Cooke Tippett 2001 Deb and Value of Firm

8/10/2019 Cooke Tippett 2001 Deb and Value of Firm

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British Accounting Review(2000) 32, 261288

doi:10.1006/bare.2000.0135, available online at http://www.idealibrary.com on

DOUBLE ENTRY BOOKKEEPING, STRUCTURAL

DYNAMICS AND THE VALUE OF THE FIRM

TERRY COOKE AND MARK TIPPETTUniversity of Exeter

We demonstrate how a parsimonious interpretation of the double entry bookkeepingsystem leads to a model under which financial variables evolve in terms of a (vector)system of stochastic differential equations. The restrictions imposed by the doubleentry bookkeeping system are reflected in a structural matrix which summarizes thesensitivity of instantaneous changes in the firms assets, liabilities, and other financialvariables to the existing levels of these variables. Given this structure and some

additional simplifying assumptions, we show that the ratio of the book value of debt tothe book value of equity will, except under catastrophic circumstances, have a tendencyto revert towards a long run normal valuea description which conforms with theempirical evidence. Further, our analysis shows that, under certain circumstances, themarket value of equity can be expressed as a linear function of the firms assets andliabilities. Under these same circumstances, the ratio of the market value of equityto the book value of equity also has a tendency to revert towards a long run norm.We also demonstrate a theoretical basis for identifying the circumstances under whichnon-linear relationships exist between a firms bookkeeping summary measures andthe market value of its equity.

2000 Academic Press

INTRODUCTION

The analysis of Garman & Ohlson (1980) and Ohlson (1983, 1990) isbased on a model under which earnings, dividends, book values, andpossibly other information variables evolve in terms of a linear structuralsystem which, with the addition of assumptions about the discountingof future dividends, links variations in financial statement and otherinformation variables to the underlying equity price.Bernard (1995,p. 733)

The authors are from the Department of Accounting in the School of Business and Economics at theUniversity of Exeter. The authors gratefully acknowledge comments made on previous drafts by DavidAshton, Haim Falk and seminar participants at the University of Manchester and the University ofWarwick, where Andy Stark and Elizabeth Whalley are worthy of particular mention. It is also a pleasureto acknowledge the perseverance and insightful comments of the anonymous referees. However, sincewe have not always followed the counsel of our advisors, none can be held responsible for what remains.

Received April 1999; revised October 1999 and March 2000; accepted April 2000

08908389/00/030261+28 $35.00/0 2000 Academic Press


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262 T. COOKE AND M. TIPPETT

suggests that the GarmanOhlson formulation, as subsequently interpretedby Ohlson (1995) and Feltham & Ohlson (1995), is among the mostimportant developments in capital markets research in the last severalyears . . . while Lundholm (1995, p. 749) describes them as landmark

works in financial accounting . . . [which] provide a logically consistentframework for thinking about . . . accounting numbers. A principal virtue ofthe GarmanOhlson framework is that whilst it is based on what manyregard as fairly innocuous assumptions (martingale equivalent pricing,clean surplus bookkeeping, linear information dynamics)1 it neverthelessleads to a powerful measurement based research paradigm which unifiesthe apparently disparate normative approaches of an earlier generation(Edwards & Bell, 1961; Mattessich, 1957, 1964; Chambers, 1966; Ijiri,1965, 1967;Sterling, 1970)2 but in such a way as to be consistent with thebasic no arbitrage requirements of finance theory (Modigliani & Miller,1958, 1963; Miller & Modigliani, 1961). Our purpose here is to contributeto the further development of this literature. Specifically, we show how aparsimonious interpretation of the double entry bookkeeping system based

on the work ofBlack (1980, 1993)leads to a structural model of stochasticdifferential equations which relates variations in bookkeeping summarymeasures (book values of equity, assets, liabilities, and their components)to variations in the market value of the firms equity. A specific versionof the model implies that there will be a simple linear relationship betweenthe market value of equity and the firms bookkeeping summary measures.Also, the ratio of the book value of debt to the book value of equity(the debt to equity ratio) will have a tendency to revert towards a longrun normalized value. Further, and again within a specific version of themodel, we demonstrate a theoretical basis for identifying the circumstancesunder which non-linear relationships exist between a firms bookkeepingsummary measures and the market value of its equity.

In the next section, bookkeeping summary measures are characterized asevolving in terms of the structure imposed by the double entry bookkeepingsystem and the production, investment, and financing technologies availableto the firm. This in turn allows the evolution of bookkeeping summarymeasures such as assets, liabilities, and stockholders equity (and theircomponents) to be defined in terms of a (vector) system of stochasticdifferential equations. Restrictions imposed by the double entry bookkeepingsystem are reflected in a structural matrix which summarizes the sensitivityof instantaneous changes in these variables to their existing levels. InSection 3, we use the dynamic specification of the evolution of bookkeepingsummary measures advanced in Section 2 to investigate the properties of thedebt to equity ratio. Section 4 then develops models of corporate valuationincluding one which is linear in the book values of the firms equity, assets,

and liabilities. We also demonstrate that under this linear interpretation ofour model, the market to book ratio has a tendency to revert towards a longrun normalized value. The circumstances leading to non-linear valuation


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DOUBLE ENTRY BOOKKEEPING, STRUCTURAL DYNAMICS 263

relationships are also briefly developed in this section. We conclude with asummary of the papers main points and some suggestions for potentiallyprofitable areas for future research.

THE STRUCTURAL DYNAMICS OF BOOKKEEPING SUMMARYMEASURES

We begin by defining the vector of balance sheet variables:3

x(t)D

S(t)A(t)L(t)

where S(t) is the book value of a firms equity, A(t) is the book value of itsassets and L(t) is the book value of its liabilities, all at time t. Furthermore,we assume that the fundamental accounting identity holds. Defining u by:

uD

111

we have, therefore, that:

xTt.uD[S(t), A(t), L(t)].

111

DS(t)CA(t)CL(t)D0,

where xTt is the transpose of the vector x(t).Now, the firms transactions and other accounting entries over the interval

from time t to time tCt may be summarized by a transactions matrix,

Mt,tCt, as follows:4

Cr

Equity Assets Liabilities

Equity a11t,tCt a12t,tCt a13t,tCt

Dr Assets a21t,tCt a22t,tCt a23t,tCt

Liabilities a31t,tCt a32t,tCt a33t,tCt

where the rows of the above matrix represent accounts which are debitedwhile the columns are the accounts which are credited. Hence, the elementa21t,tCt represents all the entries over the interval [t,tCt] which involvea debit to assets and a credit to equity. An example of such an entry wouldbe the debit to bank and credit to sales arising from a cash sales transaction.

Similarly, the element a13t,tCt involves a debit to equity and a creditto liabilities. An example of such an entry would be the debit to operatingexpenses and credit to accounts payable arising from the accrual of the


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insurance premium on a firms manufacturing plant.5 It is also important tonote here that the elements of the transactions matrix, M(t,tCt), evolvestochastically over time.

Now if we define:

x(t)DxtCtx(t)D

S(t)A(t)L(t)

as the vector of first differences in the financial variables, (i.e., S(t)DStCtS(t) is the first difference in the book value of equity and A(t)and L(t) are similarly defined), then we can show that the book valueof equity, assets, and liabilities evolve in accordance with the differenceequation (Tippett, 1978,p. 274):

x(t)D[Mt,tCtMTt,tCt].u

where Mt,tCtMTt,tCt is the difference between the transactions

matrix and its transpose6 and takes the skew symmetric form:

Mt,tCtMTt,tCtD

0 b12t,tCt b13t,tCt

b12t,tCt 0 b23t,tCtb13t,tCt b23t,tCt 0

,

with bjkt,tCtDajkt,tCtakjt,tCt for j, kD1, 2, 3. Furthermore,we can divide the first, second, and third columns of the matrixMt,tCtMTt,tCt by the opening values of equity, assets, andliabilities respectively, in which case:

x(t)D[Mt,tCtMTt,tCt].uDt,tCtx(t),

where:

t,tCt

0 b12t,tCt

A(t)

b13t,tCt

L(t)b12t,tCt

S(t) 0

b23t,tCt

L(t)b13t,tCt

S(t)

b23t,tCt

A(t) 0

is called the matrix of structural coefficients, related to the underlyingtransactions matrix, whose elements summarize the sensitivity of propor-tionate changes in the bookkeeping variables to the opening levels of thesevariables. Furthermore, the structural coefficients will evolve stochastically

through time, since the elements of the transaction matrix from which theyare derived, are all stochastic functions of time. We are now in a position todevelop the properties of the structural coefficients in greater detail.


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Properties of the structural matrix

We begin by noting that the fundamental accounting identity, S(t)CA(t)CL(t)D0, implies that the structural coefficients bear particular relationships

one to another. It will help in demonstrating this if we define g(t)D

L(t)

S(t) tobe the book value of the firms liabilities divided by the book value of itsequity at time t. We shall henceforth refer to this variable as the debt toequity ratio. Now, this definition taken in conjunction with the fundamentalaccounting identity, shows:

A(t)D1Cg(t)S(t)

and so, both the book value of the firms liabilities and the book value ofits assets can be determined once the debt to equity ratio and the bookvalue of equity are known. We can use these definitions in conjunctionwith the matrix of structural coefficients, t,tCt, given above, todetermine the following more parsimonious expression for the relationshipbetween the structural coefficients:

Proposition 1

If g(t)DL(t)

S(t)is defined as the firms debt to equity ratio at time t, then

the fundamental accounting identity, S(t)CA(t)CL(t)D0, implies that thematrix of structural coefficients takes the form:

t, tCtD

0 !12t,tCtt !13t,tCtt

!12t,tCt1Cg(t)t 0 !23t,tCtt

g(t)!13t,tCtt !23t,tCt. g(t)1Cg(t)

(t) 0

where!12t,tCttDb12t,tCt

A(t),!13t,tCttD

b13t,tCt

L(t),!23t,tC

ttDb23t,tCt

L(t) and the!jkt,tCt are the upper diagonal structural

coefficients which hold over the interval [t,tCt].

Proof

See Appendix.

Note that the structural coefficients, !jkt,tCt, defined here are statedon a per unit time basis and so their impact over the finite interval fromtime t until time tCt will be !jkt,tCtt (Tippett & Warnock, 1997,


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pp. 10761079). Without further assumptions, however, little progress canbe made beyond this point. We thus invoke a convention, followed in theliterature, of assuming that structural coefficients can be partitioned intointertemporally stationary expected and stochastic components (Garman

& Ohlson, 1980; Ohlson, 1983, 1990, 1995; Tippett & Warnock,1997). Furthermore, since continuous time models have both analyticaland computational advantages over their discrete time equivalents, wehenceforth assume that all structural coefficients evolve continuously intime.7 These considerations lead us to state the following proposition:8

Proposition 2

Suppose the structural coefficients evolve in accordance with the diffusionprocess:

!jk(t)dtDmjkdtCsjkdzjk(t)

for k>j and where m jk is the expected value or mean (per unit time) of the

jkth

structural coefficient, s2jkis its instantaneous variance (per unit time) and

dzjk(t) is a standard Gauss-Wiener process. Then instantaneous changes inthe bookkeeping summary measures evolve in accordance with the vectorsystem of stochastic differential equations:

dx(t)D(t)x(t)dtCS(t)dZ(t)

where dx(t)D

dS(t)dA(t)dL(t)

contains the instantaneous changes in the firms

bookkeeping summary measures, x(t)D

S(t)A(t)L(t)

is the vector containing the

levels of these variables, tD

0 m

12 m

131Cg(t)m12 0 m23

g(t)m13g(t)

1Cg(t).m23 0

is thematrix containing the means or expected values of the structural coefficients

(per unit time) and dZ(t)D

s121Cg(t)dz12(t)Cs13g(t)dz13(t)s121Cg(t))dz12(t)Cs23g(t)dz23(t)

g(t)s13dz13(t)g(t)s23dz23(t)

is a

vector of stochastic error terms.

Proof

See Appendix.

There are some important observations which need to be made aboutthis proposition. The first stems from the fact that the exact form of the


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vector system of stochastic differential equations hinges critically on theassumptions made about the evolution of the structural coefficients, !jk(t).The principal assumption here is that the structural coefficients evolvein terms of the stochastic constant growth technology which underpins the

standard asset pricing models of finance theory (Merton, 1969, pp. 247248;Merton, 1973,pp. 870873). However, for pedagogical and computationalefficiency we assume that the Gauss-Wiener processes, dzjk(t), on which thestructural coefficients are based, are all uncorrelated. There are, however,reasons to believe that this may not be the case in practice. For example,bad debt write offs will often be accompanied by the creation of an extraprovision, in which case we can expect the structural coefficients !12(t)and !13(t) to be correlated. Given this, it is important to note that a moregeneral model replaces the dzjk(t) with stochastic processes which dependon factors external to the firm and which have a potentially significanteffect on the evolution of the firms bookkeeping summary measures. Thesecould include such things as the rate of growth of gross national product, oilprices, interest rates, inflation etc. The stochastic terms are then serially and

cross-sectionally correlated and non-Markovian, and the vector system ofstochastic differential equations which describes the evolution of the firmsbookkeeping summary measures is not then restricted to just the doubleentry bookkeeping variables themselves. Unfortunately, the complexity ofsuch models means that we cannot develop them in any detail here. It is,however, important to make this point given that empirical work appearsto have established that there are occasions on which a firms bookkeepingsummary measures are largely value-irrelevant (Amir & Lev, 1996).

A second point arises out of the fact that our assumptions implythat the vector of stochastic terms, dZ(t), has a mean of zero. Hence,Proposition 2 implies that Et[dx(t)]D(t)x(t)dt, where Et () is theexpectations operator taken at time t, is the vector whose elements arethe expected instantaneous increments in the book values of equity, assets,and liabilities. Thus, the expected instantaneous change in the book valueof equity is Et[dS(t)]D[m12A(t)Cm13L(t)]dtDm12S(t)Cm13m12L(t)]dt.Now suppose the firm follows a convention of instantaneously marking allits assets and liabilities to market. Then in the absence of taxes, bankruptcycosts, and other market imperfections, the Modigliani & Miller (1958)theorem shows Et[dS(t)]D[puCpurg(t)]S(t)dtD[puS(t)CpurL(t)]dt,where pu is the cost of equity for an unlevered firm and r is the costof debt. Hence, these assumptions imply puDm12 is the cost of equityfor an unlevered firm and purDm13m12, or that rDm13 is the costof debt. Of course, firms do not instantaneously mark their assets andliabilities to market and the Modigliani & Miller (1958) theorem holdsonly under highly idealised conditions and so in practice, the best we

can say is that m12 approximates for the cost of equity for an unleveredfirm; likewise m13 approximates for the cost of debt. Hence, the accuracyof these approximations depends on the bookkeeping policies invoked by


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the firm and the sensitivity of the model to violations of the assumptionsunderlying the Modigliani & Miller (1958) theorem. Note also, that ouranalysis here says nothing about the parameter m23. However, we now showhow this parameter is best interpreted in terms of the crucial role it plays in

determining the firms debt to equity ratio, g(t).

DISTRIBUTIONAL PROPERTIES OF THE DEBT TO EQUITYRATIO

There is nothing in the previous analysis which implies that the debt to equityratio is an intertemporal constant. We can, however, derive the processthrough which the debt to equity ratio evolves over time. Proposition 3provides a specification of this process.

Proposition 3

If the book values of the firms equity, assets, and liabilities evolve inaccordance with the vector system of stochastic differential equations:

dx(t)D(t)x(t) dtCS(t)dZ(t),

given in Proposition 2, then instantaneous proportionate variations in thedebt to equity ratio evolve in accordance with the non-linear stochasticdifferential equation:

dg(t)

g(t)D[s212Cs

2131Cg(t)

2C1Cg(t)m12m13s213m23]dt

C1Cg(t)s12dz12s13dz13s23dz23

As a consequence, instantaneous proportionate variations in the debt toequity ratio are normally distributed with mean:

Et

dg(t)

g(t)

D[s212Cs

2131Cg(t)

2C1Cg(t)m12m13s213m23]dt

and variance:

Vart

dg(t)

g(t)

D[1Cg(t)2s212Cs

213Cs

223]dt.

where Vart() is the variance operator taken at time t.

Proof

See Appendix.


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This proposition implies that expected instantaneous proportionate changes

in the debt to equity ratio, Et[dg(t)

g(t)], are zero and thus have no tendency to

change, at the following two roots:9

1Cg(t)Ds213Cm13m12

m12m13s

213

2C4m23s212Cs

213

2s212Cs213

Now, previous analysis shows that puDm12 proxies for the (unlevered)cost of equity capital whilst rDm13 proxies for the cost of debt. Hence, wewould normally expects213Cm13m12 >0. Furthermore, ifm23

m12m13s

213

2C4m23s212Cs

213> 0

and both the above roots will be positive.Figure Idepicts the instantaneous

expected proportionate change in the debt to equity ratio, g(t), under theassumption thatm23


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than (or to the left of) the least positive root, however,

Et

dg(t)

g(t)

>0

in which case the expected instantaneous proportionate change in the debtto equity ratio will also be back towards this root. Finally, when 1Cg(t)exceeds (or is to the right of) the most positive root, then:

Et

dg(t)

g(t)

>0

and so here the expected instantaneous proportionate change in the debtto equity ratio will be to the right of and away from the root. Hence,this most positive root is an unstable equilibrium in the sense that should1Cg(t) exceed it, our expectation will be for the debt to equity ratio tobecome infinitely large, thereby significantly increasing the probability of

catastrophic events such as the firm entering bankruptcy.When m23 > 0, a similar analysis shows that g(t) will either convergetowards a long term mean of zero (in which case the firms activities arefinanced purely by equity) or become infinitely large, in the latter case againsignifying a high likelihood of catastrophic events such as the firm enteringbankruptcy. However, since firms normally finance their operations withboth equity and (non-catastrophic levels) of debt, subsequent analysis isbased on the assumption that m23


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In other words, the above Lemma shows that when the debt to equityratio,g(t), is less than a given threshold value, gu, we would normally expectit to exhibit the kind of mean reversion properties documented for ratiosbyLev (1969) and others for the United States andWhittington & Tippett

(1995) in the United Kingdom. These papers find that financial ratios dohave a tendency to revert towards a long run or industry norm, althoughconvergence can often be very slow. When the debt to equity ratio exceedsthe threshold value of gu however, its statistical behaviour becomes moreproblematic and there is a significant chance that debt levels will become solarge relative to other sources of financing, that the firm will end up havingno choice but to initiate bankruptcy proceedings.

THE STRUCTURAL MATRIX AND CORPORATE VALUATION

We begin by noting that the standard no-arbitrage and nonsatiationrequirements of finance theory imply that over the non-infinitesimal

interval [t,tCt] equity value will have to satisfy the recursive relationship(Rubinstein, 1976,pp. 408409):

B(x(t))Der(x(t))tEtfBxtCtg

where B(x(t)) is the value of the equity security at time t and r(x(t)) is thediscount rate implied by a vector, x(t) of continuously evolving informationvariables. This formula says nothing more than the price of a unit of equitytoday is the expected present value of the price of a unit of equitytomorrow.10 Analytical and pedagogical convenience dictates, however,that we let x(t)Dx(S(t), g(t) so that equity value depends exclusively on thefirms bookkeeping summary measures.11 It then follows that the valuationequation is compatible with the simpler expression:

B(S(t), g(t)Der(S(t),g(t))tEtfB(S(tCt, g(tCtg

After expanding the right hand side of this equation as a seriesexpansion, taking limits (t!0) and substituting the means, variancesand covariances implied by the components of the vector dx(t) we end upwith Proposition 4.12

Proposition 4

If the discount rate, rg, depends only on the debt to equity ratio and thebook value of the firms equity, assets and liabilities evolve in accordance

with the vector system of stochastic differential equations:

dx(t)D(t)x(t)dtCS(t)dZ(t)


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given in Proposition 2, then equity value, B(S(t),g(t)), satisfies the partialdifferential equation:

S[m121CgCgm13]@B

@S

Cg[s212Cs2131Cg

2

C1Cgm12m13s213m23]

@B

@gC

1

2S2[s2121Cg

2

Cs213g2]@2B

@S2Sg[1Cg2s212Cg1Cgs

213]

@2B

@g@S

C1

2g2[1Cg2s212Cs

213Cs

223]

@2B

@g2rgBS, gD0

Proof

Generalize the proof in Karlin & Taylor (1981, pp. 203204) from a

function of one to a function of two, variables.This result has as its nearest analogue, the Black & Scholes (1973,

pp. 642643) differential (fundamental valuation) equation in optionspricing theory. Here it is well known that riskless hedging implies thatthe value of a European call option, C(P(t),t), must satisfy the recursiverelationship C(P(t),t)DeitEtfC(P(tCt),tCtg, where P(t) is the priceof the underlying equity security at time t and i is the riskless rate of interest.In continuous time (t!0), this leads to the BlackScholes differentialequation (Cox & Ross, 1976,pp. 153155). Now, Proposition 4 is based onthe same ideas as those that lie behind the derivation of theBlack & Scholes(1973) differential equation using this recursive valuation formula.

The derivation of Proposition 4 begins by assuming that equity valuedepends on a set of continuously evolving information variables. We thenuse this assumption in conjunction with the recursion formula to show thatin continuous time, equity value will have to satisfy a differential equationbased on these information variables. This in turn gives insights into therelationship between equity value and the assumed information variableswhich would not otherwise be available.

To demonstrate this we first note that two firms, which are identical in thesense that increments in the book values of their respective equities, assets,and liabilities are perfectly correlated with each other (and are thereforebased on identical investment, production, financing, and bookkeepingstrategies or structural matrices, (t)), must also have the same market valueof equity unless the book values of their equities are different. When thebook values of equity are different, the ratio of the market values of the two

equity securities must be equal to the ratio of the book values of the equitysecurities. If this were not the case we can use the arbitrage arguments ofModigliani & Miller (1958, 1963) and Miller & Modigliani (1961)to show


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that it will be possible to construct a perfectly hedged and self financingportfolio which returns only positive cash flows in the future therebyimplying the existence of arbitrage profits. Now we can state Proposition 5which holds only when such arbitrage opportunities do not exist.13

Proposition 5

The no arbitrage condition:

BlS, gDlBS, g

where l is a real parameter, implies that every solution to the partialdifferential equation defined in Proposition 4 takes the form:14

BS, gDSJg

where J(g) is a differentiable function satisfying the auxiliary equation:

[1Cgm13m12rgCm13]JgCg[1Cgm12m13m23]J0g

C 12g2[1Cg2s212Cs

213Cs

223]J

00gD0

Proof

See Appendix.

An important application of this result stems from the fact that it provides

a rigorous basis for analysing the book to market ratio, B(S,g

S for equity.

The empirical work ofFama & French (1992, 1993, 1995, 1996) amongstothers shows this ratio to be highly correlated with equity returns andthere is, as a consequence, a view that it plays an instrumental rolein generating these returns. We now provide specific interpretations ofProposition 5 and of the auxiliary equation in particular, which results inthere being alinearrelationship between the market value of equity and thebookkeeping summary measures which underscore it. Here it is importantto note, however, that solutions to the auxiliary equation hinge critically onspecification of the functional relationship between the cost of the firmsequity capital, rg, and its (book) debt to equity ratio, g. This means it isnot possible to write down a general closed form solution for the auxiliaryequation and so the relationship between the market and book values ofequity depends on the accounting and bookkeeping practices employedby the firmas one might expect. For pedagogical and computational

simplicity the results we report are initially based on the assumption thatthe discount rate, rg, is a constant, independent of the debt to equityratio.15


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Lemma 2

Under the transversality condition rgDm12 >0 for all g,16 the auxiliary

equation is satisfied by the simple linear valuation equation:

JgD

1Cg

gm23

m13CrCm23

This result also implies:

JgDB(S, g

S D

m13Cr1CgCm23

m13CrCm23

is the ratio of the market value of equity to the book value of equity and is alinear function of the book debt to equity ratio.

The implications of this Lemma can probably best be understoodby recalling from previous analysis that if assets and liabilities are

instantaneously marked to market then puDm12 will be an unleveredfirms cost of equity capital. However, the Modigliani & Miller (1958)theorem says that pLDpuCpurg, where pL is the cost of equity for alevered firm and rDm13 is the cost of debt. Hence, under a mark tomarket bookkeeping strategy and the transversality condition imposed inthe hypothesis, Lemma 2 implies rgDpLDpuDm12 irrespective of thefirms debt to equity ratio. It thus follows that the valuation procedures arebased on risk neutral pricing, or equivalently rDm12Dm13Dr for all g.Substituting this equality into Lemma 2 then shows that under a mark tomarket bookkeeping strategy, the auxiliary function is identically equal tounity, or JgD1. This implies that the linear valuation equation collapsesto B(S,gDS, or that the book value of equity is identically equal to itsmarket valuewhich, of course, must be the case with a mark to market

bookkeeping strategy. Other bookkeeping strategies, however, will drive awedge between the market and book values of equity. Thus, if assetsare instantaneously marked to market (in which case r0Dm12Dpu) butliabilities are recorded using some other (non-market based) bookkeepingconventions (so that r6Dm13 in general), then in the risk neutral settingconsidered here the wedge between the market and book values of equitywill be summarized by the auxiliary function contained in Lemma 2. 17

This Lemma has several other important implications. For example, fromProposition 3, we know that under fairly general conditions the debt toequity ratio, g, will have a tendency to revert towards a long term normalvalue ofgL. As a consequence, Lemma 2 implies that the market to bookratio of equity must have a tendency to revert towards a long term norm

as well.

18

Furthermore, we can use the relationship between S, A, L andg, to show that the market value of equity can be restated in terms of thefollowing Lemma:


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Lemma 3

Under the transversality condition rgDm12 >0 for allg, the market valueof the firm can be stated as a linear function of assets and liabilities as givenby the valuation equation:

B(S, gDB(A,L)D

AC

m23

m13CrCm23L

Thus, Lemma 3 provides some comfort for researchers who wouldfocus on balance sheet approaches to (linear) valuation modelling.19 Weshould also recall that this result is based on risk neutral pricing and amark to market bookkeeping strategy for assets. Liabilities, however, arenot necessarily recorded on a mark to market basis. It thus follows that

m23

m13CrCm23is the coefficient which adjusts the book value of liabilities to

their market value. Note here that a necessary condition for a conservative

bookkeeping strategy (where the book value of liabilities exceeds theirmarket value) is that

m23

m13CrCm23> 1. Further, the risk neutral pricing

assumption implies that the only elements of the structural matrix, (t),which have value relevance are the expected structural coefficients,rDm12,m13 and m23. However, both intuition and previous research suggests thatthere are other characteristics of the structural matrix which ought to holdvalue relevance. Black (1980, p. 22), for example, notes that . . . at times,price-earnings ratios may depend on growth rates and measures of volatility. . . of past transactions . . . whilst Govindaraj (1992, p. 503) suggests thatthere is good reason to believe that the volatility of the of market valueof equity is related to . . . the volatility of . . . accounting variables.Unfortunately, these considerations lead in general to non-linearities in the

valuation equations.20Probably the simplest, most general, and easiest way of assessing the

value relevance of the higher moments and other (risk) parameters is toemploy power series expansions to obtain analytic solutions for the auxiliaryequation contained in Proposition 5. These expansions take the generalform:

JgDgq

1C

1kD1

ak(q)gk

where the ak are parameters and q is called the exponent of singularity.21

Now, there are two exponents of singularity corresponding to the two series

expansions which characterise the general solution of the auxiliary equationdefined in Proposition 5. They are determined by computing the roots of theindicial equation, which for the auxiliary equation defined in Proposition 5,


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takes the form (Boyce & DiPrima, 1969,pp. 183186):

F(r)Dq2C

2m12m13m23

s212Cs213Cs

223

1

q

2m12Cr0

s212Cs213Cs

223

D0

where r(0) is the cost of equity capital for an unlevered firm. Once theexponent of singularity is determined the series expansion for J(g) and itsderivatives, J0g and J00g, are substituted into the auxiliary equation; theak are then determined so that the auxiliary equation is formally satisfied.

A simple illustration of these procedures is provided by assuming thatthe cost of equity for an unlevered firm is given by r0DpuDm12, orthat the firm follows a bookkeeping policy of marking asset values tomarket. The reader will confirm that under this assumption, q1D0 and

q2D12m12m13m23

s212Cs213Cs

223

are the two roots of the above indicial equation.

Substitution then shows:

JgD1C

1kD1

akq1gk

and:

JgDgq2

1C

1kD1

akq2gk

are two linearly independent series expansions which formally satisfy theauxiliary equation. Now, the mark to market bookkeeping policy for assets,on which the above series expansions are based, implies that the valueof an all equity financed firm is B(S,0)DSJ0DS o r J0D1. Wehave also previously noted that as the debt to equity ratio grows, the

likelihood of the firm entering bankruptcy proceedings also increases.In the limit, therefore, we must also have for the value of equity thatLimitg!1

B(S, gDS[ Limitg!1

Jg]D0, or Limitg!1

JgD0. These two boundary

conditions are sufficient to ensure that there is a unique solution to theauxiliary equation and that the series expansion for this unique solutioncan be expressed as a linear combination of the two series expansions givenabove. This, in turn, means that the market value of equity is uniquelydefined in terms of the book value of equity and the firms debt to equityratio (Karlin & Taylor, 1981,pp. 203204).

In general, both the exponents of singularity, q1 and q2, as well asthe parameters,ak(q1) and ak(q2), are functions not only of the expectedstructural coefficientsm12,m13 and m23 but also of the volatility parameters

s

2

12, s

2

13 and s

2

23. Hence, whilst the variance parameters will now have aninstrumental role to play in determining the valuation equations, there willno longer be a linear relationship between the book values of the underlying


17/28


information variables and the value of the firms equity. Fortunately, thisdoes not detract from the models potential for use in empirical work. Themild regularity conditions which assure convergence of the series expansionswhich satisfy the auxiliary equation (Boyce & DiPrima, 1969,pp. 159162)

means that it will, in general, be possible to base econometric proceduresand empirical work on truncated versions of the series expansions with verylittle chance of this resulting in significant errors.

SUMMARY AND CONCLUSIONS

We demonstrate how a parsimonious interpretation of the double entrybook-keeping system can lead to the evolution of financial variables beingdefined in terms of a (vector) system of stochastic differential equations.Here, the restrictions imposed by the double entry bookkeeping systemare reflected in a structural matrix which summarizes the sensitivity ofinstantaneous changes in the firms equity, assets, liabilities, and other

financial variables to the existing levels of these variables. A specific versionof our model implies that the debt to equity ratio and the ratio of the marketvalue to the book value of equity will have a tendency to revert towardslong run normal valuesa description which conforms with the empiricalevidence. Further, and again within a specific version of the model, wedemonstrate a theoretical rationale under which there will be a simple linearrelationship between the market value of equity and the firms bookkeepingsummary measures. However, these are special cases of a more generalnon-linear pricing relationship implied by our model. Developing this moregeneral model beyond the embryonic form contained in the text has thepotential to open up fresh insights into the impact a firms production,investment, financing, and bookkeeping strategies have on the evolution ofits equity price.

Our analysis also has several other important implications which, due tospace restrictions, cannot be developed in any detail here. First, it can beshown, under quite general regularity conditions, that our analysis impliesthat all the eigenvalues of the matrix of expected structural coefficients, (t),are real. This resolves an important issue relating to the evolution of a firmsbookkeeping summary measures which was first identified by Ashton (1995)and discussed in detail by Tippett & Warnock (1997). And that relates to theissue of whether a firms bookkeeping summary measures can be expectedto evolve harmonically (in which case the eigenvalues of the structuralmatrix are complex) or exponentially (the eigenvalues of the structuralmatrix are real). Standard interpretations of the structural models found inthe literature (Ohlson, 1983, 1990, 1995)return empirical evidence which

suggests that whilst real eigenvalues are the norm, a significant proportion ofcompanies examined return structural matrices with complex eigenvalues.However, Ashton (1997) and Tippett & Warnock (1997) argue that the


18/28


modelling procedures employed in these papers are mis-specified and thatit is this which accounts for the existence of the complex eigenvalues. Ouranalysis here can only add weight to this assertion.

A second issue relates to the fact that we can use Proposition 3 of the

text in conjunction with the techniques summarized in Biekpe, Tippett &Willett (1998, pp. 116117) to determine the probability distribution of thedebt to equity ratio itself. The distributional properties of this (and other)ratios has been the focus of considerable attention over the years and is ofinterest to a variety of stakeholders in the firm (Ezzamel, Mar-Molinero &Beecher, 1987; Ezzamel & Mar-Molinero, 1990; McLeay, 1986). However,our analysis shows that it is likely the debt to equity ratio will be describedby a distribution which possesses no moments at all. In other words, neitherits mean nor its variance nor any of its higher moments will exist. Here it isinteresting to note thatWhittington & Tippett (1995) examine the empiricalevidence relating to a large number of time series models of the evolutionof financial ratios and show that non-stationarity is a problem with virtuallyall of them. Furthermore, Whittington & Tippett (1999)report empirical

evidence which suggests that the components of some commonly employedfinancial ratios do not possess a co-integrating relationship. It may be shownthat empirical evidence like this is compatible with the kind of probabilitydistribution which our analysis suggests describes the debt to equity ratio.

Probably the most obvious way in which our analysis can be extended,however, is by considering less aggregated transactions matrices than thesimple 33 matrices based on equity, assets, and liabilities considered here.This opens up the possibility of determining the role played by ratios otherthan the debt to equity ratio and the market to book ratio in the valuationprocess. There are also important issues arising from the fact that there isconsiderable flexibility in the way accounting conventions, rules, and stan-dards can be applied in practice. An interesting possibility here is to assumethat managers have a degree of discretion in the way they use the doubleentry bookkeeping system to adjust financial variables to conform with theirlong term growth rates[Lev (1969),Tippett (1990),Whittington & Tippett(1995)]. In sum, the embryonic analysis contained here can be extended tocast further light on the evolution of a firms bookkeeping summary measuresand of the relation these measures have to underlying equity value.

NOTES

1. SeeLundholm (1995,pp. 750753) for a more detailed account of these assumptions.Dixit & Pindyck (1994, pp. 121124) contains an intuitive and very readable discussionof the ideas which underscore martingale equivalent pricing. Feltham & Ohlson (1995,p. 694) discuss the meaning and significance of the clean surplus restriction.

2. Bernard (1995, p. 733) notes that:The value ofOhlson (1995)and Feltham & Ohlson (1995)can best be appreciatedwhen one recognizes where the studies fit on the evolutionary tree of research.


19/28


These studies return to issues so basic as to render them direct descendants of workdone no later than the 1960s (e.g.Edwards & Bell (1961) . . .).Ohlson (1995)andFeltham & Ohlson (1995) represent the base of a branch that capital markets researchmight have followed but did not. Instead framed within the so called informationalperspective, research since the late 1960s developed without much emphasis on the

precise structure of the relation between accounting data and firm value. In a senseOhlson (1995) andFeltham & Ohlson (1995) return to step one and attempt tobuild a more solid foundation for further work.

Thus, the normative models alluded to here [Edwards & Bell (1961), Mattessich (1957,1964), Chambers (1966), Ijiri (1965, 1967), Sterling (1970)] can all be interpreted asspecific applications of theGarman & Ohlson (1980)formulation.

3. A partial and very embryonic form of the analysis contained in this section is to be foundin the articles byMattessich (1957, 1964, pp. 7477),Tippett (1978,pp. 274275) andWillett (1987, 1988).Black (1980, p. 22; 1993, pp. 23)notes that past transactions area kind of raw material which, in the summary form of the profit and loss account andbalance sheet, constitute basic inputs for many of the valuation procedures employedby accountants in practice. Hence, models of the evolution of a firms bookkeepingnumbers have the potential to enhance our understanding of the relationship betweenthe firms accounting system and the way the value of its equity, debt and other capitalinstruments evolve over time.

4. We follow Mattessich (1957, 1964, pp. 7477) and Tippett (1978, pp. 274275).Mattessich (1964, p. 102) and Chambers (1994, p. 78) quote Cayley (1894, p. 5) asbeing of the view that the principles of Book-keeping by Double Entry constitute atheory which is mathematically by no means uninteresting: it is in fact like Euclidstheory of ratios an absolutely perfect one . . .. Since the theory of matrices [was] aninvention of Cayleys [Bell (1937, p. 425)], it is more than likely Cayley was awarethat the double entry bookkeeping system has the matrix interpretation given to it here.However, Cayleys premature death in 1895 [Bell (1937,p. 450)] meant that furtherdevelopments in the matrix interpretation of the double entry bookkeeping system hadto wait until well into the twentieth century, culminating with the pathbreaking analysisofMattessich (1957, 1964).

5. Pedagogical considerations dictate that Mt, tCt is the aggregated or reduced formof a transactions matrix containing more than just the three accounts considered here.One might consider, for example, the nn transactions matrix, n>3 which partitionsthe equity account into sales and expense categories; assets into fixed and currentcategories (stock, accounts receivable, bank, etc.) and liabilities into accounts payable,

bank overdraft, accruals etc. Call this expanded transactions matrix, N. It is easilyshown, however, that there is an adjacency matrix, A, [ Wallis, Street & Wallis (1972,p. 337)] which makes this more detailed transactions matrix, N, equivalent to thesimple 33 transactions matrix, M, in the sense that MDATNA [Finkbeiner (1966,p. 129)]. In other words, the adjacency matrix serves to close all equity accounts into oneaccount, all asset accounts into one account and all liability accounts into one account.Hence, there is not a great loss in generality, but obvious pedagogical convenience, fromconsidering the simple 33 transactions matrix, M, considered in the text. The seminaltreatment of Mattessich (1957, 1964, pp. 448465) outlines some of the importantproperties of the transactions matrices considered here. The Appendix contains furtherdetails on the relationship between the reduced form of the transactions matrix and theadjacency matrix which generates it.

6. This operation is equivalent to balancing off the accounts in the general ledger [Wood(1996, Chapter 5)].

7. Sims (1971, p. 559), Karlin & Taylor (1981,p. 356) andBergstrom (1990, pp. 13),

catalogue the merits of continuous time models relative to their discrete time equivalents.8. Willett (1987, 1988)was the first to provide a concrete demonstration of the fact that a

firms production, investment and financing technologies can be defined in terms of the


20/28


restrictions imposed on the structural matrix, t, tCt, by the requirements of thedouble entry bookkeeping system and the assumptions one makes about the stochasticprocesses which underscore the evolution of the elements of the structural matrix.

9. Both here and in the sections which follow, we assume that the roots of this equationare both real or that m12m13s213

2C4m23s212Cs213> 0.

10. The recursion formula stated here implies that equity valuation can be reduced to aterminal control problem in dynamic programming [Dreyfus (1965,pp. 2024)]. Thisis a methodology which has been extensively applied in the contingent claims valuationtheory of financial economics [Dixit & Pindyck (1994,pp. 121123)].

11. Recall that once the book value of equity [S(t)] and the debt to equity ratio [g(t)] areknown, both the book value of liabilities [L(t)] and assets [A(t)] can be determined usingthe fundamental accounting identity, S(t)CA(t)CL(t)D0. The relative importance ofaccounting (bookkeeping) information in the assessment of equity value is stressed byBlack (1993, p. 2) in the following terms:

. . . information is incorporated into [equity] prices through the accounting process.Without that process, the price changes would occur later. One way to see theimportance of accounting . . . is to note the time path of the price response to new

information. The response accelerates up to the time of the [statutory] accountingreport, on average. This suggests that the accounting numbers are finding their wayinto the market as the firm is preparing its accounting statements. The accounting

process itself is informing the market . . .. If the firm doesnt do its accountinghomework, its stock price will be noisier.

Furthermore, since our analysis implies that equity value is determined as the expectedpresent value [of dividends] conditional on ideal (but achievable and objective)accounting rules it follows that every proposed choice of accounting rules definesa different [equity] value. (Black, 1993, p. 2). Hence, as Black (1993, pp. 56)observes, restricting accounting policies to those based on the clean surplus identity(where everything goes through the profit and loss statement) will in general lead toaccounting policies which do not maximise the market value of the firms equity. This,

combined with the fact that Tippett & Warnock (1997, p. 1094) show that the cleansurplus identity does not hold up in practice are the principal reasons why our analysisplaces no special emphasis on the clean surplus restriction. For an alternative view,however, seeOhlson (1995,pp. 665667) andFeltham & Ohlson (1995,pp. 693697)or the collection of essays inBrief & Peasnell (1996).

12. The coefficients associated with the first order partial derivatives in Proposition 4 are

the means of the affected variables whilst (apart from a scaling factor), the coefficientsassociated with the second order partial derivatives are the variances or covariances

of the affected variables. Thus, from Proposition 2, Et[dS(t)]

dtD[m12A(t)Cm13L(t)]D

S[m121CgCgm13] is the coefficient associated with @B

@S. Similarly, Proposition 2

also implies that,Var t[dS(t)]

dtD

Vart[s121Cg(t)dz12(t)Cs13g(t)dz13(t)]

dt.S2D[s2121C

g2Cs213g2]S2 is the coefficient associated with the term

@2B

@S2. All other coefficients

are similarly derived. Furthermore, the partial differential equation contained in thisproposition is a special case of a very general result known as the Feynman-Kackilling formula which has been extensively applied in the financial economics literature(ksendal, 1985,pp. 9596;Dixit, 1989; Dixit & Pindyck 1994, pp. 121123).

13. SeeBarth & Kallupur (1996) for empirical evidence which also suggests that book valueacts as a proxy for scale differences.

14. It warrants emphasizing that this result holds for much more general transactionsand structural matrices. That is, if we specify a firms operations in terms of an nnstructural matrix rather than the simple 33 matrix based on equity, assets and liabilities


21/28


considered here, then it may be shown that the solution takes the form:

BS,g1,g2, ,gnDSJg1,g2, ,gn

where the gj are a more general set of financial ratios characterizing the firmsproduction, investment, financing, and bookkeeping strategies. Furthermore, if wedefine ES,g1,g2, ,gn to be the firms future sustainable, normalized orpermanent earnings, it follows that equity value takes the form B S,g1,g2, ,gnDqES,g1,g2, ,gn, where q is an intertemporally constant scaling factor appliedto the permanent earnings [Black (1980, 1993)]. Note that under this specification

permanent earnings, ES,g1,g2, ,gnDSJg1,g2, ,gn

q , turns out to be a

function of the bookkeeping variables comprising the transactions matrix (one of whichwill be the accounting earnings figure). Hence, as Black (1980, p. 20) notes, the endresult of the current system of accounting . . . is an earnings figure that usually givesa reliable estimate of value, plus other information that can be used to arrive at aneven better estimate of value. The Black (1980, 1993) specification of the relationshipbetween equity value and the book figures appearing in a firms accounting records isradically different from its better known counterpart based on the work ofOhlson (1983,1990, 1995).Biekpe, Tippett & Willett (1998)provide a simple illustration of the way

Blacks ideas may be applied at an empirical level.15. We relax this assumption below.16. The transversality condition ensures that the equity stocks price will always remain

finite. SeeDuffie (1988,p. 198) or Ingersoll (1987, p. 275) for further details.17. Note that when rgDm12m12m13g and the firm follows a mark to market

bookkeeping policy for both its assets and liabilities but with puDm12 >m13Dr sothat the more general (non-risk neutral) Modigliani & Miller (1958) theorem holds,then JgD1 also satisfies the auxiliary equation contained in Proposition 5. Hence inthis case too, we have BS,gDS or that the market and book values of equity areidentically equal. When other bookkeeping policies are used, however, a wedge is againdriven between the market and book values of equity. Further, we show below that itis unlikely the relationship will take the linear form given by the simple risk neutralexample considered in Lemma 2.

18. This result might also help to explain why the market to book ratio of equity appearsto be such an important variable in explaining equity returns [Fama & French (1992,1993, 1995, 1996)]. A market to book ratio in excess of the long term mean of:

m13Cr1CgLCm23m13CrCm23

implies that equity is currently overvalued whilst a market to book ratio below this figureimplies that equity is undervalued. Hence, if book values do not reflect opportunisticbehaviour on the part of management, then investors will expect future equity returns onhigh market to book ratio firms to be depressed whilst they will expect future returnson low market to book firms to be relatively attractive.

19. A more general version of this Lemma is based on the net book value, S(t), and the

instantaneous accounting rate of return, r(t), where dS(t)

S(t)r(t)dt. We could then use a

result similar to that given here to show that the market value of equity is a weightedaverage of the balance sheet summary measures and of the instantaneous earnings. Here,it is interesting to note that basing his analysis on a model very similar to this, Penman

(1998) reports empirical evidence which suggests that periodic earnings is the moreimportant determinant of equity value, although the relative importance of earnings andnet book value fluctuate markedly over time. Burgstahler (1998)provides an interesting


22/28


perspective on the econometric problems that arise in the empirical application of suchmodels.

20. There are, in fact, a variety of reasons which might lead to non-linearities in thevaluation equations; probably, the most obvious stems from the option firms have to atleast partially reverse their investment, production, financing and even their bookkeeping

strategies [Dixit (1989), Dixit & Pindyck (1994)].21. A parenthetical observation is in order here. Much contemporary finance (and to

a lesser extent, accounting) theory proceeds from (difference) differential equationswhich saw their earliest applications in (quantum) physics. Bell (1937, p. 480)notes that the intractability of these equations was such that without the theoryof power series most of mathematical physics . . . as we know it would not exist.Of course, the power series solution techniques demonstrated here have seldom,if ever, seen application in the accounting literature. Hence, one cannot help butwonder whether the current fixation with linear information dynamics in accountingtheory is a convenient simplifying assumption which imparts the analytical tractabilitynecessary to insure the existence of simple closed form solutions to the valuationequations.

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APPENDIX

Adjacency matrices

Suppose we expand the transactions matrix so that in addition to theequity, assets, and liabilities account categories contained in the text, wealso include an account for earnings. The earnings account records all thebookkeeping entries relating to sales, income and expenses. The expandedtransactions matrix will then take the form:

Nt,tCtD

Cr

Earnings Equity Assets Liabilities

Earnings a11t,tCt a12t,tCt a13t,tCt a14t,tCt

Dr Equity a21t,tCt a22t,tCt a23t,tCt a24t,tCt

Assets a31t,tCt a32t,tCt a33t,tCt a34t,tCt

Liabilities a41t,tCt a42t,tCt a43t,tCt a44t,tCt

Now, define the adjacency matrix AD

1 0 01 0 00 1 00 0 1

. It then follows that

the reduced form transactions matrix employed in the text, Mt,tCt,is equivalent to Nt,tCt in the sense that Mt,tCtDATNt,tCtA.Pre-multiplication by the transpose of the adjacency matrix has the effectof closing rows 1 and 2 of Nt,tCt into row 1 of Mt,tCt. Postmultiplication by the adjacency matrix has the effect of closing columns 1and 2 of Nt,tCt into column 1 of Mt,tCt. It then follows that:

Mt,tCtDATNt,tCtAD

2

jD12

kD1 ajk2

jD1 aj32

jD1 aj42

jD1

a3j a33 a34

2jD1

a4j a43 a44

or that the earnings account has been closed off to the equity account.Furthermore, this result implies:

Mt,tCtMTt,tCtD

02

jD1

aj3a3j2

jD1

aj4a4j

2

jD1aj3a3j 0 a34a43

2jD1

aj4a4j a34a43 0


26/28


or that the matrix of structural coefficients is defined by:

t,tCtD

0

2

jD1aj3a3jA(t)

2

jD1aj4a4jL(t)

2

jD1

aj3a3j

S(t)0

a34a43

L(t)

2

jD1

aj4a4j

S(t)

a34a43

A(t)0

Now, consider the first column of this matrix. If we assume that the ratioof the account balance, ajk, for earnings to the account balance for equity,S(t), evolves with the drift parameter, mjkt, and a normally distributed

stochastic component, sjkzjk(t), as in the text, then the closure propertiesof the normal distribution insure that the sum of these ratios will alsoevolve in terms of a normal process with drift. A similar conclusion appliesto the other two columns of this matrix. Hence, there is virtually no lossof generality, but obvious pedagogical convenience, from considering thesimple 33 transactions matrix, M, contained in the text.

Proof of proposition 1

By equating the elements of the structural matrix, t,tCt, contained inSection 2 of the text, we haveb12t,tCtD!21t,tCtS(t)t a n d b12t,tCtD!12t,tCtA(t)t. Now since A(t)D1Cg(t)S(t) it follows thatb12t,tCtD!12t,tCtA(t)tD!12t,tCt1Cg(t)S(t)t. We then

have that!21t,tCttD!12t,tCt1Cg(t)t. Algebraic manipulationssimilar to this also show that !31t,tCttD!13t,tCtg(t)t and

!32t,tCttD!23t,tCt.g(t)

1Cg(t)t.


We begin by approximating the upper diagonal elements of the structuralmatrix, t,tCt, defined in Proposition 1 by an intertemporally constantexpected component and a normally distributed stochastic component asfollows:

!jkt,tCttmjktCsjkzjk(t)

for j


27/28


and variance s2jkt. Substitution then shows that discrete time movementsin the firms financial variables take the form:

x(t)Dt,tCtx(t)

0 m12 m13

1Cg(t)m12 0 m23

g(t)m13g(t)

1Cg(t).m23 0

S(t)A(t)L(t)

t

C

A(t)s12z12(t)CL(t)s13z13(t)

1Cg(t)S(t)s12z12(t)CL(t)s23z23(t)

gS(t)s13z13(t)Cg(t)

1Cg(t).A(t)s23z23(t)

If we now substitute L(t)Dg(t)S(t) and A(t)D1Cg(t))S(t) for A(t) andL(t) in the stochastic component and take limits such that Limitt

t!0Ddt and

LimitS(t)t!0

DdS(t) etc., then the evolution of the bookkeeping variables may

be described by the vector system of stochastic differential equations:

dS(t)dA(t)dL(t)

D

0 m12 m131Cg(t)m12 0 m23

g(t)m13g(t)

1Cg(t).m23 0

S(t)A(t)L(t)

dt

C

s121Cg(t)dz12(t)Cs13g(t)dz13(t)s121Cg(t)dz12(t)Cs23g(t)dz23(t)

g(t)s13dz13(t)g(t)s23dz23(t)

S(t)

This is the result reported in the text.


Itos Lemma implies that the debt to equity ratio, gDL

S, will evolve in

accordance with the formula (Kloeden & Platen, 1992,pp. 9099):

dgD@g

@LdLC

@g

@SdSC

1

2(dL)2.

@2g

@L2C(dS.dL)

@2g

@L@SC

1

2(dS)2.

@2g

@S2

Now, since @g

@LD

1

S, @g

@SD

L

S2, @2g

@L2D0,

@2g

@L@SD

1

S2 and

@2L

@S2D

2L

S3, substi-

tution results in the following stochastic differential equation:

dg

gD

dL

L

dS

S

dS

S .

dL

LC

dS

S

2


28/28


Given the stochastic processes for S, A, and L contained in proposition 2 ofthe text, we have:

dSD[m121CgCgm13]Sdt[s121Cgdz12s13gdz13]S,

dLD[gm13gm23]Sdt[gs13dz13Cgs23dz23]S,

Vart(dS)D[s2211Cg

2Cs213g2]S2dt

and:CovtdS,dLDs

213g

2S2dt.

Substitution then implies:

dg

gD[s212Cs

2131Cg

2C1Cgm12m13s213m23]dt

C1Cgs12dz12s13dz13s23dz23

and by applying the mean and variance operators to this expression, weobtain the results reported in the text.


Eulers Theorem for homogeneous functions shows that BlS, gDlBS, g

if and only if S@B

@SDBS, g (Apostol, 1969, exercise 9, p. 287). Furthermore,

the method of separation of variables shows that B S, gDH(S)Jg, whereH(S) and J(g) are differentiable functions of S and g, respectively (Boyce &DiPrima, 1969, pp. 422429). This, taken in conjunction with Eulers

Theorem implies H0(S)

H(S)

D1

S

or H(S)DaS, where a is a constant of

integration. It thus follows that the equity valuation dynamics are definedby the equation BS, gDH(S)JgDSJg, where the constant term, a,is suppressed into the expression for J(g). Substituting BS, gDSJg,into Proposition 4 then shows that J(g) must satisfy the auxiliary equationspecified in Proposition 5.

Cooke Tippett 2001 Deb and Value of Firm

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Transcript of Cooke Tippett 2001 Deb and Value of Firm