Increases in risk aversion and the distribution of portfolio payoffs

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Journal of Economic Theory 147 (2012) 1222–1246

www.elsevier.com/locate/jet

Increases in risk aversion and the distribution of

portfolio payoffs

✩

Philip H. Dybvig

a,∗

_{, Yajun Wang}

b

a_{Olin Business School, Washington University in St. Louis, St. Louis, MO 63130, United States} b_{Robert H. Smith School of Business, University of Maryland, College Park, MD 20742, United States}

Received 11 October 2010; final version received 11 June 2011; accepted 26 July 2011 Available online 29 November 2011

Abstract

Oliver Hart proved the impossibility of deriving general comparative static properties in portfolio weights. Instead, we derive new comparative statics for the distribution of payoffs:Ais less risk averse than

BiffA’s payoff is always distributed asB’s payoff plus a non-negative random variable plus conditional-mean-zero noise. If either agent has nonincreasing absolute risk aversion, the non-negative part can be chosen to be constant. The main result also holds in some incomplete markets with two assets or two-fund separation, and in multiple periods for a mixture of payoff distributions over time (but not at every point in time).

JEL classification:D33; G11

Keywords:Risk aversion; Portfolio theory; Stochastic dominance; Complete markets; Two-fund separation

1. Introduction

The off between risk and return arises in many portfolio problems in finance. This trade-off is more-or-less assumed in mean-variance optimization, and is also present in the comparative statics for two-asset portfolio problems explored by Arrow[1]and Pratt[15](for a model with

✩ _{We thank An Chen, David Levine, Hong Liu, Jun Liu, Georg Pflug, Chris Rogers, Steve Ross, Walter Schachermayer,}

Jianfeng Zhang, and an anonymous referee for their helpful comments. We are grateful for the support from the Sloan Research Fellowship program and from SWUFE IFS. All remaining errors are our own responsibility.

* _{Corresponding author. Fax: +1 (314) 935 6359.}

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a riskless asset) and Kihlstrom, Romer, and Williams[10]and Ross[19](for models without a riskless asset). However, the trade-off is less clear in portfolio problems with many risky assets, as pointed out by Hart[8]. Assuming a complete market with many states (and therefore many assets), we show that a less risk-averse (in the sense of Arrow and Pratt) agent’s portfolio payoff is distributed as the payoff for the more risk-averse agent, plus a non-negative random variable (extra return), plus conditional-mean-zero noise (risk). Therefore, the general complete-markets portfolio problem, which may not be a mean-variance problem, still trades off risk and return.

If either agent has nonincreasing absolute risk aversion, then the non-negative random variable (extra return) can be chosen to be a constant. We also give a counter-example that shows that in general, the non-negative random variable cannot be chosen to be a constant. In this case, the less risk averse agent’s payoff can also have a higher mean and a lower variance than the more risk averse agent’s payoff. We further prove a converse theorem. Suppose there are two agents, such that in all complete markets, the first agent chooses a payoff that is distributed as the second’s payoff, plus a non-negative random variable, plus conditional-mean-zero noise. Then the first agent is less risk averse than the other agent.

Our main result applies directly in a multiple-period setting with consumption only at a termi-nal date, and perhaps dynamic trading is the most natural motivation for the completeness we are assuming. Our main result can also be extended to a multiple-period model with consumption at many dates, but this is more subtle. Consumption at each date may not be ordered when risk aversion changes, due to shifts in the timing of consumption. However, for agents with the same pure rate of time preference, we show there is a mixture of the payoff distributions across periods that preserves the single-period result.

Our main result also extends to some special settings with incomplete markets, for example, a two-asset world with a risk-free asset. The proof is in two parts. The first part is the standard result: decreasing the risk aversion increases the portfolio allocation to the asset with higher return. The second part shows that the portfolio payoff for the higher allocation is distributed as the other payoff plus a constant plus conditional-mean-zero noise. However, for a two-asset world without a risk-free asset, both parts of the proof fail in general and we have a counter-example. Therefore, our result is not true in general with incomplete markets. We further provide sufficient conditions under which our results still hold in a two-risky-asset world using Ross’s stronger measure of risk aversion. Each result from two assets can be re-interpreted as applying to parallel settings with two-fund separation identifying the two funds with the two assets.

The proofs in the paper make extensive use of results from stochastic dominance, portfolio choice, and Arrow–Pratt and Ross[19]risk aversion. One contribution of the paper is to show how these concepts relate to each other. We use general versions of the stochastic dominance results forL1random variables1and monotone concave preferences, following Strassen[22]and Ross[17]. To see why our results are related to stochastic dominance, note that if the first agent’s payoff equals the second agent’s payoff plus a non-negative random variable plus conditional-mean-zero noise, this is equivalent to saying that negative the first agent’s payoff is monotone-concave dominated by negative the second agent’s payoff.

Section2introduces the model setup and provides some preliminary results, Section3 de-rives the main results. Section4discusses the case with incomplete markets. Section5extends

1 _{We assume that the consumptions have unbounded distributions instead of compact support (e.g., Rothschild and} Stiglitz[20]). Compact support for consumption is not a happy assumption in finance because it is violated by most of our leading models. Unfortunately, as noted by Rothschild and Stiglitz[21], the integral condition is not available in our general setting.

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the main results in a multiple-period model. Section6 illustrates the main results using some examples and Section7concludes.

2. Model setup and some standard results

We want to work in a fairly general setting with complete markets and strictly concave increas-ing von Neumann–Morgenstern preferences. There are two agentsAandBwith von Neumann– Morgenstern utility functionsUA(c)andUB(c), respectively. We assume thatUA(c)andUB(c)

are of classC2,U_A(c) >0,U_B(c) >0,U_A(c) <0 andU_B(c) <0. Each agent’s problem has the form:

Problem 1.Choose random consumptionc˜to maxEUi(c)˜

,

s.t. E[ ˜ρc˜] =w0. (1)

InProblem 1,i=AorBindexes the agent,w0is initial wealth (which is the same for both

agents), andρ >˜ 0 is the state price density. We will assume thatρ˜ is in the classP for which both agents have optimal random consumptions with finite means, denotedc˜Aandc˜B.

The first order condition is

U_i(c˜i)=λiρ,˜ (2)

i.e., the marginal utility is proportional to the state price densityρ˜. We have

˜

ci=Ii(λiρ),˜ (3)

whereIiis the inverse function ofU_i(·). By continuity and negativity of the second order

deriva-tiveU_i(·),c˜i is a decreasing function ofρ˜.

Our main result will be thatc˜A∼ ˜cB+ ˜z+ ˜ε, where “∼” denotes “is distributed as”,z˜0,

andE[˜ε|cB+z] =0.2We firstly review and give the proofs inAppendix Aof some standard

results in the form needed for the proofs of our main results.

Lemma 1.IfBis weakly more risk averse thanA(∀c,−UB(c) U_B(c)−

U_A(c) U_A(c)), then

1. for any solution to(2)(which may not satisfy the budget constraint(1)), there exists some critical consumption levelc∗(can be±∞)such thatc˜Ac˜B whenc˜Bc∗, and such that

˜

cAc˜Bwhenc˜Bc∗;

2. assuming c˜A and c˜B have finite means, andAand B have equal initial wealthsw0, then

E[˜cA]E[˜cB]_Ew0_{[ ˜}_ρ_]. Note that_Ew0_{[ ˜}_ρ_]is the payoff to a riskless investment ofw0.

The first result inLemma 1implies that the consumptions function of the less risk averse agent crosses that of the more risk averse agent at most once and from above. This single-crossing result is due to Pratt [15], expressed in a slightly different way.Lemma 1gives us a sense in which decreasing the agent’s risk aversion takes us further from the riskless asset. In fact, we can

2 _{Throughout this paper, the letters with “tilde” denote random variables, and the corresponding letters without “tilde”} denote particular values of these variables.

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obtain a more explicit description (our main result) of how decreasing the agent’s risk aversion changes the optimal portfolio choice. The description and proof are both related to monotone concave stochastic dominance.3 The following theorem gives a distributional characterization of stochastic dominance for all monotone and concave functions of one random variable over another. The form of this result is from Ross[17]and is a special case of a result of Strassen

[22]which generalizes a traditional result for bounded random variables to possibly unbounded random variables with finite means.

Theorem 1(Monotone concave stochastic dominance). (See Strassen[22]and Ross[17].) LetX˜

andY˜ be two random variables defined inR1with finite means;thenE[V (X)˜ ]E[V (Y )˜ ], for all concave nondecreasing functionsV (·), i.e.,X˜ monotone-concave stochastically dominatesY˜, if and only ifY˜∼ ˜X− ˜Z+ ˜ε, whereZ˜0, andE[˜ε|X−Z] =0.

Rothschild and Stiglitz[20,21]popularized a similar characterization of stochastic dominance for all concave functions (which implies equal means) that is a special case of another result of Strassen’s.

Theorem 2(Concave stochastic dominance). (See Strassen[22], and Rothschild and Stiglitz

[20,21].) LetX˜andY˜be two random variables defined inR1with finite means;thenE[V (X)˜ ] E[V (Y )˜ ], for all concave functionsV (·), i.e.,X˜ concave stochastically dominatesY˜, if and only ifY˜ ∼ ˜X+ ˜ε, whereE[˜ε|X] =0.

Rothschild and Stiglitz[20]also offered an integral condition for concave stochastic domi-nance, which unfortunately does not generalize to all random variables with finite mean, as they note in Rothschild and Stiglitz[21].4

3. Main results

Suppose agentAwith utility functionUAand agentBwith utility functionUBhave identical

initial wealthw0 and solveProblem 1. Recall that we assume that UA(c) andUB(c) are of

classC2,U_A(c) >0,U_B(c) >0,U_A(c) <0 andU_B(c) <0. We have

Theorem 3. Assume a single-period setting with complete markets in which agentsA andB

solveProblem1. IfB is weakly more risk averse thanAin the sense of Arrow and Pratt(∀c,

−U_B(c) U_B(c) −

U_A(c)

U_A(c)), then for every ρ˜∈P, c˜A is distributed as c˜B+ ˜z+ ˜ε, where z˜0 and

E[˜ε|cB+z] =0. Furthermore, ifc˜A= ˜cB, neitherz˜norε˜is identically zero.

3 _{We avoid using the term “second order stochastic dominance” in this paper because different papers use different} definitions. In this paper, we follow unambiguous terminology from Ross[17]: (1) ifE[V (X)˜ ]E[V (Y )˜ ]for all nonde-creasing functions, thenX˜monotone stochastically dominatesY˜; (2) ifE[V (X)˜ ]E[V (Y )˜ ]for all concave functions, thenX˜concave stochastically dominatesY˜; (3) ifE[V (X)˜ ]E[V (Y )˜ ]for all concave nondecreasing functions, thenX˜ monotone-concave stochastically dominatesY.˜

4 _{The integration by parts used to prove the integral condition unfortunately includes a term at the lower endpoint} which needs not equal to zero in general. Therefore, the integral condition may not be sufficient or necessary condition for concave stochastic dominance under unbounded distribution. As noted by Rothschild and Stiglitz[21], the integral condition does not appear to have any natural analog in these more general cases. Ross[17]has a sufficient condition for the integral condition to be valid, but unfortunately it is hard to interpret.

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Proof. The first step of the proof5is to show that−˜cB monotone-concave stochastically

dom-inates −˜cA,i.e.,E[V (−˜cB)]E[V (−˜cA)]for any concave nondecreasing functionV (·). By Lemma 1,c˜A andc˜B are monotonely related and there is a critical valuec∗above whichc˜Ais

weakly larger and below whichc˜B is weakly larger. LetV(·)be any selection from the

sub-gradient correspondence∇V (·), thenV(·)is positive and nonincreasing and it is the derivative of V (·)whenever it exists. Recall from Rockafellar[16], the subgradient for concave6V (·)is

∇V (x1)≡ {s|(∀x), V (x)V (x1)+s(x−x1)}. By concavity ofV (·),∇V (x)is nonempty for

allx1. And ifx2> x1, thens2s1for alls2∈ ∇V (x2)ands1∈ ∇V (x1).

The definition of subgradient for concaveV (·)implies that

V (x+x)V (x)+V(x)x. (4) Lettingx= −˜cBandx= −˜cA+ ˜cB in(4), we have

V (−˜cA)−V (−˜cB)V(−˜cB)(−˜cA+ ˜cB). (5)

Ifc˜Bc∗, thenc˜Ac˜B (byLemma 1), andV(−˜cB)V(−c∗), while ifc˜Bc∗, thenc˜A ˜

cBandV(−˜cB)V(−c∗). In both cases, we always have(V(−˜cB)−V(−c∗))(c˜A− ˜cB)0.

Rewriting(5)and substituting in this inequality, we have

V (−˜cB)−V (−˜cA)V(−˜cB)(c˜A− ˜cB)V

−c∗(c˜A− ˜cB). (6)

SinceV (·)is nondecreasing andE[˜cA]E[˜cB](result 2 ofLemma 1), we have

EV (−˜cB)−V (−˜cA)

EV−c∗(c˜A− ˜cB)

=V−c∗E[˜cA] −E[˜cB]0. (7) Therefore, we have that −˜cB is preferred to−˜cA by all concave nondecreasingV (·), and by Theorem 1, this says that−˜cAis distributed as−˜cB− ˜z+ ˜ε, wherez˜0 andE[˜ε|−cB−z] =0.

This is exactly the same as saying that c˜A is distributed asc˜B+ ˜z+(−˜ε), wherez˜0 and

E[−˜ε|cB+z] =0. Relabel−˜εasε˜, and we have proven the first sentence of the theorem.

To prove the second sentence of the theorem, note that because c˜A andc˜B are monotonely

related,cÃis distributed the same asc˜B only ifcÃ= ˜cB. Therefore, ifcÃ= ˜cB, one or the other

of z˜ or ε˜ is not identically zero. Now, ifz˜ is identically zero, then ε˜ must not be identically zero, andc˜Ais distributed asc˜B+ ˜ε, by Jensen’s inequality, we haveE[UA(c˜A)] =E[UA(c˜B+ ˜

ε)] =E[E[UA(c˜B+ ˜ε)|˜cB]]< E[UA(E[˜cB|˜cB] +E[˜ε|˜cB])] =E[UA(c˜B)], which contradicts

the optimality ofc˜Afor agentA. Ifε˜is identically zero, thenz˜must not be, andc˜Ais distributed

asc˜B+ ˜z, wherez˜0 and is not identically zero. Therefore,c˜Astrictly monotone stochastically

dominates c˜B, contradicting optimality ofc˜B for agentB. This completes the proof that if c˜A

andc˜Bdo not have the same distribution, then neitherε˜norz˜is identically 0. 2

We now prove a converse result ofTheorem 3: if in all complete markets, one agent chooses a portfolio whose payoff is distributed as a second agent’s payoff plus a non-negative random variable plus conditional-mean-zero noise, then the first agent is less risk averse than the second. Specifically, we have

5 _{As noted in footnote}₄_{, the integral condition does not hold under unbounded distributions, so that a proof using} Lemma 1and the integral condition would be wrong. More specifically, becausec˜Aandc˜Bmight be unbounded, we

can-not get that−˜cBmonotone concave stochastically dominates−˜cAdirectly fromqc=−∞[F−cA(q)−F−cB(q)]dq0. 6 _{For convex}_{V (}_·_{), the inequality is reversed. We follow Rockafellar’s convention of calling the derivative} correspon-dence the subgradient in both cases.

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Theorem 4.If for all ρ˜∈P,E[˜cA]E[˜cB], thenB is weakly more risk averse thanA(∀c,

−UB(c) U_B(c)−

U_A(c)

U_A(c)). This implies a converse result ofTheorem3:if for allρ˜∈P,c˜Ais distributed

asc˜B+ ˜z+ ˜ε, wherez˜0andE[˜ε|cB+z] =0, thenBis weakly more risk averse thanA.

Proof. We prove this theorem by contradiction. IfB is not weakly more risk averse than A, then there exists a constantcˆ, such that−UB(c)ˆ

U_B(c)ˆ <− U_A(c)ˆ

U_A(c)ˆ . SinceUA andUB are of the class

ofC2, from the continuity of−Ui(c)

U_i(c), wherei=A, B, we get that there exists an intervalRA

containingcˆ,s.t.,∀c∈RA,−UB(c) U_B(c)<−

U_A(c)

U_A(c). We pick c1, c2∈RAwithc1< c2. Now from Lemma 5inAppendix A, there exist hypothetical agentsA1andB1, so thatUA1 agrees withUA

andUB1 agrees withUBon[c1, c2], butA1is everywhere strictly more risk averse thanB1(and

not just on[c1, c2]).

Fix any λB >0 and choose ρ˜ to be any random variable that takes on all the values on [U_B(c2)

λB , U_B(c1)

λB ]. Then, the correspondingc˜B solving the first order conditionU

B(c˜B)=λBρ˜

takes on all the values on[c1, c2]. BecauseUB<0, theF.O.C solution is also sufficient

(ex-pected utility exists becauseρ˜ andUB(c˜B)are bounded), c˜B solves the portfolio problem for

utility functionUB, state price densityρ˜and initial wealthw0=E[ ˜ρc˜B]. SinceUB1 =UB on

the support ofc˜B, lettingc˜B1 = ˜cB, thenc˜B1 solves the corresponding optimization forUB1 for

λB1=λB.

We now show that there existsλA1 such thatc˜A1≡IA1(λA1ρ)˜ satisfies the budget constraint

E[ ˜ρc˜A1] =w0. Due to the choice ofUA1,IA1(λA1ρ)˜ exists and is a bounded random variable

for all λA1. Letting ρ= U_B(c2) λB andρ = U_B(c1) λB (so, ρ˜∈ [ρ, ρ]), we define λ1= U_A 1(c1) ρ and λ2= U_A 1(c2) ρ , then we have c1=IA1(λ1ρ) > IA1(λ1ρ),˜ c2=IA1(λ2ρ) < IA1(λ2ρ).˜ (8)

The inequalities follow fromIA1(·)decreasing. From(8)andc1c˜Bc2, we have

EρI˜ A1(λ1ρ)˜ < E[ ˜ρc1]E[ ˜ρc˜B] =w0, EρI˜ A1(λ2ρ)˜ > E[ ˜ρc2]E[ ˜ρc˜B] =w0. (9)

Now,IA1(λρ)˜ is continuous from the assumption thatUA1(·)is in the class ofC

2_and_U A1<0.

By the intermediate value theorem, there existsλA1, such thatE[ ˜ρIA1(λA1ρ)˜ ] =w0,i.e.,c˜A1

satisfies the budget constraint forρ˜andw0.

From the second result ofLemma 6 inAppendix A, ifc˜A1 = ˜cB1, then we have thatc˜B1

has a wider range of support than that of c˜A1. Let the support of A1’s optimal consumption

be[c3, c4] ⊆ [c1, c2]. From the construction ofUA1,UA1=UA on the support ofc˜A1. Letting ˜

cA= ˜cA1, thenc˜Asolves the corresponding optimization forUAforλA=λA1.

Now, sinceB1is strictly less risk averse thanA1, fromTheorem 3,c˜B1∼ ˜cA1+ ˜z1+ ˜ε1, where ˜

z10 andE[˜ε1|cA1+z1] =0. Furthermore, ifc˜A1= ˜cB1, then neitherz˜1norε˜1is identically

zero. From the first result ofLemma 6inAppendix A, ifA1is strictly more risk averse thanB1,

thenc˜A1= ˜cB1. Thus, byTheorem 3, neitherz˜1norε˜1is identically zero. Therefore,E[˜cB1]>

E[˜cA1],i.e.E[˜cB]> E[˜cA], this contradicts the assumption that, for allρ˜∈P,E[˜cA]E[˜cB].

This therefore also contradicts the stronger condition: for allρ˜∈P,c˜Ais distributed asc˜B+ ˜z+ ˜

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Theorem 3 shows that if B is weakly more risk averse than A, then c˜A is distributed as ˜

cB plus a risk premium plus random noise. The distributions of the risk premium and the

noise term are typically not uniquely determined. Also, it is possible that the weakly less risk averse agent’s payoff can have a higher mean and a lower variance than the weakly more risk averse agent’s payoff as we will see in Example 6.2. This can happen because al-though adding condition-mean-zero noise always increases variance, adding the non-negative random variable decreases variance if it is sufficiently negatively correlated with the rest (since Var(c˜A)=Var(c˜B)+Var(ε)˜ +Var(z)˜ +2Cov(c˜B,z)˜ , ifCov(c˜B,z) <˜ −1₂(Var(z)˜ +Var(ε))˜ , then

Var(c˜A) <Var(c˜B)). This should not be too surprising, given that it is well known that in general

variance is not a good measure of risk7for von Neumann–Morgenstern utility functions,8and for general distributions in a complete market, mean-variance preferences are hard to justify.

Our second main result says that when either of the two agents has nonincreasing absolute risk aversion, we can choosez˜to be nonstochastic, in which casez=E[˜cA− ˜cB]. The basic idea

is as follows. If either agent has nonincreasing absolute risk aversion, then we can construct a new agentA∗whose consumption equals toA’s consumption plusE[˜cA− ˜cB]. We can therefore

get the distributional results for agentA∗andBsinceA∗is weakly less risk averse thanB.

Theorem 5.IfBis weakly more risk averse thanAand either of the two agents has nonincreas-ing absolute risk aversion, thenc˜Ais distributed asc˜B+z+ ˜ε, wherez=E[˜cA− ˜cB]0and

E[˜ε|cB+z] =0.

Proof. Define the utility functionUA∗(c)˜ =UA(c˜+E[˜cA− ˜cB]). In the case whenAhas

non-increasing absolute risk aversion, A∗ is weakly less risk averse than B because A is weakly less risk averse thanB and nonincreasing risk aversion ofAimplies thatA∗is weakly less risk averse than A. In the case when B has nonincreasing absolute risk aversion, B∗ with utility

UB∗=UB(c˜+E[˜cA− ˜cB])is weakly less risk averse thanBandA∗is weakly less risk averse

thanB∗. Therefore, in both cases, we have thatA∗is weakly less risk averse thanB.

Give agentA∗initial wealthwA∗=w0−E[ ˜ρ]E[˜cA− ˜cB], wherew0is the common initial

wealth of agentAandB.A∗’s problem is

max ˜ c EUA ˜ c+E[˜cA− ˜cB] , s.t. E[ ˜ρc˜] =wA∗. (10)

The first order conditions are related to the optimality ofc˜Afor agentA. To satisfy the budget

constraints, agentA∗will optimally holdc˜A−E[˜cA− ˜cB].

By Lemma 1,c˜A−E[˜cA− ˜cB]andc˜B are monotonely related and there is a critical value

c∗ above whichc˜A−E[˜cA− ˜cB]is weakly larger and below whichc˜B is weakly larger. This

implies that V(−˜cB)−V −c∗c˜A−E[˜cA− ˜cB] − ˜cB 0, (11)

7 _{See, for example Hanoch and Levy}_[9]_{, and the survey of Machina and Rothschild}_[14]_.

8 _{If von Neumann–Morgenstern utility functions are mean-variance preferences, then they have to be quadratic} util-ity functions, but quadratic preferences are not appealing because they are not increasing everywhere and they have increasing risk aversion where they are increasing. Also, Dybvig and Ingersoll[6]show that if markets are complete, mean-variance pricing of all assets implies there is arbitrage unless the payoff to the market portfolio is bounded above.

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whereV (·)is an arbitrary concave function and V(·) is any selection from the subgradient correspondence∇V (·). The concavity ofV (·)implies that

V−˜cA+E[˜cA− ˜cB] −V (−˜cB)V(−˜cB) −˜cA+E[˜cA− ˜cB] + ˜cB . (12)

(11) and (12)imply that

V (−˜cB)−V −˜cA+E[˜cA− ˜cB] V−c∗c˜A−E[˜cA− ˜cB] − ˜cB . (13) We have EV (−˜cB)−V −˜cA+E[˜cA− ˜cB] EV−c∗c˜A− ˜cB−E[˜cA− ˜cB] =0. (14) Therefore, for any concave functionV (·), we have

EV (−˜cB)

EV−˜cA+E[˜cA− ˜cB]

. (15)

ByTheorem 2, this says that−˜cA+E[˜cA− ˜cB]is distributed as−˜cB+ ˜ε, whereE[˜ε| −cB] =0.

This is exactly the same as saying thatc˜A−E[˜cA− ˜cB] is distributed asc˜B+(−˜ε), where

E[−˜ε|cB] =0. Relabel−˜εasε˜, and we have

˜

cA−E[˜cA− ˜cB] ∼ ˜cB+ ˜ε, i.e., c˜A∼ ˜cB+E[˜cA− ˜cB] + ˜ε, (16)

whereE[˜ε|cB+z] =0. 2

The nonincreasing absolute risk aversion condition is sufficient but not necessary. A quadratic utility function has increasing absolute risk aversion. But, as illustrated byExample 6.1, the non-negative random variable can still be chosen to be a constant for quadratic utility functions (which can be viewed as an implication of two-fund separation andTheorem 7). If the non-negative random variable can be chosen to be a constant, then we have the following corollary:

Corollary 1.IfBis weakly more risk averse thanAand either of the two agents has nonincreas-ing absolute risk aversion, then Var(c˜A)Var(c˜B).

Proof. From Theorem 5, the non-negative random variable z˜ can be chosen to be the con-stantE[˜cA− ˜cB]. Then we haveE(ε˜|˜cB)=0, which implies thatCov(ε,˜ c˜B)=0. Therefore,

Var(c˜A)=Var(c˜B)+Var(ε)˜ +2Cov(c˜B,ε)˜ =Var(c˜B)+Var(ε)˜ Var(c˜B). 2

4. Possibly incomplete market case

Our result still holds in a two-asset world with a risk-free asset. For a two-asset world without a risk-free asset, we have a counter-example to our result holding. Therefore, our main result does not hold in general with incomplete markets. However, our result holds in a two-risky-asset world if we make enough assumptions about asset payoffs and the risk-aversion measure. Also, each two-asset result has a natural analog for models with many assets and two-fund separation, since the portfolio payoffs will be the same as in a two-asset model in which only the two funds are traded.9Note that while this section is intended to ask to what extent our results can be extended to incomplete markets, the results also apply to complete markets with two-fund separation.

9 _{See Cass and Stiglitz}_[3]_{and Ross}_[18]_{for characterizations of two-fund separation,}_i.e._{, for portfolio choice to be} equivalent to choice between two mutual funds of assets.

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First, we show that our main result still holds in a two-asset world with a risk-free asset. The proof is in two parts. The first part is the standard result: decreasing the risk aversion increases the portfolio allocation to the asset with higher return. The second part shows that the portfolio payoff for the higher allocation is distributed as the other payoff plus a constant plus conditional-mean-zero noise. To show the second part, we use the following lemma:

Lemma 2.

1. IfE[ ˜q] =0and0m1m2, thenm2q˜∼m1q˜+ ˜ε, whereE[˜ε|m1q] =0.

2. LetE[ ˜χ]be finite,E[ ˜q|χ]0, and0m1m2. Thenχ˜+m2q˜∼ ˜χ+m1q˜+ ˜z+ ˜ε, where ˜

z=(m2−m1)E[ ˜q|χ]0andE[˜ε|χ+m1q+z] =0.

Proof. We prove 2, and 1 follows immediately by settingχ˜=0 andE[ ˜q] =0. Letz˜0≡E[ ˜q|χ]

and z˜≡(m2−m1)z˜0. By Theorem 2, we only need to show that, for any concave function

V (·),E[V (χ˜ +m2q)˜ ]E[V (χ˜ +m1q˜+ ˜z)]. FixV (·)and letV(·)be any selection from its

subgradient correspondence ∇V (·)(soV(·)is the derivative ofV (·)whenever it exists). The concavity ofV (·)and the definitions ofz˜0andz˜imply that

V (χ˜+m2q)˜ −V (χ˜+m1q˜+ ˜z)V(χ˜+m1q˜+ ˜z)(m2−m1)(q˜− ˜z0). (17)

Furthermore,V(·)nonincreasing,m2m10, and the definitions ofz˜0andz˜imply

V(χ˜+m1q˜+ ˜z)−V(χ˜+m2˜z0)

(m2−m1)(q˜− ˜z0)0. (18)

From(17), (18), and the definitions ofz˜0andz˜, we get

EV (χ˜+m2q)˜ −EV (χ˜+m1q˜+ ˜z) EV(χ˜+m1q˜+ ˜z)(m2−m1)(q˜− ˜z0) EV(χ˜+m2z˜0)(m2−m1)(q˜− ˜z0) =EEV(χ˜+m2˜z0)(m2−m1)(q˜− ˜z0)|χ =0. 2

Now, we consider the following portfolio choice problem:

Problem 2(Possibly incomplete market with two assets).Agenti’s (i=A, B) problem is max

αi∈R

EUi(w0x˜+αiw0v)˜

,

where w0is the initial wealth, αi is the proportion invested in the second asset, andv˜ is the

excess of the return on the second asset over the first asset,i.e.,v˜= ˜y− ˜x, wherex˜andy˜are the total returns on the two assets. We assume thatE[˜v]0,v˜ is nonconstant, andE[˜v]andE[ ˜x]

are finite.

Note.Problem 2is stated in terms of two possibly risky assets, but it is also formally equivalent to a problem with non-traded wealth. Suppose an agent has non-traded wealthN˜ and allocates financial wealth f0>0 between assets payingχ˜1andχ˜2per dollar invested. Then the choice

problem is max αi∈R EUi _˜ N+f0χ˜1+αif0(χ˜2− ˜χ1)

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We denote agentAandB’s respective optimal investments in the risky asset with payoffy˜

byα∗_A andα_B∗. The payoff for agent Aisc˜A=w0x˜+αA∗w0v˜ and agent B’s payoff is c˜B=

w0x˜+α∗Bw0v˜. We maintain the utility assumptions made earlier:Ui(·) >0 andUi(·) <0, sov˜

nonconstant implies thatα∗_Aandα_B∗ are unique if they exist. We have the following well-known result.

Lemma 3.Supposex˜is riskless(x˜nonstochastic), ifBis weakly more risk averse thanA, then the agents’ solutions toProblem2satisfyα∗_Aα_B∗.

Proof. The first order condition ofA’s problem is:

EU_Axw0+α∗Aw0v˜

w0v˜

=0. (19)

The analogous expression forBisϕ(α_B∗)=0, where

ϕ(α)≡EU_B(xw0+αw0v)w˜ 0v˜

. (20)

SinceUB(·)=G(UA(·)), whereG(·) >0 andG(·)0, we have:

ϕα_A∗=EGUA xw0+α_A∗w0v˜ U_Axw0+α∗_Aw0v˜ w0v˜ =GUA(xw0) EU_Axw0+α∗Aw0v˜ w0v˜ +EGUA xw0+α∗Aw0v˜ −GUA(xw0) U_Axw0+αA∗w0v˜ w0v˜ 0, (21)

where the first term in(21)is zero by(19)and the expression inside the expectation in the second term is non-positive becauseG(·)0 andU_A(·) >0. Finally, the concavity ofUB(·)implies

thatϕ(·)is decreasing, and therefore from(20) and (21), we must haveα∗_Aα_B∗. 2

Lemma 3implies that decreasing the risk aversion increases the portfolio allocation to the asset with higher return. Now, we show that our main result still holds in a two-asset world with a risk-free asset. We have

Theorem 6(Two-asset world with a riskless asset). Consider the two-asset world with a riskless asset(x˜nonstochastic)ofProblem2, ifBis weakly more risk averse thanAin the sense of Arrow and Pratt, thenc˜Ais distributed asc˜B+z+ ˜ε, wherez=E[˜cA− ˜cB]0andE[˜ε|cB+z] =0.

Proof. When the first asset inProblem 2is riskless, then we havec˜A−E[˜cA] =α∗Aw0(y˜−E[ ˜y])

andc˜B−E[˜cB] =αB∗w0(y˜−E[ ˜y]). FromLemma 3,α∗AαB∗. Letq˜≡ ˜y−E[ ˜y],m1≡αB∗w0

andm2≡α_A∗w0 in the first part ofLemma 2, we havec˜A−E[˜cA] ∼ ˜cB−E[˜cB] + ˜ε, which

implies thatc˜Ais distributed asc˜B+z+ ˜ε, wherez=E[˜cA− ˜cB]0 andE[˜ε|cB+z] =0. 2 Theorem 6generalizes in obvious ways to settings with two-fund separation since optimal consumption is the same as it would be with ordering the two funds as assets. The main re-quirement is that one of the funds can be chosen to be riskless, for example, in a mean-variance world with a riskless asset and normal returns for risky assets.10In this example, ifB is weakly

10 _{This example is a special case of two-fund separation in mean-variance worlds or the separating distributions of} Ross[18].

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more risk averse than A, Theorem 6 tells us thatc˜A∼ ˜cB+z+ ˜ε, where z0 is constant

andE[˜ε|cB+z] =0. We know thatA’s optimal portfolio is further up the frontier thanB’s,

i.e.,E[˜cA]E[˜cB]andVar[˜cA]Var[˜cB]. This result is verified by noting that we can choose

z=E[˜cA− ˜cB],ε˜∼N (0,Var[˜cA] −Var[˜cB]), andε˜is drawn independently ofc˜B.

Now, we examine the case with two risky assets inProblem 2. For a two-asset world without a riskless asset, we have a counter-example to our result holding. In the counter-example,α∗_A> α_B∗, but the distributional result does not hold.

Example 4.1.We assume that there are two risky assets and four states. The probabilities for the four states are 0.2, 0.3, 0.3 and 0.2 respectively. The payoff ofx˜ is(10 8 1 1)T and the net payoffv˜is(−1 1 1 −1)T. Agent’s utility function isUi(w˜i)= −e−δiw˜i, wherei=A, B, and

˜

wi is agenti’s terminal wealth. We assume that agentBis weakly more risk averse thanAwith

δA=1 andδB=1.5. The agents solveProblem 2with initial wealthw0=1.

The agents’ problems are: max αA 0.2e−(10−αA)+₀_.₃_e−(8+αA)+₀_.₃_e−(1+αA)+₀_.₂_e−(1−αA)_, and max αB 0.2e−1.5(10−αB)+₀_.₃_e−1.5(8+αB)+₀_.₃_e−1.5(1+αB)+₀_.₂_e−1.5(1−αB)_. First order conditions give α_A∗ = 1₂log(3+3e−7

2+2e−9)=0.2, andα∗B= 1 3log(

3+3e−10.5

2+2e−13.5)=0.135.

Therefore, agent A’s portfolio payoff is(9.8 8.2 1.2 0.8)T and agentB’s portfolio payoff is

(9.865 8.135 1.135 0.865)T. If agentA’s payoffc˜A∼ ˜cB+ ˜z+ ˜ε, whereE[˜ε|cB+z] =0, then we

havePr(ε˜0|cB+z) >0, therefore, we have maxc˜Amaxc˜B. However, in this example, we

can see that maxc˜A=9.8 and maxc˜B=9.865,i.e., maxc˜A<maxc˜B. Contradiction! Therefore,

in general, our result does not hold in a two-asset world without a riskless asset.

It is a natural question to ask whether our main result holds in a two-risky-asset world if we make enough assumptions about asset payoffs. We can, if we use Ross’s stronger measure of risk aversion (see Ross[18]) and his payoff distributional condition. We have

Theorem 7(Two risky assets with Ross’s measure). Consider the two-risky-asset world of Prob-lem2withE[˜v|x]0for allx. IfB is weakly more risk averse thanAunder Ross’s stronger measure of risk aversion, thenc˜Ais distributed asc˜B+ ˜z+ ˜ε, whereE[˜ε|cB+z] =0, andz˜0.

Proof. Our proof is in two parts. The first part is from Ross[19]: if agentAis weakly less risk averse than B under Ross’s stronger measure, thenα∗_Aα∗_B. The first order condition ofA’s problem is EU_Aw0x˜+αA∗w0v˜ w0v˜ =0. (22)

The analogous expression forBisϕ(α∗_B)=0, where

ϕα_B∗≡EU_Bw0x˜+αB∗w0v˜

w0v˜

. (23)

From Ross[19], ifB is weakly more risk averse thanAunder Ross’s stronger measure, then there existλ >0 and a concave decreasing function G(·), such that UB(·)=λUA(·)+G(·).

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Therefore, ϕα_A∗=EλU_Aw0x˜+αA∗w0v˜ +Gw0x˜+α∗Aw0v˜ w0v˜ =EGw0x˜+αA∗w0v˜ w0v˜ =EEGw0x˜+αA∗w0v˜ w0v˜|x 0, (24)

where the last inequality is a consequence of the fact thatG(·)is negative and decreasing while

E[˜v|x]0. The concavity ofUB(·)implies thatϕ(·)is decreasing. Therefore, from(22) and (24), we haveα_A∗ α∗_B.

The second part shows that the portfolio payoff for the higher allocation is distributed as the other payoff plus a constant plus conditional-mean-zero noise. Letting q˜ ≡ ˜v,χ˜ ≡w0x˜,

m1≡αB∗w0andm2≡αA∗w0inLemma 2, part 2, we havew0x˜+α∗Aw0v˜∼w0x˜+αB∗w0v˜+ ˜z+ ˜ε,

i.e.,c˜A∼ ˜cB+ ˜z+ ˜ε, wherez˜=w0(αA∗−αB∗)E[˜v|x]0 andE[˜ε|cB+z] =0. 2

Theorem 7implies that our main result holds when we use Ross’s stronger measure of risk aversion with the assumption ofE[˜v|x]0. If the conditionE[˜v|x]0 is not satisfied, then our main result may not hold even when we use Ross’s stronger measure of risk aversion as we can see in the following example.

Example 4.2.We assume that there are two risky assets and four states. The probabilities for the four states are 0.3, 0.2, 0.3 and 0.2 respectively. The payoff ofx˜is(10 8 1 1)T and the net payoff

˜

vis(−1 1 1 −1)T. AgentA’s utility function isUA(w˜A)=e6w˜A−e− ˜wA, and agentB’s utility

function isUB(w˜B)= ˜wB−e6−1.5w˜B, wherew˜i is the terminal wealth of agenti. The agents

solveProblem 2with initial wealthw0=1. We have

U_B(w) U_A(w)= 2.25e6−1.5w e−w =2.25e 6₋0.5w_, UB(w) U_A(w)= 1+1.5e6−1.5w e6₊_e−w . Therefore, infw U_B(w) U_A(w)>supw U_B(w)

U_A(w), for any 0w10, which implies that agentBis strictly

more risk aversion than agentAunder Ross’s stronger measure of risk aversion. The agents’ problems are:

max αA 0.3e6(10−αA)−e−(10−αA) +0.2e6(8+αA)−e−(8+αA) +0.3e6(1+αA)−e−(1+αA) +0.2e6(1−αA)−e−(1−αA) , and max αB 0.310−αB−e6−1.5(10−αB) +0.28+αB−e6−1.5(8+αB) +0.31+αB−e6−1.5(1+αB) +0.21−αB−e6−1.5(1−αB) .

From the first order condition,e2α∗A=3+2e− 7 2+3e−9,i.e.,αA∗ = 1 2log( 3+2e−7 2+3e−9)=0.2029, and e 3α∗_B ₌ 3e−1.5₊_2e−12 2e−1.5₊_3e−15,i.e.,α∗B= 1 3log( 3e−1.5₊_2e−12

2e−1.5₊_3e−15)=0.1352. Therefore, agentA’s portfolio payoff is

(9.7971 8.2029 1.2029 0.7971)T

and agentB’s portfolio payoff is

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If agentA’s payoffc˜A∼ ˜cB+ ˜z+ ˜ε, whereE[˜ε|cB+z] =0, then we havePr(ε˜0|cB+z) >0.

Therefore, we have maxc˜Amaxc˜B. However, in this example, we can see that maxc˜A =

9.7971 and maxc˜B=9.8648,i.e., maxc˜A<maxc˜B. Contradiction! Therefore, in a

two-risky-asset world, our main result does not hold in general even under Ross’s stronger measure of risk aversion if we don’t make the assumption thatE[˜v|x]0. 2

An alternative to the approach following Ross[19]is the approach of Kihlstrom, Romer and Williams[10]for handling random base wealth. They show that the Arrow–Pratt measure works if we restrict attention to comparisons in which (1) at least one of the utility functions has non-increasing absolute risk aversion and (2) base wealth is independent of the other gambles. Here is how their argument works. The independence implies that we can convert a problem with random base wealth x to a problem with nonrandom base wealth by using the indirect utility functionsUˆi(w)≡E[Ui(x˜+w)], and our results for nonrandom base wealth apply directly. For

this to work, the indirect utility functionsUˆAandUˆBmust inherit the risk aversion ordering from

UA andUB, which as they point out, does not happen in general. However, lettingF (·)be the

distribution function ofx˜, simple calculations tell us that provided integrals exist, we can write

−Uˆi(w) ˆ U_i(w) = _U i(x˜+w) U_i(y˜+w) dF (y˜+w) −Ui(x˜+w) U_i(x˜+w) dF (x).˜ (25)

For both agents, the risk aversion of the indirect utility function is therefore a weighted average of the risk aversion of the direct utility function, but the weights are different so the risk aversion ordering is not preserved in general (since the more risk averse agent may have relatively higher weights from wealth regions where both agents have small risk aversion). However, we do know that the more risk averse agents’ weights put relatively higher weight on lower wealth levels (since i’s absolute risk aversion is −dlog(U_i(w)/dw)), so if either agent has nonincreasing absolute risk aversion, then the risk aversion ordering of the direct utility function is inherited by the indirect utility function. Subject to existence of some integrals (ensured by compactness in their paper), their results and ourTheorem 6imply that ifB is weakly more risk averse thanA, at least one ofUAandUBhas nonincreasing absolute risk aversion, andv˜ is independent ofx˜,

then our main result holds:c˜A∼ ˜cB+ ˜z+ ˜ε, wherez˜0 andE[˜ε|cB+z] =0.

As we have shown that our main result does not hold in general in the traditional type of incomplete markets where portfolio payoffs are restricted to a subspace. In addition, as we will see in Example 6.6, our main result does not hold in incomplete markets where agents have non-traded asset. However, it is an open question whether the results extend to more interesting models of incomplete markets in which there is a reason for the incompleteness. For example, a market that is complete over states distinguished by security returns and incomplete over other private states with an insurance need (see Chen and Dybvig[5]). Another type of incompleteness comes from a non-negative wealth constraint (which is an imperfect solution to information problems when investors have private information or choices related to default), which means agents have individual incompleteness and cannot fully hedge future non-traded wealth or else they would violate the non-negative wealth constraint (see Dybvig and Liu[7]).

5. Extension to a multiple-period model

There is a trivial extension of our analysis to multiple-period models with complete markets and consumption only at the end, and indeed this is the normal motivation forProblem 1. See,

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e.g., Cox and Huang[2]to see how to restate a continuous-time model without consumption withdrawal in the form ofProblem 1. This section is concerned with a less trivial extension to consumption at multiple dates. In these models, the result ofTheorem 3typically does not hold at every date because changing risk aversion in effect changes time preference as well, and the resulting shift of consumption over time means that one agent consumes more on average at some dates and the other agent consumes more on average at other dates. However, we can still show that a form ofTheorem 3holds, for a carefully chosen mixture across dates of the distribution of consumption.

We now examine our main results in a multiple-period model. We assume that each agent’s problem is: Problem 3. max ˜ ct E _T t₌1 DtUi(c˜t) , s.t. E _T t=1 ˜ ρtc˜t =w0, (26)

wherei=Aor B indexes the agent,Dt>0 is a utility discount factor (e.g.,Dt=e−κt if the

pure rate of time discountκ is constant), andρ˜t is the state price density in periodt.

Again, we will assume that both agents have optimal random consumptions, denotedc˜At and ˜

cBt, and bothc˜At andc˜Bthave finite means. We also assume that the utility discount factors are

the same for both agents. The first order condition gives us

U_i(c˜it)=λi ˜ ρt Dt , i=A, B, we have ˜ cit =Ii λi ˜ ρt Dt ,

whereIi(·)is the inverse function ofU_i(·). By negativity of the second order derivatives,c˜itis a

decreasing function ofρ˜t.

By similar arguments in the one-period model, we have

Lemma 4.IfBis weakly more risk averse thanA, then

1. for eacht, there exists some critical consumption levelc_t∗(can be±∞)such thatc˜At c˜Bt

whenc˜Btct∗, and such thatc˜Atc˜Bt whenc˜Btc∗t;

2. if it happens that the budget shares as a function of time are the same for both agents at some timet, i.e.,E[ ˜ρtc˜At] =E[ ˜ρtc˜Bt], thenE[˜cAt]E[˜cBt], and we havec˜At ∼ ˜cBt+ ˜zt+ ˜εt,

where z˜t 0 and E[˜εt|cBt+zt] =0. And if c˜At = ˜cBt, then neither z˜t nor ε˜t is

identi-cally zero. In particular, if the budget shares are the same for allt, then this distributional condition holds for allt.

The proof ofLemma 4 is essentially the same as the proof of the corresponding parts of

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or the same for the two agents without any restriction on budget shares, then the distributional condition may not hold in any period. For example, if the weakly more risk averse agent B

spends most of the money earlier but the weakly less risk averse agentAspends more later, then the mean payoff could be higher in an earlier period for the weakly more risk averse agent,i.e.,

E[˜cBt]E[˜cAt]. The following example shows that the result ofTheorem 3typically does not

hold at every date because changing risk aversion in effect changes time preference as well.

Example 5.1.An investor choosesc˜t to maximize

E ∞ t=0 e−βtU (c˜t) dt s.t. w0=E ∞ t=0 ˜ ρtc˜tdt .

Investors’ utility functionU (c˜t)=c˜ 1−γ t

1−γ, whereγ is investors’ risk-aversion. The state price

den-sity is ρ˜t =e−(r+ 1

2κ2)t−κZt_{, where} _{κ >}_{0 is the Sharpe ratio,} _r _{is risk-free rate, and} _Z t is a

standard Wiener process. This is consistent with Merton’s model with constant means of returns. A discrete model with finite horizon, more exactly in the form of(26), can also be used to show the same result, but with messier algebra.

The first order condition gives use−βtc−_t γ =λρt,i.e.,c˜t=(λeβtρt)− 1

γ_{. Substituting}_c˜ t into

the budget constraint, we get

w0=E ∞ 0 e−(1−1γ)(r+ 1 2κ2)t−(1− 1 γ)κZt− β γt_λ− 1 γ _dt .

We getλ=w−₀γ((r+_2γκ2)(1−_γ1)+β_γ)−γ, and therefore

˜ ct=w0 r+ κ 2 2γ 1− 1 γ + β γ e −β γt+(r+12κ2)γt+κγZt_. Therefore, we haveE[˜ct]increases inγif and only if

t β−κ 2 γ − r+1 2κ 2 r+κ 2 2γ 1− 1 γ + β γ > β+ κ2 2 − κ2 γ −r.

If β < r−κ₂2 +κ_γ2,11 then there existsε >0 such thatE[˜ct]increases in γ for allt ∈ [0, ε].

Therefore, when there is no restriction on budget shares, the distributional result (ofTheorem 3) does not hold fort∈ [0, ε].12

Although our main result does not hold period-by-period, it holds “on average” for a mixture of the distributions across periods (with mixing weights proportional to the two agents’ shared utility discount factors). Thus, there is a sense in which the overall distribution of payoffs, across time as well as states of nature, represents a trade-off between risk and return, even though no such relationship exists in general period-by-period.

11 _{For the choice problem to have a solution, we also need to impose Merton’s condition}_{β > (r}₊κ2

2γ)(1−γ ), which

is consistent with this restriction.

12 _{This example can also be used to show that the wealth processes are not ordered, disproving another possible} conjec-ture.

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Theorem 8.In a multiple-period model, assume agentsAandBhave the same discount factors

Dt and solve Problem 2. Let FAt (resp. FBt) be the cumulative distribution function of c˜At

(resp.c˜Bt). Given any vectorμ=(μ1, . . . , μT)of positive weights summing to one, choosec˜A

(resp.c˜B)to be a random variable with cumulative distribution functionFA(c)=

T

t=1μtFAt(c)

(resp.FB(c)=

T

t=1μtFBt(c)). Then there exists a vectorμof weights summing to one such

that our earlier results hold “on average”, i.e.c˜Aandc˜Bsatisfy

1. ifBis weakly more risk averse thanA, then,c˜A∼ ˜cB+ ˜z+ ˜ε, wherez˜0,E[˜ε|cB+z] =0;

2. if B is weakly more risk averse than A and either of the two agents has nonincreasing absolute risk aversion, thenc˜Ais distributed asc˜B+ ˜z+ ˜ε, wherez=E[˜cA− ˜cB]0and

E[˜ε|cB+z] =0.

Proof. Suppose the original probability space has probability measure P over states Ω with filtration {Ft}. We define the discrete random variable τ on associated probability space

(Ω∗,F∗, P∗)so thatP∗(τ =t )=μt ≡Dt/(

T

t=1Dt). We then define a single-period

prob-lem on a new probability space (Ω,ˆ Fˆ,P )ˆ . Define the state of nature in the product space

(t, ω)∈ ˆΩ ≡Ω∗×Ω witht and ω drawn independently. Let Fˆ be the optional σ-algebra, which is the completion ofF∗×Fτ. The synthetic probability measure is the one consistent

with independence generated from P (fˆ ∗, f )=P∗(f∗)×P (f ) for all subsetsf∗∈F∗ and subsetsf ∈Fτ.

The synthetic probability measure assigns a probability measure that looks like a mixture model, drawing time first assigning probabilityμt to timet, and then drawing fromρ˜t using its

distribution in the original problem.

Recall that under the original probability measure, each agent’s problem is given in(26). We choosec˜A(resp.c˜B) to be a random variable with cumulative distribution functionFA(c)=

T

t=1μtFAt(c)(resp.FB(c)=

T

t=1μtFBt(c)). Now we want to write down an equivalent

prob-lem, in terms of the choice of distribution of eachc˜t, but with the new synthetic probability

measure. The consumptionc˜under the new probability space over which synthetic probabilities are defined is a function ofρ˜andt; we identifyc(˜ ρ, t)˜ with what used to bec˜t(ρ)˜ . To write the

objective function in terms of the synthetic probabilities, we can write

E _T t=1 DtU (c˜t) = T t=1 DtE U (c˜t) = T t=1 _T s=1 Ds μtEˆ U (c)˜ |t = _T s₌1 Ds _T t₌1 μtEˆ U (c)˜ |t= _T s₌1 Ds ˆ EU (c)˜ , (27) whereEˆdenotes the expectation under the synthetic probability.T_s₌₁Dsis a positive constant,

so the objective function is equivalent to maximizingEˆ[U (c)˜ ].

Now, we can write the budget constraint in terms of the synthetic probabilities,

w0=E _T t=1 ˜ ρtc˜t = T t=1 μtE _˜ ρt μt˜ ct = T t=1 μtEˆ _˜ ρ μc˜|t = ˆE _˜ ρ μc˜ . (28)

Then we can apply our single-period results (Theorem 3, 4 and 5) and the formula for the distribution of the mixture of thec˜it’s to derive that our main results hold on a mixture model of

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Therefore, if (as we expect) the budget shares are not the same for both agents at each time periodt, then the distributional result may not hold period-by-period in a multiple-period model with time-separable von Neumann–Morgenstern utility having identical weights over time. How-ever, Theorem 8 implies that our main results still hold for a weighted average distribution function in a multiple-period model. These results retain the spirit of our main results while acknowledging that changing risk aversion may cause consumption to shift over time.

6. Examples

InExample 6.1, we illustrate our main result with specific distribution ofc˜A,c˜Bandε˜. In this

example, the non-negative random variablez˜can be chosen to be a constant, and therefore from

Corollary 1in Section3, the variance of the less risk averse agent’s payoff is higher.

Example 6.1.Bis weakly more risk averse thanA,AandBhave the same initial wealthw0=12

and the utility functions are as follows

UA(c)˜ = − 1 2(40− ˜c) 2_, _U B(c)˜ = − 1 2(26− ˜c) 2_,

wherec <˜ 40 for agent A, andc <˜ 26 for agent B. We assume that the state price densityρ˜

is uniformly distributed in[2₃,4₃]. The first order conditions give usc˜A=40−λAρ˜, andc˜B=

26−λBρ˜. Because E[ ˜ρ] =1 andE[ ˜ρ2] =28₂₇, the budget constraint E[ ˜ρc˜i] =12,i=A, B,

implies that λA=27 andλB=27₂. Therefore,c˜A is uniformly distributed in[4,22]andc˜B is

uniformly distributed in[8,17]. We haveE[˜cA] −E[˜cB] =1₂. Letε˜have a Bernoulli distribution

drawn independently ofc˜B with two equally possible outcomes 9₂ and−9₂. It is not difficult to

see thatc˜Ais distributed asc˜B+ ˜z+ ˜ε, wherez˜=E[˜cA] −E[˜cB] =1₂, andε˜is independent of ˜

cB, which impliesE[˜ε|cB+z] =0.

Next, inExample 6.2, we show that in generalz˜may not be chosen to be a constant. Interest-ingly, the variance of the weakly less risk averse agent’s payoff can be smaller than the variance of the weakly more risk averse agent’s payoff.

UA(c)˜ = −

(8− ˜c)3

3 , UB(c)˜ = −

(8− ˜c)5

5 , wherec <˜ 8. The first order conditions give us

U_A(c˜A)=(8− ˜cA)2=λAρ,˜ UB(c˜B)=(8− ˜cB)4=λBρ.˜ (29) Therefore, ˜ cA=8− λAρ,˜ c˜B=8−(λBρ)˜ 1/4. (30) From(30), we get ˜ cA=8− λA λB (8− ˜cB)2. (31)

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We have:c˜Ac˜B iffc˜B8−

λB

λA. FromTheorem 3, we know thatc˜A∼ ˜cB+ ˜z+ ˜ε, where ˜

z0 andE[˜ε|cB+z] =0. To find an example that the variance of the less risk averse agent’s

payoff can be smaller, we assume thatρ˜has a discrete distribution,i.e.,ρ1=εwith probability 1 2,ρ2= 1 4 with probability 1 4, andρ3= 1 2 with probability 1

4. Ifεis very tiny (close to zero),

then from(30)and the budget constraintE[ ˜ρc˜A] =1. It is not difficult to computeλA≈17.5,

λB≈125.8,c˜A≈(8 5.91 5.045)andc˜B≈(8 5.632 5.184). Therefore,E[˜cA] ≈6.73,E[˜cB] ≈

6.70, andVar(c˜A)≈1.684<Var(c˜B)≈1.704,i.e., the variance of the weakly more risk averse

agent’s payoff is higher. In this example, both agents’ utility functions have increasing absolute risk aversion, which gets very high at the shared satiation pointc˜=8. In the high-consumption (lowρ˜), the optimal consumptions of agentAandBare both very close to 8. To haveE[˜cA]> E[˜cB],z˜is greater in the low-consumption states. Thereforez˜is large whenc˜is small and small whenc˜is large, and thusz˜is very negatively correlated withc˜B+ ˜ε. As noted in Section3, we

know that if the non-negative random variablez˜can be chosen to be a constant, thenVar(c˜A)=

Var(c˜B)+V ar(ε)˜ Var(c˜B). Therefore, in this example,z˜cannot be chosen to be a constant.

The next example shows that if the utility functions are not strictly concave, then our main result does not hold.

and the utility functions areUA(c)˜ =UB(c)˜ = ˜c. We assume there are two states withρ1=1₂

with probability 1₃, andρ2=1₂ with probability ₃2. It is not difficult to see thatc˜A=(0,3)and ˜

cB=(4,1)is an optimal consumption for agentAandBfor λA=λB=2. We haveE[˜cA] =

E[˜cB] =2 andVar[˜cA] =Var[˜cB] =2. Ifc˜A∼ ˜cB+ ˜z+ ˜ε, wherez˜0 andE[˜ε|cB+z] =0,

thenz˜=0 andε˜=0, we getc˜A∼ ˜cB. Contradiction! So, we cannot havec˜A∼ ˜cB+ ˜z+ ˜ε, where ˜

z0 andE[˜ε|cB+z] =0.

Example 6.3is degenerate with constantρ˜and linear utility. It is not difficult to construct a more general example (Example 6.4), whereρ˜is random and the utility function has two straight segments. The optimal portfolio is not unique on these two straight segments taken together and therefore our payoff distributional result may not hold.

UA(c)˜ =UB(c)˜ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −(c˜−1)4+ ˜c c <˜ 1 ˜ c 1c˜2 1 256(c˜4−16c˜3+72c˜2+128c˜+80) 2<c <˜ 6 1 2c˜+2 6c˜14 1 2e−(c˜−14)−2e−(c˜−14)/2+9 c˜14.

In this example, the utility function has two straight segments and the optimal portfolio is not unique on these two straight segments taken together. We assume thatρ1=1₂ with probability1₂

andρ2=1₄ with probability 1₂. Then, it is not difficult to see thatc˜A=(2,12)andc˜B=(1,14)

is the optimal consumption for agentAandBforλA=λB=2. So, whileAis weakly less risk

averse thanB (their risk aversion is equal everywhere),c˜Ais not distributed asc˜B+ ˜z+ ˜εwith ˜