0.75 × A′) and determine the strategic choice in year 1. Year 1 project value willbe: max (A, 0.4 × A + 0.75 × A′), or
Vu1 = max (Au1, 0.4 × Au1 + 0.75 × Au′1) = max (525, 0.4 × 525 + 0.75 × 412.5)
= max (525, 519.39) = 525
when 100 percent sport hunting is maintained, and
Vd1 = max (Ad1, 0.4 × Ad1 + 0.75 × A′d1) = max (180, 0.4 × 180 + 0.75 × 233.75)
= max (180, 247.31) = 247.31
when mix operation is chosen. The strategic NPV (value of the project plus the option to mixoperation) is
Strategic NPV = [pVu1 + (1 − p)Vd1]/(1 + r ) − I0
= (0.3913 × 525 + 0.6087 × 247.31)/(1.05) − 310
= (355.97/1.05) − 310 = 29.02
The value of the strategic option to switch and mix operation can be calculated as
Value of option to switch and mix operation = Strategic NPV − Passive NPV
= 29.02 − (−10) = 39.02
The value of the option to switch to a mixed operation accounts for about 12.6 percent ofthe gross value of the project. To the degree that mixed operation may involve higher cost inreconfiguring the operations, the higher value of the option to mix would justify the strategicchoice if the increase in cost is not more than 7.86 (= 39.02 − 31.16) over that of case 1.
We note that, for the sake of clarity and tractability, we have assumed that the optimal mix ofthe two operations is given. In practice, the optimal mix would be part of the optimal exerciseproblem and is not independent of, nor can be predetermined before, the optimal exerciseproblem. The consideration of various degrees of mixed operation makes the analysis muchmore complex. The complexity, though, comes with a higher value for the strategic flexibility.
Option to Abandon for Salvage Value
Most assets can be sold for salvage value when continued operation is not economical. Assumethat after one year the assets and land can be sold for 70 percent of the total initial costs at 217(= 310 × 70 percent). At year 1, the Brand management would compare the salvage value tothe value from current operations and decide whether to continue the operation or to abandon.Year 1 project value will be: max (A, 217), or
Vu1 = max (Au1, 217) = max (525, 217) = 525
when sport hunting is continued, and
Vd1 = max (Ad1, 217) = max (180, 217) = 217
268 Capital Asset Investment: Strategy, Tactics & Tools
when the operation is abandoned and assets and land are sold for their salvage value or convertedto “ranchettes”. The strategic NPV (value of the project plus the option to abandon) is
Strategic NPV = [pVu1 + (1 − p)V d1]/(1 + r ) − I0
= (0.3913 × 525 + 0.6087 × 217)/(1.05) − 310
= (337.52/1.05) − 310 = 11.45
The value of the strategic option to abandon can be calculated as
Value of option to abandon = Strategic NPV – Passive NPV
= (11.45) − (−10) = 21.45
The option to abandon a project for its salvage value is a valuable option when a businessprospect turns out unsatisfactory. By not accounting for this option, conventional NPV anal-ysis forces the assumption that once a project is accepted, it will be carried out to the endregardless of the realized state of economy. This is obviously an erroneous assumption, withserious implications. In our example, the abandonment option accounts for over 7 percent ofthe gross value of the project. The abandonment option also makes the project viable (strategicNPV = 21.45 > 0).
The Option to Expand Operation (Growth Option)
Practically all businesses have the flexibility to expand the scale of operation by incurringadditional investment. Growth options are particularly valuable when state of economy turnsout to be good. Assume that management can make a follow-up investment of IG = 300 anddouble the ranching operations. At year 1, management decides whether to expand the scale ofoperation by comparing the asset values from the base-case operation to those of the expandedoperation. Year 1 asset values will be: V1 = max (A1, 2 × A1 − I G
1 ), or,
Vu1 = max (525, 2 × 525 − 300) = max (525, 750) = 750
when the growth option is exercised, and,
Vd1 = max (180, 2 × 180 − 300) = max (180, 60) = 180
when base operation is maintained the strategic NPV (value of the project plus the option togrow) is:
Strategic NPV = [p Vu1 + (1 − p)Vd1]/(1 + r ) − I0
= (0.3913 × 750 + 0.6087 × 180)/(1.05) − 310
= (403.04/1.05) − 310 = 83.85
The value of the strategic option to grow can be calculated as
Value of option to grow = Strategic NPV − Passive NPV
= 73.85 − (−10) = 44.51
The strategic option to grow when business prospects are good is a highly valuable realoption that is embedded in many businesses. In many businesses, growth options account for alarge fraction of firm value. By failing to account for this type of option, conventional NPV can
Real Options 269
lead to serious underinvestment and missed opportunities. In the above example this growthoption accounts for nearly 27 percent of the gross value of the project (= 83.85/310).
Interaction among Strategic Options
It is easy to see that all the real options analyzed above interact with each other. For example,the option to grow interacts with options to abandon and to switch. For the sake of clarity,we have focused on the value of individual real options and on how they enhance the project.When multiple real options are present, which is in most real-world cases, one applies thesame analytical process for determining project cash flows from future operations. Currentvalue (the strategic NPV) of the project can then be derived by using CCA. In this section, weanalyze and illustrate the interaction between the real option to abandon and the real option togrow.
Using data given above, when a manager has both the option to abandon and the option toexpand, year 1 project value is determined by comparing incomes from base operation (A1),expanded operation (2 × A1 − I G
1 ), and salvage value (VS = 217 = 70 percent × 310), or
Vu1 = max (Au1, 2 × Au1 − 300, 217) = max (525, 2 × 525 − 300, 217)
= max (525, 750, 217) = 750
when expansion is chosen, and
AVd1 = max (Ad1, 2 × Ad1 − 300, 217) = max (180, 2 × 180 − 300, 217)
= max (180, 60, 217) = 217
when operation is abandoned. The strategic NPV (value of the project plus the option to switchoperation) is
Strategic NPV = [pVu1 + (1 − p)Vd1]/(1 + r ) − I0
= (0.3913 × 750 + 0.6087 × 217)/(1.05) − 310
= (425.56/1.05) − 310 = 95.3
The value of the portfolio of the two strategic options can be calculated as
Value of the options = Strategic NPV − Passive NPV
= 95.3 − (−10) = 105.3
Note that the growth option is exercised in the up-state, and the abandonment option is exercisedin the down-state. The recognition of the strategic flexibility has transformed the project to thedegree that it is completely different from the original base-case operation.
The combined value of these two options accounts for nearly 34 percent of the gross valueof the project (= 105.3/310)). Although in this illustration the strategic value of the jointpresence of the two options just equals the sum of the stand-alone value of the two options(105.3 = 21.45 + 83.85), it has been shown [160] that in most cases the value of a portfolioof options can be substantially higher than the sum of the values of the individual stand-aloneoptions.17 The implication is that the presence of multiple real options in a sport hunting
17However, it has been noted that, although in general a portfolio of options has greater value than an option on a portfolio of assets,in the presence of some other real options, the incremental value of an additional option is lower than its isolated value. In essence,one has diminishing marginal value increase as one adds more options.
270 Capital Asset Investment: Strategy, Tactics & Tools
operation, as analyzed above, makes the miscalculation of conventional NPV, in the form ofundervaluation, even more pronounced.
Our analyses and discussions illustrate the importance of real option in the valuation ofbusiness investment projects. We also provide illustration for the optimal exercise of thesestrategic options. Although the analyses in this example are framed in terms of a game farmand sport hunting project, the analytical tools, strategic conclusions and insights are completelygeneral and should be applied to all business strategic management.
Example 23.2 Paper Moon is a “dot.com” start-up company that has a novel Internet appli-cation. The company has a patent pending on its technological development. The company’spatent attorney estimates it will take a year for the US patent office to issue the patent. There isa probability of 0.5 that the patent will be issued, and 0.5 that the patent office will refuse thepatent because the patent examiner believes it is obvious, or a search reveals the technologyhas already been patented by others. The management of Paper Moon wants to launch theirInternet operation immediately when the patent is issued. To be ready to do that, they mustinvest $11.0 million now for programming, computer and network hardware, and marketingmaterials. If the patent is not issued, or is delayed, they will launch the application anywayafter a year. But the consequences of not having a patent will be that competitors can duplicatetheir application, and Paper Moon’s earnings will be much lower than otherwise.
The payoff diagram (conventional decisions tree, predicated on the base-scale operation)for Paper Moon is shown in Figure 23.6. The risk-free interest rate is rf = 6 percent, and thecost of capital considered appropriate to projects of this risk of k = 20 percent. A venturecapital firm has been approached to fund the $11.0 million of development. The company hastwo strategic options available, abandonment at end of year 1, and growth at end of year 2. Thecompany can abandon the project and sell the firm for $1.5 million salvage value at the endof year 1 in the down-state. At the end of year 2, the company also has the strategic flexibilityto invest an additional $5 million to increase their operation by 60 percent.
DeLong and Tinic, Paper Moon’s consulting firm, has completed the first-stage research forthe contingent claim valuation, and came up with risk-neutral probabilities of
p = 0.625 and (1 − p) = 0.375
They also generated the value tree of the firm with risk-neutral probabilities, as shown inFigure 23.3.
As senior financial staff, you are given the assignment to come up with an in-firm valuationto check against DeLong and Tinic’s number. What is the value of the project? How should thefirm proceed over time? Will the firm be able to acquire the necessary funding from the venturecapitalist? Should they invest or pass on this opportunity? What are the critical uncertaintiesin this situation?
Contingent Claim Valuation
Step 1: Risk-neutral probabilities has been estimated to be
p = 0.625 and (1 − p) = 0.375
Step 2: Valuation by applying the risk-neutral probabilities to the state-dependent projectvalues including optimal exercise of any real options.
Real Options 271
Cost � $11.0MM
Patent Granted;No Revenue Yet
Patent Refused;Competition Now
Value � $-5MM
Value � $-5MM
Value � $-5MM
Value � $-10MM
Value � $5MM
Value = $5MM
Value � $19MM
Value � $24MM
Value � $18MM
Value � $18MM
Value � $4MM
Value � $4MM
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
Figure 23.6 Sample problem: Paper Moon
The Optimal Exercise of the Growth Option and Firm Value
At the end of year 2, the value of the firm would be (1 + 60 percent)∗ S3 − I G if the firm exer-cises the growth option and expands operation. Firm value would be S3 if the firm maintainsthe base-scale of operation. Firm value at t = 3 will be V3 = max (S3, 1.6∗S3 − I G). Firmwill exercise if the former is larger:
Vuuu3 = max (Suuu3, 1.6∗Suuu3 − I G) max (24, 1.6∗24 − 5)
= max (24, 33.4) = 33.4
272 Capital Asset Investment: Strategy, Tactics & Tools
The growth option is exercised.
Vuud3 = max (Suud3, 1.6∗Suud3 − I G) = max (18, 1.6∗18 − 5)
= max (18, 23.8) = 23.8
The growth Option is exercised.Applying the risk-neutral probabilities, we have
Vuu2 = (0.625∗33.4 + 0.375∗23.8)/(1 + 6%) = 29.8/1.06 = 28.11
Similarly, at Node (t = 2, ud), we have
Vudu3 = max (Sudu3, 1.6∗Sudu3 − I G) = max (18, 1.6∗18 − 5)
= max (18, 23.8) = 23.8
The growth option is exercised.
Vudu3 = max (Sudd3, 1.6∗Sudd3 − I G) = max (4, 1.6∗4 − 5)
= max (4, 1.4) = 4
The growth option is not exercised.Applying the risk-neutral probabilities, we have
Vud2 = (0.625∗23.8 + 0.375∗4)/(1 + 6%) = 16.38/1.06 = 15.45
Applying the risk-neutral probabilities obe more round, we have
Vu1 = (0.625∗28.11 + 0.375∗15.45)/(1 + 6%) = 23.36/1.06 = 22.04
We note that the growth option will never be exercised in any of the down-states, as the valueof the firm in those stated are all worse that the “udd3” state when the growth option was shownabove not to be exercised.
The Abandon Option and Firm Value
The firm will exercise the abandon option — sell the firm for salvage value — when the realizedstate is such that salvage value is higher than the value of the firm as a going concern. This isan option that is available for all businesses, but often ignored in conventional NPV analysis.
Suppose at t = 1, Paper Moon can be sold for $1.5 million salvage value in the down-state.(It can be sold at around $22.4 million as a going concern in the up-state. The abandon optionwill not be exercised in the up-state.)
The manager of Paper Moon will sell the firm for salvage value if firm value as a goingconcern under base-scale of operation is lower.
To find Vu1, we again apply the risk-neutral provabilities to firm value in t = 2. We haveVuu2 = $5 million and Vud2 = −$5 million (Base-scale will be maintained as the growthoption will not be ecercised in the down-states.)
Applying the risk-neutral probabilities, we have
Vd1 = (0.625∗5 + 0.375∗(−5))/(1 + 6%) = 1.25/1.06 = 1.18
Real Options 273
Current Value of the Firm
As discussed above, the firm will exercise the growth option in the up-state, and will exercisethe abandon option in the down-state. Current value of the firm must incorporate these optimalexercises of strategic operational flexibility. In other words, the base-scale value-tree is not areliable basis for valuation when strategic flexibility has not been considered.
Having determined the optimal excise of these real option, we now apply the risk-neutralprobabilities one last time to derive current firm value:
V0 = (0.625∗Vu1 + 0.375∗Vd1)/(1 + 6%)
= (0.625∗22.04 + 0.375∗1.18)/(1 + 6%) = 14.22/1.06 = 13.41
Strategic NPV = 13.41 − 11 = 2.41
The Strategic Dimension of the Real-option Analysis
Our analyses above not only generate the numerical analyses necessary in the process of ac-quiring funding from the venture capitalist, but also provide the required managerial/strategicplanning for the firm to be successfully run under different realized states. As discussed in ear-lier sections, such managerial/strategic insight may be as important as the process of numericalsolution.
The critical uncertainty in this project is whether the firm will acquire the patent at t = 1.This uncertainty in turn drives the optimal exercise of the two real options. We have shown abovethat the growth option will be optimally exercised in three out of four states in the up-branch,but the abandonment option will never be excercised in the up-branch. In the down-branch,the growth option will never be exercised, but the abandonment option will immediately beexercised as soon as the patent is rejected.
With the recognition above, the project is radically transformed and substantially differentfrom the base-scale operation. Accordingly, the process of valuation is also fundamentallydifferent.
The recognition and optimal exercise of such real options create additional valur only if themanagerial/strategic dimensions are properly appreciated.
Given a current value of $13.41 million, and a fully specified optimal exercise of all theavailable strategic options, the firm should have no difficulty convincing the venture capitalistto fund the project. This would not have been possible if only the base-scale operation waspresented.
CONCLUSION
Real options provide a way of looking at capital investments and other decision situations thatoffers insights that traditional DCF methods cannot. Whether or not adopting a real optionsapproach will provide a clear numerical signal to adopt or shun a particular project is not asimportant as the fact that engaging in the exercise of framing the problem in real option terms,and focusing management attention on the framed problem, will facilitate better decision-making.
The numerical solution method used to solve a particular real option problem is not nearlyas important as careful framing of the problem and attention to including only the truly relevantsources of uncertainty and key decision variables. The Black–Scholes model works very well
274 Capital Asset Investment: Strategy, Tactics & Tools
for those problems for which it is appropriate: problems that have embedded American-styleoptions rather than the European-style option the model was designed to solve. The binomialmodel is more flexible, and can handle the latter type of option as well as option situations thathave more complex characteristics. In truly complex option evaluation computer simulation(Monte Carlo method) may be required. It is beyond the scope of this chapter and book toventure into such details, which can be found in specialized works on option pricing. The focusof this chapter is on what managers need to know to integrate real options thinking into theiroperations, not to take up technical details.
Appendix: Financial Mathematics Tablesand Formulas
Tabl
eA
.1Si
ngle
paym
entc
ompo
und
amou
nt.T
ofin
dF
for
agi
ven
PF
=P
(1+
i)N
N/i
12
34
56
78
910
11.
0100
001.
0200
001.
0300
001.
0400
001.
0500
001.
0600
001.
0700
001.
0800
001.
0900
001.
1000
002
1.02
0100
1.04
0400
1.06
0900
1.08
1600
1.10
2500
1.12
3600
1.14
4900
1.16
6400
1.18
8100
1.21
0000
31.
0303
011.
0612
081.
0927
271.
1248
641.
1576
251.
1910
161.
2250
431.
2597
121.
2950
291.
3310
004
1.04
0604
1.08
2432
1.12
5509
1.16
9859
1.21
5506
1.26
2477
1.31
0796
1.36
0489
1.41
1582
1.46
4100
51.
0510
101.
1040
811.
1592
741.
2166
531.
2762
821.
3382
261.
4025
521.
4693
281.
5386
241.
6105
10
61.
0615
201.
1261
621.
1940
521.
2653
191.
3400
961.
4185
191.
5007
301.
5868
741.
6771
001.
7715
617
1.07
2135
1.14
8686
1.22
9874
1.31
5932
1.40
7100
1.50
3630
1.60
5781
1.71
3824
1.82
8039
1.94
8717
81.
0828
571.
1716
591.
2667
701.
3685
691.
4774
551.
5938
481.
7181
861.
8509
301.
9925
632.
1435
899
1.09
3685
1.19
5093
1.30
4773
1.42
3312
1.55
1328
1.68
9479
1.83
8459
1.99
9005
2.17
1893
2.35
7948
101.
1046
221.
2189
941.
3439
161.
4802
441.
6288
951.
7908
481.
9671
512.
1589
252.
3673
642.
5937
42
111.
1156
681.
2433
741.
3842
341.
5394
541.
7103
391.
8982
992.
1048
522.
3316
392.
5804
262.
8531
1712
1.12
6825
1.26
8242
1.42
5761
1.60
1032
1.79
5856
2.01
2196
2.25
2192
2.51
8170
2.81
2665
3.13
8428
131.
1380
931.
2936
071.
4685
341.
6650
741.
8856
492.
1329
282.
4098
452.
7196
243.
0658
053.
4522
7114
1.14
9474
1.31
9479
1.51
2590
1.73
1676
1.97
9932
2.26
0904
2.57
8534
2.93
7194
3.34
1727
3.79
7498
151.
1609
691.
3458
681.
5579
671.
8009
442.
0789
282.
3965
582.
7590
323.
1721
693.
6424
824.
1772
48
161.
1725
791.
3727
861.
6047
061.
8729
812.
1828
752.
5403
522.
9521
643.
4259
433.
9703
064.
5949
7317
1.18
4304
1.40
0241
1.65
2848
1.94
7900
2.29
2018
2.69
2773
3.15
8815
3.70
0018
4.32
7633
5.05
4470
181.
1961
471.
4282
461.
7024
332.
0258
172.
4066
192.
8543
393.
3799
323.
9960
194.
7171
205.
5599
1719
1.20
8109
1.45
6811
1.75
3506
2.10
6849
2.52
6950
3.02
5600
3.61
6528
4.31
5701
5.14
1661
6.11
5909
201.
2201
901.
4859
471.
8061
112.
1911
232.
6532
983.
2071
353.
8696
844.
6609
575.
6044
116.
7275
00
211.
2323
921.
5156
661.
8602
952.
2787
682.
7859
633.
3995
644.
1405
625.
0338
346.
1088
087.
4002
5022
1.24
4716
1.54
5980
1.91
6103
2.36
9919
2.92
5261
3.60
3537
4.43
0402
5.43
6540
6.65
8600
8.14
0275
231.
2571
631.
5768
991.
9735
872.
4647
163.
0715
243.
8197
504.
7405
305.
8714
647.
2578
748.
9543
0224
1.26
9735
1.60
8437
2.03
2794
2.56
3304
3.22
5100
4.04
8935
5.07
2367
6.34
1181
7.91
1083
9.84
9733
251.
2824
321.
6406
062.
0937
782.
6658
363.
3863
554.
2918
715.
4274
336.
8484
758.
6230
8110
.834
706
261.
2952
561.
6734
182.
1565
912.
7724
703.
5556
734.
5493
835.
8073
537.
3963
539.
3991
5811
.918
177
271.
3082
091.
7068
862.
2212
892.
8833
693.
7334
564.
8223
466.
2138
687.
9880
6110
.245
082
13.1
0999
428
1.32
1291
1.74
1024
2.28
7928
2.99
8703
3.92
0129
5.11
1687
6.64
8838
8.62
7106
11.1
6714
014
.420
994
291.
3345
041.
7758
452.
3565
663.
1186
514.
1161
365.
4183
887.
1142
579.
3172
7512
.172
182
15.8
6309
330
1.34
7849
1.81
1362
2.42
7262
3.24
3398
4.32
1942
5.74
3491
7.61
2255
10.0
6265
713
.267
678
17.4
4940
2
311.
3613
271.
8475
892.
5000
803.
3731
334.
5380
396.
0881
018.
1451
1310
.867
669
14.4
6177
019
.194
342
321.
3749
411.
8845
412.
5750
833.
5080
594.
7649
416.
4533
878.
7152
7111
.737
083
15.7
6332
921
.113
777
331.
3886
901.
9222
312.
6523
353.
6483
815.
0031
896.
8405
909.
3253
4012
.676
050
17.1
8202
823
.225
154
341.
4025
771.
9606
762.
7319
053.
7943
165.
2533
487.
2510
259.
9781
1413
.690
134
18.7
2841
125
.547
670
351.
4166
031.
9998
902.
8138
623.
9460
895.
5160
157.
6860
8710
.676
581
14.7
8534
420
.413
968
28.1
0243
7
361.
4307
692.
0398
872.
8982
784.
1039
335.
7918
168.
1472
5211
.423
942
15.9
6817
222
.251
225
30.9
1268
137
1.44
5076
2.08
0685
2.98
5227
4.26
8090
6.08
1407
8.63
6087
12.2
2361
817
.245
626
24.2
5383
534
.003
949
381.
4595
272.
1222
993.
0747
834.
4388
136.
3854
779.
1542
5213
.079
271
18.6
2527
626
.436
680
37.4
0434
339
1.47
4123
2.16
4745
3.16
7027
4.61
6366
6.70
4751
9.70
3507
13.9
9482
020
.115
298
28.8
1598
241
.144
778
401.
4888
642.
2080
403.
2620
384.
8010
217.
0399
8910
.285
718
14.9
7445
821
.724
521
31.4
0942
045
.259
256
411.
5037
522.
2522
003.
3598
994.
9930
617.
3919
8810
.902
861
16.0
2267
023
.462
483
34.2
3626
849
.785
181
421.
5187
902.
2972
443.
4606
965.
1927
847.
7615
8811
.557
033
17.1
4425
725
.339
482
37.3
1753
254
.763
699
431.
5339
782.
3431
893.
5645
175.
4004
958.
1496
6712
.250
455
18.3
4435
527
.366
640
40.6
7611
060
.240
069
441.
5493
182.
3900
533.
6714
525.
6165
158.
5571
5012
.985
482
19.6
2846
029
.555
972
44.3
3696
066
.264
076
451.
5648
112.
4378
543.
7815
965.
8411
768.
9850
0813
.764
611
21.0
0245
231
.920
449
48.3
2728
672
.890
484
461.
5804
592.
4866
113.
8950
446.
0748
239.
4342
5814
.590
487
22.4
7262
334
.474
085
52.6
7674
280
.179
532
471.
5962
632.
5363
444.
0118
956.
3178
169.
9059
7115
.465
917
24.0
4570
737
.232
012
57.4
1764
988
.197
485
481.
6122
262.
5870
704.
1322
526.
5705
2810
.401
270
16.3
9387
225
.728
907
40.2
1057
362
.585
237
97.0
1723
449
1.62
8348
2.63
8812
4.25
6219
6.83
3349
10.9
2133
317
.377
504
27.5
2993
043
.427
419
68.2
1790
810
6.71
8957
501.
6446
322.
6915
884.
3839
067.
1066
8311
.467
400
18.4
2015
429
.457
025
46.9
0161
374
.357
520
117.
3908
53
511.
6610
782.
7454
204.
5154
237.
3909
5112
.040
770
19.5
2536
431
.519
017
50.6
5374
281
.049
697
129.
1299
3852
1.67
7689
2.80
0328
4.65
0886
7.68
6589
12.6
4280
820
.696
885
33.7
2534
854
.706
041
88.3
4417
014
2.04
2932
531.
6944
662.
8563
354.
7904
127.
9940
5213
.274
949
21.9
3869
836
.086
122
59.0
8252
496
.295
145
156.
2472
2554
1.71
1410
2.91
3461
4.93
4125
8.31
3814
13.9
3869
623
.255
020
38.6
1215
163
.809
126
104.
9617
0817
1.87
1948
551.
7285
252.
9717
315.
0821
498.
6463
6714
.635
631
24.6
5032
241
.315
001
68.9
1385
611
4.40
8262
189.
0591
42
561.
7458
103.
0311
655.
2346
138.
9922
2215
.367
412
26.1
2934
144
.207
052
74.4
2696
512
4.70
5005
207.
9650
5757
1.76
3268
3.09
1789
5.39
1651
9.35
1910
16.1
3578
327
.697
101
47.3
0154
580
.381
122
135.
9284
5622
8.76
1562
581.
7809
013.
1536
245.
5534
019.
7259
8716
.942
572
29.3
5892
750
.612
653
86.8
1161
214
8.16
2017
251.
6377
1959
1.79
8710
3.21
6697
5.72
0003
10.1
1502
617
.789
701
31.1
2046
354
.155
539
93.7
5654
016
1.49
6598
276.
8014
9060
1.81
6697
3.28
1031
5.89
1603
10.5
1962
718
.679
186
32.9
8769
157
.946
427
101.
2570
6417
6.03
1292
304.
4816
40
611.
8348
643.
3466
516.
0683
5110
.940
413
19.6
1314
534
.966
952
62.0
0267
710
9.35
7629
191.
8741
0833
4.92
9803
621.
8532
123.
4135
846.
2504
0211
.378
029
20.5
9380
237
.064
969
66.3
4286
411
8.10
6239
209.
1427
7836
8.42
2784
631.
8717
443.
4818
566.
4379
1411
.833
150
21.6
2349
339
.288
868
70.9
8686
512
7.55
4738
227.
9656
2840
5.26
5062
641.
8904
623.
5514
936.
6310
5112
.306
476
22.7
0466
741
.646
200
75.9
5594
513
7.75
9117
248.
4825
3544
5.79
1568
651.
9093
663.
6225
236.
8299
8312
.798
735
23.8
3990
144
.144
972
81.2
7286
114
8.77
9847
270.
8459
6349
0.37
0725
661.
9284
603.
6949
747.
0348
8213
.310
685
25.0
3189
646
.793
670
86.9
6196
216
0.68
2234
295.
2220
9953
9.40
7798
671.
9477
453.
7688
737.
2459
2913
.843
112
26.2
8349
049
.601
290
93.0
4929
917
3.53
6813
321.
7920
8859
3.34
8578
681.
9672
223.
8442
517.
4633
0714
.396
836
27.5
9766
552
.577
368
99.5
6275
018
7.41
9758
350.
7533
7665
2.68
3435
691.
9868
943.
9211
367.
6872
0614
.972
710
28.9
7754
855
.732
010
106.
5321
4220
2.41
3339
382.
3211
8071
7.95
1779
702.
0067
633.
9995
587.
9178
2215
.571
618
30.4
2642
659
.075
930
113.
9893
9221
8.60
6406
416.
7300
8678
9.74
6957
712.
0268
314.
0795
498.
1553
5716
.194
483
31.9
4774
762
.620
486
121.
9686
5023
6.09
4918
454.
2357
9486
8.72
1652
722.
0470
994.
1611
408.
4000
1716
.842
262
33.5
4513
466
.377
715
130.
5064
5525
4.98
2512
495.
1170
1595
5.59
3818
732.
0675
704.
2443
638.
6520
1817
.515
953
35.2
2239
170
.360
378
139.
6419
0727
5.38
1113
539.
6775
4710
51.1
5320
074
2.08
8246
4.32
9250
8.91
1578
18.2
1659
136
.983
510
74.5
8200
114
9.41
6840
297.
4116
0258
8.24
8526
1156
.268
519
752.
1091
284.
4158
359.
1789
2618
.945
255
38.8
3268
679
.056
921
159.
8760
1932
1.20
4530
641.
1908
9312
71.8
9537
1
cont
inue
sov
erle
af
Tabl
eA
.1(c
onti
nued
)
N/i
12
34
56
78
910
762.
1302
204.
5041
529.
4542
9319
.703
065
40.7
7432
083
.800
336
171.
0673
4134
6.90
0892
698.
8980
7413
99.0
8490
977
2.15
1522
4.59
4235
9.73
7922
20.4
9118
742
.813
036
88.8
2835
618
3.04
2055
374.
6529
6476
1.79
8900
1538
.993
399
782.
1730
374.
6861
2010
.030
060
21.3
1083
544
.953
688
94.1
5805
819
5.85
4998
404.
6252
0183
0.36
0801
1692
.892
739
792.
1947
684.
7798
4210
.330
962
22.1
6326
847
.201
372
99.8
0754
120
9.56
4848
436.
9952
1790
5.09
3274
1862
.182
013
802.
2167
154.
8754
3910
.640
891
23.0
4979
949
.561
441
105.
7959
9322
4.23
4388
471.
9548
3498
6.55
1668
2048
.400
215
812.
2388
824.
9729
4810
.960
117
23.9
7179
152
.039
513
112.
1437
5323
9.93
0795
509.
7112
2110
75.3
4131
822
53.2
4023
682
2.26
1271
5.07
2407
11.2
8892
124
.930
663
54.6
4148
911
8.87
2378
256.
7259
5055
0.48
8119
1172
.122
037
2478
.564
260
832.
2838
845.
1738
5511
.627
588
25.9
2788
957
.373
563
126.
0047
2127
4.69
6767
594.
5271
6812
77.6
1302
027
26.4
2068
684
2.30
6723
5.27
7332
11.9
7641
626
.965
005
60.2
4224
113
3.56
5004
293.
9255
4164
2.08
9342
1392
.598
192
2999
.062
754
852.
3297
905.
3828
7912
.335
709
28.0
4360
563
.254
353
141.
5789
0431
4.50
0328
693.
4564
8915
17.9
3202
932
98.9
6903
0
862.
3530
885.
4905
362.
7057
8029
.165
349
66.4
1707
115
0.07
3639
336.
5153
5174
8.93
3008
1654
.545
912
3628
.865
933
872.
3766
195.
6003
4713
.086
953
30.3
3196
369
.737
925
159.
0780
5736
0.07
1426
808.
8476
4918
03.4
5504
439
91.7
5252
688
2.40
0385
5.71
2354
13.4
7956
231
.545
242
73.2
2482
116
8.62
2741
385.
2764
2687
3.55
5461
1965
.765
998
4390
.927
778
892.
4243
895.
8266
0113
.883
949
32.8
0705
176
.886
062
178.
7401
0541
2.24
5776
943.
4398
9721
42.6
8493
848
30.0
2055
690
2.44
8633
5.94
3133
14.3
0046
734
.119
333
80.7
3036
518
9.46
4511
441.
1029
8010
18.9
1508
923
35.5
2658
253
13.0
2261
2
912.
4731
196.
0619
9614
.729
481
35.4
8410
784
.766
883
200.
8323
8247
1.98
0188
1100
.428
296
2545
.723
975
5844
.324
873
922.
4978
506.
1832
3615
.171
366
36.9
0347
189
.005
227
212.
8823
2550
5.01
8802
1188
.462
560
2774
.839
132
6428
.757
360
932.
5228
296.
3069
0015
.626
507
38.3
7961
093
.455
489
225.
6552
6454
0.37
0118
1283
.539
565
3024
.574
654
7071
.633
096
942.
5480
576.
4330
3816
.095
302
39.9
1479
498
.128
263
239.
1945
8057
8.19
6026
1386
.222
730
3296
.786
373
7778
.796
406
952.
5735
386.
5616
9916
.578
161
41.5
1138
610
3.03
4676
253.
5462
5561
8.66
9748
1497
.120
549
3593
.497
147
8556
.676
047
962.
5992
736.
6929
3317
.075
506
43.1
7184
110
8.18
6410
268.
7590
3066
1.97
6630
1616
.890
192
3916
.911
890
9412
.343
651
972.
6252
666.
8267
9217
.587
771
44.8
9871
511
3.59
5731
284.
8845
7270
8.31
4994
1746
.241
408
4269
.433
960
1035
3.57
8016
982.
6515
186.
9633
2818
.115
404
46.6
9466
411
9.27
5517
301.
9776
4675
7.89
7044
1885
.940
720
4653
.683
016
1138
8.93
5818
992.
6780
337.
1025
9418
.658
866
48.5
6245
012
5.23
9293
320.
0963
0581
0.94
9837
2036
.815
978
5072
.514
488
1252
7.82
9400
100
2.70
4814
7.24
4646
19.2
1863
250
.504
948
131.
5012
5833
9.30
2084
867.
7163
2621
99.7
6125
655
29.0
4079
213
780.
6123
40
N/i
1112
1314
1516
1718
1920
11.
1100
001.
1200
001.
1300
001.
1400
001.
1500
001.
1600
001.
1700
001.
1800
001.
1900
001.
2000
002
1.23
2100
1.25
4400
1.27
6900
1.29
9600
1.32
2500
1.34
5600
1.36
8900
1.39
2400
1.41
6100
1.44
0000
31.
3676
311.
4049
281.
4428
971.
4815
441.
5208
751.
5608
961.
6016
131.
6430
321.
6851
591.
7280
004
1.51
8070
1.57
3519
1.63
0474
1.68
8960
1.74
9006
1.81
0639
1.87
3887
1.93
8778
2.00
5339
2.07
3600
51.
6850
581.
7623
421.
8424
351.
9254
152.
0113
572.
1003
422.
1924
482.
2877
582.
3863
542.
4883
20
61.
8704
151.
9738
232.
0819
522.
1949
732.
3130
612.
4363
962.
5651
642.
6995
542.
8397
612.
9859
847
2.07
6160
2.21
0681
2.35
2605
2.50
2269
2.66
0020
2.82
6220
3.00
1242
3.18
5474
3.37
9315
3.58
3181
82.
3045
382.
4759
632.
6584
442.
8525
863.
0590
233.
2784
153.
5114
533.
7588
594.
0213
854.
2998
179
2.55
8037
2.77
3079
3.00
4042
3.25
1949
3.51
7876
3.80
2961
4.10
8400
4.43
5454
4.78
5449
5.15
9780
102.
8394
213.
1058
483.
3945
673.
7072
214.
0455
584.
4114
354.
8068
285.
2338
365.
6946
846.
1917
36
113.
1517
573.
4785
503.
8358
614.
2262
324.
6523
915.
1172
655.
6239
896.
1759
266.
7766
747.
4300
8412
3.49
8451
3.89
5976
4.33
4523
4.81
7905
5.35
0250
5.93
6027
6.58
0067
7.28
7593
8.06
4242
8.91
6100
133.
8832
804.
3634
934.
8980
115.
4924
116.
1527
886.
8857
917.
6986
798.
5993
599.
5964
4810
.699
321
144.
3104
414.
8871
125.
5347
536.
2613
497.
0757
067.
9875
189.
0074
5410
.147
244
11.4
1977
312
.839
185
154.
7845
895.
4735
666.
2542
707.
1379
388.
1370
629.
2655
2110
.538
721
11.9
7374
813
.589
530
15.4
0702
2
165.
3108
946.
1303
947.
0673
268.
1372
499.
3576
2110
.748
004
12.3
3030
414
.129
023
16.1
7154
018
.488
426
175.
8950
936.
8660
417.
9860
789.
2764
6410
.761
264
12.4
6768
514
.426
456
16.6
7224
719
.244
133
22.1
8611
118
6.54
3553
7.68
9966
9.02
4268
10.5
7516
912
.375
454
14.4
6251
416
.878
953
19.6
7325
122
.900
518
26.6
2333
319
7.26
3344
8.61
2762
10.1
9742
312
.055
693
14.2
3177
216
.776
517
19.7
4837
523
.214
436
27.2
5161
631
.948
000
208.
0623
129.
6462
9311
.523
088
13.7
4349
016
.366
537
19.4
6075
923
.105
599
27.3
9303
532
.429
423
38.3
3760
021
8.94
9166
10.8
0384
813
.021
089
15.6
6757
818
.821
518
22.5
7448
127
.033
551
32.3
2378
138
.591
014
46.0
0512
022
9.93
3574
12.1
0031
014
.713
831
17.8
6103
921
.644
746
26.1
8639
831
.629
255
38.1
4206
145
.923
307
55.2
0614
423
11.0
2626
713
.552
347
16.6
2662
920
.361
585
24.8
9145
830
.376
222
37.0
0622
845
.007
632
54.6
4873
566
.247
373
2412
.239
157
15.1
7862
918
.788
091
23.2
1220
728
.625
176
35.2
3641
743
.297
287
53.1
0900
665
.031
994
79.4
9684
725
13.5
8546
417
.000
064
21.2
3054
226
.461
916
32.9
1895
340
.874
244
50.6
5782
662
.668
627
77.3
8807
395
.396
217
2615
.079
865
19.0
4007
223
.990
513
30.1
6658
437
.856
796
47.4
1412
359
.269
656
73.9
4898
092
.091
807
114.
4754
6027
16.7
3865
021
.324
881
27.1
0927
934
.389
906
43.5
3531
555
.000
382
69.3
4549
787
.259
797
109.
5892
5113
7.37
0552
2818
.579
901
23.8
8386
630
.633
486
39.2
0449
350
.065
612
63.8
0044
481
.134
232
102.
9665
6013
0.41
1208
164.
8446
6229
20.6
2369
126
.749
930
34.6
1583
944
.693
122
57.5
7545
474
.008
515
94.9
2705
112
1.50
0541
155.
1893
3819
7.81
3595
3022
.892
297
29.9
5992
239
.115
898
50.9
5015
966
.211
772
85.8
4987
711
1.06
4650
143.
3706
3818
4.67
5312
237.
3763
14
3125
.410
449
33.5
5511
344
.200
965
58.0
8318
176
.143
538
99.5
8585
712
9.94
5641
169.
1773
5321
9.76
3621
284.
8515
7732
28.2
0559
937
.581
726
49.9
4709
066
.214
826
87.5
6506
811
5.51
9594
152.
0363
9919
9.62
9277
261.
5187
1034
1.82
1892
3331
.308
214
42.0
9153
356
.440
212
75.4
8490
210
0.69
9829
134.
0027
2917
7.88
2587
235.
5625
4731
1.20
7264
410.
1862
7034
34.7
5211
847
.142
517
63.7
7743
986
.052
788
115.
8048
0315
5.44
3166
208.
1226
2727
7.96
3805
370.
3366
4549
2.22
3524
3538
.574
851
52.7
9962
072
.068
506
98.1
0017
813
3.17
5523
180.
3140
7324
3.50
3474
327.
9972
9044
0.70
0607
590.
6682
29
3642
.818
085
59.1
3557
481
.437
412
111.
8342
0315
3.15
1852
209.
1643
2428
4.89
9064
387.
0368
0252
4.43
3722
708.
8018
7537
47.5
2807
466
.231
843
92.0
2427
612
7.49
0992
176.
1246
3024
2.63
0616
333.
3319
0545
6.70
3427
624.
0761
3085
0.56
2250
3852
.756
162
74.1
7966
410
3.98
7432
145.
3397
3120
2.54
3324
281.
4515
1538
9.99
8329
538.
9100
4474
2.65
0594
1020
.674
739
58.5
5934
083
.081
224
117.
5057
9816
5.68
7293
232.
9248
2332
6.48
3757
456.
2980
4563
5.91
3852
883.
7542
0712
24.8
096
4065
.000
867
93.0
5097
013
2.78
1552
188.
8835
1426
7.86
3546
378.
7211
5853
3.86
8713
750.
3783
4510
51.6
675
1469
.771
6
4172
.150
963
104.
2170
8715
0.04
3153
215.
3272
0630
8.04
3078
439.
3165
4462
4.62
6394
885.
4464
4712
51.4
843
1763
.725
942
80.0
8756
911
6.72
3137
169.
5487
6324
5.47
3015
354.
2495
4050
9.60
7191
730.
8128
8110
44.8
268
1489
.266
421
16.4
711
4388
.897
201
130.
7299
1419
1.59
0103
279.
8392
3740
7.38
6971
591.
1443
4185
5.05
1071
1232
.895
617
72.2
270
2539
.765
344
98.6
7589
314
6.41
7503
216.
4968
1631
9.01
6730
468.
4950
1768
5.72
7436
1000
.409
814
54.8
168
2108
.950
130
47.7
183
4510
9.53
0242
163.
9876
0424
4.64
1402
363.
6790
7253
8.76
9269
795.
4438
2611
70.4
794
1716
.683
925
09.6
506
3657
.262
0
4612
1.57
8568
183.
6661
1627
6.44
4784
414.
5941
4261
9.58
4659
922.
7148
3813
69.4
609
2025
.687
029
86.4
842
4388
.714
447
134.
9522
1120
5.70
6050
312.
3826
0647
2.63
7322
712.
5223
5810
70.3
492
1602
.269
323
90.3
106
3553
.916
252
66.4
573
4814
9.79
6954
230.
3907
7635
2.99
2345
538.
8065
4781
9.40
0712
1241
.605
118
74.6
550
2820
.566
542
29.1
603
6319
.748
749
166.
2746
1925
8.03
7669
398.
8813
5061
4.23
9464
942.
3108
1914
40.2
619
2193
.346
433
28.2
685
5032
.700
875
83.6
985
5018
4.56
4827
289.
0021
9045
0.73
5925
700.
2329
8810
83.6
574
1670
.703
825
66.2
153
3927
.356
959
88.9
139
9100
.438
2
cont
inue
sov
erle
af
Tabl
eA
.1(c
onti
nued
)
N/i
1112
1314
1516
1718
1920
5120
4.86
6958
323.
6824
5350
9.33
1595
798.
2656
0712
46.2
061
1938
.016
430
02.4
719
4634
.281
171
26.8
075
1092
0.52
5852
227.
4023
2336
2.52
4347
575.
5447
0391
0.02
2792
1433
.137
022
48.0
990
3512
.892
154
68.4
517
8480
.901
013
104.
6309
5325
2.41
6579
406.
0272
6965
0.36
5514
1037
.426
016
48.1
075
2607
.794
941
10.0
838
6452
.773
010
092.
2722
1572
5.55
7154
280.
1824
0245
4.75
0541
734.
9130
3111
82.6
656
1895
.323
630
25.0
421
4808
.798
076
14.2
721
1200
9.80
3918
870.
6685
5531
1.00
2466
509.
3206
0683
0.45
1725
1348
.238
821
79.6
222
3509
.048
856
26.2
937
8984
.841
114
291.
6666
2264
4.80
23
5634
5.21
2738
570.
4390
7893
8.41
0449
1536
.992
225
06.5
655
4070
.496
665
82.7
636
1060
2.11
2517
007.
0833
2717
3.76
2757
383.
1861
3963
8.89
1768
1060
.403
817
52.1
712
2882
.550
347
21.7
761
7701
.833
412
510.
4928
2023
8.42
9132
608.
5153
5842
5.33
6614
715.
5587
8011
98.2
563
1997
.475
133
14.9
329
5477
.260
290
11.1
451
1476
2.38
1524
083.
7306
3913
0.21
8359
472.
1236
4280
1.42
5833
1354
.029
622
77.1
216
3812
.172
863
53.6
219
1054
3.03
9717
419.
6101
2865
9.63
9446
956.
2620
6052
4.05
7242
897.
5969
3315
30.0
535
2595
.918
743
83.9
987
7370
.201
412
335.
3565
2055
5.14
0034
104.
9709
5634
7.51
44
6158
1.70
3539
1005
.308
617
28.9
604
2959
.347
350
41.5
986
8549
.433
614
432.
3671
2425
5.06
5240
584.
9154
6761
7.01
7262
645.
6909
2811
25.9
456
1953
.725
333
73.6
559
5797
.838
399
17.3
430
1688
5.86
9528
620.
9769
4829
6.04
9381
140.
4207
6371
6.71
6930
1261
.059
122
07.7
096
3845
.967
766
67.5
141
1150
4.11
7819
756.
4673
3377
2.75
2757
472.
2987
9736
8.50
4864
795.
5557
9314
12.3
862
2494
.711
843
84.4
032
7667
.641
213
344.
7767
2311
5.06
6739
851.
8482
6839
2.03
5411
6842
.205
865
883.
0669
3015
81.8
725
2819
.024
349
98.2
196
8817
.787
415
479.
9410
2704
4.62
8147
025.
1809
8138
6.52
2214
0210
.646
9
6698
0.20
4292
1771
.697
231
85.4
975
5697
.970
410
140.
4555
1795
6.73
1531
642.
2149
5548
9.71
3596
849.
9614
1682
52.7
763
6710
88.0
268
1984
.300
935
99.6
122
6495
.686
211
661.
5238
2082
9.80
8537
021.
3914
6547
7.86
1911
5251
.454
120
1903
.331
668
1207
.709
722
22.4
170
4067
.561
874
05.0
823
1341
0.75
2424
162.
5779
4331
5.02
7977
263.
8770
1371
49.2
303
2422
83.9
979
6913
40.5
578
2489
.107
045
96.3
448
8441
.793
815
422.
3653
2802
8.59
0450
678.
5827
9117
1.37
4916
3207
.584
129
0740
.797
470
1488
.019
127
87.7
998
5193
.869
696
23.6
450
1773
5.72
0032
513.
1648
5929
3.94
1710
7582
.222
419
4217
.025
134
8888
.956
9
7116
51.7
012
3122
.335
858
69.0
727
1097
0.95
5320
396.
0780
3771
5.27
1269
373.
9118
1269
47.0
224
2311
18.2
598
4186
66.7
483
7218
33.3
884
3497
.016
166
32.0
521
1250
6.88
9023
455.
4898
4374
9.71
4681
167.
4768
1497
97.4
864
2750
30.7
292
5024
00.0
980
7320
35.0
611
3916
.658
074
94.2
189
1425
7.85
3526
973.
8132
5074
9.66
8994
965.
9479
1767
61.0
340
3272
86.5
677
6028
80.1
176
7422
58.9
178
4386
.657
084
68.4
674
1625
3.95
3031
019.
8852
5886
9.61
6011
1110
.159
020
8578
.020
138
9471
.015
672
3456
.141
175
2507
.398
849
13.0
558
9569
.368
118
529.
5064
3567
2.86
8068
288.
7545
1299
98.8
861
2461
22.0
637
4634
70.5
086
8681
47.3
693
7627
83.2
126
5502
.622
510
813.
3860
2112
3.63
7341
023.
7982
7921
4.95
5315
2098
.696
729
0424
.035
255
1529
.905
210
4177
6.84
3277
3089
.366
061
62.9
372
1221
9.12
6124
080.
9465
4717
7.36
7991
889.
3481
1779
55.4
751
3427
00.3
615
6563
20.5
872
1250
132.
2118
7834
29.1
963
6902
.489
713
807.
6125
2745
2.27
9054
253.
9731
1065
91.6
438
2082
07.9
059
4043
86.4
266
7810
21.4
987
1500
158.
6542
7938
06.4
079
7730
.788
515
602.
6022
3129
5.59
8162
392.
0690
1236
46.3
068
2436
03.2
499
4771
75.9
834
9294
15.5
835
1800
190.
3850
8042
25.1
128
8658
.483
117
630.
9405
3567
6.98
1871
750.
8794
1434
29.7
159
2850
15.8
024
5630
67.6
604
1106
004.
5444
2160
228.
4620
8146
89.8
752
9697
.501
119
922.
9627
4067
1.75
9382
513.
5113
1663
78.4
704
3334
68.4
888
6644
19.8
393
1316
145.
4078
2592
274.
1544
8252
05.7
614
1086
1.20
1222
512.
9479
4636
5.80
5694
890.
5380
1929
99.0
257
3901
58.1
319
7840
15.4
103
1566
213.
0353
3110
728.
9853
8357
78.3
952
1216
4.54
5325
439.
6311
5285
7.01
8310
9124
.118
722
3878
.869
845
6485
.014
392
5138
.184
218
6379
3.51
2037
3287
4.78
2484
6414
.018
613
624.
2908
2874
6.78
3160
257.
0009
1254
92.7
365
2596
99.4
890
5340
87.4
668
1091
663.
0573
2217
914.
2792
4479
449.
7388
8571
19.5
607
1525
9.20
5732
483.
8649
6869
2.98
1014
4316
.647
030
1251
.407
262
4882
.336
112
8816
2.40
7726
3931
7.99
2353
7533
9.68
66
8679
02.7
124
1709
0.31
0436
706.
7674
7830
9.99
8416
5964
.144
034
9451
.632
473
1112
.333
315
2003
1.64
1031
4078
8.41
0864
5040
7.62
3987
8772
.010
719
141.
1476
4147
8.64
7189
273.
3981
1908
58.7
656
4053
63.8
936
8554
01.4
299
1793
637.
3364
3737
538.
2089
7740
489.
1487
8897
36.9
319
2143
8.08
5346
870.
8713
1017
71.6
739
2194
87.5
805
4702
22.1
165
1000
819.
6730
2116
492.
0570
4447
670.
4686
9288
586.
9784
8910
807.
9944
2401
0.65
5652
964.
0845
1160
19.7
082
2524
10.7
176
5454
57.6
552
1170
959.
0175
2497
460.
6272
5292
727.
8576
1114
6304
.374
190
1199
6.87
3826
891.
9342
5984
9.41
5513
2262
.467
429
0272
.325
263
2730
.880
013
7002
2.05
0429
4700
3.54
0162
9834
6.15
0513
3755
65.2
489
9113
316.
5299
3011
8.96
6367
629.
8395
1507
79.2
128
3338
13.1
740
7339
67.8
208
1602
925.
7990
3477
464.
1773
7495
031.
9191
1605
0678
.298
792
1478
1.34
8233
733.
2423
7642
1.71
8717
1888
.302
638
3885
.150
185
1402
.672
118
7542
3.18
4841
0340
7.72
9389
1908
7.98
3819
2608
13.9
585
9316
407.
2965
3778
1.23
1486
356.
5421
1959
52.6
650
4414
67.9
226
9876
27.0
997
2194
245.
1262
4842
021.
1205
1061
3714
.700
723
1129
76.7
502
9418
212.
0991
4231
4.97
9197
582.
8926
2233
86.0
381
5076
88.1
110
1145
647.
4356
2567
266.
7977
5713
584.
9222
1263
0320
.493
827
7355
72.1
002
9520
215.
4301
4739
2.77
6611
0268
.668
625
4660
.083
458
3841
.327
613
2895
1.02
5330
0370
2.15
3367
4203
0.20
8215
0300
81.3
876
3328
2686
.520
2
9622
439.
1274
5307
9.90
9812
4603
.595
529
0312
.495
167
1417
.526
815
4158
3.18
9435
1433
1.51
9479
5559
5.64
5717
8857
96.8
513
3993
9223
.824
397
2490
7.43
1459
449.
4990
1408
02.0
630
3309
56.2
444
7721
30.1
558
1788
236.
4997
4111
767.
8777
9387
602.
8619
2128
4098
.253
047
9270
68.5
891
9827
647.
2488
6658
3.43
8915
9106
.331
137
7290
.118
688
7949
.679
220
7435
4.33
9648
1076
8.41
6911
0773
71.3
771
2532
8076
.921
157
5124
82.3
070
9930
688.
4462
7457
3.45
1517
9790
.154
243
0110
.735
210
2114
2.13
1024
0625
1.03
3956
2859
9.04
7713
0712
98.2
250
3014
0411
.536
169
0149
78.7
683
100
3406
4.17
5383
522.
2657
2031
62.8
742
4903
26.2
381
1174
313.
4507
2791
251.
1994
6585
460.
8858
1542
4131
.905
535
8670
89.7
280
8281
7974
.522
0
N/i
2122
2324
2526
2728
2930
11.
2100
001.
2200
001.
2300
001.
2400
001.
2500
001.
2600
001.
2700
001.
2800
001.
2900
001.
3000
002
1.46
4100
1.48
8400
1.51
2900
1.53
7600
1.56
2500
1.58
7600
1.61
2900
1.63
8400
1.66
4100
1.69
0000
31.
7715
611.
8158
481.
8608
671.
9066
241.
9531
252.
0003
762.
0483
832.
0971
522.
1466
892.
1970
004
2.14
3589
2.21
5335
2.28
8866
2.36
4214
2.44
1406
2.52
0474
2.60
1446
2.68
4355
2.76
9229
2.85
6100
52.
5937
422.
7027
082.
8153
062.
9316
253.
0517
583.
1757
973.
3038
373.
4359
743.
5723
053.
7129
30
63.
1384
283.
2973
043.
4628
263.
6352
153.
8146
974.
0015
044.
1958
734.
3980
474.
6082
744.
8268
097
3.79
7498
4.02
2711
4.25
9276
4.50
7667
4.76
8372
5.04
1895
5.32
8759
5.62
9500
5.94
4673
6.27
4852
84.
5949
734.
9077
075.
2389
095.
5895
075.
9604
646.
3527
886.
7675
237.
2057
597.
6686
288.
1573
079
5.55
9917
5.98
7403
6.44
3859
6.93
0988
7.45
0581
8.00
4513
8.59
4755
9.22
3372
9.89
2530
10.6
0449
910
6.72
7500
7.30
4631
7.92
5946
8.59
4426
9.31
3226
10.0
8568
610
.915
339
11.8
0591
612
.761
364
13.7
8584
9
118.
1402
758.
9116
509.
7489
1410
.657
088
11.6
4153
212
.707
965
13.8
6248
015
.111
573
16.4
6216
017
.921
604
129.
8497
3310
.872
213
11.9
9116
413
.214
789
14.5
5191
516
.012
035
17.6
0535
019
.342
813
21.2
3618
623
.298
085
1311
.918
177
13.2
6410
014
.749
132
16.3
8633
818
.189
894
20.1
7516
522
.358
794
24.7
5880
127
.394
680
30.2
8751
114
14.4
2099
416
.182
202
18.1
4143
220
.319
059
22.7
3736
825
.420
707
28.3
9566
831
.691
265
35.3
3913
739
.373
764
1517
.449
402
19.7
4228
722
.313
961
25.1
9563
328
.421
709
32.0
3009
136
.062
499
40.5
6481
945
.587
487
51.1
8589
3
1621
.113
777
24.0
8559
027
.446
172
31.2
4258
535
.527
137
40.3
5791
545
.799
373
51.9
2296
958
.807
859
66.5
4166
117
25.5
4767
029
.384
420
33.7
5879
238
.740
806
44.4
0892
150
.850
973
58.1
6520
466
.461
400
75.8
6213
786
.504
159
1830
.912
681
35.8
4899
241
.523
314
48.0
3859
955
.511
151
64.0
7222
673
.869
809
85.0
7059
297
.862
157
112.
4554
0719
37.4
0434
343
.735
771
51.0
7367
659
.567
863
69.3
8893
980
.731
005
93.8
1465
810
8.89
0357
126.
2421
8314
6.19
2029
2045
.259
256
53.3
5764
062
.820
622
73.8
6415
086
.736
174
101.
7210
6611
9.14
4615
139.
3796
5716
2.85
2416
190.
0496
38
cont
inue
sov
erle
af
Tabl
eA
.1(c
onti
nued
)
N/i
1112
1314
1516
1718
1920
2154
.763
699
65.0
9632
177
.269
364
91.5
9154
610
8.42
0217
128.
1685
4315
1.31
3661
178.
4059
6221
0.07
9617
247.
0645
2922
66.2
6407
679
.417
512
95.0
4131
811
3.57
3517
135.
5252
7216
1.49
2364
192.
1683
5022
8.35
9631
271.
0027
0532
1.18
3888
2380
.179
532
96.8
8936
411
6.90
0822
140.
8311
6116
9.40
6589
203.
4803
7924
4.05
3804
292.
3003
2734
9.59
3490
417.
5390
5424
97.0
1723
411
8.20
5024
143.
7880
1017
4.63
0639
211.
7582
3725
6.38
5277
309.
9483
3237
4.14
4419
450.
9756
0254
2.80
0770
2511
7.39
0853
144.
2101
3017
6.85
9253
216.
5419
9326
4.69
7796
323.
0454
5039
3.63
4381
478.
9048
5758
1.75
8527
705.
6410
01
2614
2.04
2932
175.
9363
5821
7.53
6881
268.
5120
7133
0.87
2245
407.
0372
6649
9.91
5664
612.
9982
1675
0.46
8500
917.
3333
0227
171.
8719
4821
4.64
2357
267.
5703
6433
2.95
4968
413.
5903
0651
2.86
6956
634.
8928
9378
4.63
7717
968.
1043
6511
92.5
333
2820
7.96
5057
261.
8636
7532
9.11
1547
412.
8641
6051
6.98
7883
646.
2123
6480
6.31
3974
1004
.336
312
48.8
546
1550
.293
329
251.
6377
1931
9.47
3684
404.
8072
0351
1.95
1559
646.
2348
5481
4.22
7579
1024
.018
712
85.5
504
1611
.022
520
15.3
813
3030
4.48
1640
389.
7578
9449
7.91
2860
634.
8199
3380
7.79
3567
1025
.926
713
00.5
038
1645
.504
620
78.2
190
2619
.995
631
368.
4227
8447
5.50
4631
612.
4328
1878
7.17
6717
1009
.742
012
92.6
677
1651
.639
821
06.2
458
2680
.902
534
05.9
943
3244
5.79
1568
580.
1156
5075
3.29
2366
976.
0991
2912
62.1
774
1628
.761
320
97.5
826
2695
.994
734
58.3
642
4427
.792
633
539.
4077
9870
7.74
1093
926.
5496
1012
10.3
629
1577
.721
820
52.2
392
2663
.929
934
50.8
732
4461
.289
857
56.1
304
3465
2.68
3435
863.
4441
3311
39.6
560
1500
.850
019
72.1
523
2585
.821
533
83.1
910
4417
.117
757
55.0
639
7482
.969
635
789.
7469
5710
53.4
018
1401
.776
918
61.0
540
2465
.190
332
58.1
350
4296
.652
556
53.9
106
7424
.032
497
27.8
604
3695
5.59
3818
1285
.150
217
24.1
856
2307
.707
030
81.4
879
4105
.250
154
56.7
487
7237
.005
695
77.0
018
1264
6.21
8637
1156
.268
5215
67.8
833
2120
.748
328
61.5
567
3851
.859
951
72.6
152
6930
.070
992
63.3
671
1235
4.33
2416
440.
0841
3813
99.0
849
1912
.817
626
08.5
204
3548
.330
348
14.8
249
6517
.495
188
01.1
900
1185
7.10
9915
937.
0888
2137
2.10
9439
1692
.892
723
33.6
375
3208
.480
143
99.9
295
6018
.531
182
12.0
438
1117
7.51
1315
177.
1007
2055
8.84
4527
783.
7422
4020
48.4
002
2847
.037
839
46.4
305
5455
.912
675
23.1
638
1034
7.17
5214
195.
4393
1942
6.68
8926
520.
9094
3611
8.86
48
4124
78.5
643
3473
.386
148
54.1
095
6765
.331
794
03.9
548
1303
7.44
0818
028.
2080
2486
6.16
1834
211.
9731
4695
4.52
4342
2999
.062
842
37.5
310
5970
.554
783
89.0
113
1175
4.94
3516
427.
1754
2289
5.82
4131
828.
6871
4413
3.44
5361
040.
8815
4336
28.8
659
5169
.787
873
43.7
823
1040
2.37
4014
693.
6794
2069
8.24
1029
077.
6966
4074
0.71
9556
932.
1445
7935
3.14
6044
4390
.927
863
07.1
411
9032
.852
212
898.
9437
1836
7.09
9226
079.
7837
3692
8.67
4752
148.
1210
7344
2.46
6410
3159
.089
845
5313
.022
676
94.7
122
1111
0.40
8215
994.
6902
2295
8.87
4032
860.
5275
4689
9.41
6966
749.
5949
9474
0.78
1613
4106
.816
7
4664
28.7
574
9387
.548
913
665.
8021
1983
3.41
5828
698.
5925
4140
4.26
4659
562.
2594
8543
9.48
1412
2215
.608
317
4338
.861
747
7778
.796
411
452.
8096
1680
8.93
6524
593.
4356
3587
3.24
0752
169.
3734
7564
4.06
9510
9362
.536
215
7658
.134
722
6640
.520
248
9412
.343
713
972.
4277
2067
4.99
1930
495.
8602
4484
1.55
0965
733.
4105
‘960
67.9
683
1399
84.0
464
2033
78.9
938
2946
32.6
763
4911
388.
9358
1704
6.36
1825
430.
2401
3781
4.86
6656
051.
9386
8282
4.09
7212
2006
.319
717
9179
.579
426
2358
.902
038
3022
.479
250
1378
0.61
2320
796.
5615
3127
9.19
5346
890.
4346
7006
4.92
3210
4358
.362
515
4948
.026
022
9349
.861
633
8442
.983
649
7929
.223
0
Tabl
eA
.2Si
ngle
paym
entp
rese
ntw
orth
fact
or.T
ofin
dP
for
agi
ven
FP
=F
(1+
i)−N
N/i
12
34
56
78
910
10.
9900
990
0.98
0392
20.
9708
738
0.96
1538
50.
9523
810
0.94
3396
20.
9345
794
0.92
5925
90.
9174
312
0.90
9090
92
0.98
0296
00.
9611
688
0.94
2595
90.
9245
562
0.90
7029
50.
8899
964
0.87
3438
70.
8573
388
0.84
1680
00.
8264
463
30.
9705
901
0.94
2322
30.
9151
417
0.88
8996
40.
8638
376
0.83
9619
30.
8162
979
0.79
3832
20.
7721
835
0.75
1314
84
0.96
0980
30.
9238
454
0.88
8487
00.
8548
042
0.82
2702
50.
7920
937
0.76
2895
20.
7350
299
0.70
8425
20.
6830
135
50.
9514
657
0.90
5730
80.
8626
088
0.82
1927
10.
7835
262
0.74
7258
20.
7129
862
0.68
0583
20.
6499
314
0.62
0921
3
60.
9420
452
0.88
7971
40.
8374
843
0.79
0314
50.
7462
154
0.70
4960
50.
6663
422
0.63
0169
60.
5962
673
0.56
4473
97
0.93
2718
10.
8705
602
0.81
3091
50.
7599
178
0.71
0681
30.
6650
571
0.62
2749
70.
5834
904
0.54
7034
20.
5131
581
80.
9234
832
0.85
3490
40.
7894
092
0.73
0690
20.
6768
394
0.62
7412
40.
5820
091
0.54
0268
90.
5018
663
0.46
6507
49
0.91
4339
80.
8367
553
0.76
6416
70.
7025
867
0.64
4608
90.
5918
985
0.54
3933
70.
5002
490
0.46
0427
80.
4240
976
100.
9052
870
0.82
0348
30.
7440
939
0.67
5564
20.
6139
133
0.55
8394
80.
5083
493
0.46
3193
50.
4224
108
0.38
5543
3
110.
8963
237
0.80
4263
00.
7224
213
0.64
9580
90.
5846
793
0.52
6787
50.
4750
928
0.42
8882
90.
3875
329
0.35
0493
912
0.88
7449
20.
7884
932
0.70
1379
90.
6245
970
0.55
6837
40.
4969
694
0.44
4012
00.
3971
138
0.35
5534
70.
3186
308
130.
8786
626
0.77
3032
50.
6809
513
0.60
0574
10.
5303
214
0.46
8839
00.
4149
644
0.36
7697
90.
3261
786
0.28
9664
414
0.86
9963
00.
7578
750
0.66
1117
80.
5774
751
0.50
5068
00.
4423
010
0.38
7817
20.
3404
610
0.29
9246
50.
2633
313
150.
8613
495
0.74
3014
70.
6418
619
0.55
5264
50.
4810
171
0.41
7265
10.
3624
460
0.31
5241
70.
2745
380
0.23
9392
0
160.
8528
213
0.72
8445
80.
6231
669
0.53
3908
20.
4581
115
0.39
3646
30.
3387
346
0.29
1890
50.
2518
698
0.21
7629
117
0.84
4377
50.
7141
626
0.60
5016
40.
5133
732
0.43
6296
70.
3713
644
0.31
6574
40.
2702
690
0.23
1073
20.
1978
447
180.
8360
173
0.70
0159
40.
5873
946
0.49
3628
10.
4155
207
0.35
0343
80.
2958
639
0.25
0249
00.
2119
937
0.17
9858
819
0.82
7739
90.
6864
308
0.57
0286
00.
4746
424
0.39
5734
00.
3305
130
0.27
6508
30.
2317
121
0.19
4489
70.
1635
080
200.
8195
445
0.67
2971
30.
5536
758
0.45
6386
90.
3768
895
0.31
1804
70.
2584
190
0.21
4548
20.
1784
309
0.14
8643
6
210.
8114
302
0.65
9775
80.
5375
493
0.43
8833
60.
3589
424
0.29
4155
40.
2415
131
0.19
8655
70.
1636
981
0.13
5130
622
0.80
3396
20.
6468
390
0.52
1892
50.
4219
554
0.34
1849
90.
2775
051
0.22
5713
20.
1839
405
0.15
0181
70.
1228
460
230.
7954
418
0.63
4155
90.
5066
917
0.40
5726
30.
3255
713
0.26
1797
30.
2109
469
0.17
0315
30.
1377
814
0.11
1678
224
0.78
7566
10.
6217
215
0.49
1933
70.
3901
215
0.31
0067
90.
2469
785
0.19
7146
60.
1576
993
0.12
6404
90.
1015
256
250.
7797
684
0.60
9530
90.
4776
056
0.37
5116
80.
2953
028
0.23
2998
60.
1842
492
0.14
6017
90.
1159
678
0.09
2296
0
260.
7720
480
0.59
7579
30.
4636
947
0.36
0689
20.
2812
407
0.21
9810
00.
1721
955
0.13
5201
80.
1063
925
0.08
3905
527
0.76
4403
90.
5858
620
0.45
0189
10.
3468
166
0.26
7848
30.
2073
680
0.16
0930
40.
1251
868
0.09
7607
80.
0762
777
280.
7568
356
0.57
4374
60.
4370
768
0.33
3477
50.
2550
936
0.19
5630
10.
1504
022
0.11
5913
70.
0895
484
0.06
9343
329
0.74
9342
10.
5631
123
0.42
4346
40.
3206
514
0.24
2946
30.
1845
567
0.14
0562
80.
1073
275
0.08
2154
50.
0630
394
300.
7419
229
0.55
2070
90.
4119
868
0.30
8318
70.
2313
774
0.17
4110
10.
1313
671
0.09
9377
30.
0753
711
0.05
7308
6
cont
inue
sov
erle
af
Tabl
eA
.2(c
onti
nued
)
N/i
12
34
56
78
910
310.
7345
771
0.54
1246
00.
3999
871
0.29
6460
30.
2203
595
0.16
4254
80.
1227
730
0.09
2016
00.
0691
478
0.05
2098
732
0.72
7304
10.
5306
333
0.38
8337
00.
2850
579
0.20
9866
20.
1549
574
0.11
4741
10.
0852
000
0.06
3438
40.
0473
624
330.
7201
031
0.52
0228
70.
3770
262
0.27
4094
20.
1998
725
0.14
6186
20.
1072
347
0.07
8888
90.
0582
003
0.04
3056
834
0.71
2973
30.
5100
282
0.36
6044
90.
2635
521
0.19
0354
80.
1379
115
0.10
0219
30.
0730
453
0.05
3394
80.
0391
425
350.
7059
142
0.50
0027
60.
3553
834
0.25
3415
50.
1812
903
0.13
0105
20.
0936
629
0.06
7634
50.
0489
861
0.03
5584
1
360.
6989
249
0.49
0223
20.
3450
324
0.24
3668
70.
1726
574
0.12
2740
80.
0875
355
0.06
2624
60.
0449
413
0.03
2349
237
0.69
2004
90.
4806
109
0.33
4982
90.
2342
968
0.16
4435
60.
1157
932
0.08
1808
80.
0579
857
0.04
1230
60.
0294
083
380.
6851
534
0.47
1187
20.
3252
262
0.22
5285
40.
1566
054
0.10
9238
90.
0764
569
0.05
3690
50.
0378
262
0.02
6734
939
0.67
8369
70.
4619
482
0.31
5753
50.
2166
206
0.14
9148
00.
1030
555
0.07
1455
00.
0497
134
0.03
4703
00.
0243
044
400.
6716
531
0.45
2890
40.
3065
568
0.20
8289
00.
1420
457
0.09
7222
20.
0667
804
0.04
6030
90.
0318
376
0.02
2094
9
410.
6650
031
0.44
4010
20.
2976
280
0.20
0277
90.
1352
816
0.09
1719
00.
0624
116
0.04
2621
20.
0292
088
0.02
0086
342
0.65
8418
90.
4353
041
0.28
8959
20.
1925
749
0.12
8839
60.
0865
274
0.05
8328
60.
0394
641
0.02
6797
10.
0182
603
430.
6518
999
0.42
6768
80.
2805
429
0.18
5168
20.
1227
044
0.08
1629
60.
0545
127
0.03
6540
80.
0245
845
0.01
6600
244
0.64
5445
50.
4184
007
0.27
2371
80.
1780
463
0.11
6861
30.
0770
091
0.05
0946
40.
0338
341
0.02
2554
50.
0150
911
450.
6390
549
0.41
0196
80.
2644
386
0.17
1198
40.
1112
965
0.07
2650
10.
0476
135
0.03
1327
90.
0206
922
0.01
3719
2
460.
6327
276
0.40
2153
70.
2567
365
0.16
4613
90.
1059
967
0.06
8537
80.
0444
986
0.02
9007
30.
0189
837
0.01
2472
047
0.62
6463
00.
3942
684
0.24
9258
80.
1582
826
0.10
0949
20.
0646
583
0.04
1587
50.
0268
586
0.01
7416
20.
0113
382
480.
6202
604
0.38
6537
60.
2419
988
0.15
2194
80.
0961
421
0.06
0998
40.
0388
668
0.02
4869
10.
0159
782
0.01
0307
449
0.61
4119
20.
3789
584
0.23
4950
30.
1463
411
0.09
1563
90.
0575
457
0.03
6324
10.
0230
269
0.01
4658
90.
0093
704
500.
6080
388
0.37
1527
90.
2281
071
0.14
0712
60.
0872
037
0.05
4288
40.
0339
478
0.02
1321
20.
0134
485
0.00
8518
6
510.
6020
186
0.36
4243
00.
2214
632
0.13
5300
60.
0830
512
0.05
1215
40.
0317
269
0.01
9741
90.
0123
381
0.00
7744
152
0.59
6058
10.
3571
010
0.21
5012
80.
1300
967
0.07
9096
40.
0483
164
0.02
9651
30.
0182
795
0.01
1319
40.
0070
401
530.
5901
565
0.35
0099
00.
2087
503
0.12
5093
00.
0753
299
0.04
5581
60.
0277
115
0.01
6925
50.
0103
847
0.00
6400
154
0.58
4313
40.
3432
343
0.20
2670
20.
1202
817
0.07
1742
70.
0430
015
0.02
5898
60.
0156
717
0.00
9527
30.
0058
183
550.
5785
281
0.33
6504
20.
1967
672
0.11
5655
50.
0683
264
0.04
0567
40.
0242
043
0.01
4510
90.
0087
406
0.00
5289
4
560.
5728
001
0.32
9906
10.
1910
361
0.11
1207
20.
0650
728
0.03
8271
20.
0226
208
0.01
3436
00.
0080
189
0.00
4808
557
0.56
7128
80.
3234
374
0.18
5471
90.
1069
300
0.06
1974
10.
0361
049
0.02
1141
00.
0124
407
0.00
7356
80.
0043
714
580.
5615
137
0.31
7095
50.
1800
698
0.10
2817
30.
0590
229
0.03
4061
20.
0197
579
0.01
1519
20.
0067
494
0.00
3974
059
0.55
5954
10.
3108
779
0.17
4825
10.
0988
628
0.05
6213
30.
0321
332
0.01
8465
30.
0106
659
0.00
6192
10.
0036
127
600.
5504
496
0.30
4782
30.
1697
331
0.09
5060
40.
0535
355
0.03
0314
30.
0172
573
0.00
9875
90.
0056
808
0.00
3284
3
610.
5449
996
0.29
8806
10.
1647
894
0.09
1404
20.
0509
862
0.02
8598
40.
0161
283
0.00
9144
30.
0052
118
0.00
2985
762
0.53
9603
60.
2929
472
0.15
9989
70.
0878
887
0.04
5583
0.02
6979
70.
0150
732
0.00
8467
00.
0047
814
0.00
2714
363
0.53
4261
00.
2872
031
0.15
5329
80.
0845
084
0.04
6246
00.
0254
525
0.01
4087
10.
0078
398
0.00
4386
60.
0024
675
640.
5289
713
0.28
1571
70.
1508
057
0.08
1258
00.
0440
438
0.02
4011
80.
0131
655
0.00
7259
00.
0040
244
0.00
2243
265
0.52
3733
90.
2760
507
0.14
6413
30.
0781
327
0.04
1946
50.
0226
526
0.01
2304
20.
0067
213
0.00
3692
10.
0020
393
660.
5185
484
0.27
0637
90.
1421
488
0.07
5127
60.
0399
490
0.02
1370
40.
0114
993
0.00
6223
50.
0033
873
0.00
1853
967
0.51
3414
30.
2653
313
0.13
8008
50.
0722
381
0.03
8046
70.
0201
608
0.01
0747
00.
0057
625
0.00
3107
60.
0016
853
680.
5083
310
0.26
0128
70.
1339
889
0.06
9459
70.
0362
349
0.01
9019
60.
0100
439
0.00
5335
60.
0028
510
0.00
1532
169
0.50
3298
00.
2550
282
0.13
0086
30.
0667
882
0.03
4509
50.
0179
430
0.00
9386
80.
0049
404
0.00
2615
60.
0013
929
700.
4983
149
0.25
0027
60.
1262
974
0.06
4219
40.
0328
662
0.01
6927
40.
0087
727
0.00
4574
40.
0023
996
0.00
1266
2
710.
4933
810
0.24
5125
10.
1226
188
0.06
1749
40.
0313
011
0.01
5969
20.
0081
988
0.00
4235
60.
0022
015
0.00
1151
172
0.48
8496
10.
2403
187
0.11
9047
40.
0593
744
0.02
9810
60.
0150
653
0.00
7662
50.
0039
218
0.00
2019
70.
0010
465
730.
4836
595
0.23
5606
60.
1155
800
0.05
7090
80.
0283
910
0.01
4212
50.
0071
612
0.00
3631
30.
0018
530
0.00
0951
374
0.47
8870
80.
2309
869
0.01
1221
360.
0548
950
0.02
7039
10.
0134
081
0.00
6692
70.
0033
623
0.00
1700
00.
0008
649
750.
4741
295
0.22
6457
70.
1089
452
0.05
2783
70.
0257
515
0.01
2649
10.
0062
548
0.00
3113
30.
0015
596
0.00
0786
2
760.
4694
351
0.22
2017
40.
1057
721
0.05
0753
50.
0245
252
0.01
1933
10.
0058
457
0.00
2882
70.
0014
308
0.00
0714
877
0.46
4787
30.
2176
641
0.10
2691
30.
0488
015
0.02
3357
40.
0112
577
0.00
5463
20.
0026
691
0.00
1312
70.
0006
498
780.
4601
854
0.21
3396
20.
0997
003
0.04
6924
50.
0222
451
0.01
0620
40.
0051
058
0.00
2471
40.
0012
043
0.00
0590
779
0.45
5629
10.
2092
119
0.09
6796
40.
0451
197
0.02
1185
80.
0100
193
0.00
4771
80.
0022
884
0.00
1104
90.
0005
370
800.
4511
179
0.20
5109
70.
0939
771
0.04
3384
30.
0201
770
0.00
9452
20.
0044
596
0.00
2118
80.
0010
136
0.00
0488
2
810.
4466
514
0.20
1088
00.
0912
399
0.04
1715
70.
0192
162
0.00
8917
10.
0041
679
0.00
1961
90.
0009
299
0.00
0443
882
0.44
2229
10.
1971
451
0.08
8582
40.
0401
112
0.01
8301
10.
0084
124
0.00
3895
20.
0018
166
0.00
0853
20.
0004
035
830.
4378
506
0.19
3279
50.
0860
024
0.03
8568
50.
0174
296
0.00
7936
20.
0036
404
0.00
1682
00.
0007
827
0.00
0366
884
0.43
3515
50.
1894
897
0.08
3497
40.
0370
851
0.01
6599
60.
0074
870
0.00
3402
20.
0015
574
0.00
0718
10.
0003
334
850.
4292
232
0.18
5774
20.
0810
655
0.03
5658
80.
0158
092
0.00
7063
20.
0031
796
0.00
1442
10.
0006
588
0.00
0303
1
860.
4249
735
0.18
2131
60.
0787
043
0.03
4287
30.
0150
564
0.00
6663
40.
0029
716
0.00
1335
20.
0006
044
0.00
0275
687
0.42
0765
80.
1785
604
0.07
6412
00.
0329
685
0.01
4339
40.
0062
862
0.00
2777
20.
0012
363
0.00
0554
50.
0002
505
880.
4165
998
0.17
5059
20.
0741
864
0.03
1700
50.
0136
566
0.00
5930
40.
0025
955
0.00
1144
70.
0005
087
0.00
0227
789
0.41
2475
10.
1716
266
0.07
2025
60.
0304
813
0.01
3006
30.
0055
947
0.00
2425
70.
0010
600
0.00
0466
70.
0002
070
900.
4083
912
0.16
8261
40.
0699
278
0.02
9308
90.
0123
869
0.00
5278
00.
0022
670
0.00
0981
40.
0004
282
0.00
0188
2
910.
4043
477
0.16
4962
20.
0678
911
0.02
8181
60.
0117
971
0.00
4979
30.
0021
187
0.00
0908
70.
0003
928
0.00
0171
192
0.40
0344
30.
1617
276
0.06
5913
60.
0270
977
0.01
1235
30.
0046
974
0.00
1980
10.
0008
414
0.00
0360
40.
0001
556
930.
3963
805
0.15
8556
50.
0639
938
0.02
6055
50.
0107
003
0.00
4431
50.
0018
506
0.00
0779
10.
0003
306
0.00
0141
494
0.39
2455
90.
1554
475
0.06
2129
90.
0250
534
0.01
0190
70.
0041
807
0.00
1729
50.
0007
214
0.00
0303
30.
0001
286
950.
3885
702
0.15
2399
50.
0603
203
0.02
4089
80.
0097
055
0.00
3944
10.
0016
164
0.00
0667
90.
0002
783
0.00
0116
9
cont
inue
sov
erle
af
Tabl
eA
.2(c
onti
nued
)
N/i
12
34
56
78
910
960.
3847
230
0.14
9411
30.
0585
634
0.02
3163
20.
0092
433
0.00
3720
80.
0015
106
0.00
0618
50.
0002
553
0.00
0106
297
0.38
0913
80.
1464
817
0.05
6857
70.
0222
724
0.00
8803
10.
0035
102
0.00
1411
80.
0005
727
0.00
0234
20.
0000
966
980.
3771
424
0.14
3609
50.
0552
016
0.02
1415
70.
0083
840
0.00
3311
50.
0013
194
0.00
0530
20.
0002
149
0.00
0087
899
0.37
3408
30.
1479
360.
0535
938
0.02
0592
00.
0079
847
0.00
3124
10.
0012
331
0.00
0491
00.
0001
971
0.00
0079
810
00.
3697
112
0.13
8033
00.
0520
328
0.01
9800
00.
0076
045
0.00
2947
20.
0011
525
0.00
0454
60.
0001
809
0.00
0072
6
N/i
1112
1314
1516
1718
1920
10.
9009
009
0.89
2857
10.
8849
558
0.87
7193
00.
8695
652
0.86
2069
00.
8547
009
0.84
7457
60.
8403
361
0.83
3333
32
0.81
1622
40.
7971
939
0.78
3146
70.
7694
675
0.75
6143
70.
7431
629
0.73
0513
60.
7181
844
0.70
6164
80.
6944
444
30.
7311
914
0.71
1780
20.
6930
502
0.67
4971
50.
6575
162
0.64
0657
70.
6243
706
0.60
8630
90.
5934
158
0.57
8703
74
0.65
8731
00.
6355
181
0.61
3318
70.
5920
803
0.57
1753
20.
5522
911
0.53
3650
00.
5157
889
0.49
8668
80.
4822
531
50.
5934
513
0.56
7426
90.
5427
599
0.51
9368
70.
4971
767
0.47
6113
00.
4561
112
0.43
7109
20.
4190
494
0.40
1877
6
60.
5346
408
0.50
6631
10.
4803
185
0.45
5586
50.
4323
276
0.41
0442
30.
3898
386
0.37
0431
50.
3521
423
0.33
4898
07
0.48
1658
40.
4523
492
0.42
5060
60.
3996
373
0.37
5937
00.
3538
295
0.33
3195
40.
3139
250
0.29
5917
90.
2790
816
80.
4339
265
0.40
3883
20.
3761
599
0.35
0559
10.
3269
018
0.30
5025
50.
2847
824
0.26
6038
20.
2486
705
0.23
2568
09
0.39
0924
80.
3606
100
0.33
2884
80.
3075
079
0.28
4262
40.
2629
530
0.24
3403
70.
2254
561
0.20
8966
80.
1938
067
100.
3521
845
0.32
1973
20.
2945
883
0.26
9743
80.
2471
847
0.22
6683
60.
2080
374
0.19
1064
50.
1756
024
0.16
1505
6
110.
3172
833
0.28
7476
10.
2606
977
0.23
6617
40.
2149
432
0.19
5416
90.
1778
097
0.16
1919
00.
1475
650
0.13
4588
012
0.28
5840
80.
2566
751
0.23
0705
90.
2075
591
0.18
6907
20.
1684
628
0.15
1974
10.
1372
195
0.12
4004
20.
1121
567
130.
2575
143
0.22
9174
20.
2041
645
0.18
2069
40.
1625
280
0.14
5226
60.
1298
924
0.11
6287
70.
1042
052
0.09
3463
914
0.23
1994
80.
2046
198
0.18
0676
60.
1597
100
0.14
1328
70.
1251
953
0.11
1019
20.
0985
489
0.08
7567
40.
0778
866
150.
2090
043
0.18
2696
30.
1598
908
0.14
0096
50.
1228
945
0.10
7927
00.
0948
882
0.08
3516
00.
0735
861
0.06
4905
5
160.
1882
922
0.16
3121
70.
1414
962
0.12
2891
70.
1068
648
0.09
3040
50.
0811
010
0.07
0776
30.
0618
370
0.05
4087
917
0.16
9632
60.
1456
443
0.12
5217
90.
1077
997
0.09
2925
90.
0802
074
0.06
9317
10.
0599
799
0.05
1963
90.
0450
732
180.
1528
222
0.13
0039
60.
1108
123
0.09
4511
0.08
0805
10.
0691
443
0.05
9245
40.
0508
304
0.04
3667
10.
0375
610
190.
1376
776
0.11
6106
80.
0980
640
0.08
2948
40.
0702
653
0.05
9607
10.
0506
371
0.04
3076
60.
0366
951
0.03
1300
920
0.12
4033
90.
1036
668
0.08
6782
30.
0727
617
0.06
1100
30.
0513
855
0.04
3279
60.
0365
056
0.03
0836
20.
0260
841
210.
1117
423
0.09
2559
60.
0767
985
0.06
3826
10.
0531
307
0.04
4297
80.
0369
911
0.03
0937
00.
0259
128
0.02
1736
722
0.10
0668
70.
0826
425
0.06
7963
30.
0559
878
0.04
6200
60.
0381
878
0.03
1616
30.
0262
178
0.02
1775
40.
0181
139
230.
0906
925
0.07
3788
00.
0601
445
0.04
9112
10.
0401
744
0.03
2920
50.
0270
225
0.02
2218
50.
0182
987
0.01
5094
924
0.08
1705
00.
0658
821
0.05
3225
20.
0430
808
0.03
4934
30.
0283
797
0.02
3096
10.
0188
292
0.01
5377
00.
0125
791
250.
0736
081
0.05
8823
30.
0471
020
0.03
7790
20.
0303
776
0.02
4465
30.
0197
403
0.01
5956
90.
0129
219
0.01
0482
6
260.
0663
136
0.05
2520
80.
0416
831
0.03
3149
30.
0264
153
0.02
1090
80.
0168
720
0.01
3522
80.
0108
587
0.00
8735
527
0.05
9742
00.
0468
936
0.03
6887
70.
0290
783
0.02
2969
90.
0181
817
0.01
4420
50.
0114
600
0.00
9125
00.
0072
796
280.
0538
216
0.04
1869
30.
0326
440
0.02
5507
30.
0199
738
0.01
5673
90.
0123
253
0.00
9711
90.
0076
681
0.00
6066
329
0.04
8487
90.
0373
833
0.02
8888
50.
0223
748
0.01
7368
50.
0135
120
0.01
0534
40.
0082
304
0.00
6443
70.
0050
553
300.
0436
828
0.03
3377
90.
0255
651
0.01
9627
00.
0151
031
0.01
1648
20.
0090
038
0.00
6974
90.
0054
149
0.00
4212
7
310.
0393
539
0.02
9801
70.
0226
239
0.01
7216
70.
0131
331
0.01
0041
60.
0076
955
0.00
5911
00.
0045
503
0.00
3510
632
0.03
5454
00.
0266
087
0.02
0021
20.
0151
024
0.01
1420
10.
0086
565
0.00
6577
40.
0050
093
0.00
3823
80.
0029
255
330.
0319
405
0.02
3757
70.
0177
179
0.01
3247
70.
0099
305
0.00
7462
50.
0056
217
0.00
4245
20.
0032
133
0.00
2437
934
0.02
8775
20.
0212
123
0.01
5679
50.
0116
208
0.00
8635
20.
0064
332
0.00
4804
90.
0035
976
0.00
2700
20.
0020
316
350.
0259
236
0.01
8939
50.
0138
757
0.01
0193
70.
0075
089
0.00
5545
90.
0041
067
0.00
3048
80.
0022
691
0.00
1693
0
360.
0233
546
0.01
6910
30.
0122
794
0.00
8941
80.
0065
295
0.00
4780
90.
0035
100
0.00
2583
70.
0019
068
0.00
1410
837
0.02
1040
20.
0150
985
0.01
0866
70.
0078
437
0.00
5677
80.
0041
215
0.00
3000
00.
0021
896
0.00
1602
40.
0011
757
380.
0189
551
0.01
3480
80.
0096
165
0.00
6880
40.
0049
372
0.00
3553
00.
0025
641
0.00
1855
60.
0013
465
0.00
0979
739
0.01
7076
70.
0120
364
0.00
8510
20.
0060
355
0.00
4293
20.
0030
629
0.00
2191
60.
0015
725
0.00
1131
50.
0008
165
400.
0153
844
0.01
0746
80.
0075
312
0.00
5294
30.
0037
332
0.00
2640
50.
0018
731
0.00
1332
70.
0009
509
0.00
0680
4
410.
0138
598
0.00
9595
40.
0066
647
0.00
4644
10.
0032
463
0.00
2276
30.
0016
010
0.00
1129
40.
0007
991
0.00
0567
042
0.01
2486
30.
0085
673
0.00
5898
00.
0040
738
0.00
2822
90.
0019
623
0.00
1368
30.
0009
571
0.00
0671
50.
0004
725
430.
0112
489
0.00
7649
40.
0052
195
0.00
3573
50.
0024
547
0.00
1691
60.
0011
695
0.00
0811
10.
0005
643
0.00
0393
744
0.01
0134
20.
0068
298
0.00
4619
00.
0031
346
0.00
2134
50.
0014
583
0.00
0999
60.
0006
874
0.00
0474
20.
0003
281
450.
0091
299
0.00
6098
00.
0040
876
0.00
2749
70.
0018
561
0.00
1257
20.
0008
544
0.00
0582
50.
0003
985
0.00
0273
4
460.
0082
251
0.00
5444
70.
0036
174
0.00
2412
00.
0016
140
0.00
1083
80.
0007
302
0.00
0493
70.
0003
348
0.00
0227
947
0.00
7410
00.
0048
613
0.00
3201
20.
0021
158
0.00
1403
50.
0009
343
0.00
0624
10.
0004
184
0.00
0281
40.
0001
899
480.
0066
757
0.00
4340
50.
0028
329
0.00
1856
00.
0012
204
0.00
0805
40.
0005
334
0.00
0354
50.
0002
365
0.00
0158
249
0.00
6014
10.
0038
754
0.00
2507
00.
0016
280
0.00
1061
20.
0006
943
0.00
0455
90.
0003
005
0.00
0198
70.
0001
319
500.
0054
182
0.00
3460
20.
0022
186
0.00
1428
10.
0009
228
0.00
0598
60.
0003
897
0.00
0254
60.
0001
670
0.00
0109
9
510.
0048
812
0.00
3089
40.
0019
634
0.00
1252
70.
0008
024
0.00
0516
00.
0003
331
0.00
0215
80.
0001
403
0.00
0091
652
0.00
4397
50.
0027
584
0.00
1737
50.
0010
989
0.00
0697
80.
0004
448
0.00
0284
70.
0001
829
0.00
0117
90.
0000
763
530.
0039
617
0.00
2462
90.
0015
376
0.00
0963
90.
0006
068
0.00
0383
50.
0002
433
0.00
0155
00.
0000
991
0.00
0063
654
0.00
3569
10.
0021
990
0.00
1360
70.
0008
455
0.00
0527
60.
0003
306
0.00
0208
00.
0001
313
0.00
0083
30.
0000
530
550.
0032
154
0.00
1963
40.
0012
042
0.00
0741
70.
0004
588
0.00
0285
00.
0001
777
0.00
0111
30.
0000
700
0.00
0044
2
560.
0028
968
0.00
1753
00.
0010
656
0.00
0650
60.
0003
990
0.00
0245
70.
0001
519
0.00
0094
30.
0000
588
0.00
0036
857
0.00
2609
70.
0015
652
0.00
0943
00.
0005
707
0.00
0346
90.
0002
118
0.00
0129
80.
0000
799
0.00
0049
40.
0000
307
580.
0023
511
0.00
1397
50.
0008
345
0.00
0500
60.
0003
017
0.00
0182
60.
0001
110
0.00
0067
70.
0000
415
0.00
0025
659
0.00
2118
10.
0012
478
0.00
0738
50.
0004
392
0.00
0262
30.
0001
574
0.00
0094
80.
0000
574
0.00
0034
90.
0000
213
600.
0019
082
0.00
1114
10.
0006
536
0.00
0385
20.
0002
281
0.00
0135
70.
0000
811
0.00
0048
60.
0000
293
0.00
0017
7
cont
inue
sov
erle
af
Tabl
eA
.2(c
onti
nued
)
N/i
1112
1314
1516
1718
1920
610.
0017
190.
0009
950.
0005
780.
0003
380.
0001
980.
0001
170.
0000
690.
0000
410.
0000
250.
0000
1562
0.00
1549
0.00
0888
0.00
0512
0.00
0296
0.00
0172
0.00
0101
0.00
0059
0.00
0035
0.00
0021
0.00
0012
630.
0013
950.
0007
930.
0004
530.
0002
600.
0001
500.
0000
870.
0000
510.
0000
300.
0000
170.
0000
1064
0.00
1257
0.00
0708
0.00
0401
0.00
0228
0.00
0130
0.00
0075
0.00
0043
0.00
0025
0.00
0015
0.00
0009
650.
0011
320.
0006
320.
0003
550.
0002
000.
0001
130.
0000
650.
0000
370.
0000
210.
0000
120.
0000
07
660.
0010
200.
0005
640.
0003
140.
0001
760.
0000
990.
0000
560.
0000
320.
0000
180.
0000
100.
0000
0667
0.00
0919
0.00
0504
0.00
0278
0.00
0154
0.00
0086
0.00
0048
0.00
0027
0.00
0015
0.00
0009
0.00
0005
680.
0008
280.
0004
500.
0002
460.
0001
350.
0000
750.
0000
410.
0000
230.
0000
130.
0000
070.
0000
0469
0.00
0746
0.00
0402
0.00
0218
0.00
0118
0.00
0065
0.00
0036
0.00
0020
0.00
0011
0.00
0006
0.00
0003
700.
0006
720.
0003
590.
0001
930.
0001
040.
0000
560.
0000
310.
0000
170.
0000
090.
0000
050.
0000
03
710.
0006
050.
0003
200.
0001
700.
0000
910.
0000
490.
0000
270.
0000
140.
0000
080.
0000
040.
0000
0272
0.00
0545
0.00
0286
0.00
0151
0.00
0080
0.00
0043
0.00
0023
0.00
0012
0.00
0007
0.00
0004
0.00
0002
730.
0004
910.
0002
550.
0001
330.
0000
700.
0000
370.
0000
200.
0000
110.
0000
060.
0000
030.
0000
0274
0.00
0443
0.00
0228
0.00
0118
0.00
0062
0.00
0032
0.00
0017
0.00
0009
0.00
0005
0.00
0003
0.00
0001
750.
0003
990.
0002
040.
0001
050.
0000
540.
0000
280.
0000
150.
0000
080.
0000
040.
0000
020.
0000
01
760.
0003
590.
0001
820.
0000
920.
0000
470.
0000
240.
0000
130.
0000
070.
0000
030.
0000
020.
0000
0177
0.00
0324
0.00
0162
0.00
0082
0.00
0042
0.00
0021
0.00
0011
0.00
0006
0.00
0003
0.00
0002
0.00
0001
780.
0002
920.
0001
450.
0000
720.
0000
360.
0000
180.
0000
090.
0000
050.
0000
020.
0000
010.
0000
0179
0.00
0263
0.00
0129
0.00
0064
0.00
0032
0.00
0016
0.00
0008
0.00
0004
0.00
0002
0.00
0001
0.00
0001
800.
0002
370.
0001
150.
0000
570.
0000
280.
0000
140.
0000
070.
0000
040.
0000
020.
0000
010.
0000
00
810.
0002
132
0.00
0103
10.
0000
502
0.00
0024
60.
0000
121
0.00
0006
00.
0000
030
0.00
0001
50.
0000
008
0.00
0000
482
0.00
0192
10.
0000
921
0.00
0044
40.
0000
216
0.00
0010
50.
0000
052
0.00
0002
60.
0000
013
0.00
0000
60.
0000
003
830.
0001
731
0.00
0082
20.
0000
393
0.00
0018
90.
0000
092
0.00
0004
50.
0000
022
0.00
0001
10.
0000
005
0.00
0000
384
0.00
0155
90.
0000
734
0.00
0034
80.
0000
166
0.00
0008
00.
0000
039
0.00
0001
90.
0000
009
0.00
0000
50.
0000
002
850.
0001
405
0.00
0065
50.
0000
308
0.00
0014
60.
0000
069
0.00
0003
30.
0000
016
0.00
0000
80.
0000
004
0.00
0000
2
860.
0001
265
0.00
0058
50.
0000
272
0.00
0012
80.
0000
060
0.00
0002
90.
0000
014
0.00
0000
70.
0000
003
0.00
0000
287
0.00
0114
00.
0000
522
0.00
0024
10.
0000
112
0.00
0005
20.
0000
025
0.00
0001
20.
0000
006
0.00
0000
30.
0000
001
880.
0001
027
0.00
0046
60.
0000
213
0.00
0009
80.
0000
046
0.00
0002
10.
0000
010
0.00
0000
50.
0000
002
0.00
0000
189
0.00
0092
50.
0000
416
0.00
0018
90.
0000
086
0.00
0004
00.
0000
018
0.00
0000
90.
0000
004
0.00
0000
20.
0000
001
900.
0000
834
0.00
0037
20.
0000
167
0.00
0007
60.
0000
034
0.00
0001
60.
0000
007
0.00
0000
30.
0000
002
0.00
0000
1
910.
0000
751
0.00
0033
20.
0000
148
0.00
0006
60.
0000
030
0.00
0001
40.
0000
006
0.00
0000
30.
0000
001
0.00
0000
192
0.00
0067
70.
0000
296
0.00
0013
10.
0000
058
0.00
0002
60.
0000
012
0.00
0000
50.
0000
002
0.00
0000
10.
0000
001
930.
0000
609
0.00
0026
50.
0000
116
0.00
0005
10.
0000
023
0.00
0001
00.
0000
005
0.00
0000
20.
0000
001
4.32
66E
-08
940.
0000
549
0.00
0023
60.
0000
102
0.00
0004
50.
0000
020
0.00
0000
90.
0000
004
0.00
0000
20.
0000
001
3.60
55E
-08
950.
0000
495
0.00
0021
10.
0000
091
0.00
0003
90.
0000
017
0.00
0000
80.
0000
003
0.00
0000
10.
0000
001
3.00
46E
-08
960.
0000
446
0.00
0018
80.
0000
080
0.00
0003
40.
0000
015
0.00
0000
60.
0000
003
0.00
0000
10.
0000
001
2.50
38E
-08
970.
0000
401
0.00
0016
80.
0000
071
0.00
0003
00.
0000
013
0.00
0000
60.
0000
002
0.00
0000
14.
6983
E-0
82.
0865
E-0
898
0.00
0036
20.
0000
150
0.00
0006
30.
0000
027
0.00
0001
10.
0000
005
0.00
0000
20.
0000
001
3.94
82E
-08
1.73
88E
-08
990.
0000
326
0.00
0013
40.
0000
056
0.00
0002
30.
0000
010
0.00
0000
40.
0000
002
0.00
0000
13.
3178
E-0
81.
4490
E-0
810
00.
0000
294
0.00
0012
00.
0000
049
0.00
0002
00.
0000
009
0.00
0000
40.
0000
002
0.00
0000
12.
7881
E-0
81.
2075
E-0
8
N/i
2122
2324
2526
2728
2930
10.
8264
463
0.81
9672
10.
8130
081
0.80
6451
60.
8000
000
0.79
3650
80.
7874
016
0.78
1250
00.
7751
938
0.76
9230
82
0.68
3013
50.
6718
624
0.66
0982
20.
6503
642
0.64
0000
00.
6298
816
0.62
0001
20.
6103
516
0.60
0925
40.
5917
160
30.
5644
739
0.55
0706
90.
5373
839
0.52
4487
30.
5120
000
0.49
9906
00.
4881
900
0.47
6837
20.
4658
337
0.45
5166
14
0.46
6507
40.
4513
991
0.43
6897
50.
4229
736
0.40
9600
00.
3967
508
0.38
4401
50.
3725
290
0.36
1111
40.
3501
278
50.
3855
433
0.36
9999
30.
3552
012
0.34
1107
70.
3276
800
0.31
4881
60.
3026
784
0.29
1038
30.
2799
313
0.26
9329
1
60.
3186
308
0.30
3278
10.
2887
815
0.27
5086
90.
2621
440
0.24
9906
00.
2383
294
0.22
7373
70.
2170
010
0.20
7176
27
0.26
3331
30.
2485
886
0.23
4781
70.
2218
443
0.20
9715
20.
1983
381
0.18
7661
00.
1776
357
0.16
8217
80.
1593
663
80.
2176
291
0.20
3761
10.
1908
794
0.17
8906
70.
1677
722
0.15
7411
20.
1477
645
0.13
8777
90.
1304
014
0.12
2589
59
0.17
9858
80.
1670
173
0.15
5186
50.
1442
796
0.13
4217
70.
1249
295
0.11
6350
00.
1084
202
0.10
1086
40.
0942
996
100.
1486
436
0.13
6899
40.
1261
679
0.11
6354
50.
1073
742
0.09
9150
40.
0916
142
0.08
4703
30.
0783
615
0.07
2538
2
110.
1228
460
0.11
2212
70.
1025
755
0.09
3834
30.
0858
993
0.07
8690
80.
0721
372
0.06
6174
40.
0607
454
0.05
5798
612
0.10
1525
60.
0919
776
0.08
3394
70.
0756
728
0.06
8719
50.
0624
530
0.05
6800
90.
0516
988
0.04
7089
40.
0429
220
130.
0839
055
0.07
5391
50.
0678
006
0.06
1026
40.
0549
756
0.04
9565
90.
0447
251
0.04
0389
70.
0365
034
0.03
3016
914
0.06
9343
30.
0617
963
0.05
5122
40.
0492
149
0.04
3980
50.
0393
380
0.03
5216
60.
0315
544
0.02
8297
20.
0253
976
150.
0573
086
0.05
0652
70.
0448
150
0.03
9689
40.
0351
844
0.03
1220
60.
0277
296
0.02
4651
90.
0219
358
0.01
9536
6
160.
0473
624
0.04
1518
60.
0364
350
0.03
2007
60.
0281
475
0.02
4778
30.
0218
344
0.01
9259
30.
0170
045
0.01
5028
217
0.03
9142
50.
0340
316
0.02
9621
90.
0258
126
0.02
2518
00.
0196
653
0.01
7192
40.
0150
463
0.01
3181
80.
0115
601
180.
0323
492
0.02
7894
80.
0240
829
0.02
0816
60.
0180
144
0.01
5607
40.
0135
373
0.01
1754
90.
0102
185
0.00
8892
419
0.02
6734
90.
0228
646
0.01
9579
60.
0167
876
0.01
4411
50.
0123
868
0.01
0659
30.
0091
835
0.00
7921
30.
0068
403
200.
0220
949
0.01
8741
50.
0159
183
0.01
3538
40.
0115
292
0.00
9830
80.
0083
932
0.00
7174
60.
0061
405
0.00
5261
8
cont
inue
sov
erle
af
Tabl
eA
.2(c
onti
nued
)
N/i
2122
2324
2526
2728
2930
210.
0182
603
0.01
5361
90.
0129
417
0.01
0918
00.
0092
234
0.00
7802
20.
0066
088
0.00
5605
20.
0047
601
0.00
4047
522
0.01
5091
10.
0125
917
0.01
0521
70.
0088
049
0.00
7378
70.
0061
922
0.00
5203
80.
0043
791
0.00
3690
00.
0031
135
230.
0124
720
0.01
0321
10.
0085
543
0.00
7100
70.
0059
030
0.00
4914
50.
0040
975
0.00
3421
10.
0028
605
0.00
2395
024
0.01
0307
40.
0084
599
0.00
6954
70.
0057
264
0.00
4722
40.
0039
004
0.00
3226
30.
0026
728
0.00
2217
40.
0018
423
250.
0085
186
0.00
6934
30.
0056
542
0.00
4618
00.
0037
779
0.00
3095
50.
0025
404
0.00
2088
10.
0017
189
0.00
1417
2
260.
0070
401
0.00
5683
90.
0045
969
0.00
3724
20.
0030
223
0.00
2456
80.
0020
003
0.00
1631
30.
0013
325
0.00
1090
127
0.00
5818
30.
0046
589
0.00
3737
30.
0030
034
0.00
2417
90.
0019
498
0.00
1575
10.
0012
745
0.00
1032
90.
0008
386
280.
0048
085
0.00
3818
80.
0030
385
0.00
2422
10.
0019
343
0.00
1547
50.
0012
402
0.00
0995
70.
0008
007
0.00
0645
029
0.00
3974
00.
0031
301
0.00
2470
30.
0019
533
0.00
1547
40.
0012
282
0.00
0976
50.
0007
779
0.00
0620
70.
0004
962
300.
0032
843
0.00
2565
70.
0020
084
0.00
1575
20.
0012
379
0.00
0974
70.
0007
689
0.00
0607
70.
0004
812
0.00
0381
7
310.
0027
143
0.00
2103
00.
0016
328
0.00
1270
40.
0009
904
0.00
0773
60.
0006
055
0.00
0474
80.
0003
730
0.00
0293
632
0.00
2243
20.
0017
238
0.00
1327
50.
0010
245
0.00
0792
30.
0006
140
0.00
0476
70.
0003
709
0.00
0289
20.
0002
258
330.
0018
539
0.00
1412
90.
0010
793
0.00
0826
20.
0006
338
0.00
0487
30.
0003
754
0.00
0289
80.
0002
242
0.00
0173
734
0.00
1532
10.
0011
582
0.00
0877
50.
0006
663
0.00
0507
10.
0003
867
0.00
0295
60.
0002
264
0.00
0173
80.
0001
336
350.
0012
662
0.00
0949
30.
0007
134
0.00
0537
30.
0004
056
0.00
0306
90.
0002
327
0.00
0176
90.
0001
347
0.00
0102
8
360.
0010
465
0.00
0778
10.
0005
800
0.00
0433
30.
0003
245
0.00
0243
60.
0001
833
0.00
0138
20.
0001
044
0.00
0079
137
0.00
0864
90.
0006
378
0.00
0471
50.
0003
495
0.00
0259
60.
0001
933
0.00
0144
30.
0001
080
0.00
0080
90.
0000
608
380.
0007
148
0.00
0522
80.
0003
834
0.00
0281
80.
0002
077
0.00
0153
40.
0001
136
0.00
0084
30.
0000
627
0.00
0046
839
0.00
0590
70.
0004
285
0.00
0311
70.
0002
273
0.00
0166
20.
0001
218
0.00
0089
50.
0000
659
0.00
0048
60.
0000
360
400.
0004
882
0.00
0351
20.
0002
534
0.00
0183
30.
0001
329
0.00
0096
60.
0000
704
0.00
0051
50.
0000
377
0.00
0027
7
410.
0004
035
0.00
0287
90.
0002
060
0.00
0147
80.
0001
063
0.00
0076
70.
0000
555
0.00
0040
20.
0000
292
0.00
0021
342
0.00
0333
40.
0002
360
0.00
0167
50.
0001
192
0.00
0085
10.
0000
609
0.00
0043
70.
0000
314
0.00
0022
70.
0000
164
430.
0002
756
0.00
0193
40.
0001
362
0.00
0096
10.
0000
681
0.00
0048
30.
0000
344
0.00
0024
50.
0000
176
0.00
0012
644
0.00
0227
70.
0001
586
0.00
0110
70.
0000
775
0.00
0054
40.
0000
383
0.00
0027
10.
0000
192
0.00
0013
60.
0000
097
450.
0001
882
0.00
0130
00.
0000
900
0.00
0062
50.
0000
436
0.00
0030
40.
0000
213
0.00
0015
00.
0000
106
0.00
0007
5
460.
0001
556
0.00
0106
50.
0000
732
0.00
0050
40.
0000
348
0.00
0024
20.
0000
168
0.00
0011
70.
0000
082
0.00
0005
747
0.00
0128
60.
0000
873
0.00
0059
50.
0000
407
0.00
0027
90.
0000
192
0.00
0013
20.
0000
091
0.00
0006
30.
0000
044
480.
0001
062
0.00
0071
60.
0000
484
0.00
0032
80.
0000
223
0.00
0015
20.
0000
104
0.00
0007
10.
0000
049
0.00
0003
449
0.00
0087
80.
0000
587
0.00
0039
30.
0000
264
0.00
0017
80.
0000
121
0.00
0008
20.
0000
056
0.00
0003
80.
0000
026
500.
0000
726
0.00
0048
10.
0000
320
0.00
0021
30.
0000
143
0.00
0009
60.
0000
065
0.00
0004
40.
0000
030
0.00
0002
0
Tabl
eA
.3O
rdin
ary
annu
ityco
mpo
und
amou
ntfa
ctor
.To
find
Ffo
ra
give
nR
rece
ived
atth
een
dof
each
peri
odF
=R
(1+i
)N
−1i
N/i
12
34
56
78
910
11.
0000
000
1.00
0000
01.
0000
000
1.00
0000
01.
0000
000
1.00
0000
01.
0000
000
1.00
0000
01.
0000
000
1.00
0000
02
2.01
0000
02.
0200
000
2.03
0000
02.
0400
000
2.05
0000
02.
0600
000
2.07
0000
02.
0800
000
2.09
0000
02.
1000
000
33.
0301
000
3.06
0400
03.
0909
000
3.12
1600
03.
1525
000
3.18
3600
03.
2149
000
3.24
6400
03.
2781
000
3.31
0000
04
4.06
0401
04.
1216
080
4.18
3627
04.
2464
640
4.31
0125
04.
3746
160
4.43
9943
04.
5061
120
4.57
3129
04.
6410
000
55.
1010
050
5.20
4040
25.
3091
358
5.41
6322
65.
5256
313
5.63
7093
05.
7507
390
5.86
6601
05.
9847
106
6.10
5100
0
66.
1520
151
6.30
8121
06.
4684
099
6.63
2975
56.
8019
128
6.97
5318
57.
1532
907
7.33
5929
07.
5233
346
7.71
5610
07
7.21
3535
27.
4342
834
7.66
2462
27.
8982
945
8.14
2008
58.
3938
376
8.65
4021
18.
9228
034
9.20
0434
79.
4871
710
88.
2856
706
8.58
2969
18.
8923
360
9.21
4226
39.
5491
089
9.89
7467
910
.259
8026
10.6
3662
7611
.028
4738
11.4
3588
819
9.36
8527
39.
7546
284
10.1
5910
6110
.582
7953
11.0
2656
4311
.491
3160
11.9
7798
8712
.487
5578
13.0
2103
6413
.579
4769
1010
.462
2125
10.9
4972
1011
.463
8793
12.0
0610
7112
.577
8925
13.1
8079
4913
.816
4480
14.4
8656
2515
.192
9297
15.9
3742
46
1111
.566
8347
12.1
6871
5412
.807
7957
13.4
8635
1414
.206
7872
14.9
7164
2615
.783
5993
16.6
4548
7517
.560
2934
18.5
3116
7112
12.6
8250
3013
.412
0897
14.1
9202
9615
.025
8055
15.9
1712
6516
.869
9412
17.8
8845
1318
.977
1265
20.1
4071
9821
.384
2838
1313
.809
3280
14.6
8033
1515
.617
7904
16.6
2683
7717
.712
9828
18.8
8213
7720
.140
6429
21.4
9529
6622
.953
3846
24.5
2271
2114
14.9
4742
1315
.973
9382
17.0
8632
4218
.291
9112
19.5
9863
2021
.015
0659
22.5
5048
7924
.214
9203
26.0
1918
9227
.974
9834
1516
.096
8955
17.2
9341
6918
.598
9139
20.0
2358
7621
.578
5636
23.2
7596
9925
.129
0220
27.1
5211
3929
.360
9162
31.7
7248
17
1617
.257
8645
18.6
3928
5320
.156
8813
21.8
2453
1123
.657
4918
25.6
7252
8127
.888
0536
30.3
2428
3033
.003
3987
35.9
4972
9917
18.4
3044
3120
.012
0710
21.7
6158
7723
.697
5124
25.8
4036
6428
.212
8798
30.8
4021
7333
.750
2257
36.9
7370
4640
.544
7028
1819
.614
7476
21.4
1231
2423
.414
4354
25.6
4541
2928
.132
3847
30.9
0565
2533
.999
0325
37.4
5024
3741
.301
3380
45.5
9917
3119
20.8
1089
5022
.840
5586
25.1
1686
8427
.671
2294
30.5
3900
3933
.759
9917
37.3
7896
4841
.446
2632
46.0
1845
8451
.159
0904
2022
.019
0040
24.2
9736
9826
.870
3745
29.7
7807
8633
.065
9541
36.7
8559
1240
.995
4923
45.7
6196
4351
.160
1196
57.2
7499
95
2123
.239
1940
25.7
8331
7228
.676
4857
31.9
6920
1735
.719
2518
39.9
9272
6744
.865
1768
50.4
2292
1456
.764
5304
64.0
0249
9422
24.4
7158
6027
.298
9835
30.5
3678
0334
.247
9698
38.5
0521
4443
.392
2903
49.0
0573
9255
.456
7552
62.8
7333
8171
.402
7494
2325
.716
3018
28.8
4496
3232
.452
8837
36.6
1788
8641
.430
4751
46.9
9582
7753
.436
1409
60.8
9329
5669
.531
9386
79.5
4302
4324
26.9
7346
4930
.421
8625
34.4
2647
0239
.082
6041
44.5
0199
8950
.815
5774
58.1
7667
0866
.764
7592
76.7
8981
3188
.497
3268
2528
.243
1995
32.0
3029
9736
.459
2643
41.6
4590
8347
.727
0988
54.8
6451
2063
.249
0377
73.1
0594
0084
.700
8962
98.3
4705
94
2629
.525
6315
33.6
7090
5738
.553
0423
44.3
1174
4651
.113
4538
59.1
5638
2768
.676
4704
79.9
5441
5193
.323
9769
109.
1817
654
2730
.820
8878
35.3
4432
3840
.709
6335
47.0
8421
4454
.669
1264
63.7
0576
5774
.483
8233
87.3
5076
8410
2.72
3134
812
1.09
9941
928
32.1
2909
6737
.051
2103
42.9
3092
2549
.967
5830
58.4
0258
2868
.528
1116
80.6
9769
0995
.338
8298
112.
9682
169
134.
2099
361
2933
.450
3877
38.7
9223
4545
.218
8502
52.9
6628
6362
.322
7119
73.6
3979
8387
.346
5293
103.
9659
362
124.
1353
565
148.
6309
297
3034
.784
8915
40.5
6807
9247
.575
4157
56.0
8493
7866
.438
8475
79.0
5818
6294
.460
7863
113.
2832
111
136.
3075
385
164.
4940
227
cont
inue
sov
erle
af
Tabl
eA
.3(c
onti
nued
)
N/i
12
34
56
78
910
3136
.132
7404
42.3
7944
0850
.002
6782
59.3
2833
5370
.760
7899
84.8
0167
7410
2.07
3041
412
3.34
5868
014
9.57
5217
018
1.94
3425
032
37.4
9406
7944
.227
0296
52.5
0275
8562
.701
4687
75.2
9882
9490
.889
7780
110.
2181
543
134.
2135
374
164.
0369
865
201.
1377
675
3338
.869
0085
46.1
1157
0255
.077
8413
66.2
0952
7480
.063
7708
97.3
4316
4711
8.93
3425
114
5.95
0620
417
9.80
0315
322
2.25
1544
234
40.2
5769
8648
.033
8016
57.7
3017
6569
.857
9085
85.0
6695
9410
4.18
3754
612
8.25
8764
815
8.62
6670
119
6.98
2343
724
5.47
6698
635
41.6
6027
5649
.994
4776
60.4
6208
1873
.652
2249
90.3
2030
7411
1.43
4779
913
8.23
6878
417
2.31
6803
721
5.71
0754
727
1.02
4368
5
3643
.076
8784
51.9
9436
7263
.275
9443
77.5
9831
3895
.836
3227
119.
1208
667
148.
9134
598
187.
1021
480
236.
1247
226
299.
1268
053
3744
.507
6471
54.0
3425
4566
.174
2226
81.7
0224
6410
1.62
8138
912
7.26
8118
716
0.33
7402
020
3.07
0319
825
8.37
5947
633
0.03
9485
938
45.9
5272
3656
.114
9396
69.1
5944
9385
.970
3363
107.
7095
458
135.
9042
058
172.
5610
202
220.
3159
454
282.
6297
829
364.
0434
344
3947
.412
2509
58.2
3723
8472
.234
2328
90.4
0914
9711
4.09
5023
114
5.05
8458
118
5.64
0291
623
8.94
1221
030
9.06
6463
340
1.44
7777
940
48.8
8637
3460
.401
9832
75.4
0125
9795
.025
5157
120.
7997
742
154.
7619
656
199.
6351
120
259.
0565
187
337.
8824
450
442.
5925
557
4150
.375
2371
62.6
1002
2878
.663
2975
99.8
2653
6312
7.83
9763
016
5.04
7683
621
4.60
9569
828
0.78
1040
236
9.29
1865
148
7.85
1811
242
51.8
7898
9564
.862
2233
82.0
2319
6510
4.81
9597
813
5.23
1751
117
5.95
0544
623
0.63
2239
730
4.24
3523
440
3.52
8133
053
7.63
6992
443
53.3
9777
9467
.159
4678
85.4
8389
2311
0.01
2381
714
2.99
3338
718
7.50
7577
224
7.77
6496
532
9.58
3005
344
0.84
5664
959
2.40
0691
644
54.9
3175
7269
.502
6571
89.0
4840
9111
5.41
2877
015
1.14
3005
619
9.75
8031
926
6.12
0851
335
6.94
9645
748
1.52
1774
865
2.64
0760
845
56.4
8107
4771
.892
7103
92.7
1986
1412
1.02
9392
015
9.70
0155
921
2.74
3513
828
5.74
9310
838
6.50
5617
452
5.85
8734
571
8.90
4836
9
4658
.045
8855
74.3
3056
4596
.501
4572
126.
8705
677
168.
6851
637
226.
5081
246
306.
7517
626
418.
4260
668
574.
1860
206
791.
7953
205
4759
.626
3443
76.8
1717
5810
0.39
6500
913
2.94
5390
417
8.11
9421
824
1.09
8612
132
9.22
4386
045
2.90
0152
162
6.86
2762
587
1.97
4852
648
61.2
2260
7879
.353
5193
104.
4083
960
139.
2632
060
188.
0253
929
256.
5645
288
353.
2700
930
490.
1321
643
684.
2804
111
960.
1723
378
4962
.834
8338
81.9
4058
9710
8.54
0647
914
5.83
3734
319
8.42
6662
627
2.95
8400
637
8.99
8999
553
0.34
2737
474
6.86
5648
110
57.1
8957
1650
64.4
6318
2284
.579
4015
112.
7968
673
152.
6670
837
209.
3479
957
290.
3359
046
406.
5289
295
573.
7701
564
815.
0835
564
1163
.908
5288
5166
.107
8140
87.2
7098
9511
7.18
0773
315
9.77
3767
022
0.81
5395
530
8.75
6058
943
5.98
5954
562
0.67
1768
988
9.44
1076
512
81.2
9938
1752
67.7
6889
2190
.016
4093
121.
6961
965
167.
1647
177
232.
8561
653
328.
2814
224
467.
5049
714
671.
3255
104
970.
4907
734
1410
.429
3198
5369
.446
5811
92.8
1673
7512
6.34
7082
417
4.85
1306
424
5.49
8973
534
8.97
8307
750
1.23
0319
372
6.03
1551
310
58.8
3494
3015
52.4
7225
1854
71.1
4104
6995
.673
0722
131.
1374
949
182.
8453
586
258.
7739
222
370.
9170
062
537.
3164
417
785.
1140
754
1155
.130
0878
1708
.719
4770
5572
.852
4573
98.5
8653
3713
6.07
1619
719
1.15
9173
027
2.71
2618
339
4.17
2026
657
5.92
8592
684
8.92
3201
412
60.0
9179
5718
80.5
9142
47
5674
.580
9819
101.
5582
643
141.
1537
683
199.
8055
399
287.
3482
492
418.
8223
482
617.
2435
941
917.
8370
575
1374
.500
0573
2069
.650
5672
5776
.326
7917
104.
5894
296
146.
3883
814
208.
7977
615
302.
7156
617
444.
9516
891
661.
4506
457
992.
2640
221
1499
.205
0625
2277
.615
6239
5878
.090
0597
107.
6812
182
151.
7800
328
218.
1496
720
318.
8514
448
472.
6487
904
708.
7521
909
1072
.645
1439
1635
.133
5181
2506
.377
1863
5979
.870
9603
110.
8348
426
157.
3334
338
227.
8756
588
335.
7940
170
502.
0077
178
759.
3648
443
1159
.456
7554
1783
.295
5348
2758
.014
9049
6081
.669
6699
114.
0515
394
163.
0534
368
237.
9906
852
353.
5837
179
533.
1281
809
813.
5203
834
1253
.213
2958
1944
.792
1329
3034
.816
3954
6183
.486
3666
117.
3325
702
168.
9450
399
248.
5103
126
372.
2629
038
566.
1158
717
871.
4668
102
1354
.470
3595
2120
.823
4249
3339
.298
0350
6285
.321
2302
120.
6792
216
175.
0133
911
259.
4507
251
391.
8760
490
601.
0828
240
933.
4694
869
1463
.827
9883
2312
.697
5331
3674
.227
8385
6387
.174
4425
124.
0928
060
181.
2637
928
270.
8287
541
412.
4698
514
638.
1477
935
999.
8123
510
1581
.934
2273
2521
.840
3111
4042
.650
6223
6489
.046
1869
127.
5746
622
187.
7017
066
282.
6619
043
434.
0933
440
677.
4366
611
1070
.799
2155
1709
.488
9655
2749
.805
9391
4447
.915
6845
6590
.936
6488
131.
1261
554
194.
3327
578
294.
9683
805
456.
7980
112
719.
0828
608
1146
.755
1606
1847
.248
0828
2998
.288
4736
4893
.707
2530
6692
.846
0153
134.
7486
785
201.
1627
406
307.
7671
157
480.
6379
117
763.
2278
324
1228
.028
0219
1996
.027
9294
3269
.134
4362
5384
.077
9783
6794
.774
4755
138.
4436
521
208.
1976
228
321.
0778
003
505.
6698
073
810.
0215
024
1314
.989
9834
2156
.710
1637
3564
.356
5355
5923
.485
7761
6896
.722
2202
142.
2125
251
215.
4435
515
334.
9209
123
531.
9532
977
859.
6227
925
1408
.039
2823
2330
.246
9768
3886
.148
6236
6516
.834
3537
6998
.689
4424
146.
0567
756
222.
9068
580
349.
3177
488
559.
5509
626
912.
2001
600
1507
.602
0320
2517
.666
7350
4236
.901
9998
7169
.517
7891
7010
0.67
6336
814
9.97
7911
123
0.59
4063
736
4.29
0458
858
8.52
8510
796
7.93
2169
616
14.1
3417
4327
20.0
8007
3846
19.2
2317
9878
87.4
6956
80
7110
2.68
3100
215
3.97
7469
423
8.51
1885
637
9.86
2077
161
8.95
4936
210
27.0
0809
9817
28.1
2356
6429
38.6
8647
9750
35.9
5326
5986
77.2
1652
4872
104.
7099
312
158.
0570
188
246.
6672
422
396.
0565
602
650.
9026
831
1089
.628
5858
1850
.092
2161
3174
.781
3980
5490
.189
0599
9545
.938
1773
7310
6.75
7030
516
2.21
8159
125
5.06
7259
541
2.89
8822
668
4.44
7817
211
56.0
0630
1019
80.5
9867
1234
29.7
6390
9959
85.3
0607
5310
501.
5319
950
7410
8.82
4600
816
6.46
2522
326
3.71
9277
343
0.41
4775
571
9.67
0208
112
26.3
6667
9021
20.2
4057
8237
05.1
4502
2765
24.9
8362
2011
552.
6851
945
7511
0.91
2846
817
0.79
1772
827
2.63
0855
644
8.63
1366
575
6.65
3718
513
00.9
4867
9822
69.6
5741
8740
02.5
5662
4571
13.2
3214
8012
708.
9537
140
7611
3.02
1975
317
5.20
7608
228
1.80
9781
346
7.57
6621
279
5.48
6404
413
80.0
0560
0624
29.5
3343
8043
23.7
6115
4577
54.4
2304
1313
980.
8490
853
7711
5.15
2195
117
9.71
1760
429
1.26
4074
748
7.27
9686
083
6.26
0724
614
63.8
0593
6626
00.6
0077
8746
70.6
6204
6884
53.3
2111
5115
379.
9339
939
7811
7.30
3717
018
4.30
5995
630
1.00
1996
950
7.77
0873
587
9.07
3760
815
52.6
3429
2827
83.6
4283
3250
45.3
1501
0692
15.1
2001
5416
918.
9273
933
7911
9.47
6754
218
8.99
2115
531
1.03
2056
852
9.08
1708
492
4.02
7448
916
46.7
9235
0329
79.4
9783
1554
49.9
4021
1410
045.
4808
168
1861
1.82
0132
680
121.
6715
217
193.
7719
578
321.
3630
185
551.
2449
767
971.
2288
213
1746
.599
8914
3189
.062
6797
5886
.935
4283
1095
0.57
4090
320
474.
0021
459
8112
3.88
8236
919
8.64
7397
033
2.00
3909
157
4.29
4775
810
20.7
9026
2418
52.3
9588
4934
13.2
9706
7363
58.8
9026
2611
937.
1257
584
2252
2.40
2360
482
126.
1271
193
203.
6203
449
342.
9640
264
598.
2665
668
1072
.829
7755
1964
.539
6379
3653
.227
8620
6868
.601
4836
1301
2.46
7076
724
775.
6425
965
8312
8.38
8390
520
8.69
2751
835
4.25
2947
262
3.19
7229
511
27.4
7126
4320
83.4
1201
6239
09.9
5381
2374
19.0
8960
2314
184.
5891
136
2725
4.20
6856
184
130.
6722
744
213.
8666
068
365.
8805
356
649.
1251
187
1184
.844
8275
2209
.416
7372
4184
.650
5792
8013
.616
7705
1546
2.20
2133
829
980.
6275
417
8513
2.97
8997
121
9.14
3939
037
7.85
6951
767
6.09
0123
512
45.0
8706
8923
42.9
8174
1444
78.5
7611
9786
55.7
0611
2116
854.
8003
259
3297
9.69
0295
9
8613
5.30
8787
122
4.52
6817
839
0.19
2660
270
4.13
3728
413
08.3
4142
2324
84.5
6064
5947
93.0
7644
8193
49.1
6260
1118
372.
7323
552
3627
8.65
9325
587
137.
6618
750
230.
0173
541
402.
8984
400
733.
2990
775
1374
.758
4935
2634
.634
2847
5129
.591
7995
1009
8.09
5609
120
027.
2782
672
3990
7.52
5281
8814
0.03
8493
723
5.61
7701
241
5.98
5393
276
3.63
1040
614
44.4
9641
8127
93.7
1234
1754
89.6
6322
5410
906.
9432
579
2183
0.73
3311
243
899.
2777
839
8914
2.43
8878
724
1.33
0055
242
9.46
4955
079
5.17
6282
315
17.7
2123
9029
62.3
3508
2258
74.9
3965
1211
780.
4987
185
2379
6.49
9309
248
290.
2055
623
9014
4.86
3267
524
7.15
6656
344
3.34
8903
782
7.98
3333
515
94.6
0730
1031
41.0
7518
7262
87.1
8542
6812
723.
9386
160
2593
9.18
4247
053
120.
2261
185
9114
7.31
1900
125
3.09
9789
445
7.64
9370
886
2.10
2666
916
75.3
3766
6033
30.5
3969
8467
28.2
8840
6713
742.
8537
053
2827
4.71
0829
358
433.
2487
303
9214
9.78
5019
125
9.16
1785
247
2.37
8851
989
7.58
6773
617
60.1
0454
9335
31.3
7208
0372
00.2
6859
5114
843.
2820
017
3082
0.43
4803
964
277.
5736
034
9315
2.28
2869
326
5.34
5020
948
7.55
0217
493
4.49
0244
518
49.1
0977
6837
44.2
5440
5177
05.2
8739
6816
031.
7445
618
3359
5.27
3936
370
706.
3309
637
9415
4.80
5698
027
1.65
1921
450
3.17
6724
097
2.86
9854
319
42.5
6526
5639
69.9
0966
9482
45.6
5751
4617
315.
2841
268
3661
9.84
8590
577
777.
9640
601
9515
7.35
3755
027
8.08
4959
851
9.27
2025
710
12.7
8464
820
40.6
9352
8942
09.1
0424
9688
23.8
5354
0618
701.
5068
569
3991
6.63
4963
785
556.
7604
661
cont
inue
sov
erle
af
Tabl
eA
.3(c
onti
nued
)
N/i
12
34
56
78
910
9615
9.92
7292
628
4.64
6659
053
5.85
0186
510
54.2
9603
421
43.7
2820
5444
62.6
5050
4694
42.5
2328
8420
198.
6274
054
4351
0.13
2110
494
113.
4365
197
162.
5265
655
291.
3395
922
552.
9256
920
1097
.467
876
2251
.914
6156
4731
.409
5349
1010
4.49
992
2181
5.51
7597
947
427.
0440
004
1035
25.7
802
9816
5.15
1831
129
8.16
6384
057
0.51
3462
811
42.3
6659
123
65.5
1034
6450
16.2
9410
7010
812.
8149
123
561.
7590
057
5169
6.47
7960
411
3879
.358
299
167.
8033
494
305.
1297
117
588.
6288
667
1189
.061
254
2484
.785
8637
5318
.271
7534
1157
0.71
196
2544
7.69
9726
256
350.
1609
768
1252
68.2
9410
017
0.48
1382
931
2.23
2305
960
7.28
7732
712
37.6
2370
526
10.0
2515
6956
38.3
6805
8612
381.
6617
927
484.
5157
043
6142
2.67
5464
713
7796
.123
4
N/i
1112
1314
1516
1718
1920
11.
0000
000
1.00
0000
01.
0000
000
1.00
0000
01.
0000
000
1.00
0000
01.
0000
000
1.00
0000
01.
0000
000
1.00
0000
02
2.11
0000
02.
1200
000
2.13
0000
02.
1400
000
2.15
0000
02.
1600
000
2.17
0000
02.
1800
000
2.19
0000
02.
2000
000
33.
3421
000
3.37
4400
03.
4069
000
3.43
9600
03.
4725
000
3.50
5600
03.
5389
000
3.57
2400
03.
6061
000
3.64
0000
04
4.70
9731
04.
7793
280
4.84
9797
04.
9211
440
4.99
3375
05.
0664
960
5.14
0513
05.
2154
320
5.29
1259
05.
3680
000
56.
2278
014
6.35
2847
46.
4802
706
6.61
0104
26.
7423
813
6.87
7135
47.
0144
002
7.15
4209
87.
2965
982
7.44
1600
0
67.
9128
596
8.11
5189
08.
3227
058
8.53
5518
78.
7537
384
8.97
7477
09.
2068
482
9.44
1967
59.
6829
519
9.92
9920
07
9.78
3274
110
.089
0117
10.4
0465
7510
.730
4914
11.0
6679
9211
.413
8733
11.7
7201
2412
.141
5217
12.5
2271
2712
.915
9040
811
.859
4343
12.2
9969
3112
.757
2630
13.2
3276
0213
.726
8191
14.2
4009
3114
.773
2546
15.3
2699
5615
.902
0281
16.4
9908
489
14.1
6397
2014
.775
6563
15.4
1570
7216
.085
3466
16.7
8584
1917
.518
5080
18.2
8470
7819
.085
8548
19.9
2341
3520
.798
9018
1016
.722
0090
17.5
4873
5118
.419
7492
19.3
3729
5120
.303
7182
21.3
2146
9222
.393
1082
23.5
2130
8624
.708
8621
25.9
5868
21
1119
.561
4300
20.6
5458
3321
.814
3165
23.0
4451
6424
.349
2760
25.7
3290
4327
.199
9366
28.7
5514
4230
.403
5458
32.1
5041
8512
22.7
1318
7224
.133
1333
25.6
5017
7727
.270
7487
29.0
0166
7430
.850
1690
32.8
2392
5834
.931
0701
37.1
8021
9639
.580
5022
1326
.211
6378
28.0
2910
9329
.984
7008
32.0
8865
3534
.351
9175
36.7
8619
6139
.403
9932
42.2
1866
2845
.244
4613
48.4
9660
2714
30.0
9491
8032
.392
6024
34.8
8271
1937
.581
0650
40.5
0470
5143
.671
9874
47.1
0267
2050
.818
0221
54.8
4090
8959
.195
9232
1534
.405
3590
37.2
7971
4740
.417
4644
43.8
4241
4147
.580
4109
51.6
5950
5456
.110
1262
60.9
6526
6066
.260
6816
72.0
3510
79
1639
.189
9485
42.7
5328
0446
.671
7348
50.9
8035
2155
.717
4725
60.9
2502
6366
.648
8477
72.9
3901
3979
.850
2111
87.4
4212
9417
44.5
0084
2848
.883
6741
53.7
3906
0359
.117
6014
65.0
7509
3471
.673
0305
78.9
7915
1887
.068
0364
96.0
2175
1210
5.93
0555
318
50.3
9593
5555
.749
7150
61.7
2513
8268
.394
0656
75.8
3635
7484
.140
7154
93.4
0560
7610
3.74
0283
011
5.26
5883
912
8.11
6666
419
59.9
3948
8463
.439
6808
70.7
4940
6278
.969
2348
88.2
1181
1098
.603
2298
110.
2845
609
123.
4135
339
138.
1664
019
154.
7399
997
2064
.202
8321
72.0
5244
2480
.946
8290
91.0
2492
7710
2.44
3582
611
5.37
9746
613
0.03
2936
314
6.62
7970
016
5.41
8018
318
6.68
7999
6
2172
.265
1437
81.6
9873
5592
.469
9167
104.
7684
175
118.
8101
200
134.
8405
060
153.
1385
354
174.
0210
046
197.
8474
417
225.
0255
995
2281
.214
3095
92.5
0258
3810
5.49
1005
912
0.43
5996
013
7.63
1638
015
7.41
4987
018
0.17
2086
420
6.34
4785
523
6.43
8455
727
1.03
0719
523
91.1
4788
3510
4.60
2893
912
0.20
4836
713
8.29
7035
415
9.27
6383
718
3.60
1384
921
1.80
1341
124
4.48
6846
828
2.36
1762
232
6.23
6863
324
102.
1741
507
118.
1552
411
136.
8314
654
158.
6586
204
184.
1678
413
213.
9776
065
248.
8075
691
289.
4944
793
337.
0104
971
392.
4842
360
2511
4.41
3307
313
3.33
3870
115
5.61
9555
918
1.87
0827
221
2.79
3017
524
9.21
4023
529
2.10
4855
934
2.60
3485
540
2.04
2491
547
1.98
1083
2
2612
7.99
8771
115
0.33
3934
517
6.85
0098
220
8.33
2743
024
5.71
1970
129
0.08
8267
334
2.76
2681
440
5.27
2112
947
9.43
0564
956
7.37
7299
927
143.
0786
359
169.
3740
066
200.
8406
110
238.
4993
271
283.
5687
656
337.
5023
901
402.
0323
372
479.
2210
933
571.
5223
722
681.
8527
598
2815
9.81
7285
919
0.69
8887
422
7.94
9890
427
2.88
9232
932
7.10
4080
439
2.50
2772
547
1.37
7834
556
6.48
0890
168
1.11
1622
981
9.22
3311
829
178.
3971
873
214.
5827
539
258.
5833
762
312.
0937
255
377.
1696
925
456.
3032
161
552.
5120
664
669.
4474
503
811.
5228
313
984.
0679
742
3019
9.02
0877
924
1.33
2684
329
3.19
9215
135
6.78
6847
043
4.74
5146
453
0.31
1730
764
7.43
9117
779
0.94
7991
396
6.71
2169
211
81.8
8156
90
3122
1.91
3174
527
1.29
2606
533
2.31
5113
040
7.73
7005
650
0.95
6918
361
6.16
1607
675
8.50
3767
793
4.31
8629
811
51.3
8748
1414
19.2
5788
2832
247.
3236
237
304.
8477
192
376.
5160
777
465.
8201
864
577.
1004
561
715.
7474
648
888.
4494
082
1103
.495
9831
1371
.151
1029
1704
.109
4594
3327
5.52
9222
334
2.42
9445
542
6.46
3167
853
2.03
5012
566
4.66
5524
583
1.26
7059
210
40.4
8580
7613
03.1
2526
0116
32.6
6981
2420
45.9
3135
1234
306.
8374
368
384.
5209
790
482.
9033
796
607.
5199
142
765.
3653
532
965.
2697
886
1218
.368
3949
1538
.687
8069
1943
.877
0767
2456
.117
6215
3534
1.58
9554
843
1.66
3496
554
6.68
0819
069
3.57
2702
288
1.17
0156
111
20.7
1295
4814
26.4
9102
2118
16.6
5161
2123
14.2
1372
1329
48.3
4114
58
3638
0.16
4405
848
4.46
3116
161
8.74
9325
479
1.67
2880
510
14.3
4567
9613
01.0
2702
7616
69.9
9449
5821
44.6
4890
2327
54.9
1432
8435
39.0
0937
4937
422.
9824
905
543.
5986
900
700.
1867
377
903.
5070
838
1167
.497
5315
1510
.191
3520
1954
.893
5601
2531
.685
7047
3279
.348
0508
4247
.811
2499
3847
0.51
0564
460
9.83
0532
879
2.21
1013
710
30.9
9807
613
43.6
2216
1217
52.8
2196
8322
88.2
2546
5329
88.3
8913
1639
03.4
2418
0450
98.3
7349
9939
523.
2667
265
684.
0101
967
896.
1984
454
1176
.337
806
1546
.165
4854
2034
.273
4833
2678
.223
7944
3527
.299
1753
4646
.074
7747
6119
.048
1999
4058
1.82
6066
476
7.09
1420
310
13.7
0424
313
42.0
2509
917
79.0
9030
8223
60.7
5724
0631
34.5
2183
9541
63.2
1302
6855
29.8
2898
1973
43.8
5783
98
4164
6.82
6933
786
0.14
2390
811
46.4
8579
515
30.9
0861
320
46.9
5385
427
39.4
7839
936
68.3
9055
249
13.5
9137
265
81.4
9648
888
13.6
2940
842
718.
9778
964
964.
3594
777
1296
.528
948
1746
.235
819
2354
.996
933
3178
.794
943
4293
.016
946
5799
.037
819
7832
.980
821
1057
7.35
529
4379
9.06
5465
1081
.082
615
1466
.077
712
1991
.708
833
2709
.246
473
3688
.402
134
5023
.829
827
6843
.864
626
9322
.247
177
1269
3.82
635
4488
7.96
2666
212
11.8
1252
916
57.6
6781
422
71.5
4807
3116
.633
443
4279
.546
475
5878
.880
897
8076
.760
259
1109
4.47
414
1523
3.59
162
4598
6.63
8559
513
58.2
3003
218
74.1
6463
2590
.564
835
85.1
2846
4965
.273
911
6879
.290
6595
31.5
7710
513
203.
4242
318
281.
3099
4
4610
96.1
6880
115
22.2
1763
621
18.8
0603
229
54.2
4387
241
23.8
9772
957
60.7
1773
780
49.7
7006
111
248.
2609
815
713.
0748
321
938.
5719
347
1217
.747
369
1705
.883
752
2395
.250
816
3368
.838
014
4743
.482
388
6683
.432
575
9419
.230
971
1327
3.94
796
1869
9.55
905
2632
7.28
631
4813
52.6
9958
1911
.589
803
2707
.633
422
3841
.475
336
5456
.004
746
7753
.781
787
1102
1.50
024
1566
4.25
859
2225
3.47
527
3159
3.74
358
4915
02.4
9653
321
41.9
8057
930
60.6
2576
743
80.2
8188
362
75.4
0545
889
95.3
8687
312
896.
1552
818
484.
8251
426
482.
6355
737
913.
4922
950
1668
.771
152
2400
.018
249
3459
.507
117
4994
.521
346
7217
.716
277
1043
5.68
477
1508
9.50
167
2181
3.09
367
3151
5.33
633
4549
7.19
075
5118
53.3
3597
926
89.0
2043
839
10.2
4304
256
94.7
5433
583
01.3
7371
912
106.
3525
817
655.
7169
625
740.
4505
337
504.
2502
354
597.
6289
5220
58.2
0293
730
12.7
0289
144
19.5
7463
764
93.0
1994
195
47.5
7977
714
044.
3689
920
658.
1888
430
374.
7316
244
631.
0577
765
518.
1546
853
2285
.605
2633
75.2
2723
849
95.1
1934
7403
.042
733
1098
0.71
674
1629
2.46
803
2417
1.08
094
3584
3.18
331
5311
1.95
875
7862
2.78
562
5425
38.0
2183
837
81.2
5450
656
45.4
8485
484
40.4
6871
612
628.
8242
518
900.
2629
128
281.
1647
4229
5.95
631
6320
4.23
091
9434
8.34
274
5528
18.2
0424
4236
.005
047
6380
.397
885
9623
.134
336
1452
4.14
789
2192
5.30
498
3308
9.96
2749
910.
2284
475
214.
0347
911
3219
.011
3
5631
29.2
0670
747
45.3
2565
372
10.8
4961
1097
1.37
314
1670
3.77
008
2543
4.35
377
3871
6.25
636
5889
5.06
957
8950
5.70
1413
5863
.813
557
3474
.419
445
5315
.764
731
8149
.260
0612
508.
3653
819
210.
3355
929
504.
8503
845
299.
0199
469
497.
1820
910
6512
.784
716
3037
.576
358
3857
.605
583
5954
.656
499
9209
.663
867
1426
0.53
654
2209
2.88
593
3422
6.62
644
5300
0.85
333
8200
7.67
486
1267
51.2
137
1956
46.0
915
5942
82.9
4219
866
70.2
1527
910
407.
9201
716
258.
0116
525
407.
8188
239
703.
8866
762
011.
9984
9677
0.05
634
1508
34.9
444
2347
76.3
098
6047
55.0
6583
974
71.6
4111
211
761.
9497
918
535.
1332
829
219.
9916
446
057.
5085
372
555.
0381
311
4189
.666
517
9494
.583
828
1732
.571
8
cont
inue
sov
erle
af
Tabl
eA
.3(c
onti
nued
)
N/i
1112
1314
1516
1718
1920
6152
79.1
2308
283
69.2
3804
613
292.
0032
721
131.
0519
433
603.
9903
853
427.
7099
8489
0.39
461
1347
44.8
064
2135
99.5
547
3380
80.0
861
6258
60.8
2662
193
74.5
4661
115
020.
9636
924
090.
3992
238
645.
5889
461
977.
1434
899
322.
7616
915
8999
.871
625
4184
.470
140
5697
.103
363
6506
.517
549
1050
0.49
2216
974.
6889
727
464.
0551
144
443.
4272
871
894.
4864
411
6208
.631
218
7620
.848
530
2480
.519
448
6837
.524
6472
23.2
3447
911
761.
5512
719
182.
3985
431
310.
0228
251
110.
9413
783
398.
6042
713
5965
.098
522
1393
.601
235
9952
.818
158
4206
.028
865
8018
.790
272
1317
3.93
742
2167
7.11
035
3569
4.42
601
5877
8.58
258
9674
3.38
095
1590
80.1
652
2612
45.4
494
4283
44.8
535
7010
48.2
346
6689
01.8
5720
214
755.
8099
124
496.
1346
940
692.
6456
667
596.
3699
711
2223
.321
918
6124
.793
330
8270
.630
350
9731
.375
784
1258
.881
567
9882
.061
494
1652
7.50
7127
681.
6322
4639
0.61
605
7773
6.82
546
1301
80.0
534
2177
67.0
082
3637
60.3
438
6065
81.3
371
1009
511.
658
6810
970.
0882
618
511.
8079
531
281.
2443
952
886.
3023
8939
8.34
928
1510
09.8
6225
4788
.399
642
9238
.205
772
1832
.791
212
1141
4.98
969
1217
7.79
797
2073
4.22
491
3534
8.80
616
6029
1.38
462
1028
09.1
017
1751
72.4
399
2981
03.4
275
5065
02.0
827
8589
82.0
215
1453
698.
987
7013
518.
3557
423
223.
3319
3994
5.15
096
6873
3.17
846
1182
31.4
669
2032
01.0
302
3487
82.0
102
5976
73.4
576
1022
189.
606
1744
439.
785
7115
006.
3748
826
011.
1317
345
139.
0205
878
356.
8234
513
5967
.187
2357
14.1
951
4080
75.9
519
7052
55.6
812
1640
6.63
120
9332
8.74
272
1665
8.07
611
2913
3.46
753
5100
8.09
326
8932
7.77
873
1563
63.2
6527
3429
.466
347
7449
.863
783
2202
.702
414
4752
4.89
2511
995.
4973
1849
1.46
448
3263
0.48
364
5764
0.14
538
1018
34.6
678
1798
18.7
548
3171
79.1
809
5586
17.3
406
9820
00.1
888
1722
555.
6230
1439
5.58
874
2052
6.52
558
3654
7.14
167
6513
4.36
428
1160
92.5
212
2067
92.5
6836
7928
.849
965
3583
.288
411
5876
1.22
320
4984
2.18
736
1727
5.70
575
2278
5.44
339
4093
3.79
867
7360
2.83
163
1323
46.4
742
2378
12.4
532
4267
98.4
658
7646
93.4
475
1367
339.
243
2439
313.
203
4340
731.
847
7625
292.
8421
645
846.
8545
183
172.
1997
515
0875
.980
627
3485
.321
149
5087
.220
489
4692
.333
616
1346
1.30
729
0278
3.71
152
0887
9.21
677
2807
6.05
4851
349.
4770
693
985.
5857
117
1999
.617
931
4509
.119
357
4302
.175
610
4679
1.03
1903
885.
342
3454
313.
617
6250
656.
059
7831
165.
4208
357
512.
4143
1062
04.7
119
1960
80.5
644
3616
86.4
872
6661
91.5
237
1224
746.
505
2246
585.
703
4110
634.
204
7500
788.
271
7934
594.
6171
264
414.
9040
212
0012
.324
422
3532
.843
441
5940
.460
377
2783
.167
514
3295
4.41
126
5097
2.13
4891
655.
703
9000
946.
925
8038
401.
025
7214
5.69
2513
5614
.926
625
4828
.441
547
8332
.529
389
6429
.474
316
7655
7.66
131
2814
8.11
358
2107
1.28
610
8011
37.3
1
8142
626.
1377
580
804.
1756
1532
45.8
6729
0505
.423
355
0083
.408
710
3985
9.19
1961
573.
464
3691
215.
774
6927
075.
8312
9613
65.7
782
4731
6.01
291
9050
1.67
667
1731
68.8
297
3311
77.1
825
6325
96.9
201
1206
237.
661
2295
041.
952
4355
635.
613
8243
221.
238
1555
3639
.93
8352
521.
7743
310
1362
.877
919
5681
.777
637
7542
.988
172
7487
.458
113
9923
6.68
626
8520
0.08
451
3965
1.02
398
0943
4.27
318
6643
68.9
184
5830
0.16
9511
3527
.423
222
1121
.408
743
0400
.006
483
6611
.576
816
2311
5.55
631
4168
5.09
960
6478
9.20
711
6732
27.7
922
3972
43.6
985
6471
4.18
815
1271
51.7
1424
9868
.191
849
0657
.007
396
2104
.313
318
8281
5.04
536
7577
2.56
671
5645
2.26
513
8911
42.0
626
8766
93.4
3
8671
833.
7488
514
2410
.919
728
2352
.056
855
9349
.988
411
0642
0.96
2184
066.
452
4300
654.
902
8444
614.
672
1653
0460
.06
3225
2033
.12
8779
736.
4612
215
9501
.23
3190
58.8
241
6376
59.9
867
1272
385.
104
2533
518.
085
5031
767.
235
9964
646.
313
1967
1248
.47
3870
2440
.74
8888
508.
4719
517
8642
.377
736
0537
.471
372
6933
.384
914
6324
3.87
2938
881.
978
5887
168.
665
1175
8283
.65
2340
8786
.68
4644
2929
.89
8998
245.
4038
720
0080
.463
4074
08.3
425
8287
05.0
588
1682
731.
4534
0910
4.09
568
8798
8.33
813
8747
75.7
127
8564
57.1
555
7315
16.8
790
1090
53.3
983
2240
91.1
185
4603
72.4
271
9447
24.7
6719
3514
2.16
839
5456
1.75
8058
947.
355
1637
2236
.33
3314
9185
6687
7821
.24
9112
1050
.272
125
0983
.052
852
0221
.842
610
7698
7.23
422
2541
4.49
345
8729
2.63
9428
969.
406
1931
9239
.87
3944
7531
.15
8025
3386
.49
9213
4366
.802
2811
02.0
191
5878
51.6
821
1227
766.
447
2559
227.
667
5321
260.
451
1103
1895
.222
7967
04.0
546
9425
63.0
796
3040
64.7
993
1491
48.1
503
3148
35.2
614
6642
73.4
008
1399
654.
7529
4311
2.81
761
7266
3.12
312
9073
18.3
926
9001
11.7
855
8616
51.0
611
5564
878.
894
1655
55.4
468
3526
16.4
927
7506
29.9
429
1595
607.
415
3384
580.
7471
6029
0.22
315
1015
63.5
231
7421
32.9
6647
5365
.76
1386
7785
5.5
9518
3767
.545
939
4931
.471
984
8212
.835
518
1899
3.45
338
9226
8.85
183
0593
7.65
817
6688
30.3
137
4557
17.8
279
1056
86.2
516
6413
427.
6
9620
3982
.976
4423
24.2
485
9584
81.5
041
2073
653.
536
4476
110.
179
9634
888.
684
2067
2532
.47
4419
7748
.03
9413
5767
.64
1996
9611
4.1
9722
6422
.103
349
5404
.158
310
8308
5.1
2363
966.
031
5147
527.
705
1117
6471
.87
2418
6863
.99
5215
3343
.68
1120
2156
4.5
2396
3533
7.9
9825
1329
.534
755
4853
.657
312
2388
7.16
326
9492
2.27
659
1965
7.86
112
9647
08.3
728
2986
31.8
661
5409
46.5
413
3305
662.
728
7562
406.
599
2789
76.7
835
6214
37.0
962
1382
993.
494
3072
212.
394
6807
607.
5415
0390
62.7
133
1094
00.2
872
6183
17.9
215
8633
739.
734
5074
888.
810
030
9665
.229
769
6010
.547
715
6278
3.64
835
0232
3.12
978
2874
9.67
117
4453
13.7
538
7379
99.3
385
6896
16.1
418
8774
151.
241
4089
867.
6
N/i
2122
2324
2526
2728
2930
11.
0000
000
1.00
0000
01.
0000
000
1.00
0000
01.
0000
000
1.00
0000
01.
0000
000
1.00
0000
01.
0000
000
1.00
0000
02
2.21
0000
02.
2200
000
2.23
0000
02.
2400
000
2.25
0000
02.
2600
000
2.27
0000
02.
2800
000
2.29
0000
02.
3000
000
33.
6741
000
3.70
8400
03.
7429
000
3.77
7600
03.
8125
000
3.84
7600
03.
8829
000
3.91
8400
03.
9541
000
3.99
0000
04
5.44
5661
05.
5242
480
5.60
3767
05.
6842
240
5.76
5625
05.
8479
760
5.93
1283
06.
0155
520
6.10
0789
06.
1870
000
57.
5892
498
7.73
9582
67.
8926
334
8.04
8437
88.
2070
313
8.36
8449
88.
5327
294
8.69
9906
68.
8700
178
9.04
3100
0
610
.182
9923
10.4
4229
0710
.707
9391
10.9
8006
2811
.258
7891
11.5
4424
6711
.836
5664
12.1
3588
0412
.442
3230
12.7
5603
007
13.3
2142
0613
.739
5947
14.1
7076
5114
.615
2779
15.0
7348
6315
.545
7508
16.0
3243
9316
.533
9269
17.0
5059
6617
.582
8390
817
.118
9190
17.7
6230
5518
.430
0411
19.1
2294
4619
.841
8579
20.5
8764
6121
.361
1979
22.1
6342
6422
.995
2697
23.8
5769
079
21.7
1389
2022
.670
0127
23.6
6895
0524
.712
4513
25.8
0232
2426
.940
4340
28.1
2872
1329
.369
1858
30.6
6389
7932
.014
9979
1027
.273
8093
28.6
5741
5530
.112
8091
31.6
4343
9633
.252
9030
34.9
4494
6936
.723
4760
38.5
9255
7940
.556
4282
42.6
1949
73
1134
.001
3092
35.9
6204
6938
.038
7552
40.2
3786
5142
.566
1287
45.0
3063
3147
.638
8146
50.3
9847
4153
.317
7924
56.4
0534
6512
42.1
4158
4244
.873
6973
47.7
8766
8950
.894
9527
54.2
0766
0957
.738
5977
61.5
0129
4565
.510
0468
69.7
7995
2274
.326
9504
1351
.991
3168
55.7
4591
0759
.778
8328
64.1
0974
1468
.759
5761
73.7
5063
3179
.106
6440
84.8
5285
9991
.016
1384
97.6
2503
5514
63.9
0949
3469
.010
0110
74.5
2796
4380
.496
0793
86.9
4947
0293
.925
7977
101.
4654
379
109.
6116
607
118.
4108
185
127.
9125
462
1578
.330
4870
85.1
9221
3492
.669
3961
100.
8151
384
109.
6868
377
119.
3465
050
129.
8611
061
141.
3029
257
153.
7499
559
167.
2863
100
1695
.779
8893
104.
9345
004
114.
9833
572
126.
0107
716
138.
1085
472
151.
3765
964
165.
9236
048
181.
8677
449
199.
3374
431
218.
4722
031
1711
6.89
3666
012
9.02
0090
514
2.42
9529
315
7.25
3356
817
3.63
5683
919
1.73
4511
421
1.72
2978
123
3.79
0713
525
8.14
5301
628
5.01
3864
018
142.
4413
359
158.
4045
104
176.
1883
211
195.
9941
624
218.
0446
049
242.
5854
844
269.
8881
822
300.
2521
133
334.
0074
391
371.
5180
232
1917
3.35
4016
419
4.25
3502
721
7.71
1634
924
4.03
2761
427
3.55
5756
230
6.65
7710
334
3.75
7991
438
5.32
2705
143
1.86
9596
448
3.97
3430
120
210.
7583
598
237.
9892
733
268.
7853
109
303.
6006
241
342.
9446
952
387.
3887
150
437.
5726
490
494.
2130
625
558.
1117
794
630.
1654
592
cont
inue
sov
erle
af
Tabl
eA
.3(c
onti
nued
)
N/i
2122
2324
2526
2728
2930
2125
6.01
7615
429
1.34
6913
433
1.60
5932
537
7.46
4773
942
9.68
0869
489.
1097
809
556.
7172
643
633.
5927
272
0.96
4195
482
0.21
5096
922
310.
7813
147
356.
4432
343
408.
8752
969
469.
0563
196
538.
1010
862
617.
2783
239
708.
0309
256
811.
9986
815
931.
0438
121
1067
.279
626
2337
7.04
5390
743
5.86
0745
950
3.91
6615
258
2.62
9836
367
3.62
6357
877
8.77
0688
290
0.19
9275
610
40.3
5831
212
02.0
4651
813
88.4
6351
424
457.
2249
228
532.
7501
099
620.
8174
367
723.
4609
971
843.
0329
473
982.
2510
671
1144
.253
0813
32.6
5864
1551
.640
008
1806
.002
568
2555
4.24
2156
665
0.95
5134
176
4.60
5447
289
8.09
1636
410
54.7
9118
412
38.6
3634
514
54.2
0141
217
06.8
0305
920
02.6
1561
2348
.803
338
2667
1.63
3009
479
5.16
5263
694
1.46
4700
011
14.6
3362
913
19.4
8898
1561
.681
794
1847
.835
793
2185
.707
916
2584
.374
137
3054
.444
3427
813.
6759
414
971.
1016
216
1159
.001
581
1383
.145
716
50.3
6122
519
68.7
1906
123
47.7
5145
727
98.7
0613
233
34.8
4263
639
71.7
7764
228
985.
5478
891
1185
.743
978
1426
.571
945
1716
.100
668
2063
.951
531
2481
.586
016
2982
.644
3535
83.3
4384
943
02.9
4700
151
64.3
1093
429
1193
.512
946
1447
.607
654
1755
.683
492
2128
.964
828
2580
.939
414
3127
.798
381
3788
.958
324
4587
.680
126
5551
.801
631
6714
.604
214
3014
45.1
5066
417
67.0
8133
721
60.4
9069
526
40.9
1638
732
27.1
7426
839
42.0
2595
948
12.9
7707
258
73.2
3056
271
62.8
2410
487
29.9
8547
9
3117
49.6
3230
421
56.8
3923
226
58.4
0355
532
75.7
3632
4034
.967
835
4967
.952
709
6113
.480
882
7518
.735
119
9241
.043
095
1134
9.98
112
3221
18.0
5508
826
32.3
4386
332
70.8
3637
340
62.9
1303
750
44.7
0979
362
60.6
2041
377
65.1
2072
9624
.980
953
1192
1.94
559
1475
5.97
546
3325
63.8
4665
632
12.4
5951
240
24.1
2873
850
39.0
1216
663
06.8
8724
278
89.3
8172
198
62.7
0331
412
320.
9756
215
380.
3098
119
183.
7681
3431
03.2
5445
439
20.2
0060
549
50.6
7834
862
49.3
7508
678
84.6
0905
299
41.6
2096
812
526.
6332
115
771.
8487
919
841.
5996
624
939.
8985
335
3755
.937
8947
83.6
4473
860
90.3
3436
877
50.2
2510
698
56.7
6131
512
527.
4424
215
909.
8241
720
188.
9664
525
596.
6635
632
422.
8680
8
3645
45.6
8484
658
37.0
4658
174
92.1
1127
396
11.2
7913
212
321.
9516
415
785.
5774
520
206.
4767
2584
2.87
706
3302
0.69
599
4215
0.72
851
3755
01.2
7866
471
22.1
9682
992
16.2
9686
611
918.
9861
215
403.
4395
619
890.
8275
925
663.
2254
133
079.
8826
442
597.
6978
354
796.
9470
638
6657
.547
183
8690
.080
131
1133
7.04
514
1478
0.54
279
1925
5.29
944
2506
3.44
276
3259
3.29
627
4234
3.24
978
5495
2.03
0271
237.
0311
839
8056
.632
092
1060
2.89
776
1394
5.56
553
1832
8.87
306
2407
0.12
4331
580.
9378
841
394.
4862
754
200.
3597
270
889.
1189
692
609.
1405
340
9749
.524
831
1293
6.53
527
1715
4.04
5622
728.
8026
3008
8.65
538
3979
2.98
172
5257
1.99
756
6937
7.46
044
9144
7.96
346
1203
92.8
827
4111
797.
9250
515
783.
5730
321
100.
4760
928
184.
7152
237
611.
8192
350
140.
1569
766
767.
4369
8880
4.14
936
1179
68.8
729
1565
11.7
475
4214
276.
4893
119
256.
9590
925
954.
5855
934
950.
0468
847
015.
7740
363
177.
5977
884
795.
6448
611
3670
.311
215
2180
.846
2034
66.2
718
4317
275.
5520
623
494.
4900
931
925.
1402
743
339.
0581
358
770.
7175
479
604.
7732
110
7691
.469
1454
98.9
983
1963
14.2
913
2645
07.1
533
4420
904.
4179
928
664.
2779
139
268.
9225
353
741.
4320
873
464.
3969
310
0303
.014
213
6769
.165
618
6239
.717
825
3246
.435
834
3860
.299
345
2529
5.34
577
3497
1.41
905
4830
1.77
472
6664
0.37
577
9183
1.49
616
1263
82.7
979
1736
97.8
403
2383
87.8
388
3266
88.9
022
4470
19.3
89
4630
608.
3683
842
666.
1312
459
412.
1829
8263
5.06
596
1147
90.3
702
1592
43.3
254
2205
97.2
572
3051
37.4
337
4214
29.6
838
5811
26.2
058
4737
037.
1257
452
053.
6801
273
077.
9849
710
2468
.481
814
3488
.962
720
0647
.59
2801
59.5
166
3905
76.9
151
5436
45.2
922
7554
65.0
675
4844
815.
9221
563
506.
4897
489
886.
9215
112
7061
.917
417
9362
.203
425
2816
.963
435
5803
.586
149
9939
.451
470
1303
.426
998
2105
.587
749
5422
8.26
5877
478.
9174
811
0561
.913
515
7557
.777
622
4203
.754
331
8550
.373
945
1871
.554
463
9923
.497
890
4682
.420
712
7673
8.26
450
6561
7.20
162
9452
5.27
933
1359
92.1
536
1953
72.6
442
2802
55.6
929
4013
74.4
711
5738
77.8
741
8191
03.0
771
1167
041.
323
1659
760.
743
Tabl
eA
.4O
rdin
ary
annu
itypr
esen
twor
thfa
ctor
.To
find
Pfo
ra
give
nR
rece
ived
atth
een
dof
each
peri
odP
=R
1−(
1+i
)−N
i
N/i
12
34
56
78
910
10.
9900
990
0.98
0392
20.
9708
738
0.96
1538
50.
9523
810
0.94
3396
20.
9345
794
0.92
5925
90.
9174
312
0.90
9090
92
1.97
0395
11.
9415
609
1.91
3469
71.
8860
947
1.85
9410
41.
8333
927
1.80
8018
21.
7832
647
1.75
9111
21.
7355
372
32.
9409
852
2.88
3883
32.
8286
114
2.77
5091
02.
7232
480
2.67
3011
92.
6243
160
2.57
7097
02.
5312
947
2.48
6852
04
3.90
1965
63.
8077
287
3.71
7098
43.
6298
952
3.54
5950
53.
4651
056
3.38
7211
33.
3121
268
3.23
9719
93.
1698
654
54.
8534
312
4.71
3459
54.
5797
072
4.45
1822
34.
3294
767
4.21
2363
84.
1001
974
3.99
2710
03.
8896
513
3.79
0786
8
65.
7954
765
5.60
1430
95.
4171
914
5.24
2136
95.
0756
921
4.91
7324
34.
7665
397
4.62
2879
74.
4859
186
4.35
5260
77
6.72
8194
56.
4719
911
6.23
0283
06.
0020
547
5.78
6373
45.
5823
814
5.38
9289
45.
2063
701
5.03
2952
84.
8684
188
87.
6516
778
7.32
5481
47.
0196
922
6.73
2744
96.
4632
128
6.20
9793
85.
9712
985
5.74
6638
95.
5348
191
5.33
4926
29
8.56
6017
68.
1622
367
7.78
6108
97.
4353
316
7.10
7821
76.
8016
923
6.51
5232
26.
2468
879
5.99
5246
95.
7590
238
109.
4713
045
8.98
2585
08.
5302
028
8.11
0895
87.
7217
349
7.36
0087
17.
0235
815
6.71
0081
46.
4176
577
6.14
4567
1
1110
.367
6282
9.78
6848
09.
2526
241
8.76
0476
78.
3064
142
7.88
6874
67.
4986
743
7.13
8964
36.
8051
906
6.49
5061
012
11.2
5507
7510
.575
3412
9.95
4004
09.
3850
738
8.86
3251
68.
3838
439
7.94
2686
37.
5360
780
7.16
0725
36.
8136
918
1312
.133
7401
11.3
4837
3710
.634
9553
9.98
5647
89.
3935
730
8.85
2683
08.
3576
507
7.90
3775
97.
4869
039
7.10
3356
214
13.0
0370
3012
.106
2488
11.2
9607
3110
.563
1229
9.89
8640
99.
2949
839
8.74
5468
08.
2442
370
7.78
6150
47.
3666
875
1513
.865
0525
12.8
4926
3511
.937
9351
11.1
1838
7410
.379
6580
9.71
2249
09.
1079
140
8.55
9478
78.
0606
884
7.60
6079
5
1614
.717
8738
13.5
7770
9312
.561
1020
11.6
5229
5610
.837
7696
10.1
0589
539.
4466
486
8.85
1369
28.
3125
582
7.82
3708
617
15.5
6225
1314
.291
8719
13.1
6611
8512
.165
6689
11.2
7406
6210
.477
2597
9.76
3223
09.
1216
381
8.54
3631
48.
0215
533
1816
.398
2686
14.9
9203
1313
.753
5131
12.6
5929
7011
.689
5869
10.8
2760
3510
.059
0869
9.37
1887
18.
7556
251
8.20
1412
119
17.2
2600
8515
.678
4620
14.3
2379
9113
.133
9394
12.0
8532
0911
.158
1165
10.3
3559
529.
6035
992
8.95
0114
88.
3649
201
2018
.045
5530
16.3
5143
3314
.877
4749
13.5
9032
6312
.462
2103
11.4
6992
1210
.594
0142
9.81
8147
49.
1285
457
8.51
3563
7
2118
.856
9831
17.0
1120
9215
.415
0241
14.0
2915
9912
.821
1527
11.7
6407
6610
.835
5273
10.0
1680
329.
2922
437
8.64
8694
322
19.6
6037
9317
.658
0482
15.9
3691
6614
.451
1153
13.1
6300
2612
.041
5817
11.0
6124
0510
.200
7437
9.44
2425
48.
7715
403
2320
.455
8211
18.2
9220
4116
.443
6084
14.8
5684
1713
.488
5739
12.3
0337
9011
.272
1874
10.3
7105
899.
5802
068
8.88
3218
424
21.2
4338
7318
.913
9256
16.9
3554
2115
.246
9631
13.7
9864
1812
.550
3575
11.4
6933
4010
.528
7583
9.70
6611
88.
9847
440
2522
.023
1557
19.5
2345
6517
.413
1477
15.6
2207
9914
.093
9446
12.7
8335
6211
.653
5832
10.6
7477
629.
8225
796
9.07
7040
0
2622
.795
2037
20.1
2103
5817
.876
8424
15.9
8276
9214
.375
1853
13.0
0316
6211
.825
7787
10.8
0997
809.
9289
721
9.16
0945
527
23.5
5960
7620
.706
8978
18.3
2703
1516
.329
5857
14.6
4303
3613
.210
5341
11.9
8670
9010
.935
1648
10.0
2657
999.
2372
232
2824
.316
4432
21.2
8127
2418
.764
1082
16.6
6306
3214
.898
1273
13.4
0616
4312
.137
1113
11.0
5107
8510
.116
1284
9.30
6566
529
25.0
6578
5321
.844
3847
19.1
8845
4616
.983
7146
15.1
4107
3613
.590
7210
12.2
7767
4111
.158
4060
10.1
9828
299.
3696
059
3025
.807
7082
22.3
9645
5619
.600
4413
17.2
9203
3315
.372
4510
13.7
6483
1212
.409
0412
11.2
5778
3310
.273
6540
9.42
6914
5
cont
inue
sov
erle
af
Tabl
eA
.4(c
onti
nued
)
N/i
12
34
56
78
910
3126
.542
2854
22.9
3770
1520
.000
4285
17.5
8849
3615
.592
8105
13.9
2908
6012
.531
8142
11.3
4979
9410
.342
8019
9.47
9013
232
27.2
6958
9523
.468
3348
20.3
8876
5517
.873
5515
15.8
0267
6714
.084
0434
12.6
4655
5311
.434
9994
10.4
0624
039.
5263
756
3327
.989
6925
23.9
8856
3620
.765
7918
18.1
4764
5716
.002
5492
14.2
3022
9612
.753
7900
11.5
1388
8410
.464
4406
9.56
9432
434
28.7
0266
5924
.498
5917
21.1
3183
6718
.411
1978
16.1
9290
4014
.368
1411
12.8
5400
9411
.586
9337
10.5
1783
549.
6085
749
3529
.408
5801
24.9
9861
9321
.487
2201
18.6
6461
3216
.374
1943
14.4
9824
6412
.947
6723
11.6
5456
8210
.566
8215
9.64
4159
0
3630
.107
5050
25.4
8884
2521
.832
2525
18.9
0828
2016
.546
8517
14.6
2098
7113
.035
2078
11.7
1719
2810
.611
7628
9.67
6508
237
30.7
9950
9925
.969
4534
22.1
6723
5419
.142
5788
16.7
1128
7314
.736
7803
13.1
1701
6611
.775
1785
10.6
5299
349.
7059
165
3831
.484
6633
26.4
4064
0622
.492
4616
19.3
6786
4216
.867
8927
14.8
4601
9213
.193
4735
11.8
2886
9010
.690
8196
9.73
2651
439
32.1
6303
3026
.902
5888
22.8
0821
5119
.584
4848
17.0
1704
0714
.949
0747
13.2
6492
8511
.878
5824
10.7
2552
269.
7569
558
4032
.834
6861
27.3
5547
9223
.114
7720
19.7
9277
3917
.159
0864
15.0
4629
6913
.331
7088
11.9
2461
3310
.757
3602
9.77
9050
7
4133
.499
6892
27.7
9948
9523
.412
4000
19.9
9305
1817
.294
3680
15.1
3801
5913
.394
1204
11.9
6723
4610
.786
5690
9.79
9137
042
34.1
5810
8128
.234
7936
23.7
0135
9220
.185
6267
17.4
2320
7615
.224
5433
13.4
5244
9012
.006
6987
10.8
1336
609.
8173
973
4334
.810
0081
28.6
6156
2323
.981
9021
20.3
7079
4917
.545
9120
15.3
0617
2913
.506
9617
12.0
4323
9510
.837
9505
9.83
3997
544
35.4
5545
3529
.079
9631
24.2
5427
3920
.548
8413
17.6
6277
3315
.383
1820
13.5
5790
8112
.077
0736
10.8
6050
509.
8490
887
4536
.094
5084
29.4
9015
9924
.518
7125
20.7
2003
9717
.774
0698
15.4
5583
2113
.605
5216
12.1
0840
1510
.881
1973
9.86
2807
9
4636
.727
2361
29.8
9231
3624
.775
4491
20.8
8465
3617
.880
0665
15.5
2436
9913
.650
0202
12.1
3740
8810
.900
1810
9.87
5279
947
37.3
5369
9130
.286
5820
25.0
2470
7821
.042
9361
17.9
8101
5715
.589
0282
13.6
9160
7612
.164
2674
10.9
1759
729.
8866
181
4837
.973
9595
30.6
7311
9625
.266
7066
21.1
9513
0918
.077
1578
15.6
5002
6613
.730
4744
12.1
8913
6510
.933
5755
9.89
6925
549
38.5
8807
8731
.052
0780
25.5
0165
6921
.341
4720
18.1
6872
1715
.707
5723
13.7
6679
8512
.212
1634
10.9
4823
449.
9062
959
5039
.196
1175
31.4
2360
5925
.729
7640
21.4
8218
4618
.255
9255
15.7
6186
0613
.800
7463
12.2
3348
4610
.961
6829
9.91
4814
5
5139
.798
1362
31.7
8784
8925
.951
2272
21.6
1748
5218
.338
9766
15.8
1307
6113
.832
4732
12.2
5322
6510
.974
0210
9.92
2558
652
40.3
9419
4232
.144
9499
26.1
6624
0021
.747
5819
18.4
1807
3015
.861
3925
13.8
6212
4512
.271
5060
10.9
8534
049.
9295
987
5340
.984
3507
32.4
9504
8926
.374
9903
21.8
7267
4918
.493
4028
15.9
0697
4113
.889
8359
12.2
8843
1510
.995
7251
9.93
5998
954
41.5
6866
4132
.838
2833
26.5
7766
0521
.992
9567
18.5
6514
5615
.949
9755
13.9
1573
4512
.304
1033
11.0
0525
249.
9418
171
5542
.147
1922
33.1
7478
7526
.774
4276
22.1
0861
2218
.633
4720
15.9
9054
3013
.939
9388
12.3
1861
4111
.013
9930
9.94
7106
5
5642
.719
9922
33.5
0469
3626
.965
4637
22.2
1981
9418
.698
5447
16.0
2881
4113
.962
5596
12.3
3205
0111
.022
0120
9.95
1915
057
43.2
8712
1033
.828
1310
27.1
5093
5722
.326
7494
18.7
6051
8816
.064
9190
13.9
8370
0612
.344
4908
11.0
2936
889.
9562
864
5843
.848
6347
34.1
4522
6527
.331
0055
22.4
2956
6818
.819
5417
16.0
9898
0214
.003
4585
12.3
5601
0011
.036
1181
9.96
0260
359
44.4
0458
8834
.456
1044
27.5
0583
0622
.528
4296
18.8
7575
4016
.131
1134
14.0
2192
3812
.366
6760
11.0
4231
029.
9638
730
6044
.955
0384
34.7
6088
6727
.675
5637
22.6
2349
0018
.929
2895
16.1
6142
7714
.039
1812
12.3
7655
1811
.047
9910
9.96
7157
3
6145
.500
0380
35.0
5969
2827
.840
3531
22.7
1489
4218
.980
2757
16.1
9002
6114
.055
3095
12.3
8569
6111
.053
2028
9.97
0143
062
46.0
3964
1635
.352
6400
28.0
0034
2822
.802
7829
19.0
2883
4016
.217
0058
14.0
7038
2712
.394
1631
11.0
5798
429.
9728
573
6346
.573
9026
35.6
3984
3228
.155
6726
22.8
8729
1219
.075
0800
16.2
4245
8314
.084
4698
12.4
0200
2911
.062
3708
9.97
5324
864
47.1
0287
3835
.921
4149
28.3
0647
8322
.968
5493
19.1
1912
3816
.266
4701
14.0
9763
5312
.409
2619
11.0
6639
529.
9775
680
6547
.626
6078
36.1
9746
5528
.452
8915
23.0
4668
2019
.161
0703
16.2
8912
2714
.109
9396
12.4
1598
3211
.070
0874
9.97
9607
3
6648
.145
1562
36.4
6810
3528
.595
0403
23.1
2180
9619
.201
0194
16.3
1049
3114
.121
4388
12.4
2220
6711
.073
4747
9.98
1461
267
48.6
5857
0536
.733
4348
28.7
3304
8823
.194
0477
19.2
3906
6116
.330
6539
14.1
3218
5812
.427
9692
11.0
7658
239.
9831
465
6849
.166
9015
36.9
9356
3528
.867
0377
23.2
6350
7419
.275
3010
16.3
4967
3514
.142
2298
12.4
3330
4811
.079
4333
9.98
4678
669
49.6
7019
9537
.248
5917
28.9
9712
4023
.330
2956
19.3
0981
0516
.367
6165
14.1
5161
6612
.438
2452
11.0
8204
899.
9860
715
7050
.168
5143
37.4
9861
9329
.123
4214
23.3
9451
5019
.342
6766
16.3
8454
3914
.160
3893
12.4
4281
9611
.084
4485
9.98
7337
7
7150
.661
8954
37.7
4374
4429
.246
0401
23.4
5626
4419
.373
9778
16.4
0051
3114
.168
5882
12.4
4705
5211
.086
6500
9.98
8488
872
51.1
5039
1537
.984
0631
29.3
6508
7523
.515
6388
19.4
0378
8316
.415
5784
14.1
7625
0612
.450
9770
11.0
8866
979.
9895
353
7351
.634
0510
38.2
1966
9729
.480
6675
23.5
7272
9719
.432
1794
16.4
2979
0914
.183
4118
12.4
5460
8411
.090
5227
9.99
0486
674
52.1
1292
1838
.450
6566
29.5
9288
1123
.627
6247
19.4
5921
8516
.443
1990
14.1
9010
4512
.457
9707
11.0
9222
269.
9913
515
7552
.587
0512
38.6
7711
4329
.701
8263
23.6
8040
8319
.484
9700
16.4
5584
8114
.196
3593
12.4
6108
4011
.093
7822
9.99
2137
7
7653
.056
4864
38.8
9913
1729
.807
5983
23.7
3116
1919
.509
4952
16.4
6778
1214
.202
2050
12.4
6396
6711
.095
2131
9.99
2852
577
53.5
2127
3639
.116
7958
29.9
1028
9623
.779
9633
19.5
3285
2616
.479
0389
14.2
0766
8212
.466
6358
11.0
9652
589.
9935
022
7853
.981
4590
39.3
3019
1930
.009
9899
23.8
2688
7819
.555
0977
16.4
8965
9314
.212
7740
12.4
6910
7211
.097
7300
9.99
4093
079
54.4
3708
8239
.539
4039
30.1
0678
6323
.872
0075
19.5
7628
3516
.499
6786
14.2
1754
5812
.471
3956
11.0
9883
499.
9946
300
8054
.888
2061
39.7
4451
3630
.200
7634
23.9
1539
1819
.596
4605
16.5
0913
0814
.222
0054
12.4
7351
4411
.099
8485
9.99
5118
1
8155
.334
8575
39.9
4560
1630
.292
0033
23.9
5710
7519
.615
6767
16.5
1804
7914
.226
1733
12.4
7547
6311
.100
7785
9.99
5561
982
55.7
7708
6740
.142
7466
30.3
8058
5823
.997
2188
19.6
3397
7816
.526
4603
14.2
3006
8512
.477
2929
11.1
0163
169.
9959
654
8356
.214
9373
40.3
3602
6130
.466
5881
24.0
3578
7319
.651
4074
16.5
3439
6514
.233
7089
12.4
7897
4911
.102
4143
9.99
6332
284
56.6
4845
2840
.525
5158
30.5
5008
5624
.072
8724
19.6
6800
7016
.541
8835
14.2
3711
1112
.480
5323
11.1
0313
249.
9966
656
8557
.077
6760
40.7
1129
0030
.631
1510
24.1
0853
1219
.683
8162
16.5
4894
6714
.240
2908
12.4
8197
4411
.103
7912
9.99
6968
7
8657
.502
6495
40.8
9342
1630
.709
8554
24.1
4281
8419
.698
8726
16.5
5561
0114
.243
2624
12.4
8330
9611
.104
3956
9.99
7244
387
57.9
2341
5441
.071
9819
30.7
8626
7324
.175
7869
19.7
1321
2016
.561
8963
14.2
4603
9612
.484
5459
11.1
0495
019.
9974
948
8858
.340
0152
41.2
4704
1130
.860
4537
24.2
0748
7419
.726
8686
16.5
6782
6714
.248
6352
12.4
8569
0711
.105
4588
9.99
7722
689
58.7
5249
0341
.418
6677
30.9
3247
9424
.237
9687
19.7
3987
4816
.573
4214
14.2
5106
0912
.486
7506
11.1
0592
559.
9979
296
9059
.160
8815
41.5
8692
9231
.002
4071
24.2
6727
7619
.752
2617
16.5
7869
9414
.253
3279
12.4
8773
2011
.106
3537
9.99
8117
8
9159
.565
2292
41.7
5189
1331
.070
2982
24.2
9545
9219
.764
0588
16.5
8367
8714
.255
4467
12.4
8864
0811
.106
7465
9.99
8288
992
59.9
6557
3541
.913
6190
31.1
3621
1824
.322
5569
19.7
7529
4116
.588
3762
14.2
5742
6812
.489
4822
11.1
0710
699.
9984
445
9360
.361
9539
42.0
7217
5431
.200
2057
24.3
4861
2419
.785
9944
16.5
9280
7714
.259
2774
12.4
9026
1311
.107
4375
9.99
8585
994
60.7
5440
9842
.227
6230
31.2
6233
5624
.373
6658
19.7
9618
5116
.596
9884
14.2
6100
6912
.490
9827
11.1
0774
089.
9987
145
9561
.142
9800
42.3
8002
2531
.322
6559
24.3
9775
5619
.805
8906
16.6
0093
2414
.262
6233
12.4
9165
0611
.108
0191
9.99
8831
3
cont
inue
sov
erle
af
Tabl
eA
.4(c
onti
nued
)
N/i
12
34
56
78
910
9661
.527
7030
42.5
2943
3931
.381
2193
24.4
2091
8819
.815
1339
16.6
0465
3214
.264
1339
12.4
9226
9111
.108
2744
9.99
8937
697
61.9
0861
6842
.675
9155
31.4
3807
7024
.443
1912
19.8
2393
7016
.608
1634
14.2
6554
5712
.492
8418
11.1
0850
869.
9990
342
9862
.285
7592
42.8
1952
5031
.493
2787
24.4
6460
6919
.832
3210
16.6
1147
4914
.266
8651
12.4
9337
2011
.108
7235
9.99
9122
099
62.6
5916
7642
.960
3187
31.5
4687
2524
.485
1990
19.8
4030
5716
.614
5990
14.2
6809
8312
.493
8630
11.1
0892
079.
9992
018
100
63.0
2887
8843
.098
3516
31.5
9890
5324
.504
9990
19.8
4791
0216
.617
5462
14.2
6925
0712
.494
3176
11.1
0910
159.
9992
743
N/i
1112
1314
1516
1718
1920
10.
9009
009
0.89
2857
10.
8849
558
0.87
7193
00.
8695
652
0.86
2069
00.
8547
009
0.84
7457
60.
8403
361
0.83
3333
32
1.71
2523
31.
6900
510
1.66
8102
41.
6466
605
1.62
5708
91.
6052
319
1.58
5214
41.
5656
421
1.54
6501
01.
5277
778
32.
4437
147
2.40
1831
32.
3611
526
2.32
1632
02.
2832
251
2.24
5889
52.
2095
850
2.17
4272
92.
1399
168
2.10
6481
54
3.10
2445
73.
0373
493
2.97
4471
32.
9137
123
2.85
4978
42.
7981
806
2.74
3235
02.
6900
618
2.63
8585
52.
5887
346
53.
6958
970
3.60
4776
23.
5172
313
3.43
3081
03.
3521
551
3.27
4293
73.
1993
462
3.12
7171
03.
0576
349
2.99
0612
1
64.
2305
379
4.11
1407
33.
9975
498
3.88
8667
53.
7844
827
3.68
4735
93.
5891
848
3.49
7602
63.
4097
772
3.32
5510
17
4.71
2196
34.
5637
565
4.42
2610
44.
2883
048
4.16
0419
74.
0385
654
3.92
2380
13.
8115
276
3.70
5695
13.
6045
918
85.
1461
228
4.96
7639
84.
7987
703
4.63
8863
94.
4873
215
4.34
3590
94.
2071
625
4.07
7565
83.
9543
657
3.83
7159
89
5.53
7047
55.
3282
498
5.13
1655
14.
9463
718
4.77
1583
94.
6065
439
4.45
0566
24.
3030
218
4.16
3332
54.
0309
665
105.
8892
320
5.65
0223
05.
4262
435
5.21
6115
65.
0187
686
4.83
3227
54.
6586
036
4.49
4086
34.
3389
349
4.19
2472
1
116.
2065
153
5.93
7699
15.
6869
411
5.45
2733
05.
2337
118
5.02
8644
44.
8364
134
4.65
6005
34.
4864
999
4.32
7060
112
6.49
2356
16.
1943
742
5.91
7647
05.
6602
921
5.42
0619
05.
1971
072
4.98
8387
54.
7932
249
4.61
0504
14.
4392
167
136.
7498
704
6.42
3548
46.
1218
115
5.84
2361
55.
5831
470
5.34
2333
85.
1182
799
4.90
9512
64.
7147
093
4.53
2680
614
6.98
1865
26.
6281
682
6.30
2488
16.
0020
715
5.72
4475
65.
4675
291
5.22
9299
15.
0080
615
4.80
2276
84.
6105
672
157.
1908
696
6.81
0864
56.
4623
788
6.14
2168
05.
8473
701
5.57
5456
25.
3241
872
5.09
1577
64.
8758
628
4.67
5472
6
167.
3791
618
6.97
3986
26.
6038
751
6.26
5059
65.
9542
349
5.66
8496
75.
4052
882
5.16
2353
94.
9376
998
4.72
9560
517
7.54
8794
47.
1196
305
6.72
9093
06.
3728
593
6.04
7160
85.
7487
040
5.47
4605
35.
2223
338
4.98
9663
74.
7746
338
187.
7016
166
7.24
9670
16.
8399
053
6.46
7420
56.
1279
659
5.81
7848
35.
5338
507
5.27
3164
25.
0333
309
4.81
2194
819
7.83
9294
27.
3657
769
6.93
7969
36.
5503
688
6.19
8231
25.
8774
554
5.58
4487
85.
3162
409
5.07
0025
94.
8434
957
207.
9633
281
7.46
9443
67.
0247
516
6.62
3130
66.
2593
315
5.92
8840
95.
6277
673
5.35
2746
55.
1008
621
4.86
9579
7
218.
0750
704
7.56
2003
27.
1015
501
6.68
6956
66.
3124
622
5.97
3138
75.
6647
584
5.38
3683
55.
1267
749
4.89
1316
422
8.17
5739
17.
6446
457
7.16
9513
36.
7429
444
6.35
8662
76.
0113
265
5.69
6374
75.
4099
012
5.14
8550
34.
9094
304
238.
2664
316
7.71
8433
77.
2296
578
6.79
2056
56.
3988
372
6.04
4247
05.
7233
972
5.43
2119
75.
1668
490
4.92
4525
324
8.34
8136
67.
7843
158
7.28
2883
06.
8351
373
6.43
3771
46.
0726
267
5.74
6493
35.
4509
489
5.18
2226
14.
9371
044
258.
4217
447
7.84
3139
17.
3299
850
6.87
2927
46.
4641
491
6.09
7092
05.
7662
336
5.46
6905
85.
1951
480
4.94
7587
0
268.
4880
583
7.89
5659
97.
3716
681
6.90
6076
76.
4905
644
6.11
8182
75.
7831
056
5.48
0428
75.
2060
067
4.95
6322
527
8.54
7800
27.
9425
535
7.40
8555
96.
9351
550
6.51
3534
36.
1363
644
5.79
7526
25.
4918
887
5.21
5131
74.
9636
021
288.
6016
218
7.98
4422
87.
4411
999
6.96
0662
36.
5335
081
6.15
2038
35.
8098
514
5.50
1600
65.
2227
997
4.96
9668
429
8.65
0109
88.
0218
060
7.47
0088
46.
9830
371
6.55
0876
66.
1655
503
5.82
0385
95.
5098
310
5.22
9243
54.
9747
237
308.
6937
926
8.05
5184
07.
4956
534
7.00
2664
16.
5659
796
6.17
7198
55.
8293
896
5.51
6806
05.
2346
584
4.97
8936
4
318.
7331
465
8.08
4985
77.
5182
774
7.01
9880
86.
5791
127
6.18
7240
15.
8370
851
5.52
2716
95.
2392
087
4.98
2447
032
8.76
8600
48.
1115
944
7.53
8298
67.
0349
832
6.59
0532
86.
1958
966
5.84
3662
55.
5277
262
5.24
3032
54.
9853
725
338.
8005
409
8.13
5352
17.
5560
164
7.04
8230
86.
6004
633
6.20
3359
85.
8492
842
5.53
1971
35.
2462
458
4.98
7810
434
8.82
9316
18.
1565
644
7.57
1696
07.
0598
516
6.60
9098
56.
2097
924
5.85
4089
15.
5355
689
5.24
8946
14.
9898
420
358.
8552
398
8.17
5503
97.
5855
716
7.07
0045
36.
6166
074
6.21
5338
35.
8581
958
5.53
8617
75.
2512
152
4.99
1535
0
368.
8785
944
8.19
2414
27.
5978
510
7.07
8987
16.
6231
369
6.22
0119
25.
8617
058
5.54
1201
55.
2531
220
4.99
2945
837
8.89
9634
68.
2075
127
7.60
8717
77.
0868
308
6.62
8814
76.
2242
407
5.86
4705
85.
5433
911
5.25
4724
44.
9941
215
388.
9185
897
8.22
0993
57.
6183
343
7.09
3711
26.
6337
519
6.22
7793
75.
8672
699
5.54
5246
75.
2560
709
4.99
5101
339
8.93
5666
48.
2330
299
7.62
6844
57.
0997
467
6.63
8045
16.
2308
566
5.86
9461
55.
5468
192
5.25
7202
44.
9959
177
408.
9510
508
8.24
3776
77.
6343
756
7.10
5040
96.
6417
784
6.23
3497
15.
8713
346
5.54
8151
95.
2581
533
4.99
6598
1
418.
9649
106
8.25
3372
07.
6410
404
7.10
9685
06.
6450
247
6.23
5773
45.
8729
355
5.54
9281
35.
2589
524
4.99
7165
142
8.97
7397
08.
2619
393
7.64
6938
47.
1137
588
6.64
7847
56.
2377
357
5.87
4303
95.
5502
384
5.25
9623
84.
9976
376
438.
9886
459
8.26
9588
77.
6521
579
7.11
7332
36.
6503
022
6.23
9427
35.
8754
734
5.55
1049
55.
2601
881
4.99
8031
344
8.99
8780
18.
2764
185
7.65
6776
97.
1204
669
6.65
2436
76.
2408
565.
8764
730
5.55
1736
85.
2606
623
4.99
8359
445
9.00
7910
08.
2825
165
7.66
0864
57.
1232
166
6.65
4292
86.
2421
428
5.87
7327
35.
5523
193
5.26
1060
74.
9986
329
469.
0161
351
8.28
7961
17.
6644
819
7.12
5628
66.
6559
068
6.24
3226
55.
8780
576
5.55
2813
05.
2613
956
4.99
8860
747
9.02
3545
28.
2928
225
7.66
7683
17.
1277
444
6.65
7310
26.
2441
608
5.87
8681
75.
5532
314
5.26
1676
94.
9990
506
489.
0302
209
8.29
7162
97.
6705
160
7.12
9600
36.
6585
306
6.24
4966
25.
8792
151
5.55
3585
95.
2619
134
4.99
9208
849
9.03
6235
08.
3010
383
7.67
3023
07.
1312
284
6.65
9591
96.
2456
605
5.87
9671
05.
5538
864
5.26
2112
14.
9993
407
509.
0416
532
8.30
4498
57.
6752
416
7.13
2656
56.
6605
147
6.24
6259
15.
8800
607
5.55
4141
05.
2622
791
4.99
9450
6
519.
0465
344
8.30
7587
97.
6772
049
7.13
3909
26.
6613
171
6.24
6775
15.
8803
938
5.55
4356
85.
2624
194
4.99
9542
152
9.05
0931
98.
3103
464
7.67
8942
47.
1350
080
6.66
2014
96.
2472
199
5.88
0678
45.
5545
396
5.26
2537
34.
9996
185
539.
0548
936
8.31
2809
37.
6804
800
7.13
5972
06.
6626
216
6.24
7603
35.
8809
217
5.55
4694
65.
2626
364
4.99
9682
054
9.05
8462
78.
3150
083
7.68
1840
77.
1368
175
6.66
3149
26.
2479
339
5.88
1129
75.
5548
259
5.26
2719
74.
9997
350
559.
0616
781
8.31
6971
77.
6830
449
7.13
7559
26.
6636
080
6.24
8218
95.
8813
074
5.55
4937
25.
2627
896
4.99
9779
2
569.
0645
749
8.31
8724
77.
6841
105
7.13
8209
86.
6640
070
6.24
8464
65.
8814
593
5.55
5031
65.
2628
484
4.99
9816
057
9.06
7184
68.
3202
899
7.68
5053
67.
1387
806
6.66
4353
96.
2486
763
5.88
1589
25.
5551
115
5.26
2897
84.
9998
467
589.
0695
356
8.32
1687
47.
6858
881
7.13
9281
26.
6646
556
6.24
8858
95.
8817
002
5.55
5179
25.
2629
394
4.99
9872
259
9.07
1653
78.
3229
352
7.68
6626
67.
1397
204
6.66
4917
96.
2490
163
5.88
1795
05.
5552
366
5.26
2974
34.
9998
935
609.
0735
619
8.32
4049
37.
6872
802
7.14
0105
66.
6651
460
6.24
9152
05.
8818
761
5.55
5285
35.
2630
036
4.99
9911
3
cont
inue
sov
erle
af
Tabl
eA
.4(c
onti
nued
)
N/i
1112
1314
1516
1718
1920
619.
0752
810
8.32
5044
07.
6878
586
7.14
0443
56.
6653
443
6.24
9269
05.
8819
454
5.55
5326
55.
2630
282
4.99
9926
162
9.07
6829
78.
3259
321
7.68
8370
47.
1407
399
6.66
5516
86.
2493
698
5.88
2004
65.
5553
614
5.26
3048
94.
9999
384
639.
0782
250
8.32
6725
17.
6888
234
7.14
0999
96.
6656
668
6.24
9456
75.
8820
552
5.55
5391
15.
2630
663
4.99
9948
664
9.07
9482
08.
3274
332
7.68
9224
27.
1412
280
6.66
5797
26.
2495
317
5.88
2098
55.
5554
162
5.26
3080
94.
9999
572
659.
0806
144
8.32
8065
37.
6895
790
7.14
1428
16.
6659
106
6.24
9596
35.
8821
354
5.55
5437
45.
2630
932
4.99
9964
3
669.
0816
346
8.32
8629
77.
6898
929
7.14
1603
66.
6660
092
6.24
9651
95.
8821
670
5.55
5455
45.
2631
036
4.99
9970
367
9.08
2553
78.
3291
337
7.69
0170
77.
1417
575
6.66
6095
06.
2496
999
5.88
2194
15.
5554
707
5.26
3112
24.
9999
752
689.
0833
817
8.32
9583
77.
6904
166
7.14
1892
66.
6661
696
6.24
9741
35.
8822
171
5.55
5483
75.
2631
195
4.99
9979
469
9.08
4127
78.
3299
854
7.69
0634
17.
1420
110
6.66
6234
46.
2497
770
5.88
2236
95.
5554
946
5.26
3125
64.
9999
828
709.
0847
997
8.33
0344
17.
6908
267
7.14
2114
96.
6662
908
6.24
9807
85.
8822
537
5.55
5503
95.
2631
308
4.99
9985
7
719.
0854
051
8.33
0664
47.
6909
970
7.14
2206
16.
6663
398
6.24
9834
35.
8822
681
5.55
5511
85.
2631
351
4.99
9988
172
9.08
5950
68.
3309
503
7.69
1147
87.
1422
860
6.66
6382
46.
2498
571
5.88
2280
55.
5555
185
5.26
3138
84.
9999
900
739.
0864
419
8.33
1205
77.
6912
813
7.14
2356
26.
6664
195
6.24
9876
85.
8822
910
5.55
5524
15.
2631
418
4.99
9991
774
9.08
6884
68.
3314
336
7.69
1399
37.
1424
177
6.66
6451
86.
2498
938
5.88
2300
05.
5555
289
5.26
3144
44.
9999
931
759.
0872
835
8.33
1637
27.
6915
038
7.14
2471
76.
6664
798
6.24
9908
55.
8823
077
5.55
5533
05.
2631
465
4.99
9994
2
769.
0876
428
8.33
1818
97.
6915
963
7.14
2519
06.
6665
042
6.24
9921
15.
8823
143
5.55
5536
45.
2631
484
4.99
9995
277
9.08
7966
48.
3319
812
7.69
1678
27.
1425
605
6.66
6525
46.
2499
320
5.88
2319
95.
5555
393
5.26
3149
94.
9999
960
789.
0882
581
8.33
2126
07.
6917
506
7.14
2597
06.
6665
438
6.24
9941
45.
8823
247
5.55
5541
85.
2631
512
4.99
9996
779
9.08
8520
88.
3322
554
7.69
1814
77.
1426
289
6.66
6559
86.
2499
495
5.88
2328
85.
5555
439
5.26
3152
24.
9999
972
809.
0887
575
8.33
2370
97.
6918
714
7.14
2656
96.
6665
738
6.24
9956
45.
8823
323
5.55
5545
75.
2631
531
4.99
9997
7
819.
0889
707
8.33
2474
07.
6919
216
7.14
2681
56.
6665
859
6.24
9962
45.
8823
353
5.55
5547
25.
2631
539
4.99
9998
182
9.08
9162
88.
3325
661
7.69
1966
07.
1427
031
6.66
6596
46.
2499
676
5.88
2337
95.
5555
485
5.26
3154
54.
9999
984
839.
0893
358
8.33
2648
37.
6920
053
7.14
2722
06.
6666
056
6.24
9972
15.
8823
401
5.55
5549
65.
2631
551
4.99
9998
784
9.08
9491
78.
3327
217
7.69
2040
17.
1427
386
6.66
6613
56.
2499
759
5.88
2341
95.
5555
505
5.26
3155
54.
9999
989
859.
0896
322
8.33
2787
27.
6920
709
7.14
2753
26.
6666
205
6.24
9979
35.
8823
435
5.55
5551
25.
2631
559
4.99
9999
1
869.
0897
587
8.33
2845
77.
6920
981
7.14
2765
96.
6666
265
6.24
9982
15.
8823
449
5.55
5551
95.
2631
562
4.99
9999
287
9.08
9872
78.
3328
980
7.69
2122
27.
1427
771
6.66
6631
76.
2499
846
5.88
2346
15.
5555
525
5.26
3156
54.
9999
994
889.
0899
754
8.33
2944
67.
6921
436
7.14
2787
06.
6666
363
6.24
9986
75.
8823
471
5.55
5552
95.
2631
567
4.99
9999
189
9.09
0068
08.
3329
863
7.69
2162
57.
1427
956
6.66
6640
36.
2499
885
5.88
2347
95.
5555
533
5.26
3156
94.
9999
996
909.
0901
513
8.33
3023
57.
6921
792
7.14
2803
16.
6666
437
6.24
9990
15.
8823
486
5.55
5553
75.
2631
571
4.99
9999
6
919.
0902
264
8.33
3056
77.
6921
940
7.14
2809
86.
6666
467
6.24
9991
55.
8823
493
5.55
5554
05.
2631
572
4.99
9999
792
9.09
0294
18.
3330
863
7.69
2207
07.
1428
156
6.66
6649
36.
2499
927
5.88
2349
85.
5555
542
5.26
3157
34.
9999
997
939.
0903
550
8.33
3112
87.
6922
186
7.14
2820
76.
6666
516
6.24
9993
75.
8823
503
5.55
5554
45.
2631
574
4.99
9999
894
9.09
0409
98.
3331
364
7.69
2228
97.
1428
252
6.66
6653
56.
2499
945
5.88
2350
65.
5555
546
5.26
3157
54.
9999
998
959.
0904
594
8.33
3157
57.
6922
379
7.14
2829
16.
6666
552
6.24
9995
35.
8823
510
5.55
5554
75.
2631
575
4.99
9999
8
969.
0905
040
8.33
3176
37.
6922
460
7.14
2832
56.
6666
567
6.24
9995
95.
8823
513
5.55
5554
95.
2631
576
4.99
9999
997
9.09
0544
18.
3331
932
7.69
2253
17.
1428
356
6.66
6658
06.
2499
965
5.88
2351
55.
5555
550
5.26
3157
64.
9999
999
989.
0905
803
8.33
3208
27.
6922
593
7.14
2838
26.
6666
592
6.24
9997
05.
8823
517
5.55
5555
15.
2631
577
4.99
9999
999
9.09
0612
98.
3332
216
7.69
2264
97.
1428
405
6.66
6660
16.
2499
974
5.88
2351
95.
5555
551
5.26
3157
74.
9999
999
100
9.09
0642
28.
3332
336
7.69
2269
87.
1428
426
6.66
6661
06.
2499
978
5.88
2352
05.
5555
552
5.26
3157
74.
9999
999
N/i
2122
2324
2526
2728
2930
10.
8264
463
0.81
9672
10.
8130
081
0.80
6451
60.
8000
000
0.79
3650
80.
7874
016
0.78
1250
00.
7751
938
0.76
9230
82
1.50
9459
71.
4915
345
1.47
3990
31.
4568
158
1.44
0000
01.
4235
324
1.40
7402
81.
3916
016
1.37
6119
21.
3609
467
32.
0739
337
2.04
2241
42.
0113
743
1.98
1303
11.
9520
000
1.92
3438
41.
8955
928
1.86
8438
71.
8419
529
1.81
6112
94
2.54
0441
02.
4936
405
2.44
8271
82.
4042
767
2.36
1600
02.
3201
892
2.27
9994
32.
2409
678
2.20
3064
32.
1662
407
52.
9259
843
2.86
3639
82.
8034
730
2.74
5384
42.
6892
800
2.63
5070
82.
5826
727
2.53
2006
12.
4829
955
2.43
5569
8
63.
2446
152
3.16
6917
83.
0922
545
3.02
0471
32.
9514
240
2.88
4976
82.
8210
021
2.75
9379
72.
6999
965
2.64
2746
07
3.50
7946
43.
4155
064
3.32
7036
13.
2423
156
3.16
1139
23.
0833
149
3.00
8663
12.
9370
154
2.86
8214
42.
8021
123
83.
7255
755
3.61
9267
63.
5179
156
3.42
1222
23.
3289
114
3.24
0726
13.
1564
276
3.07
5793
32.
9986
158
2.92
4701
89
3.90
5434
33.
7862
849
3.67
3102
13.
5655
018
3.46
3129
13.
3656
557
3.27
2777
73.
1842
135
3.09
9702
23.
0190
013
104.
0540
780
3.92
3184
33.
7992
700
3.68
1856
33.
5705
033
3.46
4806
13.
3643
919
3.26
8916
83.
1780
637
3.09
1539
5
114.
1769
239
4.03
5397
03.
9018
455
3.77
5690
63.
6564
026
3.54
3496
93.
4365
290
3.33
5091
33.
2388
091
3.14
7338
112
4.27
8449
54.
1273
746
3.98
5240
33.
8513
634
3.72
5122
13.
6059
499
3.49
3329
93.
3867
900
3.28
5898
53.
1902
601
134.
3623
550
4.20
2766
14.
0530
409
3.91
2389
83.
7800
977
3.65
5515
83.
5380
551
3.42
7179
73.
3224
019
3.22
3277
014
4.43
1698
34.
2645
623
4.10
8163
33.
9616
047
3.82
4078
13.
6948
538
3.57
3271
73.
4587
342
3.35
0699
23.
2486
746
154.
4890
069
4.31
5215
04.
1529
783
4.00
1294
13.
8592
625
3.72
6074
53.
6010
013
3.48
3386
13.
3726
350
3.26
8211
2
164.
5363
693
4.35
6733
64.
1894
132
4.03
3301
73.
8874
100
3.75
0852
73.
6228
357
3.50
2645
43.
3896
396
3.28
3239
417
4.57
5511
84.
3907
653
4.21
9035
24.
0591
143
3.90
9928
03.
7705
180
3.64
0028
13.
5176
917
3.40
2821
43.
2947
995
184.
6078
610
4.41
8660
14.
2431
180
4.07
9930
93.
9279
424
3.78
6125
43.
6535
654
3.52
9446
63.
4130
398
3.30
3692
019
4.63
4595
94.
4415
246
4.26
2697
64.
0967
184
3.94
2353
93.
7985
123
3.66
4224
83.
5386
302
3.42
0961
13.
3105
323
204.
6566
908
4.46
0266
14.
2786
159
4.11
0256
83.
9538
831
3.80
8343
13.
6726
179
3.54
5804
83.
4271
016
3.31
5794
1
cont
inue
sov
erle
af
Tabl
eA
.4(c
onti
nued
)
N/i
2122
2324
2526
2728
2930
214.
6749
511
4.47
5627
94.
2915
577
4.12
1174
83.
9631
065
3.81
6145
33.
6792
267
3.55
1410
03.
4318
617
3.31
9841
622
4.69
0042
24.
4882
196
4.30
2079
44.
1299
797
3.97
0485
23.
8223
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266
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Index
abandon, real option to, 272abandonment analysis, 149–158accounting method of investment return
measurement, 43adverse minimum, defined, 80algorithm, defined, 41American Council for Capital Formation, 2annuity factor, 48, 56 footnotesAPT and CAPM, 236arbitrage pricing theory, APT, 234–237average discounted rate of return, 142average return on average investment, 43
Bacon, P., 143Balas, E., 162basic valuation model, 11Baum, 221, 222, 248Bell, E.J., 163 footnoteBell, M.D., 162beta, 228binomial option pricing model, 260, 261Black, F., 262 footnoteBlack-Scholes model, 258, 262“borrowing rate”, two-stage method, 117Bower, D.H. and R.S., 237
Campanella, J.A., 131Canada, 1capacity disparities, 82capital budgeting, defined, 5capital cost, defined, 77capital market line, 227capital rationing, defined, 11capital recovery, 118capital recovery factor, 53 footnote, 78capital structure, optimal, 36capital, defined, 1CAPM, capital asset pricing model, 33, 38, 202,
222, 225–237, 252 footnote
CAPM, compared with portfolio approaches, 232CAPM, some criticisms, 233–234Carlson, 221, 222cash flows, 6, 12, 19cash flows, quarterly, 64cash outflows, types of, 22certainty equivalents, 192–194challenger, defined, 77change scale, option to, 254Childs and Gridley leveraged lease, 112coefficient of variation, 186, 242component projects, defined, 20computer simulation, 195–214conflicting rankings, NPV vs. IRR, 68constraint specification, 164constraints, post-optimization, 175contingent claim analysis, CCA, 260contingent projects, 161contingent relationships, 215Corel Corporation, 259correlation coefficient, 239cost minimization, 13cost of capital, 6, 31
and CAPM, 231common stock, 33debt, 31marginal, 47marginal, 35preferred stock, 32
cost, initial, 15cost, opportunity, 66cost, sunk, 15cost-benefit analysis, defined, 5cumulative variance, 254Cyert, R.M., 9
Daellenbach, H.G., 163 footnotedecision trees, 203defender, defined, 77
316 Index
delay, real option to, 254depreciation, 23
ACRS, 24, 27, 28ADR, 23, 26Canadian, 28double-declining balance, 24general guidelines (GG), 23modified accelerated cost recovery (MACRS),
24straight line, 24sum-of-year’s-digits, 24
Descartes’ rule, 55, 59 footnote, 85deterioration, physical, 18deviational variables, 168Dias, M.A.G., 251, footnotedividend capitalization model, 33dominated point, 221, 222dual problem formulation, 13duality, defined, 75Durand, D., 146duration, Macaulay’s, 146Dyl, E.A., 149dynamic programming, 156
efficient frontier, 247efficient set, 220, 225evolution, of real option, 260Excel, answer report, 174Excel, solution model, 173excess present value, defined, 70existing firm assets, need to include, 221
factor loadings, 236factor rotation, 236factors, factor analysis, 235Federal Reserve Bank of St. Louis, 1first standard assumption, MAPI, 80Fisher’s intersection, rate, 68, 70, 71Flatto, J., 254forecasting, 19framing, real option, 255, 257France, 1future value, project, 87
Galbraith, J.K., 9geometric mean rate of return, 141Germany, 1goal programming, 165Gomory, R.E., 162Gordon model, 33Gross Domestic Product (GDP), 1
Haessler, R., 143Harlow, C.V., 184Hasan, S., 207 footnoteHirschleifer, J., 94
Holthausen, D.M., 233Horvath, P.A., 162housing, 3 footnoteHughes, J.S., 233
IIM, 99, 100immunization, 146independence, project, 217, 242indivisibility, project, 217, 243inflation, 30initial cost, components, 16initial investment method rate of return, 99, 100integer linear programming, 162interaction of financing and investment, 38internal rate of return
adjusted or modified IRR, 73deficiencies, 85IRR, 49, 145, 147IRR, calculating, 60IRR, caution and rule for, 58IRR, defined, 55IRR, uniqueness, 94
interval bisection method for IRR, 60investment curve, 184investment tax credit, 29investment, unrecovered, 43Italy, 1
Japan, 1Jean, W.H., 94Jucker, J.V., 221, 222
Koopmans, T.C., 146
lattice or tree, binomial option model, 261Lawler, E.L., 162lease
alternative analysis, 122derivation of generalized valuation, 128financial, 119leveraged, analysis, 132–139leveraged, defined, 131operating, 119traditional analysis, 120
leasing, 119–129alleged advantages, 119practical perspective, 127
Lee, J.A., 9 footnotelevel annuities, 53leveraged lease, see lease, leveragedLevy, R.E., 234Lewellen, W.G., 200–202linear programming formulation, 76linear programming, general approach, 159Linux, 259“loan rate”, two-stage method, 112
Index 317
Logue, D.E., 237Long, H.W., 149Long, M.S., 200–202Luehrman, T.A., 256
Magee, J.F., 203magnitude of capital investment, 1major projects, defined, 20Mao, J.C.T., 33, 72, 73, 87, 94, 95, 187, 188, 190,
220–222MAPI, 75, 151, 158
basic assumptions, 79, 80capacity disparities, 82
March, J.G., 9market portfolio, 227Markov chain, 203Markowitz, H.M., 207, 218–219, 221–223,
232–233, 246Masse, P., 203Microsoft, 33MISFM, 99, 107mixed cash flows, defined, 87mixed project, defined, 88Modigliani and Miller (MM), 36Monte Carlo simulation, 258, 274multiple investment sinking fund method,
MISFM, 99, 107multiple project selection, 218–219mutual exclusivity, 161, 192Myers, Dill, and Bautista lease model, 122, 123,
127Myers, S., 251Myers, S.C., 234
net present value, see NPVNewton-Raphson method for IRR, 61normative model, for capital budgeting, 10NPV, 47
as a function of cost of capital, 105relationship to two-stage method, 115
NPVq, 254
obsolescence, technological, 19operating inferiority, 17, 77, 78optimizer, spreadsheet, 171option exercise, decision rule, 255option, real, see real optionown risk, 202owner trustee, 131
Pappas, J.L., 157payback, 39, 191payback, two-stage method, 112Perg, W.F., 123 footnote, 127perpetuity, 33 footnotePettway, R.H., 162
phase diagram, option, 256physical deterioration, 79polynomial equation, 86portfolio return, overall, 247portfolio risk, 247portfolio selection approaches to selection,
215–224“price of risk”, 228principal components, 236profitability index, 51profitability index, 253project balance equations, defined, 87project
characteristics, 14financing rate, 88investment rate, 88life, 17types, 13types, defined, 20
public sector, cash flows, 12pure investments, defined, 88
qualified investment, under tax code, 29qualitative considerations, 14quotient, 254
rate of return, naıve, 41rates: nominal vs. effective, 62real option, defined, 251real options, types of, 254reinvestment rate, assumptions, 65Rendelman, R.J., 233return on invested capital, RIC, 88risk, 7
attitudes toward, 178, 187–188aversion, 177measures of, 185other considerations, 207of ruin, 188–189
risk-adjusted discount rate, 194–195, 249risk-free asset, 226riskiness, relative, 216risk-neutral asset, 260risk-neutral probability, 260, 263, 264rmin defined, 88Robichek, R.R., 149Robinson, E.A.G., 5Roll, R., 234Ross, S.A., 235Roy, A.D., 220
“satisficing”, 9saving & investment, 1second standard assumption, MAPI, 80securities, 247security market line, 228
318 Index
semi-variance, 186Sharpe, W.F., 222Simon, H., 9simple investments, defined, 88simulation, oilfield example, 207–214sinking fund earnings rate, 99sinking fund method rate of return, 99, 100sinking fund methods, 99solver, spreadsheet, 170spreadsheet optimization, 170standard error of the estimate, 186stochastic independence, 215Stone, H.L., 184strategic investments, 252sunk cost, 15switch production function, mix, etc., 254switch production, option to, 254systematic risk, 229
taxes, 23techological obsolescence, 79, 19Teichroew, D., 157Teichroew, Robicheck & Montalbano, TRM, 87,
97, 111Terborgh, G., 14, 19, 42, 75, 79, 80, 158Teweles, R.J., 184time spread, Boulding’s, 145time-related measures, 145–148tracking, asset or instrument, 260trade-off functions, 167, 168Trippi, R.R., 142, 144, 148TRM algorithm, defined, 89TSFM, 99, 100Turnbull, S.W., 234
two-stage method of analysis, 111, 112two-stage method, relationship to NPV, 115
Uncertainty, 7, 8uncertainty, a brief digression, 118undepreciated balance, 26unequal project lives, 51unequal project size, 51United Kingdom, 1unrecovered investment, 147unrecovered investment, 43unsystematic risk, 229useful life, 17utility, 177
enterprise, 184investor, 223personal, calculating, 180
valuation model, basic, 11valuation, and CAPM, 230value at risk, VAR, 252 footnotevalue of the enterprise, 11Van Horne, J.C., 149Varian, H., 263 footnotevariance, 185
wealth, maximization of as goal, 11Weil, R.L., 146Wiar method, 97Wiar, R., 97–99Williams, R., 143working capital requirements, 16
zero-one integer programming, 162