0195380614

Financial Management Association Survey and Synthesis Series

R

VIII Preface

I thank Shmuel Baruch, David Chapman, Larry Epstein, Mike GaUrneyer, Bob Goldstein, Philipp Illeditsch, Martin Schneider, Jessica Wachter. and Guofu Zhou for helpful advice.

Kerry Back Rice University

August, 2009

Contents

Single-Period Models

Utility Functions and Risk Aversion Coefficients

1.1 Uniqueness of Utility Functions 1.2 Concavity and Risk Aversion 1.3 Coefticients of Risk Aversion 1.4 Risk Aversion and Risk Premia 1.5 Constant Absolute Risk Aversion 1.6 Constant Relative Risk Aversion !.7 Linear Risk Tolerance 1.8 Condirioning and Aversion to Noise 1.9 Notes and References

Exercises

2 Portfolio Choi,ce and Stochastic Discount Factors

2.1 The First-Order Condition 2.2 Stochastic Discount Factors 2.3 A Single Risky Asset 2.4 Linear Risk Tolerance 2.5 Constant Absolute Risk Aversion with Multivariate Normal Returns 2.6 Mean-Variance Preferences

3

4 4 5 6 8

10 10 13 14 17

21

26 27 31 34 36

x Contents

2.7 Complete \;larkets 2.8 Beginning-of-Period Consumption 2.9 Time-Additive Utility 2.! 0 ~Oles and References

Exercises

3 Equilibrium and Efficiency

3.1 Pareto Optima 3.2 Social Pianners Problem 3.3 Pareto Optima and Sharing Rules 3.4 Competitive Equilibria 3.5 Complete Markets 3.6 Linear Risk Tolerance 3.7 Beginning-or-Period Consumption 3.8 !\otes and References

Exercises

4 Arbitrage and Stochastic Discount Factors

4.! Fundamental Theorem on Existence of Stochastic Discollnt Factors 4.2 La\1i of One Price and Stochastic Discount Factors 4.3 Risk-Neutral Probabilities 4.4 Projecting SDFs omo the Asset Span 4.5 Projecting onto a Constant and the Asset Span 4.6 Hansen-Jagannathan Bound with a Risk-Free Asset 4.7 Hansen-Jagannathan Bound \vith No Risk-Free Asset 4.8 Hilbert Spaces and Gram-Schmidt Orthogonalization 4.9 :\otes and References

Exercises

5 Mean-Variance Analysis

5.1 The Calculus Approach for Risky Assets 5.2 Two-Fund Spanning 5.3 The Mean-Standard Deviation Tr

xii Contents

9.7 "-'otes and Refercnt:es

10 Conditional Beta Pricing tl:1ode!s

10.\ From Conditional to Cnconditional \tJodels

10.2 The ConditiOnal Capital Asset Pricing Model

10.3 The Consumption-Based Capital Asset Pricing Model

10.4 The Intertemporal Capiml Asset Pricing Model

10.5 An Approximate Capital Asset Pricing Model

10.6 :\otes and References

Exercises

11 Some Dynamic Equilibrium Nlode!s

11.1 Rt":pre~('ntiltivc Imestors

11.2 Valuing the Market Portfolio

11.3 The Risk-Free Return

11.4 The Equit:y Premium PUllk

11.5 The Risk-Free Rate Puzzle

1!.6 Uninsurable Idiosyncratic Income Risk

1 ). 7 Exten1a! ibbits 11.8 Notes and References

Exercises

12 Brownian Motion and Stochastic Calculus

\ 2.1 Brov,,'nian \l1otian 12.2 Quadratic Vilriatioll 12.3 1to Integral 12,4 Local Martingales and Doubling Strategies

12.5 116 Proccs.->es 12.6 A:-;sd and Portfolio Returns

12.7 Maningaie Represenwtion Theorem

12.8 ho's Formul:J: Version 1 12.9 Geometric BrO\vnian \lotion

12.10 Co\ariZltions of It6 Processes

12.11 Ito's Formula: Ver~ion II 12,12 Conditional Variances and Covariances

12.\3 Transformations of \lodeis

12,14 Notes and References

Exercises

13 Securities Markets in Continuous Time

13.1 Dividend-Reinvested Asset Prices

13.2 Securilico:; ).1arkets

171 173

177

178 179 181 183 187 187 188

189

i80 190 j 91 193 194 194 198 101 202

204

104 205 207 108 209 210 112 212 216 218 2i9 220 222 224

225

231

231

132 t t

~.

Contents

13.3 Self-Financing Wealth Processes

13.4 Conditional Mean-Variance Frontier

13,5 Stochastic Discount Factor Processes

13.6 Properties of Stochastic Discount Factor Processes

\3.7 Sufficient Conditions for MW to be a Martingale

\3,8 Valuing Consumption Streams

13.9 Risk !\'eutral Probabilities 13,10 Complete Markets

13.11 Markets without a Risk-Free Asset

13.12 Inflation and Foreign Exchange

13.13 Notes and References Exercises

14 Continuous-Time Portfolio Choice and Beta Pricing

14.) The Static Budget Constraint \4.2 Complete Markets

14.3 Constant Capita! ?vIarket Line

!..1..4 Dynamic Programming Example

14.5 General Markovian Portfolio Choice

\4.6 The Consumption-Based Capital Asset Pricing Model

14.7 The Intenemporal Capital Asset Pricing Model

14,8 The Capital Asset Pricing Mode!

i4.9 Infin.ite-Horizon Dynamic Programming

14.10 Value Function for Constant Relative Risk Aversion

14,11 Verification Theorem

14,12 Notes and References

Exercises

III Derivative Securities

15 Option Pricing

15.1 Introduction to Options

15.2 Put-Call Parity and Option Bounds

15.3 Stochastic Discount Factor Processes

15.4 Changes of Measure

15.5 Market Completeness

15.6 The Black-Scholes Formula

15.7 Delta Hedging

15.8 The Fundamental Partial Differential Equation

15.9 American Options

IS.! 0 Smooth Pasting

15.11 European Options on Dividend-Paying Assets

15.12 Notes and References

Exercises

234 235 236 237 241 242 243 245 246 247 248 249

256

256 257 259 260 263 265 267 268 269 269 271

273

275

283

284

286 286 287 289 290 293 296 297 298 301 301 304

.... Ullltlll) LOments xv

16 Forwards, Futures, and More Option Pricing 310 19.7 The Kyle Model in Continuous Time 387 19.8 Notes and References 390

16.1 Forward :\treasures 310 Exercises 392 ]6.1 Forward Contracts 311

16.3 Futures Contracts 313 20 Alternative Preferences in Single-Period Models 394 16.4 Exchange Options 314 16.5 Options on Forwards and Futures 316 20.1 The ElIsberg Paradox 395

16.6 Dividends and Random Interest Rates 318 20.2 The Sure Thing Principle 396

16.7 Implied Volatilities and Local Volatilities 319 20.3 Multiple Priors and Max-Min Utility 396

16.8 Stochastic Volatility 321 2004 NonAdditive Set Functions 398

i6.9 Notes and References 315 10.5 The A!lais Paradox 399

Exercises 316 10.6 The Independence Axiom 401 20.7 Betweenness Preferences 402

17 Term Structure Models 334 10.8 Rank-Dependent Preferences 406 20.9 First-Order Risk Aversion 408

17.1 Vasicek Model 335 ] 7.1 Cox-Ingersoll-Ross Ylodel

10.10 Framing and Loss Aversion 409 337

17.3 :vlultifactor Cox-Ingersoll--Ross \1odels 10.11 Prospect Theory 410

339 10.11 !"otes and References 410 17.4 Affine Models 340

Exc:rciscs 411 17.5 Completely Affine 0.'lodels 342 17.6 Quadratic Models 343 21 Alternative Preferences in Dynamic Models 414 17.7 Forw

xvi Contents

Appendices

A Some Probability and Stochastic Process Theory

A, I Rnndom Variables A.2 Probabilities A.3 Distribution Functions and Densities A.4 Expectations A.5 Convergence of Exp,xtations ...\.6 Interchange of Differentiation and Expectation :\.7 Random Vectors ...\.8 Conditioning A.9 !ndependence A. i 0 Equivalent Probability \.kasures A.11 Filtrations. ?vbrtingales. and Stopping Times A.l:? ;-v1artingales under Equivalent y1easures A.13 Local Martingales A.14 The l'sual Conditions

Bibliography Index

456

456 457 457 458 459 459 460 461 461 463 464 464-465 465

467 481

Part I

Si ngle-Period Models

1


The first part of this book addresses the decision problem of an investor in a one-period framework. We suppose the investor makes celtain decisions at the beginning of the period (how much of his wealth to spend and how to invest what he does not spend) and the generally random investment returns and any other income (such as labor income) determine his wealth at the end of the period. The investor has preferences for spending at the beginning of the period and wealth at the end of the period, and these preferences, in conjunction with the available investment opportunities, determine his choices.

This is a simplification in that we are ignoring the allocation decision for beginning-of-period spending and end-of-period wealth across the different consumption goods available. Likewise, we are ignoring the prices of different consumption goods at the beginning and end of the period. This is the common practice in finance, and it is followed throughout this book. To put this fonnally, we can say we are assuming there is only a single consumption good and we are using it as the numeraire (meaning the unit in which prices are measured, so the price of the consumption good is always 1 ).1

We are also simplifying here in assuming the investor consumes all of his end-of-period wealth. This assumption is relaxed later, and multi-period models are studied. Introducing multiple periods typically introduces "state-dependence" in the investor's preferences for end-of-period wealth. Specifically, the investor

1. Occasionally, where it se'~ms conveniemand unlikely to cause confusion, we will discuss consumption and wealth as if they were denominated in dollars. The reader is free to substitute any other currency or to translate "doBar" as "unit of the consumption good."

4 Single-Period Models

carcs about the investment opportunities available at the end of the period as well as his \vealth. because it is the combination of wealth and investment opportunities that determine the possibilities for future \vealth (and consumption). The correlation of wealth with changes in investment opportunities is usually an inlponant consideration in multi-period settings; however. in the single-period model it is assumed that preferences depend only On the (marginal) probability distribution of wealth.

It is assumed in most of the book that each investor satisfies certain axioms of rationality, \I/hich imply that his choices are those that maximize the expected value of a utility function. Specifically, letting Co denote beginning-of-period consumption and cj denote end-of-period consumption (\V'hlch equals end-of-period wealth). assume there is a function v such that the investor maximizes the expected value of deo. Cj ).2 In many parts of the book-in particular, in this chapter and the next-the probabilities with respect to I,vllich expected values are computed can be "subjective probabilities:" that is, we do not need to assume the investor knows the "true probabilities" of the outcomes implied by his choices.

In this chapter and in many other places in the book, we simplify our model even further and focus on end~of-period \vealth. Vie can do this by assuming Co ie-; optimally chosen and by considering the derived utility function Hi wo), and strict concavity is equivalent to decreasing marginal utility (u'(I-Vj) < u'CyVo) if >,v'j > wo). For a twice differentiable function [t, concavity is equivalent to U"(Hi) ::: 0 for all w, and strict concavity is implied by ul/(w) < 0 for all w.

1.3 COEFFICIENTS OF RISK AVERSION

The coefficient of absolute risk aversion at a wealth level IV is defined as ul/(1.


absolute risk m/ersion (Xl! and coefficients of risk tolerance Til = 1/0:'11' then the aggregate risk tolerance is defined as r = Lf;=l TJr The aggregate absolute risk aversion is defined to be the reciprocal of the aggregate risk tolerance:

ex = H . Lh~l l/a"

This isequaJ to the harmonic mean of the absolute risk aversion coefficients divided by H.'

The next section describes the sense in which a(w) and pew) measure risk aversion. \Ve will also see why o'(\v) is called the coefficient of absolute risk aversion and \vhy p(I.V) is called the coefficient of relative risk aversion.

1.4 RISK AVERSION AND RISK PREMIA

Let Ii' be the mean of a random It!. so It,' = It" + , where E[El = O. A constant x is said to be the certaimy equivalent of II> for an individual with utility function u if

u(x) = E[u(w + Ell. A constant 7T is said to be the risk premium of}t> ifw - IT is the certainty equivalent, that is,

lI(W -;r) = E[lI(w + ell. In other \vords, starting at wealth }V, ]I is the largest amount the individual would pay to avoid the gamble E. 5 One can show that, for 'small gambles' (and assuming u is (I,.vice continuously differentiable and the gamble is a bounded random variable)

1 , IT ~ 2a-a(w). (13)

where a 2 is the variance of E. Thus, the amount one would pay to avoid the gamble is approximately proportional to the coefficient of absolute risk aversion. Equation (1.3) is derived. and the meaning of the approximation explained, at the end of this section.

The distinction between absolute and relative risk aversion can be seen by contra~ting (1.3) with the following: Let;r = VH/ be the risk premium of w + }VE, v,:here H-' is a constant and E is a zero-mean random variable with variance a 2 . Then

I , u'" -a-pew) 2 . ( 1.4)

4. The hmmonic mc:m of nl.lmhers x, . .... xn is the reciprocal of the aver,\ge reciprocal:

" L;'", , I/x, .

5. This use of the word "premium" is from insurance. The term "risk premium" is used in a different way in most of the book. meaning the extra expected relUlTl an investor cams from holding a risky asset.

Utility Fu nctions and Risk Aversion Coefficients 7

Thus, the proportion 1) of initial wealth w that one would pay to avoid a gamble equal to the proportion $ of initial wealth depends on relative risk aversion and the variance of e. The result 0.4) follows immediately from (1.3): Let we be the gamble we considered when discussing absolute risk aversion; then the variance of the gamble is w 2a 2 ; thus,

I? 1 ") u'" -wa-a(w) = -a-p(w).

2 2 To make (1.3) more concrete, consider flipping a coin for SI. In other words,

take E = 1 with equal probabilities. The standard deviation of this gamble is I, and so the variance is 1 also. Condition (1.3) says that one would pay approximately Sl x a(w)/2 to avoid it. If one would pay 10 cents to avoid it, then a(w) '" 0.2.

To make (1.4) more concrete, let w be your wealth and consider flipping a coin where you win 10% of w if the coin comes up heads and lose 10% of}v if it comes up tails. This is a large gamble, so the approximation in 0.4) may not be very good. Nevertheless, it can help us interpret (1.4). The standard deviation of the random variable e defined as e = 0.1 with equal probabilities is 0.1, and its variance is 0.01 = I %. According to 0.4), one would pay approximately

1 2: p (w) x 1%

of one's wealth to avoid the gamble. If one would pay exactly 2% of one's wealth to avoid this 10% gamble, then 0.4) says that pew) '" 4.

The remainder of this section is the proof of (1.3). For a simpler, perhaps more intuitive proof (but which requires auxiliary assumptions) and another interpretation of (1.3), see Exercise 1.2.

Let u be twice continuously differentiable, and let IV be in the interior of the domain of [( with u' (1-v) -::f- O. Take E to be a bounded zero-mean random variable with unit variance and define n = O"IlE for a sequence of numbers 0",1 converging to zero. For sufficiemly large n, lot" + 11 is in the domain of u with probability 1. Moreover, the variance of n is or

Let ~v - Kn be the certainty equivalent ohv + Ell' We will show that

(1.5)

This is the meaning of the approximation (1.3). To establish (1.5), take exact Taylor series expansions of u(\1.l - ]Tn) and u(w + E,J We

have

for some numbers xI! between IV and w - Trw Likewise,


for some random numbers ,::;'n between wand Hi + En" Csing the fact that w ~ "J(n is the certainty equivalent of w + ill and the fact that Ell has zero mean, we have

I 'l'" '"J :;::::;II(W)+ -. E U (Y )E: .., 11), Thus,

which implies

_~ E[u"(i,,)c'] 2 [{I(X/])

The random variables [/" (5:'/l)2 are bounded (because u' ! is continuous and hence bounded on bounded sets and because the SOn are bounded) and converge to 1I1/(,v)i2, Hence. from the dominated convergence theorem (Appendix A.5).

using the fact [hal 6: has zero rne.;m and unit variance for the last equality. Moreover, [/'(Xn ) -} u'(,v), Therefore.

2 u'(w)

1,5 CONSTANT ABSOLUTE RISK AVERSION

I = -a(w).

2

If absolute risk aversion is the same at every wealth level. then one says that the investor has CARA (Constant Absolute Risk Aversion) utility. It is left as an exercise (Exercise 1.9) to demonstrate that every CARA utility function is a monotone afl1ne transform of the utility function

u(w) = _e-O:'It'

\vhere 0' is a constant and equal to the absolute risk aversion. This is called "negative exponential utility" (or sometimes just "exponential utility").

CARA utility is characterized by an absence of \vealth effects. This "absence" applies to the risk premium discussed in the previous section and also to portfolio choice. For the risk premium, note that

U(H' - rr) = _e-O:'wemr ,

and

Utility Functions and Risk Aversion Coefficients 9

so

u(w -;r) = E[u(w + i')] implying

J[ = log E [ e -a,] , (1.6) which is independent of w, Thus, an individual with CARA utility will pay the same to avoid a fair gamble no matter what his initial wealth might be. This seems some\vhat unreasonable, as is discussed further below.

If the gamble i' is normally distributed, then the risk premium 0,6) can be calculated more explicitly. We use the fact, which has many applications in finance, that if.r is normally distributed with mean f1. and variance a 2 , then6

(1.7)

In the case at hand, -,t- = -0'8, which has mean zero and variance O' 2a 2. Thus

and 0,6) implies ,

;r=2:O'a-. ( 1.8)

This shows that the approximate formula (1.3) is exact \vhen absolute risk aversion is constant and the gamble is normally distributed.

Consider flipping a fair coin for SI,OOO. Formula (1.6) says that the amount an individual with CARA utility would pay to avoid the gamble is the same whether he starts with wealth of S I ,000 or wealth of S I ,000,000,000, One might think that in the latter case the gamble would seem much more trivial, and, since it is a fair gamble. the individual would pay very little to avoid it. On the other hand, one might pay a significant amount to avoid gambling all of one's wealth. If so-that is, if one would pay less with an initial wealth of S 1,000,000,000 than with an initial wealth ofS1,000 to avoid agiven gamble-then one has decreasing absolute risk aversion, meaning that absolute risk aversion is smaller when initial wealth is higher.

6. !t is \lsefullO compare (J .7) to Jensen's inequality. Jensen's inequality states that iff is a concave function and.t is a random variable (not necessarily normal) with mean M. then EU(x)] Sf(M). On the other hand, if/ is convex, then we have the opposite inequality: E[fex)] :::: /(p,). The exponential function is convex. so Jensen's inequality tells us that

So, we can ask: By how much must we scale up the right-hand side ~o make it equal the left-hand side? The formula (1.7) says that we must multiply the right-hand side by the numberer: /2 (which jslargerthan one) when.~ is normally distributed. Formula (1.7) is encounlered in statistics as the moment generating function of the normal distribution.


1.6 CONSTANT RELATIVE RISK AVERSION

One says that an individual has eRR A (Constant Relative Risk Aversion) utility if the relative risk aversion is the same at all wealth levels. Note that any eRRA utility function (with positive risk aversion) has decreasing absolute risk aversion, because a(w) = p(w)/w.

Any monotone CRRA utility function is a monotone affine transform of one of the following functions (see Exercise 1.9): (i) It(w) = log ~V, where log is the natural logarithm, (ii) u(w) equals a positive power, less than one. ofw, or (iii) u(w) equals minus a negative power of w. The last two cases (power utility) can be consolidated by v'"Titing

I ., Lt(1,-I/) = -H.!'

Y

where y < I and y :j:. O. A slightly more convenient formulation, which we will adopt, is to write

1 I li(\-V) = --w -p l-p

(1.9)

where p = I - y is a positive constant different from 1. One can easily check that p is the coefficient of relative risk aversion of the utility function (1.9). Logarithmic utility has constant relative risk aversion equal to L and an investor \\lith power utility (l.9) is said to be more risk averse than a log-utility investor if p > I and to be less risk averse than a log-utility investor if p < 1.

The fraction of wealth an individual with CRRA utility would pay to avoid a gamble that is proportional to initial wealth is independent of the individual's \vealth. To see this, let be a zero-mean gamble. An individual will pay rrw to avoid the gamble w if

u((1- ;r)w) = E[u((1 -'- E)W)]. (1.10) One can confirm (see Exercise 1.4) that rr is independent of w for CRRA utility by using the facts that log(xy) = logx + logy and (xy)Y = xYyY.

Logarithmic utility is a limiting case of power utility obtained by taking p --+ 1, in the sense that a monotone affine transform of power utility converges to the natural logarithm function as p --+ 1. Specifically.

I __ }V 1- p - -- --+ loaH! I-p l-p ~

as p --+ I for each \V > 0 (by I'H6pital's IUle).

1.7 LINEAR RISK TOLERANCE

Many finance papers use one or more of the following special utility functions, the first three of which have already been introduced. All of these are concave functions. The lisk tolerance formulas below are all straightforward calculations.


Negative Exponential For every real number w,

u(w) = _e-aw , for a constant ex > 0, where e is the natural exponentiaL The risk tolerance is

Logarithmic For every w > 0,

1 r(w) = - .

a

u(w) = logw, where log is the natural logarithm function. The risk tolerance is

r(w) = w.

11

Power For a constant p with p > 0 and p i=- 1 and for every w > 0, and including w=Oifp < 1,

The risk tolerance is

1 u(\-v) = __ w 1- p .

l-p

w r(w) = -.

p

Shifted Logarithmic For some constant I; and every w > 1;,

u(w) = log(w -I;), where log is the natural logarithm function. The risk tolerance is

r(IV) = IV - 1;.

Shifted Power For a constant I; and a constant p with p t= 0 and p t= 1 and for w such that (w - 1;)/ p > 0, and including IV = I; if p < 1,

The risk tolerance is

P (W_I;)I-P u(w)=-- --

l-p p

IV-I; r(IV) = --.

p

Obviously, the shifted log utility function includes logarithmic utility as a special case (I; = 0). Also, the shifted power utility function includes power utility as a special case (when p > 0, the additional factor pP in the definition of shifted power

12 Single-Period IvlodeJs

utility is irrelevant). For the shifted utility functions with p > 0, one can interpret the constant ( as a "subsistence level of consumption" and interpret the utility as the utility of consumption in excess of the subsistence !evel. This interpretation probably makes more sense when ( > 0. but we do not require ( > 0 to use the utility functions. For rhe shifted pmver utility function with p < 0, ( is a satiation Cbliss") level of wealth: For H' > ( either the utility function is not defined (in the real numbers) or it is decreasing in wealth, as in the quadratic case discussed below.

Each of these special utility functions has linear fisk tolerance (LRT), meaning that

T(W) =.4 +Bw (I.ll) for some constants A and E.7 The paraITleter B is called the cautiousness parameter. It can be shown (see Exercise 1.9) that any monotone utility function with linear risk tolerance is a monotone affine transform of one of these functions. One also sas"s that these utility functions have hyperbolic absolute risk aversion (HARA), due to the fact that the graph of the function

. I H' ,......,.. a(.\") = --- .

A +Bw is a hyperbola.

There afe three different cases for the shifted pO\ver utility function, the first {\Vo of \vhich parallel the cases for po\ver utility.

(i) p > 1. The utility is proportional to ~(H' ~ ()-I). where rJ = P ~ 1 > O. It is defined for 1-\-" > ( and is monotone increasing up to zero as IV --+ ce.

(ii) 0 < p < I. The utility is proportional to (n' ~ 1;)/1, where rJ = I - P E (0. 1). It is zero at 2; ane! is monotone increasing up to infinity as w --+ 00.

(iii) p < O. The utility is proportional to -(1'-, = t;E \1'. - 2E[H"j- - 2 varU\l). where var(\-I:) denotes the variance of Ii":. Thus, preferences over gambles depend onl yon their means and variances \vhen an investor has quadratic utility. Quadratic utility is defined for IV> (. but it is decreasing in \vealth for J.-1-' > t;.

7 Genera!!y, in thi, hook. rl di~(inction is made hetwcen lim::"r and :1l"fint: functions. a linear function being of ,he ]'Onl,13w and an affine i"uncti(lIl including a constant (intercept): A..;.. 8\ '. HOwc\'cr, we make an exception in l;lC term

"iinc~,r risk tolerance." WhlCh I~ rirmly entrencheu in the Ii,eraturc.


Any utility function with linear risk tolerance r(w) = A + Bw with B > 0 has decreasing absolute risk aversion. On the other hand, quadratic utility, in addition to being a decreasing function of wealth for w > ;;, also has the unattractive propel1y of increasing absolute risk aversion, even for w < (. This property of increasing absolute risk aversion (decreasing risk tolerance) is shared by every shifted power utility function with p < O.

1.8 CONDITIONING AND AVERSION TO NOISE

Given random variables.~ and y, the conditional expectation Eli I}'] is defined in Appendix A. It depends on the realization of y and hence is a random variable. Intuitively, one can think of it as the probability-weighted average value of X, given that one knows y. Observing}' will generally cause one to update the probabilities of various events. and this produces the dependence of E[.t I yJ on }'.

Some important facts about conditional expectations are:

(i) The "law of iterated expectations" states that the expectation of the conditional expectation is just the unconditional expectation; that is,

E[E[.t I yll = E[i] . (ii) If _t and 5' are independent, then E[i i5'] = E[i]. The interpretation is

that knowing y tells you nothing about the average value of x when S-is indepensIent of X.

(iii) If z depends only on y in the sense that z = g(y) for some function g, then E[zx I yJ = zE[i I YJ. The interpretation is that if y is known, then z is known, so it is like a constant, pulling out of the expectation.

(iv) Jensen's inequality applies to conditional expectations. Recall that Jensen's inequality states that

E[u(iv)] :': u(E[w]) for any concave function u. This generalizes to conditional expecta-tions as

E[u(ie) in:,: "(E[i,, 15;]) Calling u a utility function, the left-hand side is the conditional expected utility and the right-hand side is the utility of the conditional expectation.

One concept of a risk being "noise" is that of mean independence. A random variable is said to be mean independent of another random variable }' if observing.y does not change the expectation of e, that is, if E[ely] = E[e]. Mean independence implies that has a zero mean even when one knows the realization of y and regardless of what realization of y occurs. Mean independence is an intermediate concept between independence and zero correlation: If and y are independent then " is mean independent of y, and if is mean independent of y,


then cov(,s, 5') = O,s The latter fact is the subject of Exercise 1.8, and the former is fact (ii) above.

To show that risk-averse individuals dislike this type of noise, suppose }v = }' + E where E[z: S] = E[s] = O. Thus, i~' equals 5' plus noise. This implies 'E[ \1: i \'] = }', so 5" is the conditional mean of l,t.,', Assuming a concave utility function Ll, Jensen's inequality states that

E[u(l'i ! 51 :oS u( 5) Taking expectations and applying the law of iterated expectations on the left-hand side yields

(LI2)

Thus. 5' is preferred to S' plus noise. Other results of this type are described in the next section.

1.9 NOTES AND REFERENCES

\Vhether probabilities can ever be regarded as objective is a point of contention. The classic reference on this issue is Savage (1954), \\'ho argues for the personalistic (subjective) poim of view. This view motivates Bayesian statistics, and the objective view underlies the frequentist approach to statistics.

Axioms of rationality implying expected utility maximization were first presented by von Neumann and Morgenstern (1947), assuming objective ~r.obabilities. The formulation of von Neumann and Morgenstern masks a cntlcal axiom, which has come to be known as the independence axiom. In Herstein and Milnor (1953), which is a fairly definitive formulation and extension of the von Neumann-Momenstern result, this axiom takes the form: If gamble A is preferred ~o gamble B, and C is any other gamble, then the compound lottery consistin~ ~f a one~half chance of A and a one-half chance of C should be preferred to recelvmg a one-half chance of B and a one~half chance of C. This axiom is consistently violated in some experimental settings, as is discussed in Chapter 20.

Savage (1954) extends the von Neumann-Morgenstern result to the setting of subjecti~e probabilities. Naturally, this also requires a version of the independence axiom (Savage's "sure thing principle").

Arrow (1971) argues that the utility function should be bounded on the set of possible outcomes (boundedness follows from his "monotone continuity" axiom). Note that all of the LRT utility functions (on their maximal domains) are either unbounded above or unbounded below or both. Based in part on the argument that a utilitv function should be bounded, Arrow (1965) suggests that utility functions sl;ould have increasing relative risk aversion (see Exercise 1.3). An unbounded utility function is somewhat problematic conceptually. For example, if

S, We usc the not 0.9 Dybvig and Lippman (1983) show that this is equivalent to thefollowing: If an individual will

9. This is ccrtainly suggested by the characterization of risk premia in Section 1.4; however, the result in Section 1.4 is only an approximate result for small gambles. To go from the "local" result of Section 1.4 to global results, one has to integrate the risk aversion coefficient, as in Exercise 1.9.

16 Single-Period iVlodels

accept a gamble (having necessarily a positive expected value if the individual is fisk averse) at any \vealth level, then he \\Iill also accept the gamble at any higher wealth ]evel.

These implications for risk premia or choices at different wealth levels also apply to the risk premia or choices of different individuals: Assuming the absolute risk aversion of individual] at each \vealth level is at least as large as that of individual 2 and both start at the same initial wealth, then the risk premium required by individual I for any zero mean gamble is at least as large as that required by individual 2, and if individual I will accept a particular gamble, then individual 2 \vill also. Pratt (! 964) shmvs that individual I being more risk averse in the :"ense of having a (weakly) higher absolute risk aversion at each wealth level is equivalent to the utility function of individual 1 being a concave transformation of that of individual 2: III (\1") = f(U .. JIF)) for a concave functionf, In this sense, "more risk averse' is equivalent to the utility function being more concave."

Ross (1981) defines a stronger concept of nonincreasing risk aversion. involving the premia for gambles when uncertainty is unavoidable. Let \t,' and l: be any gambles SLlch that E[l: j 1-"\:J = O. Let a > 0 be a constant. Let ;TO and 7r] be the risk premia for s when initial wealth is the random amounts }I' and }I' + a respectively, meaning

Ell/(II - :roll = E[u(I; + E)l and E[u(,;;' + a - iTl)] = E[uC'v + a + E)l. Then an individual exhibits nonincreasing risk aversion in Ross' sense if IT) .::s ;roo Machina (1982) proposes an even stronger concept of non increasing risk aversion, requiring 7r] .::s ;Yo whenever a is a positive random variable. He shO\vs, surprisingly, that this is inconsistent with expected utility maximization. Epstein (1985) proposes a yet stronger concept, suggesting we should have 7T) .::s 7To if we replace \t,' + a by any gamble that is weakly preferred to lll. He shows, under some technical conditions. that this implies mean-variance preferences.

Pratt and Zeckhauser (1987) consider yet another concept that is stronger [han llonincreasing risk aversion: they define preferences to exhibit "proper risk aversion" if. \vhenever each of two independent gambles is independent of initial wealth and individually undesirable, then the sum of the gambles is undesirable. Assuming expected utility maximization, this means that for any gambles it'. i and s: which are mutually independent, if E[uUj] ::::. E[u(}I> + x)] and E[I/(';)I '" E!I/(,' + .v)] then E[I/(")I '" E[I/(,,' +., + DI. The interpretation is that adding "background risk" in the form of}' cannot make the unattractive risk.'t mtractivc. In Terms of risk premia, proper risk aversion is equivalent to either of the follO\:ving: (j) adding background risk y' to ~~l (weakly) increases the risk premium oLi:. or (Ii) the risk premium oLi: + S: is at least as large as the sum of the separate risk premia of .t and S', Pratt and Zeckhauser shmv that CARA and CRRA utilities are proper in this sense.

Kimball (1990) defines -u"'(w)/I1"(,v) (0 be the absolute prudence of a utility function II at wealth H' and -Ho{{f!I(\v)/u"(.v) to be relative prudence. There are many parallels between prudence and risk aversion (prudence is the risk aversion of the marginal utility function). Kimball relates prudence to "precautionary premia,"


paralleling the relation of risk aversion to risk premia. One of Kimball's results is that nonincreasing prudence implies a nonincreasing precautionary premium, as initial wealth is increased (for the definition of "precautionary premium," see Exercise 2.9). Kimball (1993) introduces a strengthening of proper risk aversion (which he calls standard risk aversion) and shows that it is equivalent to the combination of nonincreasing risk aversion and non increasing prudence.

Some pairs of gambles are ranked the same way by all investors with monotone preferences or by all monotone risk-averse investors. Let F denote the cumulative distribution function of a random variable x and G the cumulative distribution function of a random variable y. Then x is said to first-order stochastically dominate }' if F(a) .::s G(a) for every constant a. This means that .i has "more mass in the upper tail" than }' at whatever level a we choose to define the tail. First-order stochastic dominance is equivalent to E[uCr)] ::::. E[u(y)] for every monotone function u (Quirk and Saposnik (1962)). The random variable ,i' is said to second-order stochastically dominate S if

{boc F(a)da:" l:oc G(a)da for each h. This is equivalent to either of the following: (i) E[u(X)] '" E[uCY)] for every monotone concave function u (Hadar and Russell (1969), and (ii) the distribution of y equals that of x + Z + l: where Z is a non positive random variable, and E[e I x + z] = 0 (Stmssen (196S)-that (ii) implies (i) is the subject of Exercise 1.7.

Rothschild and Stiglitz (1970) give related results, establishing the equivalence of the following: (i) E[u(x)] '" E[u(y)] for every concave-not necessarily monotone-function u, (ii) the distribution of5' equals the distribution of adding a "mean-preserving spread" tox, and (iii) the distribution ofyequals the distribution of x +;: where E[E I x] = O-that (iii) implies (i) is shown in Section 1.8. For more on these equivalences, see Leshno, Levy, and Spector (1997) and Machina and Pratt (1997).

EXERCISES

1.1 Calculate the risk tolerance of each of the five special utility functions in Section 1.7 to verify the formulas given in the text.

1.2 Let" be a random variable with zero mean and variance equal to 1. Let n(a) be the risk premium for the gamble a l: at wealth w, meaning

u(W - iT(O') = E [u(w + O'E)] . (Ll3)

Assuming 7r is a sufficiently differentiable function, we have the Taylor series approximation

! 1 II 2 iT(O') "" ][(0) +][ (0)0' + ZiT (0)0'


for small a. ObviouslY, ;reO) = O. Assuming differentiation and expectation can be interchanged, differentiate both sides of (1.13) to show that n/(O) = 0 and Jr"(O) is the coefficient of absolute risk aversion.

1.3 Consider the five special utility functions in Section 1.7 (the utility functions with linear risk tolerance), Which of these utility functions, for some parameter values, have decreasing absolute risk aversion and increasing relative risk aversion? Which of these utility functions are monotone increasing and bounded on the domain w :::. 07

1.4 Consider a person with constant relative risk aversion p.

LS

(a) Verify that the fraction of wealth he will pay to avoid a gamble that is proportional to wealth is independent of initial wealth (i,e" show that;r defined in (l.lO) is independent of}V' for logarithmic and power utility).

(b) Consider a gamble E. Assume I + lis lognormally distributed; specifi-cally, assume 1 + E = e=:, \vhere z is normally distributed with yariance rJ 2. ~nd mean -rJ 1 /2, iVore that by the rule for means of exponentials of normals, E[e] = 0, Show that Jr defined in (UO) equalS

/'.)ote: This is consistent >i-'ith the approximation (1.4), because a first-order Ta,rlor series expansion of the exponential function eX around x = 0 shows that eX ~ 1 + x when !xi is small.

Consider a person with constant relative risk aversion p.

(a) Suppose the person has wealth of 5 100,000 and faces a gamble in which he wins or loses x with equal probabilities. Calculate the amount he would pay to avoid the gamble, for yarious values of p (say, between 0.5 and 40\ and for x = 5100, x = SI,OOO, x = SIO,OOO, and x = $:25,000. For large gambles, do large values of p seem reasonable? What about small gambles?

(b) Suppose p > I and the person has wealth w. Suppose he is offered a gamble in which he loses x or wins y with equal probabilities, Show that he will reject the gamble no maner how large}' is if

2. > 1 - 0.5!j(P-1) w

10g(0.5) + 10g(1 - X/IV) P ":. -''''-c-'---:c--''::'''--,--'--'-

log (I - x/w)

For example, ifw is $]00,000, then the person would reject a gamble in Hi/zich he loses $]0,000 or wins 1 trillion dollars with equal probabilities when p satisfies this inequality for x/w = 0.1. What values of p (if any) seem reasonable?

1.6 This exercise is a ver}' simple version of a model of the bid-ask spread presented by Stoll (1978).

1.7

1.8 1.9


Consider an individual with conStant absolute risk aversion a. Starting from a random wealth w,

(a) Compute the maximum amount the individual would pay to obtain a random payoff x; that is, compute BID satisfying

E[u(\v)] = E[u(\v + X - BIDl]. (b) Compute the minimum amount the individual would require to accept

the payoff -x; that is, compute ASK satisfying E[u(w)] = E[u(w - x + ASK)].

Show that condition (ii) in the discussion of second-order stochastic dominance in the end-of-chapter notes implies condition 0); that is, assume y = x + z + l where Z is a nonpositive random variable and E[l) x + z] = 0 and show that E[u(x)] ":. E[u(y)] for every monotone concave function u. Note: The statement of (ii) is that y has the same distribution as x + Z + E, which is a weaker condition than S' = x + Z + , but if y has the same dislribution as): + z + e and j/ = x + z + e, then E[u(y)] = E[u( jil] so one can without loss of generality take y = x + Z + s (though [his is not true for the reverse implication (i) :::::} (iij). Show that if E is mean independent of y, then cov(j, e) = O. Show that any monotone utility function with linear risk tolerance is a monotone affine transform of one of the five utility functions: negative exponential, log, power, shifted log, or shifted power, Hint.' Consider first the special cases (i) risk tolerance = A and (ii) risk tolerance = Bw. In case Ii) use the fact that

u"(w) d log U'(IV) --=

U'(\1/) dw and in case (iij use the fact that

d log U'(IV) ---=

u'(w) dlogw to derive formulas for log u'(w) and hence u'(w) and hence u(w), For the case A :j:: 0 and B :j=. 0, define

V(W)=UC";A) show that the risk tolerance ofv is BIV, apply the results from case Iii) 10 v, and then derive the form of u,

1.10 Suppose an investor has log utility: u(w) = log IV for each w > O. (a) Construct a gamble w such that E[u(wl] = 00. Verify that E[w] = 00, (b) Construct a gamble w such that vv > 0 in each state of the world and

E[u(w)] = -00.

-""Ci,c:"r 0, then the return on asset i is defined as

Ri = 3.. Pi

For each unit of the consumption good invested, the investor obtains Ri . The tenn "return" is used in this book more generally for the payoff of a portfolio with a unit price. The rate of return is defined as l

- x ~ p. ri = Ri - 1 = -'--' .

Pi If there is a risk-free asset, then Rf denotes its return. The risk premium of a risky asset is defined to be E[Rd - Rf . This extra average return is an investor's compensation for bearing the risk of the asset. Explaining why different assets have different risk premia is the main goal of asset pricing theory.

EXcept fo;-Sections-:fs aoifi-:-9,rhischapter ad-dresses the optima I investment problem, assuming consumption at the beginning of the period is already determined. 2 Let Wo denote the amount invested at the beginning of the period,

1. What we are calling the "return" is often called the "gross retum," and one will often encounter the tem "return" being used for what we are calling the "rate of return." In this book, gross returns appear more frequently than rates of return, so we usc the shorter name "returns" for them.

2. We can nevertheless assume, for concreteness, that the consumption good is the numeraire for the beginning-of-period as~et prices.

and let Bi denote the number of shares the investor chooses to hold of asset i. The investor may have some possibly random endowment}' at the end of the period (e.g., labor income) which he consumes in addition to the end-of-period portfolio value. Letting n denote the number of assets, the investor's choice problem is:

n n

max E [u(,v)] subject to (8 1 .... ,('),,)

"'e d - --1- "'e-L iPi = wa an w = Y , L iXi' i=1 i=1

(2.1 ) In (2.1), we have represented a portfolio in terms of the number of shares held

of each asset. Alternatively, we can represent it in terms of the amount i = BiPi of the consumption good invested in each asset. Assuming the asset prices are positive, the choice problem is

11 Jl

max E [,,(w)] subject to (OJ .... i,,)

L i = Wo and ~t,' = }' + L /?i . i=1 i=!

(2.1') Yet another way to represent a ponfolio that is often convenient is in terms of the fraction J[i = ).1/0 of initial wealth invested in each asset. Assuming the asset prices are positive, the choice problem is

n 11

max E [u(,v)] (,"f! ... ,;t"l

subject to L ITi = 1 and ~,= 5; + 14'0 L ;r/{i . i=1 i=]

(2.1") Note that ei < 0 is allowed in problem (2.1 )-likewise,

24 SinglePeriod ~vlodels

So. the first~order condition (2.4c) can be expressed as: Marginal utility evaluated at theoptimal \vea!th is orthogonal to each excess retllr~-:--------'-'-- -- --" .. ----~-... ----

"-Th~-s~mp-leintuj-t-jon -fm:T?:4c) is that the expec-iution

is the marginal value of adding the zero-cost portfolio to the optimal portfolio. If the expectation were positive, then adding a little of the zero-cost portfolio to the optima! portfolio e would yield a portfolio even better than the optimal portfolio, which is of course impossible. If the expectation vy'ere negative, then reducing the holding of the zero-cost portfolio by a little (i.e., adding a little of the portfolio v,:ith payoff Rj - Ri ) would lead to an improvement in utility. Because it is impossible to improve upon the optimum. the expectation must be zero (at the oplimal wealth >to).

The key assumption needed to derive the first-order condition (see (2.5) and (2.6) below) is that it is actually feasible to add to and subtract from the optimal portfolio a little of the zero-cost portfolio with pa).!off Ri - Rj . A simple example in \.vhich it is infeasible to do this and the first-order condition (2.4) may fail is if there is a risk-free asset. the utility! function is only defined for nonnegative wealth (as with CRRA utility). there is no end-of-period endo\.vment )', and the risky asset returns are normally distributed. In this case, regardless of the expected returns on the risky assets. the only feasible P0l1folio is to invest all of one's wealth in the risk-free asset. This produces a constant optimal wealth >t', so E[ur(~t')(Ri - Rj )] = u'(lt')E[Rj - Rjl. which need not be zero.

Thus, there are cases in which the first-order condition (2.4) does not hold. However. those cases are generally ignored in this book. We could qualify each re.-,ult that depends on the fir5t-order condition by stating that it holds under the assumptions that are needed to derive the first-order condition, but that would be too cumbersome. Instead, we wilL unless otherwise noted, simply assume that the first-order condition holds.

We \vill prove CAc) when Pi> 0 and Pj > O. Let e denote the optimal ponfolio, so

" \\o;:;:.\~LRI:l:i i=l

is the optimal wealth. Suppose the utility function is defIned for all \l' > ~. where ~ is some constant, possibly equal to -ex:. Assume the utility function is concave and differentiable.

Assume there exists E > 0 such that

(2.5)

in all states of the \vorld OJ and al! 0 such that (of::: E. Assume funher that

(26)

I I l

Portfolio Choice and Stochastic Discount Factors 25

for all 0 such that ]oi ::: E. The optimality of ).~, implies

[U(';; + 8(R - R) - [l(iV)] E I < 0 8 - (2.7)

for all 0 > O. We will use the following property of any concave function u: For any w > !y and any

real a.

u(w + oa) - u(w) t aso~, (2.8)

t.1.king 0 > 0 and sufficiently small that ~v + oa > ~. To prove (2.8), consider 0 < 0] < OJ. Define ), = 02/1' Apply the definition of concavity in footnote 3 of Chapter I with 1,

J"'5'-: covariances with an"y SDF. ItTs-worthwhile'to' point --C;utOne-;:dditional implication of the first-order

condition (2.4a), equivalently (2.11). Concavity of utility implies marginal utility is a decreasing function of wealth. Therefore, the first-order condition (2.11) implies that optimal wealth must be inversely related to an SDF m. This is intuitive: Investors consume less in states that are more expensive.

2.3 A SINGLE RISKY ASSET

Returning to the derivation of optimal portfolios, this section addresses the special case in which there is a risk-free asset, a single risky asset with return R, and no end-of-period endowment CY = 0). Let fJ.. denote the mean and 17 2 the variance ofR.

The investor chooses an amount to invest in the risky asset, leaving Wo - to invest in the risk-free asset. This leads to wealth

w=R+(wO-)Rj = woRj + (R - Rj ). (2.14)

- _ _- ~~~'"

The first-order condition is

E[u,(.t> (R-Rr)] =0. (215) Investment is Positive if the Risk Premium is Positive If the risk premium is nonzero and the investor has strictly monotone utility, then it cannot be optimal for him to invest 100% of his wealth in the risk-free asset. If it were, then }Ii would be nonrandom, which means that u'(w) could be taken out of the expectation in (2.15). leading to

u'(.,')E[R - Rf ] \,.:hich is nonzero by assumption. Therefore, putting 100% of wealth in the risk-free asset contradicts the first-order condition (2.15).

In fact, if fi., > F!J., then it is optimal to invest a positive amount in the risky asset-=... and ifjI.~ JT for sufficiently sIllall > 0, so w - ;r = >voRf + (p, - R,) - ;r > l'oRr

for sufficiently small 1; > 0. and consequently

~. w~ uc !lOW u~ing (;2 to dCIlO!c the variance of k so the variantc of ,0 j~ "2(}2


for sufficiently small > O. Therefore, some investment in the risky asset is better than none, when its risk premium is positive.

Constant Absolute Risk Aversion with Normal Returns This subsection considers the example of a normally distributed asset return and an investor with CARA utility. Given an amount if; invested in the risky asset, the realized utility of the investor is

The random variable

is normally distributed with mean

and variance a?La?. Therefore, using the fact that the expectation of the exponential of a normally distributed random variable is the exponential of the mean plus one-half the variance, the expected utility is

E [- exp (-a.v)] = - exp (-a [WoRf + (I" - Rf ) - ~a2cr2]) (2.16) Equation (2.16) states that

(2.17) is the certainty equivalent of the random wealth w woRf + (R - Rf ). _Maxi.rY!!?:i_~_K~?~p~_~ted ~tility is equivalent_to maximizing the utility of the

5U )Ingle-Penod Models

margin requirements. However. as is shown in Section 2.4, the absence of wealth effects does not depend on the return being normally distributed.

This is another example of mean-variance preferences. The certainty equiva-lent (2.17) can be expressed as

. [ ,(f,l-Rr) I 2a 2] ''"'0 R/ T - -awo--

Wo 2 1-\.15 [ - I _ ]

= Wo E[R] - 2:"wo YareR) , (2.19)

\vhere the investor's portfolio return is

- ~t,' 4;_ R=-=Rr+-CR-R,).

Wo }Va J

Thus, in this circumstance expected utility depends only on mean and variance (and the parameters Ci and lVO)' This is a general property of normal distributions (Section 2.6).

Decreasing Absolute Risk Aversion

~f an investor has decreasing absolute risk aversion (which, as noted before, JOcludes CRRA utilities), then his investment in the risky asset is laro-er when his .injti~l wealth is larger. Assume the asset has a positive risk premiu~, so the optimal Investment is positive. The dependence of investment on initial wealth is derived by differentiating the first-order condition (2.15), with the random wealth .. t" being defined in (2.14) and assuming the optimal investment 1; is a continuously differentiable function of wo. Because the first-order condition holds

~or all ).1"0, the derivative of (2.15) with respect to 1170 must be zero; thus, using the tormula (2.14) (0 compute the derivative, we have

Therefore.

implying

d -RrE [CR - Rr)u"(I'v)] E [CR - Rr )2 u"(I'i] C2.20)

The denominator in (2.20) is negative, due to risk aversion. Our claim is that the numerator is also negative, leading to d jdlvo > O. This is obviously equivalent to

E[(R-Rf)U"(;V)] >0, (2.21) \vhich is established below.

,UI UU!IU \.IIUH .. C


Y_;19J.lO_1.}_~~~_~._p __ a_~~~~~~~~}~~~~ -~_~~~~~!2_~ ~~


the s~cond step if this is optimal. The first investment produces (, so the total wealth achieved is

\vhcrc k denotes the return on the second investment. This can be \vritten as

where \ve deill1c

(226)

This implies that the utility achieved is, for shifted log.

and. ror shifted power.

In either case, the logic of the previous paragraph leads to the conclusion that the optimal ;[i and hence ~i are inJependent of ,1'0 and A.

2.5 CONSTANT ABSOLUTE RISK AVERSION WITH MULTIVARIATE NORMAL RETURNS

To illustrate the result of the previous section, consider a CARA investor who chooses among multiple normally distributed assets. Continue to assume there is no end-of-period endowment Suppose there is a risk-free aS5et with return Rf and H risky assets with returns R; that are joint normally distributed. Let Rvec denote the n-dimensional column vector with Ri as its ith element, fL the vector of expected returns (the n-dimensional column vector with ith element E[Rd), and 1 the II-dimensional column vector of ones. Let f denote the investment in the risk-free asset. let i denote the investment in risky' asset i, and let P denote the n-dimensional column vector with Pi as its ith element.

The budget constraint of the investor is

" m'~m=wo. 'P/ T L. 'P,

i=1

Portfolio Choice and Stochastic Discount Factors

where Wo is the given initial wealth. This can also be written as

/ = Wo - l',

where f denotes the transpose operator. The end-of-period wealth is n

fRf + I: ,R, = fRf + ' R''' ;=1

= woRf + 'CR''' - Rfl), and the expected end-of-peliod wealth is

woRf + '(fJ- - Rfl).

35

Let L denote the covariance matrix of the risky asset returns. The (i, j)th element of L: is cov(k i R.). Of course, the diagonal elements are variances. In matrix notation, L is given by E[(Rvec - fJ-)(kec - fJ-)']. The variance of end-of-period wealth is

var (t ,R,) = 'L. 1=1

To see this. note that, because the square of a scalar equals the scalar multiplied by its transpose, the variance is

E [('(Rvec - fJ-))2] = E [' (Rvec - fJ-)CR'"c - fJ-)' ] = 'E[(R'" - fJ-)(Rvec - fJ-l'l = 'L.

Assume L is nonsingular, which we can ensure simply by eliminating redundant assets.s

As was the case with a single normally distributed risky asset, maximizing expected CARA utility with multiple normally distributed assets is equivalent to solving a mean-variance problem: Choose P to maximize

, 1 , (fJ- - Rfl) - 'ia L. Differentiating with respect to and equating the derivative to zero produces:

fJ--R-l-a"I:=O .I

5. If'L is singular, thcre is a nonzero vector,p such that 'L,p "" O. Of course, this implies ,p'"L "" 0, so in this circumstance the portfolio,p of risky assets is risk-free. One can scale,p such thall' "" 1, meaning that 1> represents a unit-cost portfolio. Hencc, in the absence of arbitrage opportunities. we must have ,'R\'cC "" Rj , showing that the risk-free asset is redundant. One can also rearrange the equation ,p'R,ec = Rj to sec. for any i such that 1>; i 0, that the return of asset i is equal to the return of a portfolio of the other risky assets and the risk free asset; thus, there is a redundant risky asset. If it were eliminated, the opportunities available to investors would be unchanged.

36 Single-Period ,\-lodeJs

\vith solution

(2.27) This is a straightforward generalization of the formula (2.18) for the optimal portfolio of a CARA investor with a single normally-distributed asset. As asserted in (2.23a), the optimal investments are independent of initial wealth. As will be seen in Chapter 5, the portfolio E-J(j.J., - Rrl) has a special significance in mean-variance analysis even when asset returns"are not normally distributed.

The formula (2.27) implies the single-asset formula (2.18) if the return of the asset is independent of all other asset returns. For such an asset i, (2.27) implies

po'; - Rr i = -:> (2.28)

eto!-

1.vhere (J? is the variance of Hi' In generaL (2.27) states that the demand for each asset i depends on the entire vector of risk premia and the covariances between asset i and the other assets.

I

JZ.6 MEAN-VARIANCE PREFERENCES

\Y~.~_


It should be apparent that true completeness is a rare thing. For example, if [here are infinitely many states of the world, then (2.29) is an infinite number of constraints. l,vhich we are supposed to satisfy by choosing a finite-dimensional vector (8 1 , , On)' This is impossible. Note that there must be infinitely many states if we '.vant the security payoffs xi to be normally distributed, or to be log-normally distributed, or to have any other continuous distribution. Thus, single-period markets with finitely many continuously distributed assets are not

comp~'-"~~---"~---~~------"- .----.-.-- .... -.----.--.---------~~---

On the other hand, if significant gains are possible by improving risk sharing, then one would expect assets to be created to enable those gains to be realized. Also, as is shown later, dynamic trading can dramatically increase the "span"

,of securities markets. '"['he real impediiTIents ~o achIe-Ylng ~iTeaiit 'approX'jffiaie~y co-mplete markets are mora-] hazai'ci"a'rid ad\;ers-e"serecdoJl-_-'F\::;l:-example~t"tl-ere are very limited op-portuni"tles--for"'-6hr-atlli"il'g 'lnsiirance' against employment risk, due

: to moral hazard. In any case. completeness is a useful benchmark against \vhich to compare

actual security markets. As remarked above. to have complete markets in a one-period model with a finite number of securities, there must be only finitely many possible states of the world_ For the remainder of this section, suppose there are k possible states and index the states as (j)). for j = 1. _ .. , k. Set xij = xJw) and ''-j = 11(W). Then. the definition of market completeness is equivalent to: For each IV E ~S'.k. there exists BERn such that

" (V j = I ..... k) :z::: Bi-'j = wj . (2.30) i=1

More succinctly, letX' be then x k-matrix (xU), where the prime denotes transpose. Then. market cornpleteness is equivalent [0: For each ... 1/ E G?.t:, there exists () E Rn such that

Px,l i\Xi b"tj X'() = w. (2.31 )

This system of equations has a solution for each Hi E Rk if and only if X has rank k. Thus, in panicuhlr.--marke-t- compf;;-re-ne-ss implies~-:::: k; tEat is, there must b~ at

------~------. - , .. --- -----._,. __ .- --- --"._.-_ .. - ------_._.,,- -----

least as many securities as states of the world. '-Con1pYetene's's-;-nea11sthe"exTsteJ;'ce o(fi-sofu-ti-on () E Il(1! to (2.30); it does not require that the solution be unique. However, if there are multiple solutions (for the same IV) having different costs 2:7=1 PiBj, then an investor cannot have an optimum, because buying the cheaper solution and shorting the more expensive solution is an arbitrage (this concept is discussed in more detail in Chapter 4). If the cost is unique, then one says that the "law of one price" holds.

The law of one price can be expressed as:

(veJj) x'e = x'e '* p'8 = p'i}, (2.32)


where p = (jJ 1 - . PI1)" The law of one price is equivalent to the existence. of an SDF This is true and important in both incomplete and complete mar~ets.ThIS fact i-s a consegLlenceorstrrugtitforward linear algebra, as is discussed further below.

In thi's finite-state model, an SDP can be identified with a vector m = (m! - . mk)' having the property that

k

Pi = L miti) prob) , )=1

for each asset i, where prob) denotes the probability of state wr Setting qj = m.prob. andq = (ql" qk)" one can write this as } }

AXI I\;(~ .IV\} P =Xq_ (2.3,) {X)l ,FI IV"f

40 Single-Period Mode!s

p~riod."da[e 0" and the end ofrhe period "date I." Now let }Vo denote the beginning-~f-~~!!:J~-.d wealth b.efore consl.~!l1ing. This includes the value of any shares held pl~s dn)- uah:::-O end()\J,i111ent. Letung v(co. c]) denote the utility function, the choice problem IS:

" max E [v{e(), c, ),] subj"ect to C ,'\' e 0' ~ iPi = Wo

i=l

" and (V ca) c, (w) = S'(w) + L O/x,(w) , (2.36)

i=J

Substituting in the second constraint. the Lagrangean for this problem is

and the first-order conditions arc:

E [a~o v(co' C,)J = y, (2.37a) (Vi) E[a~"V(CO'C")i']=Yp" (2.37b)

The system (2,37) is equivalent to:

(Vi) [3 --1 [3 J E 'a-v(co'c,)x, =p,E -, -I,(CO,C,) ci ..J aco

(2,38)

As ~efor~, this is a necessary condition for optimality provided it is feasible, startll~g from the optimal portfolio, to add a little or subtract a little of each asset l.

" ~he I~e\V featUl~e. relati,,:e to the problem considered previously in this chapter, IS thelt we have a tormula for the Lagrange multiplier y. We can write (2.38) as

(V,') E[--] . mXi =Pi' (238')

where

rn = 3v(co, Z,)/ilc,

E [av(co, c, )/3co] , (2,39)

This looks I~ore complicated than the con'esponding formula (2.11), but it savs the same thmg: The. ~arginal. utility of end-of~period wealth is proportional ~o an SDF The only difference IS that here there is a formula for the canstant of proportIonalIty.


2.9 TIME-ADDITIVE UTILITY

To obtain strong results in the model \.vith optimal beginning-of-period consump-tion. one can assume the investor has time-additive utility, meaning that there are functions Uo and Uj such that v(co, c!) = uo(Cj) + III (el)' In this circumstance,

3 , ~v(co, c,) = "o(co) and aco

Therefore, the SDF ", in (2.39) is _ LI~(Cl) In=--,

"o(co) (2.39')

Thus, with time-additive utility, the investor's marginal rate of substitution between date-O consumption and date-l consumption is an SDF. The first-order condition (2.38) with time-additive utility can be expressed as:

(2.38") This equation is called the Euler equation.

A leading special case is when the functions Uo and ul are the same except for a discounting of future utility ul' So suppose there is a function u and discount factor 0 < 0 < 1 such that Llo = " and LI, = ou, Then the SDF 111 in (2,39) is

_ ou'(c,) m=---,

u'(co) (2.39")

As was remarked, time-additive utility leads to strong results. For example, in continuous time it produces the Consumption-Based Capital Asset Pricing Model (CCAPM) of Breeden (1979), However, it is also a strong assumption, In particular, it links the wayan investor trades off consumption at different dates with the investor's tolerance for risk, which one might think should be distinct aspects of an investor's preferences. A precise statement of the link is that, with constant relative risk aversion, the ela

\t'oRf + (R - R/) is 2 var(R). hence propol1ionat to cp2, and the risk premium (j>(rL - R/) is proportional to . Thus, the compensation required is less than the compensation received, and the gamble is desirable, for all sufficiently small . As noted in Section 1.9, this reflects the approximate risk neutrality of expected utility maximizers with regard to small gambles. An application of this result to insurance markets is as follows. Suppose that an uninsured individual has final wealth w -l. Suppose the individual can buy insurance at a cost y per unit, meaning that, if an amount x of insurance is chosen, then the final wealth is w -l -.ry + xl. We can write this as w - .v + (x - 1 )(l - y). If the insurance is actually unfair, then it is optimal to choose less than full insurance. because in that circumstance the choice problem is equivalent to a portfolio choice problem in \vhich the risky asset has a negative risk premium (E[l] - y < 0), implying a shon position (x - 1 < 0) is optimal.

Mossin (1968) establishes the result (2.23) that optimal investments are affine in vvealth when an investor has LRT utility. Cass and Stiglitz (1970) establish the two-fund separation result of Section 2.4 when investors have linear risk tolerance \vith the same cautiousness parameter. They also shO\:y' that this condition is a neces~ary condition on preferences for two-fund separation to hold with a risk-free asset and for all distributions of risky asset returns. They give other conditions on preferences that are necessary and sufficient for two-fund separation in complete markets and in markets without a risk-free asset.

The CARA/nonnai model is a special case of mean-variance optimization, studied by Markowitz (1952, 1959) and addressed further in Chapter 5, The fact that investors have mean-variance preferences when returns are elliptically distributed is shown by Owen and Rabinovitch (1983) and Chamberlain (1983a). Owen and Rabinovitch (1983) give several examples of elliptical distributions. Chamberlain (1983a) also gives necessary and sufficient conditions for mean-variance preferences in the absence of a risk-free asset.

More on the concepts of precautionary savings and precautionary premia, illustrated in Exercises 2.8-2.9, can be found in Kimball (1990). Chamberlain (1988) is the source of Exercise 2.10.

EXERCISES

2.1 Consider the portfolio choice problem of a CARA investor with n risky assets having normally distributed returns studied in Section 2.5, but suppose there is no risk-free asset, so the budget constraint is 1! = H-'O' Show that the optimal portfolio is

Note: As }vi/l he seen in Section5.i, the nvo vectors 2::- I,u and 2::- 11 play all impOrUlnf role in rnean-variance analysis even \vithout the CARAlnonnal assumption.

t'OrtrO!lO UlOlle dilU ')LUUld':>lH. UI':>LUUllt r''' .. lVI:)

2,2 Suppose there is a risk-free asset and n risky assets. Adopt the notation of Section 2.5. but do not assume the risky asset retul11S are normally distributed. Consider an investor with quadratic utility who seeks to maximize

_ 1 -, } c-) ("E[w] - -E[wl- - - var w . 2 2

Show that the optimal portfolio for the investor is

where

It is shown in Chapter 5 that K is the maximum Sharpe ratio of any portfolio. Hint: In the first-order conditions, define y = (IL - Rj lY, solve for in lerms ofy, and then compute y.

2.3 This exercise provides another illustration of the absence of wealth effects for CARA utility. The investor chooses how much to consume at date 0 and hmv much to invest, but the investment amount does not affect the optimal portfolio of risky assets. Consider the portfolio choice problem in which there is consumption at date 0 and date 1. Suppose there is a risk-free asset with return Rf and n riskv assets the returns of which are joint normally distributed with mean vec;or /.L and nonsingular covariance matrix L:. Consider an investor who has time-additive utility and CARA utility for date-) consumption:

Show that (2.27) is the investor's optimal portfolio of risky assets. 2.4 This exercise repeats the previous one, but using asset payoffs and prices

instead of returns and solving for the optimal number of shares to hold of each asset instead of the optimal amount to invest. Suppose there is a risk-free asset with return Rj and n risky assets with payoffs Xi and prices Pi. Assume the vector x = (Xl" ,x,Y is normally distributed with mean /.Lx and nonsingular covariance matrix :EX' Let p = (PI ... P,Y Suppose there is consumption at date 0 and consider an investor with initial wealth Wo and CARA utility at date):

Let f). denote the number of shares the investor considers holding of asset i , and set e = (e , .' . eny. The investor chooses consumption Co at date 0 and


a portfolio fj" producing \vealth (wo ~ Co ~ e'p)R{ + fJ'.\: at date 1. Show that the oprimal vector of share holdings is .'

I _, IJ = -Z, (I", -RIP).

(X

\./2.5 Consider a utility function v(co, CI)' The marginal rate of substitution is defined to be the negative of the slope of an indifference curve and is equal to

f2.6

1.7

'II! R S( . ) _ .J '..c(...:c 0.:... _c,-,' ):.../o_c-,,o - CO'(I - :-

av(co, c) )/ac) .

The elasticity of intertemporal substitution is defined as

d log MRS(co. c,) .

\vhere the marginal rate of substitution is varied holding utility constant. Show that. if

then the elasticity of intertemporal substitution is I/p, This exercise shows thm an imlJrOl'eJnel1! in fhe investment opportunitv .'lei leads to higher saving (fhe suhsritution effecl dominates) }Vhell th~ elasticilY (~f interlemporaJ substitUTion is high and higher consUlnption (the ~-vealth e.fTecr dominates) ~j)hen the elasticity of inrertemporal substitution is lmv,

Consider the portfolio choice problem \vith only a risk-free asset and with

consumption at both the beginning and end of the period. Assume the investor has time-additive po\ver utility', so he solves

max subject to

~ho\V: that the ?ptimal consumption-to-\vealth ratio co/wo is a decreasing i"unctlon of R/ If P < 1 and an increasing function of Rr if p > l. Each pan qf this exercise illustrates [he absence qf wecllth effectsfor CARA wi liry a fld is not tcuefor genera/uTility functions. The assumption. in Parr (c) ThaI," = aRr + bR + E: >I'here E: has :.em meon and is uncorrelared with k is' H'ithollf fos-,)~ (~f"gene!-alilY (even \virhoUf the normality assumption): define h = cov(~. R)/var(R). a = (EL"]- hEIR])/Rr and E = .,. - aRf - hR. This IS a specwl case qf WI 0I1hogoilai prqjection (linear regression), ~vhich is discllssed ill lI10re generaliT), in Section 4.5. The optimal porr/olio in Part (c) can be interpreted as fhe oprimal por(/olio in rhe absence of an endowment plus a hedge (~b)for .y.

Suppose there is a risk-free asset with return RJ and a risky asset with retum

k Consider an investor who maximizes expected end-of-period utility of wealth and who has CARA utility and invests woo Suppose the investor has

a random endowment 5' at the end of the period, so his end-of-period wealth is is the same as if there were no end-of-period endowment.

(b) Show that if y and R are independent, then the optimal 1> is the same as if there were no end-of-period endowment, regardless of whether y and k are normally distributed. Hint: Use the law of iterated expectations as in Section 1.8 and the fact that if v and x are independent random variables then E[v,,] = E[v]E[,t].

(c) Suppose)' and R have ajoim normal distribution and)' = aRI + hR + l for constants a and b and some l that has zero mean and is uncorrelated with R. Show that the optimal 1> is 1>' - b, where 1>' denotes the optimal investment in the risky asset when there is no end-of-period

endowment.

2.8 This exercise illustrates the concept of "precautionary savings ))-the risk imposed by y results in higher savings Wo ~ co. Consider the portfolio choice problem with only a risk-free asset and with

consumption at both the beginning and end of the period. Suppose the

investor has time-additive utility with Uo = u and ul = Ciu for a common

function u and discount factor o. Suppose the investor has a random endowment y at the end of the period: so he chooses Co to maximize

,,(co) + oE[,,(wo - co)Rf + 5')].

Suppose the investor has convex marginal utility (ufl! > 0) and suppose that ELy] = O. Show that the optimal Co is smaller than if y = O.

2.9 Letting Co denote optimal consumption in the previous problem, define the "precautionary premium" ;r by

,,'((lVo - IT - c~)Rf) = E[u'((wo - co)Rj + 5')]

(a) Show that Co would be the optimal consumption of the investor ifhe had no end-of-period endowment and had initial wealth \-1/0 - J(,

(b) Assume the investor has CAR A utility. Show that the precautionary premium is independent of initial wealth (again, no wealth effects with CARA utility).

2.10 The assumption in this exercise is a weak form of market completeness. The conclusion fallows in a complete market from thefonnulation (2.35) of the portfolio choice problem,

46 Single-Period ivlodels

Su,ppose there ~s an SDF in with the property that for everv function g there eXIsts a portfoho e (depending on g) such that ~

" L 8;.t; = ge,),) . i=l

~onSider ,an inv:stor ~'ith no labor income y. Show that his optimal wealth IS a functIOn of m. Hmt: For anyfeosible 'rt,., define \1:,,'" = E[w l,n] and u~e the resull oj Section 1.B. ' .

3 Equilibrium and Efficiency

This chapter presents the definitions of "competitive equilibrium" and "Pareto

optimum." Competitive equilibria in complete markets are Pareto optimal. Also,

competitive equilibria are Pareto optimal, regardless of the completeness of

markets, if all investors have linear risk tolerance with the same cautiousness

parameter. "Gorman aggregation" means that equilibrium prices are independent of the

initial distribution of wealth across investors. Gorman aggregation is possible for

all asset payoff distributions if (and only if) investors have linear risk tolerance with the same cautiousness parameter.

It is assumed except in Section 3.7 that beginning-of-period consumption is

already determined, so the focus is upon the investment problem. Section 3.7

shows that the results also hold when investors choose both beginning-of-period

consumption and investments optimally. It is assumed throughout the chapter that

all investors agree on the probabilities of the different possible states of the world.

3.1 PARETO OPTIMA

Suppose there are H investors, indexed as h = 1, ... , H, with utility functions till'

A social objective is to allocate the aggregate end-of-period wealth wm ("m" for "market") to investors in such a way that it is impossible to further increase the expected utility of any investor without reducing the expected utility of another.

An allocation with this property is called Pareto optimal. As is discussed further

below, Pareto optimality in our securities market model is an issue of efficient risk

sharing.


Formally, an allocation (it>j, .... 11)H) is defined to be Pareto optimal if (i) it is feasible~that is, )'~i=1 )~'h(V) = H!m(tJ) in each state of the \vorJd w, and Oi) there docs not exist any other feasible allocation (It';, ... , 1{:~/) SLlch that

E[UhC,,'\!;)] ::: E[Uh(l~'h)] for all h, with

E[uhC>;>;])] > E[u,J}t,'h)] for some h. For the sake of brevity, the term "allocation" will mean "feasible allocation" in the remainder of the chapter.

A simpJe example of an allocation thar does not in\'olve efficient risk sharing and hence is not Pareto optimal is as follows. Suppose there are t\VO risk-averse investors and t\\i{) possible states of the world. \vith It"iII being the same in both ."lares, say. i1.'!/) = 6. and \vith the t\\:o states being equally likely. The allocation

{~ in state I lV I = in state :2 {~ in state I IV,., = in state 2

is not Pareto optimal, because both investors \vould prefer to receive 3 in each stj, ... , WH) is Pareto optimal, then each individual must be allocated higher wealth in states in which market wealth is higher. This says nothing about which individuals get higher wealth than others, only th~t all individuals must share in market prosperity, and all must suffer (relatlvely speaking) \vhen market wealth is low. As we will see, this is a simple consequence of the first-order condition (3.4) and risk aversion,

For any two investors j and h, the first-order condition (3.4) implies

(3.5)


Considering two different states wI and w2 and dividing (3.5) in state w1 by (3.5) in state w2 yields

U;I(~{;'h(W! u;1(~'h(W2 . (3.6)

This is the familiar result from microeconomics that marginal rates of substitution must be equalized across individuals at a Pareto optimum~ Here, wealth (consump-tion) in different states of the world plays the role of different commodities in the usual consumer choice problem.

Assuming strict risk aversion (strictly diminishing marginal utilities), the equality (3.6) of marginal rates of substitutions produces the following chain of implications: ~

=? It'h(w\) > }t,'h(w2)'

Because this is true for every pair of investors, it follows that at a Pareto optimum all investors must have higher wealth in states in which market wealth is hiaher Thus. each investor's wealth is related in a one-to-one fashion with market we~lth; that is, letting fll denote the one-to-one relationship for investor h, we have l,t,'h (w) = fh( l,Vm (w)) in each state of the world (J). The functionsfll are called sharing rules.

A particular consequence of these sharing rules is that if market wealth is the same in t\'\10 different states of the world, then each investor's wealth in a Pareto-optimal allocation must be constant across the two states of the world. This shows that the example in Section 3.1 is inconsistent with Pareto optimalitv \vhen investors are strictly risk averse. ~

If investors have linear risk tolerance with the same cautiousness parameter, then the sharing rules fh must be affine (linear plus a constant) functions. Specifically, if an allocation (M.l j , ... , l,,'iH ) is Pareto optimal, then there exist constants ah and bh > 0 for each h such that

(3.7)

for each (I). Thus. the wealths of different investors move together and do so in a linear way. This is shown in Section 3.6.

3.4 COMPETITIVE EQUILIBRIA

A competitive equilibrium is characterized by two conditions: (i) markets clear, and (ii) each agent optimizes, taking prices as given. \Ve take production decisions

Equi!ibrium and Efficiency 51

as given and model the economy as an exchange economy. Thus, part (ii) means that each investor chooses an optimal portfolio.

To define a competitive equilibrium formally, let elli denote the number of shares of asset i owned by investor h before trade at date O. The value of the shares, which of course depends on the asset prices, is the investor's wealth at date 0. 1 Assume investor h invests his date-O wealth in a portfolio eh = (8h1 , .. , Bhn ) of the n assets. Let xi denote the payoff of asset i, and set ei = L~=l {jhl' which is the total supply of asset i. One can allow investors to have (possibly random) endowments at date 1, which they consume in addition to their portfolio values. Let Yh denote the endowment of investor h at date 1.

A competitive equilibrium is a set of prices (PI' ... , Pn) and a set of portfolios (8, ..... 8H ) such that markets clear,' that is,

(V i) H

I:8hi = ai' h=l

. and such that each investor's portfolio is optimal, i.e, for each h, ell solves

(3.8) II 11

subject to L 8lriPi = L ehiPi n

wh(w) = Yh(w) + I: 8hixJw). i=!

and (V w) i=! i=!

3.5 COMPLETE MARKETS

Suppose there are only finitely many states of the world and the market is complete and satisfies the law of one price. As explained in Section 2.7, in this circumstance the portfolio choice problem (3.8) can be expressed as a standard consumer choice problem. Moreover, standard results about competitive equilibria in pure exchange economies apply to competitive equilibria in the securities market model, including the First Welfare Theorem.

I. If \\'e included consumption at date O. we would allow investors to have other wealth at date 0, for example labor income, which we would call "consumption-good endowments" and/or for the assets to have paid dividends in the consumption good before trade at date O. Because in equilibrium all of the assets must be held by investors, total consumption of all investors at date 0 must equallotal consumption-good endowments plus dividends. This variation of the model is discussed in Section 3.7. 2. As is standard in microeconomics. equilibrium prices can be scaled by a positive constant and remain equilibrium prices because the set of budget-feasible choices for each investor (in this model, the set of feasible portfolios GiJl is unaffected by the scaling-technically. budget equations are "homogeneous of degree zero" in prices. In microeconomics. it is common to resolve this indetenninacy by requiring the price vector to lie in the unit simplex. As mentioned in Chapter 1. it is customary to resolve it in finance by setting the price of the consumption good equal to I (making the consumption good the 'numeraire"). Choosing a different numeraire scales all of the returns \/Pi by the same factor.


To provide some support for this claim, adopt the notation of Section 2.7 and deline

k

Uj,(W) = L probjuj,(wi) i=1

for any W E Rk. Thus, Uh(w) is the expected utility of the random wealth defined by >\', Let ,vh = G,JWj)'" S',Jwk))' denote the date-l endowment of investor h. Then. (3.8) can be expressed as:

max o

U,JX' B + ,\';}) subject to / e = pi eli . (39) In a complete market satisfying the !a\v of one price. for any H: E sJ:. there exists a portfolio e such that X'O + Yli = It". Moreover. the cost of this portfolio is p'P = q'X'f} = q/I\.- - q'Yh' where q is the vector of state prices satisfying p = Xq defined in (2.34). Hence. (3.9) can be expressed as:

(3.10)

Set \~.,/! = K'(~h -+- YII' This is the end-of-period wealth of the investor in the absence of trade. \Ve have p'Rh = q'X'Bh = q'.;t:h - q',vlr Therefore, (3.9) can be expressed as:

max

"

(3.11 )

The economy' in which investors have endowments \Vh E IRk and solve the consumer choice problem (3.11), with the price vector q being determined by market clearing, is called an AITow-Debreu econorny, in recognition of Arrow and Debreu (1954) and other \\:ork of those authors. There is a one~to-one relationship between equilibria of the Arrow-Debreu economy and equilibria of the securities market, \\'ith the price vector pERil in [he securities market and the price vector q E ?,.k in the Arrow-Debreu economy being related by p = Xq. More precisely,

(a) Suppose (q. wI' ... , 'vi';!) is such that }vt: solves (3.11) for each hand markets clear:

!-f Ii

L .v"J; = L 11'11 . (312) h~1 h=1

Then, settingp = Xq. there exists a portfolio G!~ for each investor solving (3.8) such that w'h = X'Gj;' + Yh for each h and the securities market clears:

H H

L 8,; = Le". (3.13) h~J 11=1

(b) Suppose (I'. e~ . .... eli) issuch thate,: solves (3.8)foreach hand (3.13) holds. Then. wI, = X'8,: + y" solves (3.11) for each hand (3.12) holds. when q = (X'X)~IX'p.

Equilibrium and Efficiency 53

To prove (a), letel~ beany solution to w~ = x'el~ + Yh for II < Hand setBH = e - Lh


and log, and power utility are special cases of shifted log and shifted power utility respectively. Thus, we are assuming one of the follmving conditions holds:

(a) Each investor has CARA utility, with possibly different absolute fisk aversion coefficients Cil!_

(b) Each investor has shifted CRRA utility with the same coefficient p > 0 (p = I meaning shifted log and p f:: 1 meaning shifted power) and with possibly different (and possibly zero) shifts Sh.

Sharing Rules with Linear Risk Tolerance

fn this subsection, it is shown that any Pareto-optima! allocation involves an affine sharing rule as in (3.7). Moreover. Ltf=1 all = 0 and L~=l hh = 1, \vhere all and hI; are the ,coefficients in (3.7). The converse-that affine sharing rules produce Pareto-optimal allocations-is left as an exercise (Exercises 3.7 and 3.8). Given a P~reto optimum Uv], .... }\"fj), let )., I, .... AH be weights such that the Pareto optImum solves the social planning problem (3.2). We can show:

(a) If each investor h has CARA utility with some absolute risk aversion coefficient ct", then (3.7) holds with

(b)

and (3.14)

wl:ere T~! ~ l/cxh is th,e c~efficient ofris~.toler~nce, and r = L.~l rj' If ~ach mvestor h has shIfted CRRA utlltty with the same coefficient p > 0, then. setting S = L~~I s,,, (3.7) holds with

and (3. I 5)

~ote tha~ the t\I/O cases are somewhat different, because the weights Ai! in the .social planning problem affect only the intercepts ah in the CARAcase, whereas for shifted ~RRA utility, an investor with a higher weight AI! has a higher coefficient hh' that IS, an_ a:loca~ion }"\;h wi~h a greater sensitivity to market wealth i"i,'m' Note also that (3.1) ImplIes the shanng rule }\!h = ah + hI; h-'m in the shifted CRRA case can be written in the perhaps more transparent form:

(3 16)

The CARA case will be proven. The shifted CRRA case, which is similar, is left as an exercise.

Equilibrium and t:.ttloency

-v-,Te need to solve the social planning problem (3.3) in each state of the world. Specializing the first-order condition (3.4) to the case of CARA utility, it becomes

We need to find ij, which we can do by (i) solving for wh:

- 1 1 - logO'hail) wil = -- og7]+

ah ah

(ii) adding over investors to obtain H

wm = -! log ij + L!e log(Aeae), =1

and then (iii) solving for ij as

Substituting this back into (3.17) yields

This establishes the affine sharing rule (3.14),

Gorman Aggregation

(3.17)

A price vector (Po, PI' ... ,Pn) is called an equilibrium price vector if there exist portfolios e" ... , eH such that the prices and portfolios fonn an equilibrium. Consider equilibrium price vectors in which Pi =1= 0 for each i, so we can apply the portfolio choice results of Section 2.4, which are expressed in terms of returns. We will show that the set of such equilibrium price vectors does not depend on the initial wealth distribution.

Walras' Law implies that the market for the risk-free asset clears if the markets for the other n assets clear,4 so markets clear if and only if

H

L


In Section 2.4. it was shO\vn that

(3.19)

where ~i is independent of AI! and \.I"M)- In Ollr current modeL investors differ only with regard to A" and whO_ so t is the same for each investor h. Consequently. the aggregate investment in risky asset i is

H

L


First Welfare Theorem with Linear Risk Tolerance

In this subsection, it is shown that any competitive equilibrium in this economy is Pareto optimal. The key fact is the effective completeness of markets established in the previous subsection. We will show that any allocation that is Pareto dominated is Pareto dominated by a Pareto optimum. Therefore, if a competitive equilibrium were not Pareto optimal, it would be Pareto dominated by an allocation that can be implemented in the securities market. But this dominant allocation could not be budget feasible for each investor; because, if it were, it would have been chosen instead of the supposed competitive equilibrium allocation. Adding budget constraints across investors shows that the Pareto-dominant allocation is not feasible, which is a contradiction. The remainder of this section provides the details of the proof.

We argue by contradiction. Consider a competitive equilibrium allocation

and suppose there is a feasible Pareto superior allocation. Without loss ofgenernlity. suppose that the first investor's expected utility can be feasibly increased without reducing the expected lltility of the other investors. Let lilt = Elu!/ll};)l for h > I. We now define a Pareto optimum thm increases the expected utility of the first investor without changing the expected utilities of other investors. ~ ~

(a) If each investor h has CARA utility \vith some

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