Math Finan Econ (2014) 8:71–108DOI 10.1007/s11579-013-0109-6

Nonmyopic optimal portfolios in viable markets

Jakša Cvitanic · Semyon Malamud

Received: 27 May 2013 / Accepted: 9 October 2013 / Published online: 24 October 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract We provide a representation for the nonmyopic optimal portfolio of an agent con-suming only at the terminal horizon when the single state variable follows a general diffusionprocess and the market consists of one risky asset and a risk-free asset. The key term of ourrepresentation is a new object that we call the “rate of macroeconomic fluctuation” whoseproperties are fundamental for the portfolio dynamics. We show that, under natural cycli-cality conditions, (i) the agent’s hedging demand is positive (negative) when the product ofhis prudence and risk tolerance is below (above) two and (ii) the portfolio weights decreasein risk aversion. We apply our results to study a general continuous-time capital asset pric-ing model and show that under the same cyclicality conditions, the market price of risk iscountercyclical and the price of the risky asset exhibits excess volatility.

Keywords Heterogeneous agents · Nonmyopic optimal portfolios · Hedging demand ·Equilibrium

JEL Classification D53 · G11 · G12

1 Introduction

The problem of optimal portfolio allocation for a long-run investor, in a market with onerisky asset and a risk-free asset, is of fundamental importance in financial economics. Whenthe market price of risk (MPR) and/or interest rates (IR) are stochastic, the optimal port-folio is inherently nonmyopic and depends on not only current asset returns but also their

J. Cvitanic (B)Division of Humanities and Social Sciences, Caltech, M/C 228-77, 1200 E. California Blvd., Pasadena,CA 91125, USAe-mail: cvitanic@hss.caltech.edu

S. MalamudSwiss Finance Institute, EPF Lausanne, Lausanne, Switzerlande-mail: semyon.malamud@epfl.ch

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future fluctuations.1 As [56,57] indicates, the optimal portfolio can be decomposed into twocomponents: a myopic, instantaneously mean-variance efficient component and a strategic,nonmyopic component responsible for hedging against future changes in the investmentopportunity set. Several recent empirical studies indicate that the hedging component mayrepresent a huge fraction (40–100 %) of total portfolio holdings.2 However, since it is gen-erally not possible to derive closed–form expressions for nonmyopic optimal portfolios,very little is known about their dependence on the agent’s risk preferences and the dynamicproperties of asset returns.

This paper provides representations for nonmyopic optimal portfolios of an agent con-suming only at the terminal horizon when the single state variable follows a general diffusionprocess. The key additional assumption we make is that the state price density is a function ofthe underlying state variable. This assumption holds true if and only if the price of the riskyasset in the economy and the IR of the risk-free asset are viable; that is, they can be realizedin a complete market equilibrium within the capital asset pricing model (CAPM) if the repre-sentative agent is chosen appropriately. This makes our results perfectly suited to equilibriumanalysis, which we present in the second part of the paper. We use the obtained represen-tations to generate empirical predictions and establish various properties of the nonmyopicportfolio strategies.

We introduce a new object that we call the “rate of macroeconomic fluctuation” andshow that the rate of macroeconomic fluctuation is the key element of all representationsderived in this paper, both in the single-agent partial equilibrium setting and in the generalequilibrium setting. The rate of macroeconomic fluctuation is an intrinsic characteristic ofthe state variable dynamics and has the following interesting property: in equilibrium, therisk-neutral drift of the MPR is independent of the form of the agents’ utility functions and isalways equal to the rate of macroeconomic fluctuation. As a result, the rate of macroeconomicfluctuation can be directly estimated from the data as the difference between the drift of theMPR and the product of the MPR and its volatility.3

We show that the intertemporal hedge against the MPR fluctuations has two components:a wealth hedge and a risk tolerance hedge. The wealth (risk tolerance) hedge is given by thecovariance of the agent’s future wealth (risk tolerance) with the future MPR, discounted bythe rate of macroeconomic fluctuation.

Our representation allows us to determine the precise dependence of the optimal portfo-lios on the market’s and the agent’s characteristics. In particular, we show that the optimalinvestment in the risky asset is monotonically decreasing in risk aversion in the sense ofRoss [63] as long as the rate of macroeconomic fluctuation is constant. This result is quitestriking in its generality, as it holds for arbitrary utilities, arbitrary dividend volatility andarbitrary stochastic IRs. It also generates this empirical prediction: if the volatility of the rateof macroeconomic fluctuation is relatively low, more risk averse agents should invest less inthe risky asset.

1 We use the terminology “myopic” and “nonmyopic” in the following sense: we call myopic all quantitiesthat are determined by the present market value of their (possibly discounted) end-of-horizon values. On theother hand, we call nonmyopic all quantities whose current value also depends on their future co-movementwith other market variables.2 See [14] and [3].3 That is, the MPR λ follows the dynamics dλt = λt (μ

λt dt + σλt d Bt ) with the drift μλt = ρt + λt σ

λt

where ρt is the rate of macroeconomic fluctuation.

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Math Finan Econ (2014) 8:71–108 73

We show that, when the MPR is countercyclical, the hedging demand is positive if andonly if the product of the agent’s prudence and risk tolerance is below 2.4 We provide anintuitive explanation of this non-obvious result and relate it to the precautionary savingsmotive.

If we adopt the common hypothesis that agents have hyperbolic absolute risk aversion(HARA) utilities, our results generate the prediction that the variation in the MPR hedgingdemand of an agent is explained by the variation in the co-movement of agent’s wealth andthe discounted MPR. This prediction can be tested directly using Panel Study of IncomeDynamics (PSID) data. It would allow us to gain a deeper understanding of the joint fluc-tuations of household wealth and optimal portfolios. For example, Brunnermeier and Nagel[12] find that the fraction of wealth that households invest in risky assets does not changewith the level of wealth. They use this finding to conclude that households’ relative riskaversion is constant. However, if this risk aversion is different from 1, the optimal portfoliomust be nonmyopic and contain a hedging component. Given our representation of hedgingdemand, the findings of Brunnermeier and Nagel [12] imply, in our model, that the degree ofco-movement of households’ wealth with the MPR is stable and does not fluctuate over time.It would be of interest to test this prediction and determine the degree to which investmentby households is driven by the nonmyopic demand.

In the section on equilibrium dynamics, we provide tight conditions for MPR counter-cyclicality and for the positivity of the nonmyopic (excess) volatility. We show that, whenIR volatility is small, MPR is countercyclical if the product of the aggregate risk aversionand dividend volatility is countercyclical and the rate of macroeconomic fluctuation is pro-cyclical. This result extends the characterization of He and Leland [42] to general dividendprocesses and stochastic IRs. In particular, we show, in contrast to the sufficient conditions inthe framework of that paper, that when the state variable follows a general diffusion processeven countercyclicality of both dividend volatility and aggregate risk aversion may not besufficient for MPR to be countercyclical, as its equilibrium behavior depends on the dynamicsof the rate of macroeconomic fluctuation.

In complete analogy with the optimal portfolios, we show that the volatility of the riskyasset can be decomposed into myopic and nonmyopic components. The latter plays therole of “excess” volatility and is positive under the same conditions for which MPR iscountercyclical. As above, these conditions are tight.

In the special case when the log-dividend follows an autoregressive process and all agentshave constant relative risk aversion (CRRA) preferences, our results are strengthened fur-ther. We find explicit bounds on MPR, excess volatility, risk-return tradeoff and the size ofequilibrium optimal portfolios in terms of the spread between the highest and the lowest riskaversion in the economy. This is a natural measure of heterogeneity, and our bounds can beused to estimate this measure from data.

We conclude the introduction with a discussion of the related literature. Our paper is mostclosely related to the fundamental work by Detemple et al. [22] and Detemple and Rindis-bacher [23,24], who were the first to use Malliavian calculus to derive general representationsfor intertemporal hedging demand with general utilities and a general multi-dimensionaldiffusion state variable. In our paper, the assumption that the market is viable leads to a par-ticularly simple and intuitive representation of optimal portfolios and allows us to establishseveral important economic properties of the latter.

4 We call an economic variable countercyclical if it is monotonically decreasing in the state of the econ-omy, represented in our model by the aggregate dividend process. We call an economic variable procyclicalotherwise.

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Aura et al. [2], Gollier and Zeckhauser [37] and Gollier [38] investigate nonmyopic port-folios in a discrete time two-period model. In particular, Gollier [38] studies the effect offirst- and second-order predictability in returns in the sign of the hedging demand. Our intro-duction of the rate of macroeconomic fluctuation allows us to determine the sign of hedgingdemand for very general continuous-time dynamic settings with an arbitrary time horizon.

A substantial body of literature is devoted to the partial equilibrium continuous-timenonmyopic optimal portfolio behavior in complete and incomplete markets when utilitiesbelong to the HARA class and stock prices follow an affine process. This specificationallows for closed-form solutions for optimal portfolios (see [48] and [65]).

Most existing work devoted to standard CAPM with heterogeneous risk preferences hasbeen done in special models with two CRRA agents only. Dumas [27] considers a produc-tion economy of that type. In this case, the risky asset returns coincide with the exogenouslyspecified returns on production technology, and only the risk-free rate is determined endoge-nously. Wang [66] studies the term structure of IRs in an economy populated by two CRRAagents, maximizing time-additive utility from intermediate consumption and the aggregatedividend following a geometric Brownian motion. Basak and Cuoco [5] study equilibria withtwo agents and limited stock market participation. Bhamra and Uppal [6,7] also consider aneconomy with two agents and derive conditions under which market volatility is higher thandividend volatility.5 Cvitanic and Malamud [17,18], Yan [69] and Cvitanic et al. [19] studyasymptotic equilibrium dynamics with an arbitrary number of CRRA agents maximizingutility from terminal consumption as the horizon increases.

Only a few papers have studied general properties of equilibria with non-CRRA prefer-ences and/or a general dividend process.6 Our paper is partially related to [42].7 They alsoallow for an arbitrary dividend process and describe the set of viable price processes, thatis, all price processes that can be attained by varying the utility of the representative agent.One of their main results declares that when the dividend is a geometric Brownian motion,MPR is countercyclical if and only if the risk aversion is countercyclical. Our representationallows us to prove the most general result of this kind. More precisely, we show that counter-cyclical risk aversion generally is not sufficient, and countercyclicality of dividend volatilityand procyclicality of the rate of macroeconomic fluctuation are needed.

In a number of models, heterogeneity arises from beliefs and asymmetric information, asin [4,10,46] and [29]. In particular, the latter paper studies excess volatility and nonmyopicoptimal portfolios in a model with two CRRA agents with identical preferences but hetero-geneous beliefs. Methodologically, our paper is close to [22,24,25,29], and [6], as we useMalliavin calculus to obtain the main results.

The paper is organized as follows. Section 2 describes the setup and notation. In Sect. 3,we introduce the rate of macroeconomic fluctuation and in Sect. 4 we use the latter toderive representations for the MPR. Section 5 studies monotonicity of optimal portfoliosin risk aversion. Section 6 is devoted to the properties of hedging demand. Sections 7and 8 study a general continuous-time CAPM with only terminal consumption. Section 9concludes.

5 The main message of Bhamra and Uppal [6] is that allowing the agents to trade in an additional derivativesecurity, making the market complete, may actually increase market volatility. Because of completeness, theirequilibrium coincides with the Arrow–Debreu equilibrium of Wang [66].6 Some papers study static, one-period economies with heterogeneous preferences. See [8,36,40]. There arealso conditions for zero equilibrium trading volume. See [9].7 See also [11] and [67].

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2 Setup and notation

2.1 The model

The economy has a finite horizon and evolves in continuous time. Uncertainty is describedby a one-dimensional, standard Brownian motion Bt , t ∈ [0 , T ] on a complete probabilityspace (�,FT , P), where F is the augmented filtration generated by Bt . There is a singlestate variable Dt evolving according to

D−1t d Dt = μD(Dt ) dt + σ D(Dt ) d Bt .

This diffusion process lives on (0,+∞) and σ D(Dt ) > 0.8

We make the following key assumption throughout the paper:

Assumption 2.1 The market is viable; that is, the unique state price density ξT is given byξT = g(DT ) for some smooth and strictly decreasing function g, and the IR process rt isgiven by rt = r(Dt ) for some smooth function r .

Assumption 2.1 is natural because it holds in any general equilibrium model with thesingle state variable Dt . Indeed, in Merton’s [58] intertemporal CAPM (ICCAPM), ξt =e−ρt U ′(Dt ), where Dt is the aggregate dividend and ρ and U are respectively the discountrate and the utility function of the representative agent, maximizing utility from intermediateconsumption. Furthermore, denoting

γU (x) = − x U ′′(x)U ′(x)

the relative risk aversion of the agent, in this case rt = r(Dt ) with

r(x) = ρ + γU (x) μD(x) − 0.5 γU (x) (γU (x)+ 1) (σ D(x))2. (1)

For the case of terminal consumption CAPM (see [11,42], and Sect. 8 below), ξT = U ′(DT ),where U is the utility function of the representative agent.9 In this case, the IR process isnot determined in equilibrium and can be specified exogenously because agents need notsubstitute consumption between periods. For this reason, everywhere in the sequel, we willwrite ξT = U ′(DT ) where we have defined U (c) = ∫ c

1 g(x) dx .Given the IR process rt , the unique state price density process ξ = (ξt ) is given by

ξt = e− ∫ t0 rs ds Mt ,

where Mt is the density process of the equivalent martingale measure Q,(

d Q

d P

)

t= Mt = Et [ MT ].

Given a state price density process, we can always construct a corresponding financial marketby assuming there are two types of assets available for trading at any moment: an instanta-neously risk-free bond with rate of return rt and a risky asset, a stock, with the price processSt given by either

8 We assume that μD and σ D are such that a unique strong solution exists and also that μD ∈ C1(R+),σ D ∈ C2(R+). In general, whenever we use a derivative of a function, we implicitly assume it exists.9 In general, we could consider an economy populated by two types of agents: those deriving utility fromintermediate consumption with a possibly infinite time horizon and those deriving utility from terminal con-sumption (wealth) at some prescribed time horizon. For example, we could think of these agents as fundmanagers. See [39] and [16]. In this case, we would need to assume that the aggregate dividend has both acontinuous rate component and a lumpy component, paid at those prescribed time periods.

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St = M−1t Et [MT DT ] = E Q

t [ e− ∫ Tt rs ds DT ] (2)

or

St = M−1t Et

⎡

⎣∞∫

t

Mτ Dτ dτ

⎤

⎦ = E Qt

⎡

⎣∞∫

t

e− ∫ τt rθ dθ Dτ dτ

⎤

⎦ . (3)

The instantaneous drift and volatility of stock price St are denoted by μSt and σ S

t respec-tively, so that10

d St

St= μS

t dt + σ St d Bt .

For simplicity, we always assume that stock volatility σ St is almost surely nonnegative.11

The MPR λt is given by

λt = μSt − rt

σ St

.

It also can be calculated directly from the state-price density process using the equation

ξ−1t dξt = −rt dt − λt d Bt .

An economic agent k chooses portfolio strategy πk t , the portfolio weight in the risky assetat time t , as to maximize the expected utility12

E [uk(Wk T )]

of the terminal wealth WkT , where the wealth Wkt of agent k evolves as

dWkt = Wkt[rt dt + πk t (S

−1t d St − rt dt)

].

Introduce the inverse of the marginal utility

Ik(x) := (u′k)

−1(x). (4)

It is well known that in this complete market setting the optimal terminal wealth is given by

Wk T = Ik(yk ξT )

where yk is determined via the budget constraint13

E[Ik(yk ξT ) ξT ] = Wk 0.

10 It is possible to show that the market is viable if and only if the diffusion coefficients of St satisfy certaindifferential equations. See [11] and [67].11 This is always true if the IR rt is not excessively procyclical. See Appendix.12 We assume that uk is C3(R+) and satisfies standard Inada conditions.13 We assume that a unique value of yk satisfying the budget constraint exists.

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Math Finan Econ (2014) 8:71–108 77

3 Rate of macroeconomic fluctuations

In this section we introduce a new object, the rate of macroeconomic fluctuation. This ratewill play a crucial role in the sequel as it will appear in expressions for all quantities in bothpartial and general equilibrium (MPR, stock volatility and optimal portfolios) and will havea fundamental impact on their static and dynamic properties.

Two forces drive the stochastic dynamics of the state variable Dt : stochastic volatilityσ D(Dt ) and stochastic driftμD(Dt ).The volatility σ D determines the size of macroeconomicfluctuation. To understand the role of μD , we will “extract” all stochastic volatility from Dt .

Fix an x0 ∈ R+ and introduce the function

F(x)de f=

x∫

x0

1

y σ D(y)dy , x > 0.

Then a direct calculation shows that the diffusion coefficient of the process

Atde f= F(Dt )

is equal to 1. Since At is also an Ito diffusion, its dynamics are given by

d At = C(At ) dt + d Bt .

Definition 3.1 The negative instantaneous expected change of the volatility-extractedprocess At will be referred to as the rate of macroeconomic fluctuation and denoted ρt .More precisely, we define

ρtde f= − C ′(At ). (5)

By definition, we have

ρt = cD(Dt ).

with

c(x) = cD(x)de f= −C ′(F(x)).

It follows directly from the definition that the function c is invariant under functional trans-formations. That is, if Dt = g(Dt ) for some process Dt and a smooth, increasing functiong, then

cD(g(x)) = cD(x).

In particular, the following is true

Corollary 3.2 The rate of macroeconomic fluctuation ρt is constant if and only if the statevariable Dt is a smooth and strictly increasing function of an autoregressive process

d At = (a − b At ) dt + d Bt . (6)

In this case, ρt = b.

As our benchmark example throughout the paper, we consider a simple generalization ofGeometric Brownian Motion (GBM) as the dividend process by having At = log Dt satisfy(6). Then if b > 0, the growth rate of the economy (the log dividend) is mean reverting. On

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the other hand, if b < 0, the growth rate of the economy departs from the “mean” a at therate equal to −b.

Using a direct calculation based on Ito’s formula, it is possible to verify that function c(x)is given by14

c(x) = −x(μD)′(x)+ x(σ D)′(x)σ D(x)−1μD(x)+ (σ D)′(x)σ D(x)x

+0.5(σ D)′′(x)x2σ D(x). (7)

This expression was introduced in the fundamental work of Detemple et al. [22], who used itto derive representations for optimal portfolios in general diffusion-driven models. Our newformula (5) provides a very intuitive economic interpretation of this object and makes someof its properties (such as functional invariance) particularly clear.

4 The MPR

Recall that by assumption the market is viable and the state price density ξT is given byξT = U ′(DT ) for some strictly concave function U. Let

γU (x) = − x U ′′(x)U ′(x)

.

In an equilibrium setting,15 U is the utility of the representative agent and γU is the relativerisk aversion of the representative agent. We also will need the following:

Definition 4.1 We will refer to

σ D(t , τ )de f= e− ∫ τ

t ρs ds σ D(Dτ )

as the discounted aggregate volatility. We will refer to

σ r (t , τ )de f= e− ∫ τ

t ρs ds σ r (Dτ )

as the discounted IR volatility. We will refer to

λ(t , τ )de f= e− ∫ τ

t ρs ds λτ

as the discounted MPR, and we will refer to

σmyopict

de f= E Qt

[σ D(t , T )

](8)

as the myopic volatility.

In general, we call myopic all quantities that are determined by the present market valueof their (possibly discounted) end-of-horizon values. In other words, their current value doesnot depend on their future co-movement with other market variables but only on the “myopicestimate” of their future value.

The next result provides a representation of the MPR in terms of the quantities introducedin Definition 4.1 and highlights the role of myopic volatility.

14 See Appendix.15 Either with terminal or with intermediate consumption.

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Math Finan Econ (2014) 8:71–108 79

Theorem 4.2 The MPR is given by the present market value of future discounted MPR netof the present cumulative value of the future discounted IR volatility,

λt = E Qt [ λ(t , T ) ] − E Q

t

⎡

⎣T∫

t

σ r (t, s) ds

⎤

⎦ (9)

where

λT = γU (DT ) σD(DT ). (10)

In particular, if γU = γ is constant, then

λt = γ σmyopict − E Q

t

⎡

⎣T∫

t

σ r (t, s) ds

⎤

⎦ . (11)

Theorem 4.2 shows that the assumed market viability leads to very interesting properties ofthe MPR dynamics. To gain economic intuition behind the representation (9), it is instructiveto compare the result of Theorem 4.2 with the analogous results for a terminal consumptionCAPM. When the interest rate is constant, the dividend follows a GBM with volatility σ andthe representative agent’s risk aversion γU = γ is constant, we have

λt = γ σ.

Formula (11) shows that when the dividend is not a GBM, the role of the fundamental volatilityσ is played by the myopic volatility σmyopic

t . The latter is given by the present market valueof future dividend volatility, discounted at the rate of macroeconomic fluctuation ρt , and thusdepends on the dynamic properties of Dt in a nontrivial way. In particular, dividend volatilityby itself is not enough for determining the size of the equilibrium MPR. We need to know therate ρt . For example, if ρt is always positive (negative), the level of fundamental volatilityleads to overestimating (underestimating) the size of the MPR.

To understand the intuition behind this discounting, consider a simple modification ofMerton’s CAPM, for which the log-dividend follows (6) with b > 0 and both γU = γ andr are constant. Then ρt = b and

λt = γ σ e−b (T −t).

That is, the equilibrium riskiness of the stock is given by σ e−b (T −t). On the other hand, thestock riskiness is determined by its sensitivity to changes in the state variable Dt : If thestock price moves significantly in response to a change in the dividend, the stock is risky;if the sensitivity is small, stock price fluctuations are small and the stock is less risky. Asrecalled in (2), the stock price is the expectation (under the risk-neutral measure) of futuredividends. If b > 0, log dividends are mean reverting, and therefore, stock price sensitivity todividend changes is smaller than in the case b = 0. This drives down the stock riskiness andthe equilibrium MPR. However, as time t approaches terminal horizon T, insufficient timeremains for the dividends to mean-revert, and the sensitivity of expected future dividends tochanges in the current value of the dividend grows exponentially at the rate b, equal to therate of mean-reversion. Thus, stock riskiness also grows exponentially, and converges to theriskiness of the dividend when t ↑ T .

The situation is opposite when the rate ρt = b is negative. Then the log-dividend isnonstationary, and every small fluctuation around the mean may force the dividend to deviatesubstantially. Hence, stock is riskier than in the case b = 0, and the MPR goes up. However,

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as time t approaches T, there is insufficient time remaining for the dividend to deviate,riskiness decreases at the rate b, and the equilibrium MPR goes down.

When the IR is stochastic, another term, E Qt

[∫ Tt σ r (t, s) ds

], appears in (11). To under-

stand the intuition behind this term, suppose that the IR is procyclical.16 Then bond pricesare countercyclical. Thus, bonds are expensive when stocks are cheap. This drives up thedemand for stocks, pushing up the equilibrium stock price, and, consequently, equilibriumstock returns and the MPR decrease.

In Merton’s [58] ICCAPM, the MPR process is given by γU (Dt ) σ (Dt ) for the represen-tative agent’s utility U and the IR is given by (1). This specific form of the IR dynamics leadsto the following interesting identity: in Merton’s ICCAPM, we always have

E Qt [ λ(t , T ) ] − E Q

t

⎡

⎣T∫

t

σ r (t, s) ds

⎤

⎦ = γU (Dt ) σ (Dt ) (12)

for any T ≥ t. To the best of our knowledge, identity (12) has never previously appeared inthe literature.

The following corollary is a direct consequence of Theorem 4.2.

Corollary 4.3 The process

e− ∫ t0 ρs ds λt +

t∫

0

e− ∫ s0 ρθdθ σ r

s ds

is a martingale under the equilibrium risk-neutral measure. In particular, if the IR is constant,the drift of the MPR is independent of U and is always equal to ρt .

We believe that formulae (9) and (12) and the result of Corollary 4.3 are of fundamentalimportance because they are model-independent, in the sense that they do not depend on thefunctional form of μD(x) or σ D(x). Therefore, they generate empirical predictions that canbe used to test various versions of complete market CAPM.

For example, if we ignore the IR volatility term, as is justified by its very low empiricalvalues, we get that MPR evolves under the physical measure as

dλt = λt (μλt dt + σλt d Bt ) ,

with the drift μλt given by

μλt = ρt + λt σλt .

If we estimate the MPR λt , its volatility σλt and driftμλt from the data, we immediately obtainan estimate for the rate ρt :

ρt = λt σλt − μλt .

In this sense, the rate of macroeconomic fluctuation is directly observable and various statis-tical hypotheses about ρt can be directly tested.

16 We call an economic variable countercyclical if it is monotonically decreasing in Dt and procyclicalotherwise.

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Math Finan Econ (2014) 8:71–108 81

5 Optimal portfolios

Let

Uk t (x)de f= sup

πEt [ uk(WkT ) | Wkt = x ]

be the value function of agent k.17 The effective relative risk aversion of agent k at time t isdefined via

γk t = γk t (Wkt )de f= − Wkt U ′′

k t (Wkt )

U ′k t (Wkt )

.

When the investment opportunity set is nonstochastic, the optimal portfolio is myopic, instan-taneously mean-variance efficient, and is given by18

πmyopick t

de f= λt

γk t σSt.

It is known [43] that the effective absolute risk tolerance

Rkt = Wk t

γk t

(i.e., the absolute risk tolerance of the value function) is a discounted martingale under therisk-neutral measure. Consequently, the following result holds:

Proposition 5.1 We have

Wk tπmyopick t = λt

σ St

Rkt = λt

σ St

E Qt [ e− ∫ T

t rs ds RkT ]. (13)

We denote

πhedgingk t

de f= πk t − πmyopick t

and refer to it as the hedging portfolio. This is the nonmyopic component of the optimalportfolio that the agent uses to hedge against (or, take advantage of) future fluctuations in thestochastic opportunity set.

Recall that by Definition 4.1,

λ(t , T )de f= e− ∫ T

t ρs ds λT , σr (t, τ ) = e− ∫ τ

t ρs ds σ rτ

are respectively the MPR and the IR volatility, discounted at the rate of macroeconomicfluctuation. The following result then holds:

Theorem 5.2 We have

πhedgingk t = π

MPR hedgek t + π

IR hedgek t

with the MPR hedging component

πMPR hedgek t = − 1

σ St Wk t

× CovQt

(λ(t , T ) , e− ∫ T

t rs ds (Wk T − Rk T )). (14)

17 Note that Uk t depends on Dt but we suppress this dependence.18 See [56,57].

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and the IR hedging component

πIR hedgek t = 1

σ St

(1

γk t− 1

)

E Qt

⎡

⎣T∫

t

σ r (t, τ ) dτ

⎤

⎦ . (15)

Formulae (14) and (15) provide closed-form expressions for hedging portfolios. Theyshow that the nature of the MPR hedge is quite different from that of the IR hedge. Thelatter is a product of two simple terms: one that depends on the current level of the agent’seffective risk aversion, another given by the market value of the (discounted) cumulative IRvolatility, normalized by the stock volatility. In particular, if the IR is procyclical, the IRhedge is positive (negative) when effective risk aversion is below (above) 1.19 Furthermore,the IR hedges of two CRRA agents differ only by a constant multiple.20

The MPR hedge is more complex. It is determined by joint fluctuations of the discountedfuture MPR with agent-specific characteristics: his future wealth net of his absolute risktolerance. Suppose for simplicity that the IR is constant. We can then write (14) as

er(T −t) σ St Wk t π

MPR hedgek t = −CovQ

t (λ(t , T ) , Wk T )︸ ︷︷ ︸wealth hedge

+ CovQt (λ(t , T ) , Rk T )︸ ︷︷ ︸

risk tolerance hedge

. (16)

The first term on the right-hand side is the covariance of agent’s wealth with the (discounted)MPR. We will refer to it as the wealth hedge. Similarly, the second term will be referred toas the risk tolerance hedge.

To understand the nature of the wealth hedge, suppose that the future MPR and the agent’swealth are positively correlated. When the agent’s wealth is low, any additional unit of wealthis valuable. Since MPR is also low in those states, the agent invests more in the risk-freeasset because it offers a better hedge against the low-wealth states. This generates a negativehedging demand. When the agent’s wealth is high, the utility of receiving an additional unitof wealth is low. therefore, even though the MPR is high in those states, it does not make therisky asset that much more attractive.21

To illustrate the intuition behind the risk tolerance hedge, suppose that the MPR and risktolerance are positively correlated. Then when risk tolerance is high, the agent is willing totake more risk, and this effect is magnified since the MPR is high in those states, driving upthe demand for stock. When risk tolerance is low, the MPR is also low. This generates anincentive to hedge against those states and to reduce the long position in the stock, but notto sell short excessively, because of low risk tolerance. As a consequence, stock investmentis altogether more attractive (the impact of the states with high MPR and high risk toleranceprevails) and this generates a positive hedging demand.22

If we adopt Arrow’s hypothesis that absolute risk aversion is monotonically decreasing inwealth, risk tolerance Rk T = Rk(Wk T ) is increasing in wealth and, generally speaking, thewealth hedge and the risk tolerance hedge will have opposite signs and represent competingeffects for determining the size of the hedging demand. Note also that if the agent’s utilityis of the CARA (exponential) class, Rk T is constant and, consequently, the risk tolerancehedge vanishes.

19 Detemple et al. [22] derive an analogous result in a partial equilibrium setting for a class of multi-dimensional diffusion state variables.20 By (13), γkt = γk if agent k has a CRRA utility. This also follows from a simple homogeneity argument.21 The argument for the case of negative correlation between wealth and the MPR is analogous.22 The case in which the MPR and risk tolerance are negatively correlated is analogous.

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Math Finan Econ (2014) 8:71–108 83

If agent k has a HARA utility, we have

Rk T = a Wk T + b

for some constants a, b and the MPR hedge takes the form

er(T −t) σ St Wk t π

MPR hedgek t = (a − 1)CovQ

t ( λ(t , T ) , Wk T ) .

This formula generates the empirical prediction that MPR hedges are driven by the co-movement of the agent’s wealth with the discounted MPR. This prediction can be testedusing PSID data (see [12]).

One also could use this formula together with the empirical state price density method ofRosenberg and Engle [62] to estimate the empirical optimal hedging portfolio.

6 Monotonicity of optimal portfolios in risk aversion

A cornerstone of the Arrow–Pratt theory of risk aversion is that in the standard one-periodportfolio allocation problem with one risky and one risk-free asset, the fraction of wealthinvested in the risky asset is monotonically decreasing in risk aversion.23

Monotonicity of optimal portfolios in risk aversion is also crucial for understanding thenature of risk-sharing in dynamic general equilibrium models. In fact, almost all existingpapers on heterogeneous equilibria conjecture this monotonicity property and use it as thebasis for their economic intuition, arguing that more-risk-averse agents will hold less stockand lend money to the less-risk-averse ones. (See [5,4,6,27,66]). It is therefore surprising thatover the past 30 years, there has been no progress in extending the Arrow–Pratt monotonicityresult to a multi-period setting, neither in partial nor in general equilibrium. The goal of thissection is to address this deficiency. We obtain strong sufficient conditions for the monotonic-ity of optimal portfolios in risk aversion that are generally applicable and, we believe, are tight.

Of course, one cannot expect the Arrow–Pratt monotonicity result to hold in a generalmulti-period setting. Even in the case when returns are independent over time and the portfoliochoice problem is myopic, stochastic future returns have a “background risk” effect on theagent’s value function.24 In particular, an ordering of risk aversions γi and γ j by no meansimplies the same ordering of the effective risk aversions γi t and γ j t . In order to formulateconditions guaranteeing this ordering of risk aversions, we will need a definition due to [63].

Definition 6.1 [63] For agent i, we denote

γ infi

de f= infxγi (x) , γ

supi

de f= supxγi (x).

Agent i is more risk averse than agent j in the sense of Ross, if

γ infi ≥ γ

supj .

In this case we write γi ≥R γ j .

Now, by (13),

γi t = Wit

E Qt [e− ∫ T

t rs ds γ−1iT WiT ]

= E Qt [e− ∫ T

t rs ds WiT ]E Q

t [e− ∫ Tt rs ds γ−1

iT WiT ]∈ [γ inf

i , γsupi ].

We arrive at

23 See [60] and [1].24 See [61], [35] and [37].

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84 Math Finan Econ (2014) 8:71–108

Proposition 6.2 If γi ≥R γ j then γi t ≥ γ j t .

Thus, when stock returns are independent, the ordering of risk aversions in the senseof Ross implies the corresponding ordering of optimal myopic portfolio weights. However,when the investment opportunity set is stochastic, optimal portfolios are nonmyopic andcontain a highly nontrivial hedging component, whose dependence on risk aversion is byno means obvious. The presence of this hedging demand may completely change portfoliobehavior.25 It is therefore crucial to identify conditions concerning the nature of stochasticfluctuations in the investment opportunity under which the natural risk-taking behavior ispreserved. There are three fundamental sources of these stochastic fluctuations: stochasticvolatility σ D

t , stochastic IRs rt , and the stochastic rate of macroeconomic fluctuation ρt .

When ρt is constant, it turns out that optimal portfolio weights are always monotonicallydecreasing in risk aversion. This follows from the following theorem.

Theorem 6.3 Suppose that the rate of macroeconomic fluctuation ρt is constant. Then opti-mal portfolio weights are monotonically decreasing in risk aversion in the sense of Ross.

The result of Theorem 6.3 is quite general, as it imposes absolutely no constraints on thenature of the agent’s utility function, MPR λT , and dividend volatility σ(DT ). If the MPRλT is stochastic, Theorem 5.2 implies that optimal portfolios have nonmyopic componentsthat are not necessarily monotonic in risk aversion. Nevertheless, the monotonicity of themyopic component fully compensates for the possible nonmonotonicity of the nonmyopiccomponents.

Another important case in which the monotonicity result holds is when risk aversions ofthe two agents are on different sides of 1. Using the fact that the log agent is myopic andalways holds an instantaneously mean-variance efficient portfolio πlog t = (σ S

t )−1λt , it is

possible to show that the following is true.26

Proposition 6.4 Suppose that γi ≤R 1 ≤R γ j . Then

πi t ≥ πlog t ≥ π j t .

Thus, when risk aversions are on different sides of 1, the monotonicity result holds inde-pendently of the model parameters. However, when risk aversions are both on the same sideof 1, and the rate of macroeconomic fluctuation ρt is stochastic, stronger conditions about thejoint variation of the rate of macroeconomic fluctuation and the MPR λT need to be imposed,as in the following theorem.

Theorem 6.5 Suppose that one of the following is true:

(i) γi ≥R γ j ≥ 1 and both −ρt and λT are countercyclical;(ii) 1 ≥ γi ≥R γ j and both −ρt and λT are procyclical.

Then agent i’s portfolio weight is smaller than agent j’s.

To gain intuition about how optimal portfolio behavior depends on the joint dynamics ofρt and λT , recall that by (16), the nonmyopic value of the stock investment for an agent isdetermined by the future co-movement of the MPR with the agent’s wealth and risk tolerance.

25 A simple Example 11.8 in the Appendix shows that nonmyopic portfolios are generally not monotonic inrisk aversion even in simple two-period binomial models.26 The proof follows directly from Proposition 11.6 in the appendix.

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Math Finan Econ (2014) 8:71–108 85

If agent i ′s wealth net of his risk tolerance (i.e., WiT − RiT ) co-moves more strongly withthe future MPR than that of agent j, he will hold a larger fraction of wealth in the nonmyopicportfolio. The latter may happen even if agent i is more risk-averse than agent j. Thus, therewill be two competing effects, determining the total size of investment and it is a priori unclearwhich effect dominates. Suppose for simplicity that both agents k = i , j are CRRA agentswith γi > γ j > 1. Introduce an agent-specific probability measure27

d Qk de f= e− ∫ Tt r(Ds )ds Wk T

E Q[

e− ∫ Tt r(Ds )ds Wk T

] d Q.

By construction, this measure assigns weights to various states of the world according to the

size of the agent’s discounted wealth e− ∫ Tt r(Ds )ds Wk T in those states. Using Theorem 5.2,

it is possible to show that

σ St πk t = −

(1 − γ−1

k

)E Qk

t [ λ(t, T ) ] + λt . (17)

Thus, the size of agent i’s optimal portfolio is driven by two effects: the size of his risk aversionγk and the size of the expectation of the discounted MPR λ(t , T ) under the agent-specificmeasure Qk .This expectation precisely captures the nonmyopic nature of the portfolio weightπk t . Since measure Qk assigns larger weight to those states in which agent i ′s wealth islarge, it will be larger for an agent whose wealth more strongly co-varies with the discounted

MPR. Since the markets are complete, wealth Wk T = Ik(yk ξT ) = (yk U ′(DT ))−γ−1

k ismonotonically increasing in DT , and the sensitivity of Wk T to changes in DT is determinedby the risk tolerance γ−1

k . Thus, when γk is low, the wealth Wk T will be very sensitiveto changes in DT and, consequently, is highly procyclical. Therefore, agent k will bet onthe realization of states with very high DT and the measure Qk will put great weight onthose states. For this reason, if λ(t , T ) is countercyclical, its expectation E Qk [λ(t , T )] ismonotonically increasing in γk

28 and therefore

1 − γ−1i

1 − γ−1j

> 1 >E

Q jt [ λ(t, T ) ]

E Qit [ λ(t, T ) ]

. (18)

The claim of Theorem 6.5 follows then from (18) and (17).By contrast, when λ(t , T ) is procyclical, its expectation E Qk [λ(t , T )] is monotonically

decreasing in γk because agents with low risk aversion bet on the realization of states withhigh DT and, consequently, high λ(t , T ).

7 The sign of the hedging demand

As discussed above, the exact behavior of nonmyopic optimal portfolios depends on thecyclicality of the rate of macroeconomic fluctuation ρt and the MPR λT . In this sectionwe provide necessary and sufficient conditions for the positivity of MPR hedging demand,depending only on the properties of the agent’s utility function and cyclicality of ρt and λT .

27 Cvitanic and Malamud [18] call this measure the agent k’s “wealth forward measure”.28 This heuristic argument is made rigorous in the proof of Theorem 6.5 in the Appendix.

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Recall that the absolute prudence and absolute risk tolerance of agent k are defined as

Pk(x) = −u′′′k (x)

u′′k (x)

and Rk(x) = −u′k(x)

u′′k (x)

respectively. The following is true:

Theorem 7.1 Suppose that both −ρt and λT are countercyclical (procyclical) and that r ′is not too large. Then the MPR hedging component πMPR hedge

t is positive if and only if theproduct of prudence and risk tolerance is below (above) 2 29, that is

Pk(x) Rk(x) ≤ 2 for all x .

The result of Theorem 7.1 is somewhat unexpected at first glance. Since the optimalportfolio of a log investor is always myopic, one would expect that the sign of the hedgingcomponent depends only on whether risk aversion is above or below 1 (as it does for theIR hedging component). However, Theorem 7.1 shows that the hedging motive depends onproperties of three derivatives of the utility function.

To understand the economic mechanisms behind Theorem 7.1, consider the case whenboth −ρt and λT are countercyclical and suppose for simplicity that rt = r is constant. Recallthe expression (16):

er(T −t) σ St Wk t π

MPR hedgek t = −CovQ

t(λ(t , T ) , (Wk T − Rk T )

).

Since markets are complete, Wk T is increasing in DT and, consequently, the wealth hedge−CovQ

t (λ(t , T ) , Wk T ) is positive. The size of the risk tolerance hedge depends on thecyclical properties of the agent’s risk tolerance Rk T . A direct calculation shows that

d

dxRk(x) = −1 + Pk(x) Rk(x).

If the product Pk Rk is large, the risk tolerance Rk T is highly sensitive to wealth fluctuationsand, consequently, is highly procyclical. Then the risk tolerance hedge

CovQt

(λ(t , T ) , Rk T

)

is very negative and dominates the wealth hedge,

− CovQt

(λ(t , T ) , Wk T

)

and the MPR hedge is negative. By contrast, if Pk Rk is small, the risk tolerance hedge is notvery negative. Therefore, the wealth hedge dominates and the MPR hedge is positive. Thethreshold of 2 appears because Wk T − Rk(Wk T ) is monotonically increasing in Wk T if andonly if Pk Rk ≤ 2.

The assumption of countercyclical MPR is natural and strong empirical evidence sup-ports it (See [13]). Furthermore, the countercyclical MPR can be generated from simplemicroeconomic assumptions about individual agents’ preferences.30 Since this study is thefirst to introduce the rate of macroeconomic fluctuation, there are no empirical data aboutthe behavior of ρt . We believe this is an important question deserving a serious empiricalanalysis. Note also that, in the benchmark case when ρt is constant (see Corollary 3.2), thisissue does not arise.

29 Note that the product of prudence and risk tolerance is exactly 2 for the log investor, for whom the hedgingportfolio is zero.30 See Proposition 9.3 below.

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Math Finan Econ (2014) 8:71–108 87

The appearance of prudence is also related to precautionary savings. As [47] showed ina static, one-period model, the strength of the precautionary savings motive for an agentanticipating stochastic fluctuations in future income is determined by his prudence Pk .Here,Pk plays a similar role, determining the strength of the savings/investment motive for anagent who anticipates future changes in the stochastic investment opportunity set.

If we consider the [1] hypothesis that γk(x) is increasing, a direct calculation showsthat it holds if and only if x Pk(x) ≤ γk(x) + 1. If γk(x) ≥ 1, we get Pk(x)Rk(x) =x Pk(x)/γk(x) ≤ 2. Theorem 7.1 then implies the following

Corollary 7.2 Let −ρt and λT be countercyclical. Suppose that γk(x) ≥ 1 and is increasingand r ′ is not excessively large. Then the MPR hedge is positive.

For the benchmark CRRA utility, Pk Rk = 1 + γ−1k and the results take a simpler form

Corollary 7.3 Suppose that γk = const, r ′ is not excessively large, and both −ρt and λT

are countercyclical. Then the MPR hedge is positive if and only if γk ≥ 1.

The intuition behind Corollary 7.3 is as follows. The marginal utility u′k(x) = x−γk

of an agent with high risk aversion γk > 1 is high in bad states (low Dt ). Therefore, anyadditional unit of consumption in those states is highly valuable for him. Since the MPR iscountercyclical, it is high in bad states. This makes the stock a highly attractive instrumentfor agent k to hedge against those states and induces him to buy additional shares. On theother hand, an agent with low risk aversion γk < 1 is not “afraid” of the bad states and betson the realization of good states with high Dt . Since the MPR is low in those states, it isoptimal for him to sell some of his stock holdings, creating a negative hedging demand.

8 Equilibrium dynamics

8.1 Setup

In this section, we study implications of the agent’s nonmyopic portfolio behavior for equi-librium dynamics. We assume that DT is the terminal dividend of the single risky asset inpositive net supply and there are no intermediate dividends.

There are K competitive agents behaving rationally. Agent k is initially endowed withψk > 0 shares of stock, and the total supply of the stock is normalized to 1,

K∑

k=1

ψk = 1.

Agent k chooses portfolio strategy πk t , the portfolio weight at time t in the risky asset, as tomaximize expected utility31

E [uk(Wk T )]

of the terminal wealth WkT . As usual, we say that the market is in equilibrium if the agentsbehave optimally and both the risky asset market and the risk-free market clear.

It is well known that the above financial market is dynamically complete if the volatilityprocess σ S

t of the stock price is almost surely nonzero.32 Because of the market completeness,

31 We assume that uk is C3(R+) and satisfies standard Inada conditions.32 This can be verified under some technical regularity conditions on the model primitives. See [45].

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any equilibrium allocation is Pareto-efficient and can be characterized as an Arrow–Debreuequilibrium (See [26,66]).33

Since in equilibrium the final wealth amounts of all agents must sum up to the aggregatedividend, equilibrium state price density ξT is uniquely determined by the consumptionmarket clearing equation

K∑

k=1

Ik(yk ξT ) = DT . (19)

It is important to mention that the model with a finite terminal horizon should be seen asan abstraction and that we should be careful interpreting it. St is seen as the price of a stockwhose first dividend is paid at time T , or at a later period. In equilibrium, ST = DT andtherefore DT is just the stock price at time T . Furthermore, σ D(DT ) is the local volatilityand can be estimated directly from the prices of options with maturity T, using Dupire’sformula (see [28]). Finally, the agent’s utilities uk from terminal wealth could be thought ofas their effective utilities (value functions), derived from consumption after time T .

8.2 Equilibrium stock price volatility

Unlike the MPR, the stock price volatility is a nonmyopic object. It depends not only on thesize of the aggregate risk but also on its future fluctuations relative to other variables. Moreprecisely, the following is true:

Theorem 8.1 The equilibrium stock price volatility is given by

σ St = σ

myopict + σ

nonmyopict − E Q

t

⎡

⎣T∫

t

σ r (t , s) ds

⎤

⎦ ,

where34

σnonmyopict = − 1

StCovQ

t

( (λ(t , T ) − σ D(t , T )

), e− ∫ T

t r(Ds )ds DT

). (20)

Furthermore, σ St is positive if r ′ is sufficiently small (the IR is not highly procyclical).

To understand the intuition behind Theorem 8.1, we first discuss our benchmark extensionof Merton’s CAPM when r, γU are constant, and the log dividend follows the autoregressiveprocess (6). Then σ D and the rate ρt = b are constant and the nonmyopic volatility σ nonmyopic

tvanishes. Consequently, by Theorems 4.2 and 8.1, the risk-return tradeoff coincides with thatof Merton’s CAPM:

λt = γ σ St .

33 Since endowments are colinear (all agents hold shares of the same single stock), it can be shown thatunder some conditions on the agents’ utility functions, the equilibrium is in fact unique up to a multiplicativefactor, and also is unique if we fix the risk-free rate. See [20,21]. If the dividend is neither bounded away fromzero nor from infinity, additional care is needed to verify the existence of equilibrium. See [21] and [52]. Weimplicitly assume throughout the paper that an equilibrium exists.34 We use CovQ

t (X, Y ) to denote conditional covariance of random variables X and Y under the equilibriumrisk-neutral measure Q.

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However, the stock volatility does not coincide with the fundamental dividend volatility.Rather, we have

σ S = σmyopict = e−b(T −t)σ D .

Stock volatility is higher than the dividend volatility if the log dividend is nonstationary(b < 0) and lower if the log dividend is stationary and mean-reverting (b > 0), because stockprice volatility is the sensitivity of expected future dividends to changes in Dt , as seen from(22) below. This sensitivity is large (small) if the log dividend is nonstationary (stationary).See also the discussion after Theorem 4.2.

If the IR is constant, expression (20) implies that the spread between the stock price volatil-ity and the myopic volatility is given by the covariance of the discounted future aggregatedividend with the future discounted MPR net of the discounted future dividend volatility.The term “nonmyopic” is most easily justified if we recall that

λ(t , T ) − σ D(t , T ) = (γU (DT ) − 1

)σ D(t , T ). (21)

Formulae (21) and (20) imply that, generically, the nonmyopic volatility vanishes if andonly if γU ≡ 1. That is, precisely when the representative agent is completely myopic.Hence, the nature of the nonmyopic volatility is necessarily driven by the agents’ nonmyopicequilibrium behavior. When MPR is stochastic, nonmyopic agents increase or decrease theirstock holdings depending on the expected future fluctuations of the MPR. The equilibriumhedging demand raises or decreases the total equilibrium demand for stocks and thereforedrives the equilibrium stock price up or down. Since Dt is the single state variable in ourmodel, standard results imply that St = S(t, Dt ) is a smooth function35 of Dt , and therefore,by Ito’s formula, we get

σ St = σ D(Dt ) Dt

∂

∂Dtlog S(t, Dt ). (22)

Thus, stock price volatility is nothing but the sensitivity of the stock price to changes in thedividend. Since equilibrium optimal portfolios respond to changes in Dt in a nonmyopicway, so does the equilibrium stock price, giving rise to nonmyopic volatility.

Formula (20) for the nonmyopic volatility is similar to the formula for the forward-futuresspread, which says that the forward-futures spread is given by the covariance under the risk-neutral measure of the future payoff with the cumulative return on short-term bonds (thebank account). As has been recognized in the literature, such a representation immediatelygenerates two empirical predictions for the spread: (1) a weak prediction that the sign of thespread should coincide with the sign of the covariance, and (2) a strong prediction that thevariation in the spread is explained by the covariance (see [59]). Based on formula (20), wecan make analogous predictions for the dynamics of the nonmyopic volatility and the co-movement of the dividends with the discounted MPR λ(t, T ) net of the discounted dividendvolatility σ D(t, T ).

There is also a clear similarity between expressions (14) and (20) for the MPR hedgeand the nonmyopic volatility respectively. In fact, the hedging demand (14) is the source ofnonmyopic volatility (20). Both arise as a nonmyopic response to the future fluctuations of theMPR. Similar to (20), expression (14) does not directly depend on the utility function of therepresentative agent. Therefore, Theorem 8.1 also has direct implications for the dynamics ofthe risk-return tradeoff. Empirical evidence suggests that expected stock returns are weaklyrelated to volatility. The results of [33,34] and [41] suggest that the relation between expected

35 Under some technical conditions. See [45].

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90 Math Finan Econ (2014) 8:71–108

returns and volatility is highly complex and nonlinear. One possible explanation of thisphenomenon is equilibrium hedging behavior. Suppose for simplicity that the risk aversionγU = γ of the representative agent and the IR r are constant. Then by a fundamental resultof [58], we have

λt = γ (σ St + λ

hedget )

where λhedget is the so-called hedge component appearing due to the equilibrium hedging

demand. Theorems 4.2 and 8.1 imply that λhedget = −σ nonmyopic

t , providing a direct economicinterpretation for Merton’s hedge component. Furthermore, when IR r is constant, the risk-return relation is driven by the nonlinear relation between myopic and nonmyopic volatility,

λt

σ St

= γσ

myopict

σmyopict + σ

nonmyopict

. (23)

Thus, calculating the risk-return tradeoff amounts to calculating the ratio of the nonmyopicand the myopic volatility. One can explore empirically this ratio using representations (8)and (20). It is particularly convenient that the latter does not explicitly depend on the utilityof the representative agent and can be directly estimated from data using the empirical stateprice density (see [62]).

9 MPR counter-cyclicality and excess volatility

To address further dynamic properties of equilibrium, we will need to make additionalassumptions about the evolution of Dt . It has become common in the literature to use con-sumption volatility as a measure of macroeconomic uncertainty. There is empirical evidencethat macroeconomic uncertainty is countercyclical (see [32] and [49]).

Definition 9.1 We say that our model exhibits countercyclical macroeconomic uncertaintyif the dividend volatility σ D

t is countercyclical and the rate of macroeconomic fluctuation ρt

is procyclical.

This definition is natural because volatility σ D and −ρt determine respectively the sizeand the speed of macroeconomic fluctuation. From now on we make the following:

Assumption 9.2 The economy exhibits countercyclical macroeconomic uncertainty, and theaggregate risk aversion γU is countercyclical.

By Definition 3.1, the economy exhibits countercyclical macroeconomic uncertainty ifand only if log Dt = f (At ) where f is increasing and concave and the process At hasconstant volatility and concave drift. For example, a simple family of such processes ariseswhen

d At = (a − b At ) dt + σ d Bt .

and the dividend process is given by Dt = e f (At ) for some increasing and concave functionf.

The assumption of countercyclical risk aversion is natural, and strong empirical evidencesupports it (See [13]). Furthermore, countercyclical aggregate risk aversion can be generatedfrom simple microeconomic assumptions about individual agents’ preferences. In fact, itis known that if all agents in the economy have (heterogeneous) CRRA preferences the

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Math Finan Econ (2014) 8:71–108 91

representative agent’s utility will have a decreasing (i.e., countercyclical) relative risk aversion(DRRA) (see [8] and [18]). A modification of the arguments from the latter paper also impliesthat the following slight generalization is true:

Proposition 9.3 If all agents have DRRA preferences, the aggregate risk aversion is coun-tercyclical.

In fact, it is possible to show that Proposition 9.3 still holds if individual risk aversionsare sufficiently heterogeneous and “not too increasing”.

It is a conventional wisdom that countercyclical risk aversion generates countercyclicalMPR.36 The following result confirms this intuition.

Proposition 9.4 Under Assumption 9.2, the MPR is countercyclical if both r ′ and r ′′ aresufficiently small.37

When the IR rt = r is constant and the dividend follows a geometric Brownian motion,[42] showed that the MPR is countercyclical if the aggregate risk aversion is countercyclical.Proposition 9.4 provides general sufficient conditions for the MPR countercyclicality. Theseconditions are tight in that a sufficiently countercyclical rate of macroeconomic fluctuationcan generate procyclical behavior in MPR even if the dividend volatility is constant and riskaversion is countercyclical.

The requirement that r ′ and r ′′ be small is not overly restrictive, given that, empirically,IR volatility is very small, especially relative to stock volatility. However, from a theoreticalperspective, it is interesting that the effect of stochastic IRs on the MPR might lead tounexpected equilibrium dynamics if countercyclicality of the aggregate risk aversion is notsufficiently strong. Consider, for example, the case when γU = γ is constant and the dividendis GBM. In this case the MPR is given by

λt = σ

⎛

⎝γ − E Qt

⎡

⎣T∫

t

r ′(Dτ ) Dτ dτ

⎤

⎦

⎞

⎠ .

Consequently, the dynamics of the MPR are determined exclusively by the IR process. Ifr ′(Dt ) Dt is larger than γ with high probability, the MPR may become negative because bondprices become so countercyclical that investors are eager to buy stock for hedging purposeseven if the stock offers a negative excess return. Furthermore, if r ′(Dt ) Dt is decreasing inDt , the MPR becomes procyclical.

By Proposition 9.4, countercyclical macroeconomic uncertainty together with counter-cyclical risk aversion generates countercyclical dynamics of MPR. The latter dynamics, inturn, determine the sign of nonmyopic volatility, as can be seen from expression (20) and thedefinition of λ(t, T ). More precisely, we arrive at the following result:

Proposition 9.5 Under Assumption 9.2, the nonmyopic volatility is positive if γU ≥ 1 andr ′ is sufficiently small.

The reasoning behind this result is as follows. Under Assumption 9.2, the MPR is low ingood states, but the future aggregate risk aversion is also (relatively) low and the expected

36 See [13]. There is also empirical evidence supporting countercyclicality of both MPR and aggregate riskaversion. See [30] and [31].37 In fact, we show that the following is true: for any interval [d1, d2] ⊂ R+ there exists an ε > 0 such thatλt is monotonically decreasing in Dt in the interval, as long as r ′ < ε and |r ′′| < ε.

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future dividends are high. Therefore, the agent is willing to hold the stock despite low MPR.This makes the price rise in good states and, by similar arguments, decline in bad states,driving up the nonmyopic part of the price volatility. Put differently, the tension between themovements in future dividends and the MPR creates nonmyopic (excess) volatility, becausethe larger the dividend, the larger the demand for the stock, while at the same time thelower the MPR , the lower the demand. The requirement that r ′ be not excessively large is

important: if the IR is highly procyclical, discounted dividends e− ∫ Tt rs ds DT may exhibit

countercyclical behavior, which, by (20), gives rise to a negative nonmyopic volatility.The nonmyopic volatility is the natural analog of excess volatility for the case of stochas-

tic dividend risk.38 Our representation illustrates that dynamical properties of aggregate riskaversion and dividend volatility by themselves are not sufficient to generate excess volatilityand the outcome is strongly influenced by the dynamics of the rate of macroeconomic fluctu-ation. This result should be contrasted with a related result of [6]. They analyze equilibriumwith two CRRA agents and a GBM dividend and show that the stock price volatility is higherthan the dividend volatility if the agent’s elasticity of intertemporal substitution (EIS) is nottoo large. Their model differs from ours because they allow for intermediate consumption,therefore, the IR is determined endogenously by the agents’ EIS.39 Our volatility represen-tation for the nonmyopic (excess) volatility (see Theorem 8.1) allows us to view the result of[6] from a different perspective. Two sources generate deviations from fundamental (myopic)volatility in their model. Countercyclical MPR drives up the nonmyopic volatility, whereasthe procyclical risk-free rate drives down the volatility, and the constraint of EIS being nottoo large diminishes the second effect.

Another important consequence of Proposition 9.5 and Theorem 4.2 is that, under theirassumptions, the following risk-return inequality40 holds:

λt < σ St max

xγU (x). (24)

The reason is that MPR is driven by myopic volatility, which is smaller than the stockvolatility by Theorem 8.1 and Proposition 9.5. Because of its myopic nature, equilibriumMPR “underestimates” the true stock price volatility and is unable to account for future stockprice and return fluctuations. Since MPR is countercyclical, the stock is cheap in bad states(those with low Dt ) and offers a high instantaneous return. This motivates the agents to buymore shares in those states, and the stock becomes more volatile, which, in turn, stabilizes thedemand. A similar argument applies in good states. Risk-return inequality is a much weakerrequirement than the risk-return equality of [58], and we can use it together with empiricaldata to estimate the maximal risk aversion in the economy.

9.1 Example: mean-reverting dividends and CRRA agents

We now illustrate the results from the previous sections in our benchmark example. That is,we make the following assumption:

38 Stock prices are said to exhibit excess volatility if they are substantially more volatile than the underlyingdividends. Excess volatility is a well known stylized fact. See [51,53,54,64] and [68].39 The simplifying assumption of no intermediate consumption is frequently used in equilibrium asset pricingliterature, including the Sharpe and Lintner’s CAPM. See [11,42,39,50]. Since agents do not have to substituteconsumption for investment, the IR process is not determined in equilibrium and can be specified exogenously.This allows the isolation of IR effects on the equilibrium dynamics. One can contrast this to productioneconomies, in which the MPR is specified exogenously and only the IR is determined in equilibrium. See [27].40 This is indirectly related to the state price density inequality of [15].

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Math Finan Econ (2014) 8:71–108 93

Assumption 9.6 The log dividend At = log Dt follows an autoregressive process

d At = (a − bAt )dt + σ d Bt ,

IR r is 0, and all agents in the economy have CRRA preferences, γk(x) = γk .

Under Assumption 9.6, σ D = σ, c = b. Consequently, all the formulae substantiallysimplify and all the dynamic properties of equilibrium are determined solely by the propertiesof the aggregate risk aversion γU . The following proposition summarizes our main results:

Proposition 9.7 Under Assumption 9.6,

(1) MPR

λt = E Qt [ γU (DT ) ] e−b (T −t) σ

is countercyclical;(2) the stock volatility is given by

σ St = σ e−b (T −t) + σ

nonmyopict ,

where the nonmyopic (excess) volatility

σnonmyopict = −e−b (T −t) σ

StCovQ

t (λT , DT )

is always positive;(3) optimal portfolios are monotonically decreasing in risk aversion. They are given by

πk t = λt

γk σSt

+ πhedgingk t ,

where the hedging component

πhedgingk t = 1

Wk t(γ−1

k − 1)e−b (T −t) σ

σ St

CovQt (λT , Wk T )

is positive if and only if γk > 1.

We conclude this section with another interesting result related to heterogeneity in riskaversions, showing how the size of the risk-return tradeoff, the size of the ratio σ S

t /σmyopict

and the size of the equilibrium optimal portfolios depends on the magnitude of heterogeneity.Let

γde f= min

kγk , �

de f= maxkγk .

Proposition 9.8 The following is true:

(1) the MPR satisfies

γ σ e−b(T −t) ≤ λt ≤ � σ e−b (T −t) ;(2) the stock volatility satisfies

σ e−b(T −t) ≤ σ St ≤ σ e−b(T −t) (1 + � − γ

) ;

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94 Math Finan Econ (2014) 8:71–108

(3) the risk-return tradeoff satisfies

γ

1 + � − γ≤ λt

σ St

≤ � ; (25)

(4) the optimal portfolios satisfy

1

γk

γ

1 + � − γ≤ πk t ≤ 1

γk

(� + (γk − 1) (� − γ )

)if γk > 1 ,

1

γk

γ

1 + � − γ≤ πk t ≤ 1

γk� if γk < 1.

These results imply that the size � − γ of heterogeneity plays a crucial role for deter-mining the size of excess volatility, risk-return tradeoff, and the size of equilibrium optimalportfolios. In the terminology of [29], investors are taking advantage of heterogeneous riskattitudes in the economy, which generates excess volatility and stochastic fluctuations of theMPR.

It follows from the asymptotic results of [18] that the bounds (25) and (26) are tight.Therefore, they are especially useful for empirical analysis because they do not depend onthe parameters of the dividend process. For example, finding the empirical range

[

minλt

σ St, max

λt

σ St

]

of the risk-return tradeoff allows us to gauge the size � − γ of heterogeneity in the economy.Moreover, given the estimates for γ and � from risk-return tradeoffs, data on empiricalportfolio holdings together with bounds (26) could be used to study the cross-sectionaldistribution of risk aversions in the economy.

10 Conclusions

We study nonmyopic optimal portfolios in viable markets, characterized by the fact that thestate price density and the IR are deterministic functions of the single diffusion state variable.Our analysis is based on new representations for optimal portfolios in terms of expectedvalues and covariances of directly observable quantities under the risk-neutral measure. Inthese representations, macroeconomic risks are priced discounted at a specific rate, that wecall the “rate of macroeconomic fluctuation”. Our representations are universal because theyhold in any exchange economy with an arbitrary Markovian dividend, arbitrary stochasticIRs, and, in the equilibrium context, arbitrary rational agents maximizing utility from terminalconsumption.

We derive tight conditions for the monotonicity of optimal portfolios in risk aversionand the positivity of the MPR hedge. In particular, we show that the MPR hedge consistsof two components, the wealth hedge and the risk tolerance hedge. Those are determinedrespectively by the covariances of agent’s future wealth and future risk tolerance with futureMPR, discounted at the rate of macroeconomic fluctuation. The latter can be directly estimatedfrom data, because the drift of the MPR is equal to the product of the MPR and its volatilityplus the rate of macroeconomic fluctuation.

Our results highlight general mechanisms behind the phenomena of MPR countercycli-cality, nonmyopic (excess) volatility, and nonmyopic optimal portfolios. In particular, weshow that the nonmyopic volatility is determined by the interplay between the MPR and IR

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Math Finan Econ (2014) 8:71–108 95

cyclicality, that the size and the sign of the IR hedge are determined by the IR volatility andthe size of agent’s risk aversion, whereas the size and the sign of MPR hedge are determinedby the interplay between prudence and risk tolerance.

We believe our results are important for understanding single-agent (partial equilibrium)nonmyopic portfolio behavior and for understanding continuous-time equilibrium CAPM.They can be used in theory for testing various versions of CAPM and in practice for calculatingoptimal portfolios for long-run investors.

Acknowledgments The research of J. Cvitanic was supported in part by NSF grant DMS 10-08219. Theresearch of S. Malamud was supported in part by NCCR FINRISK, Project D1. We thank Patrick Bolton,Jerome Detemple, Bernard Dumas, Campbell Harvey, Julien Hugonnier, Elyes Jouini, Loriano Mancini,Rajnish Mehra, and Erwan Morellec for helpful comments, as well as audiences at a number of seminarsand conferences. Existing errors are our sole responsibility. A previous version of this paper was titled “Equi-librium Asset Pricing and Portfolio Choice with Heterogeneous Preferences”.

Appendix

Proofs: stock price dynamics

Denote by Dt the Malliavin derivative operator.41 The following is the main technical resultof the paper.

Proposition 11.1 The drift and volatility of the stock price are given by

μSt = r(Dt ) + σ S

t

(

E Qt

[γU (DT ) (Dt DT )/DT

]

−T∫

t

E Qt

[r ′(Ds) (Dt Ds)

]ds

)

,

σ St = 1

Et [ξT DT ] Et [(1 − γU (DT )ξT Dt DT )] (26)

−Et

[e∫ T

0 r(Ds )ds(∫ T

t r ′(Ds) (Dt Ds) ds)ξT

]+ Et

[e∫ T

0 r(Ds )ds DtξT

]

Et

[e∫ T

0 r(Ds )ds ξT

] ,

and the optimal portfolio of agent k is given by

πk t = 1

σ St

Et[Dt ξT

(yk ξT I ′

k(yk ξT )+ Ik(yk ξT ))]

Et [ξT Ik(yk ξT )] (27)

−Et

[e∫ T

0 r(Ds )ds(∫ T

t r ′(Ds) (Dt Ds) ds)ξT

]+ Et

[e∫ T

0 r(Ds )ds DtξT

]

Et

[e∫ T

0 r(Ds )ds ξT

] ,

where

Dt DT = DtσD(Dt )e

δT −δt (28)

41 For an expedient introduction to Malliavin derivatives, see [22].

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96 Math Finan Econ (2014) 8:71–108

with

δt =t∫

0

[Ds(μD)′(Ds)− 0.5(Ds(σ

D)′(Ds))2 − Ds(σ

D)′(Ds)σD(Ds)]ds

+t∫

0

Ds(σD)′(Ds)d Bs (29)

and

DtξT = − 1

DTγU (DT )ξT Dt DT .

Proof of Proposition 11.1 By definition,

ξt = e− ∫ t0 r(Ds )ds Mt = e− ∫ t

0 r(Ds )ds Et [MT ] = Et [e∫ T

t r(Ds )ds ξT ].The price St and the wealth of agent k satisfy

log St =t∫

0

r(Ds) ds + log Et [ξT DT ] − log Et [e∫ T

0 r(Ds )ds ξT ]

and

log Wk t =t∫

0

r(Ds) ds + log Et [ξT Ik(yk ξT )] − log Et [e∫ T

0 r(Ds )ds ξT ].

We get the volatilityσ St as the Malliavin derivative Dt log St and we getσ S

t πk t as the Malliavinderivative Dt log Wk t . Thus, we have

πk t = Dt log Wk t

Dt log St. (30)

We will now calculate the Malliavin derivatives. For process D, it is well known that theMalliavin derivative

Yt := Dt Du , u ≥ t

satisfies the linear stochastic differential equation

dYu = (Du(μD)′(Du)+ μD(Du)) Yu du + (Du(σ

D)′(Du)

+σ D(Du)) Yu d Bu , u ≥ t

with

Yt = DtσD(Dt )

and (28) follows. Therefore,

DtξT = U ′′(DT )Dt DT . (31)

Using the identity

Dt Et [X ] = Et [Dt X ]

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Math Finan Econ (2014) 8:71–108 97

we arrive at

Dt log Wk t = 1

Et [ξT Ik(yk ξT )] Et[ykξT I ′

k(yk ξT )DtξT + Ik(yk ξT )DtξT]

−Et

[e∫ T

0 r(Ds )ds(∫ T

t r ′(Ds) (Dt Ds) ds)ξT

]+ Et

[e∫ T

0 r(Ds )ds DtξT

]

Et

[e∫ T

0 r(Ds )ds ξT

]

(32)

and

Dt log St = 1

Et [ξT DT ] Et [DT DtξT + ξT Dt DT ]

−Et

[e∫ T

0 r(Ds )ds(∫ T

t r ′(Ds) (Dt Ds) ds)ξT

]+ Et

[e∫ T

0 r(Ds )ds DtξT

]

Et

[e∫ T

0 r(Ds )ds ξT

] .

(33)

It remains to show the expression for the drift. By the martingale property, we have

d Et [ξT DT ]Et [ξT DT ] = Ut dWt ,

d Et [e∫ T

0 r(Ds )ds ξT ]Et [e

∫ T0 r(Ds )ds ξT ]

= Vt dWt ,

where, by the Clarke–Ocone formula and (31),

Ut = Dt Et [ξT DT ]Et [ξT DT ] = 1

Et [ξT DT ] E[ξT (1 − γU (DT ))Dt DT ]

and

Vt = Dt Et [e∫ T

0 r(Ds )dsξT ]Et [e

∫ T0 r(Ds )dsξT ]

= −Et [e

∫ T0 r(Ds )ds (γU (DT )ξT Dt DT )/DT ] − Et

[e∫ T

0 r(Ds )ds(∫ T

t r ′(Ds) (Dt Ds) ds)ξT

]

Et [e∫ T

0 r(Ds )ds ξT ].

(34)

Applying Ito’s formula, we get

d log St = r(Dt ) dt + d logEt [ξT DT ]

Et [ξT ] = 1

2(2 r(Dt ) + V 2

t − U 2t ) dt

+ (Ut − Vt ) dWt .

Therefore,

μSt = r(Dt ) + 1

2(V 2

t − U 2t + (Ut − Vt )

2) = r(Dt ) + Vt (Vt − Ut )

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98 Math Finan Econ (2014) 8:71–108

and thus, by (33),

μSt = r(Dt )

+Et [e

∫ T0 r(Ds )ds(γU (DT )ξT Dt DT )/DT ] − Et

[e∫ T

0 r(Ds )ds(∫ T

t r ′(Ds) (Dt Ds) ds)ξT

]

Et [e∫ T

0 r(Ds )ds ξT ]× Dt log St

= r(Dt ) + σ St

⎛

⎝ E Qt

[γU (DT ) (Dt DT )/DT

]−

T∫

t

E Qt

[r ′(Ds) (Dt Ds)

]ds

⎞

⎠ .

(35)

�The following result allows us to rewrite the Malliavin derivative Dt D without involving

stochastic integrals. It has also been proved by Detemple et al. [22] in a slightly differentform, but we present a derivation here for the reader’s convenience.

Lemma 11.2 We have

Dt DT = DT σD(DT ) e− ∫ T

t ρs ds . (36)

Proof By Ito’s formula,

log(DT σD(DT )) − log(Dt σ

D(Dt )) =T∫

t

(σ D(Ds)+ Ds (σD)′(Ds)) d Bs

+T∫

t

((σ D(Ds) + Ds (σ

D)′(Ds)) σD(Ds)

−1 μD(Ds))

ds

+ 1

2

T∫

t

((2(σ D)′(Ds)+ Ds(σ

D)′′(Ds)) σD(Ds)

− (σ D(Ds) + (Ds (σD)′(Ds))

2))

ds.

(37)

It remains to show that

ρs = c(Ds)

where

c(x) = −x (μD)′(x) + x (σ D)′(x) σ D(x)−1 μD(x)

+ (σ D)′(x) σ D(x) x + 0.5 (σ D)′′(x) x2 σ D(x).(38)

This claim can be verified by direct calculation. �Proof of Theorem 4.2 The proof follows directly by substituting (36) into (26). �

We will need the following known

Lemma 11.3 For any one-dimensional diffusion, the function

G(t , x) = E[g(DT ) | Dt = x]

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Math Finan Econ (2014) 8:71–108 99

is monotonically increasing (decreasing) in x for all t ∈ [0, T ] if and only if g(x) is monoton-ically increasing (decreasing) in x. Furthermore, if both g(x) and h(x) are increasing (orboth decreasing), then

Et [g(DT )] Et [h(DT )] ≤ Et [g(DT ) h(DT )].If both g and h are strictly increasing (or both strictly decreasing), then the inequality isalso strict unless DT is constant. If one function is increasing and the other is decreasing,the inequality reverses.

Proof See [44]. �Lemma 11.4 Suppose that F and G1, . . . ,G N are monotonically increasing functions. Thenfor any N ∈ N and any {t1 ≤ · · · ≤ tN } ⊂ [t , T ],

E[F(XT )G1(Xt1) . . . G N (XtN ) | Xt = x]is monotonically increasing in x and

Et [F(XT )G1(Xt1) . . . G N (XtN )] ≥ Et [F(XT ) ] Et[

G1(Xt1) . . . G N (XtN )].

Proof The proof is by induction. For N = 1, we have

Et [F(XT )Gt1(Xt1)] = Et [ Et1 [F(XT )] Gt1(Xt1)].By Lemma 11.3, the function inside the expectation is increasing in Xt1 and another applica-tion of Lemma 11.3 provides monotonicity of Et [F(XT )Gt1(Xt1)]. Now, by Lemma 11.3,

Et [ Et1 [F(XT )] G1(Xt1)] ≥ Et [ Et1 [F(XT )] ] Et [Gt1(Xt1)]= Et [F(XT )] Et [Gt1(Xt1)] ,

(39)

and we are done. Suppose now that the claim has been proved for N . Then

Et [F(XT )G1(Xt1) · · · G N (XtN )]= Et [ G1(Xt1) Et1 [ F(XT )G2(Xt2) · · · G N (XtN )] ]≥ Et [Et1 [ F(XT ) ] Et1 [G1(Xt1)G2(Xt2) · · · G N (XtN )] ]

(40)

and the claim follows from Lemma 11.3 and the induction hypothesis. �Lemma 11.5 Suppose that f and g are both increasing (decreasing). Then the following istrue:

(1) the function42

E[

f (DT ) e∫ T

t g(Ds ) ds | Dt = x]

is also monotonically increasing (decreasing);(2) we have

Covt

(h(DT ) , f (DT ) e

∫ Tt g(Ds ) ds

)≥ 0

if h has the same direction of monotonicity as f and g, and the inequality reverses if hhas the opposite direction of monotonicity.

42 Item (1) of Lemma 11.5 is contained in [55].

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100 Math Finan Econ (2014) 8:71–108

Proof of Lemma 11.5 The claim follows from Lemma 11.4, approximating the integral∫ Tt g(Ds) ds by discrete integral sums. �

Proof of Theorem 8.1 The proof follows directly from Proposition 11.1 and (36). The factthat σ S

t > 0 when r ′ is sufficiently small follows from (2), Lemma 11.5 and

σ St = ∂

∂DtS(t , Dt ) σ

D(Dt ).

�Proofs: optimal portfolios

We start with the following auxiliary result:

Proposition 11.6 The optimal portfolio πk t is given by

σ St πk t =

E Qt

[λ(t, T ) e− ∫ T

t r(Ds )ds Wk T

(γ−1

k (Wk T ) − 1)]

Wkt+ λt . (41)

Proof of Proposition 11.6 The claim follows directly from Proposition 11.1 and (36). �Proof of Proposition 5.1 Let t = 0 for simplicity. By definition, the value function is

Uk(x) = E[ uk(Ik(ξT yk))]where yk = yk(x) solves

x = E[ξT Ik(ξT yk)].Differentiating this identity, we get

y′k(x) = E[ξ2

T I ′k(ξT yk)]−1

and therefore

U ′k(x) = E [u′

k(Ik(ξT yk)) I ′k(ξT yk) ξT ] y′

k(x) = E [ ξT yk I ′k(ξT yk) ξT ] y′

k(x) = yk(x).

Consequently,

U ′′k (x) = y′

k(x) = 1

E[ξ2T I ′

k(ξT yk)]and

γk0(x) = − x

yk E[ξ2T I ′

k(ξT yk)]= − E[ξT Ik(ξT yk)]

E[ξ2T yk I ′

k(ξT yk)].

Differentiating the identity

u′k(Ik(x)) = x

we get

I ′k(x) = (u′′

k (x))−1

and therefore

ykξT I ′k(ykξT ) = − γ−1

kT Wk T .

�

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Math Finan Econ (2014) 8:71–108 101

Proof of Theorem 5.2 By Propositions 11.6 and 5.1,

σ St π

hedgingk t = σ S

t

(πk t − π

myopick t

)

=E Q

t

[γU (DT ) σ

D(t , T ) e− ∫ Tt rs ds Wk T

(γ−1

kT − 1)]

E Qt [e− ∫ T

t rs ds WkT ]+ λt

− λtE Q

t [e− ∫ Tt rs ds γ−1

kT Wk T ]E Q

t [e− ∫ Tt rs ds Wk T ]

=E Q

t

[γU (DT ) σ

D(t , T ) e− ∫ Tt rs ds Wk T

(γ−1

kT − 1)]

E Qt [e− ∫ T

t rs ds WkT ]

− λtE Q

t [e− ∫ Tt rs ds (γ−1

kT − 1)WkT ]E Q

t [e− ∫ Tt rs ds WkT ]

= 1

E Qt [Wk T ]

(E Q

t

[γU (DT ) σ

D(t , T ) e− ∫ Tt rs ds Wk T

(γ−1

kT − 1)]

− E Qt [γU (DT ) σ

D(t , T )] E Qt [(γ−1

kT − 1) e− ∫ Tt rs ds Wk T ]

)

+ E Qt [e− ∫ T

t rs ds (γ−1kT − 1)WkT ]

E Qt [e− ∫ T

t rs ds WkT ]E Q

t

⎡

⎣T∫

t

σ r (t, τ ) dτ

⎤

⎦ ,

(42)

Q.E.D. �Lemma 11.7 Suppose γi (x) < a for some a > 0. Then there exists a smooth one-parameterfamily of utilities u(x, β) such that u(x, 0) = ui (x), u(x, 1) = x1−a/(1−a) and the function

I (x, β)de f= (u′(x, β))−1 is such that

−Ix (x, β) x

I (x, β)

is monotonically decreasing in β.

Proof A direct calculation shows that Ii (x) = (u′i (x))

−1 satisfies

− I ′i (x) x

Ii (x)= − u′

i (Ii (x))

u′′i (Ii (x)) Ii (x)

= 1/γi (Ii (x)). (43)

Defining

I (x, β) = exp

⎛

⎝−x∫

1

1/((1 − β)γi (Ii (y))+ β a) y−1 dy

⎞

⎠ ,

we get

−Ix (x, β) x

I (x, β)= 1/((1 − β)γi (Ii (x))+ β a)

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102 Math Finan Econ (2014) 8:71–108

which clearly satisfies the required conditions, and

u′(x, β) = (I (x, β))−1

is the required marginal utility family. �

Proof of Theorem 6.3 Suppose that supx γi (x) ≤ inf x γ j (x) and let a = supx γi (x). Thenit suffices to show that the optimal portfolio holding of agent i is smaller than that of anagent with CRRA a and the portfolio holdings of agent j are larger than that of an agentwith constant relative risk aversion a. We only prove the first claim, as the second claim isanalogous.

Recall that Wit = Ii (yi U ′(DT )) where yi is the Lagrange multiplier for agent i andtherefore, by (43),

(1 − γ−1i (Wi T ))Wi T = I (yi U ′(DT ) , β) + Ix (yi U ′(DT ) , β) yi U ′(DT ).

Consider the family I (x, β) of Lemma 11.7 and define the function

φ(β) = Et [σ(DT ) γU (DT )U ′(DT )

(I (yi U ′(DT ) , β)+ Ix (yi U ′(DT ) , β) yi U ′(DT )

)]Et [U ′(DT ) I (yi U ′(DT ) , β)] .

Then by Proposition 11.6, we have

φ(0) = Et [σ(DT ) γU (DT )U ′(DT )

(Ii (yi U ′(DT ))+ I ′

i (yi U ′(DT )) yi U ′(DT ))]

Et [U ′(DT ) Ii (yi U ′(DT ))]= eρ (T −t) (−σ S

t πi t + λt ).

(44)

Furthermore, a direct calculation shows that

φ(1) =(1 − a−1) Et

[σ(DT ) γ

U (DT )((U ′(DT ))

1−a−1)]

Et [U ′(DT ) (U ′(DT ))1−a−1 ] (45)

is independent of yi . Therefore, by Proposition 11.6,

eρ (T −t) (σ St )

−1(−φ(1)+ λt )

is the optimal portfolio holdings of an agent with risk aversion a. Here, the rate of macro-economic fluctuation ρt = ρ is constant by assumption. Thus, it suffices to show that φ(β)is monotonically increasing.

Differentiating φ(β) with respect to β, we get(Et [U ′(DT ) I (y U ′(DT ) , β)]

)2 × φ′(β)

= Et

[σ(DT ) γ

U (DT )U ′(DT )(

Iβ(yi U ′(DT ) , β)+ Ixβ(yi U ′(DT ) , β) yi U ′(DT ))]

× Et [U ′(DT ) I (yi U ′(DT ) , β)]− Et

[σ(DT ) γ

U (DT )U ′(DT )(

I (yi U ′(DT ) , β)+ Ix (yi U ′(DT ) , β) yi U ′(DT ))]

× Et

[U ′(DT ) Iβ(yi U ′(DT ) , β)

]

(46)

Now, by Corollary 3.2, the assumption that ρt is constant implies that DT = f (AT ) whereAT has a Gaussian distribution. Modifying the function f if necessary, we may assume

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Math Finan Econ (2014) 8:71–108 103

without loss of generality that AT ∼ N (0, 1). Then expression (46), multiplied by 2π, isgiven by

∫

R

e−x2/2 f ′(x) (−U ′′( f (x)))(

Iβ(yi U′( f (x)), β)+ Ixβ(yi U

′( f (x)), β)yi U ′( f (x)))

dx

×∫

R

e−x2/2 U ′( f (x)) I (yi U′( f (x)), β)) dx

−∫

R

e−x2/2 f ′(x) (−U ′′( f (x)))(

I (yi U′( f (x)), β)+ Ix (yi U

′( f (x)), β)yi U′( f (x))

)dx

×∫

R

e−x2/2 U ′( f (x)) Iβ(yi U′( f (x)), β) dx

(47)

Here, we often suppress the dependence on β. Now, we have

Iβ(x, β) + Ixβ(x, β) x dx = d

dx(Iβ(x, β) x)

and therefore, integrating by parts, we get

+∞∫

−∞e−x2/2 f ′(x) (−U ′′( f (x)))

(Iβ(yi U

′( f (x)), β) + Ixβ(yi U′( f (x)), β)yi U ′( f (x))

)dx

= −+∞∫

−∞e−x2/2 d

(Iβ(yi U

′( f (x)), β) yi U ′( f (x)))

= −+∞∫

−∞x e−x2/2 Iβ(yi U ′( f (x)), β) yi U ′( f (x)) dx ,

(48)

where the minus sign appears because U ′ is monotonically decreasing. Similarly,

+∞∫

−∞e−x2/2 f ′(x) (−U ′′( f (x)))

(I (yi U

′( f (x)), β) + Ix (yi U′( f (x)), β)yi U ′( f (x))

)dx

= −+∞∫

−∞e−x2/2 d

(I (yi U

′( f (x)), β) yi U ′( f (x)))

= −+∞∫

−∞x e−x2/2 I (yi U ′( f (x)), β) yi U ′( f (x)) dx ,

(49)

Therefore, the expression in (47) is equal to

123

104 Math Finan Econ (2014) 8:71–108

−∫

R

x e−x2/2 Iβ(yi U′( f (x)), β) yi U

′( f (x)) dx∫

R

e−x2/2 U ′( f (x)) I (yi U′( f (x)), β) dx

+∫

R

x e−x2/2 I (yi U′( f (x)), β)yi U

′( f (x)) dx∫

R

e−x2/2 U ′( f (x)) Iβ(yi U′( f (x)), β) dx .

(50)

By Lemma 11.7,

0 > − ∂

∂β

Ix (x, β)

I (x, β)= − Ixβ I − Ix Iβ

I 2 x ,

that is

Ixβ I > Ix Iβ ⇔ ∂

∂x

IβI> 0.

Consequently, the function

ψ(x) = Iβ(yi U ′( f (x)), β)

I (yi U ′( f (x)), β)

is monotonically decreasing in x .Introduce a new probability measure

dμ = e−x2/2 U ′( f (x)) I (yi U ′( f (x)), β) dx∫

Re−x2/2 U ′( f (x)) I (yi U ′( f (x)), β) dx

.

Then we can rewrite expression (50) in the form

⎛

⎝∫

R

e−x2/2 U ′( f (x)) I (yi U′( f (x)), β) dx

⎞

⎠

2(−E μ[AT ψ(AT )] + E μ[AT ] E μ[ψ(AT )]

)

which is nonnegative by Lemma 11.3. �Proof of Theorem 6.5 By (41), we need to show that

E Qt

[γU (DT ) σ

D(DT ) e− ∫ Tt (ρs+rs )ds WkT

(1 − γ−1

k (Wk T ))]

E Qt [e− ∫ T

t rs ds WkT ]

≥E Q

t

[γU (DT ) σ

D(DT ) e− ∫ Tt (ρs+rs ) ds W jT

(1 − γ−1

j (W j T ))]

E Qt [e− ∫ T

t rs ds W j T ]. (51)

We only prove case (1). Case (2) is analogous. Since, by assumption,

inf (1 − γ−1k T ) ≥ sup (1 − γ−1

j T ),

it suffices to show that

E Qt

[γU (DT ) σ

D(DT ) e− ∫ Tt (ρs+rs )ds WkT

]

E Qt [e− ∫ T

t rs ds WkT ]≥

E Qt

[γU (DT ) σ

D(DT ) e− ∫ Tt (ρs+rs ) ds W j T

]

E Qt [e− ∫ T

t rs ds W jT ].

123

Math Finan Econ (2014) 8:71–108 105

Introduce a new probability measure

d Qk = e− ∫ Tt rs ds Wk T

E Q[e− ∫ Tt rs ds Wk T ]

d Q

and let

f (x) = I j (y j x)

Ik(yk x).

and zi = Ii (λi x) , i ∈ { j , k}. Then

f ′(x) = I j (y j x)

x Ik(yk x)

(

y j xI ′

j (y j x)

I j (y j x)− yk x

I ′k(y j x)

Ik(yk x)

)

= I j (y j x)

x Ik(yk x)

(u′

j (z1)

z1 u′′j (z1)

− u′k(z2)

z2 u′′k (z2)

)

= z1

x z2(γ−1

k (z2) − γ−1j (z1)) ≤ 0.

(52)

That is, f is decreasing. Therefore, f (U ′(DT )) is increasing and, by Lemma 11.5,

E Qt

[γU (DT ) σ

D(DT ) e− ∫ Tt (ρs+rs ) ds W j T

]

E Qt [e− ∫ T

t rs ds W jT ]

=E Qk

t

[γU (DT ) σ

D(DT ) e− ∫ Tt ρs ds f (U ′(DT ))

]

E Qkt [ f (U ′

(DT ))]≤ E Qk

t

[γU (DT ) σ

D(DT ) e− ∫ Tt ρs ds

]

=E Q

t

[γU (DT ) σ

D(DT ) e− ∫ Tt (ρs+rs ) ds WkT

]

E Qt [e− ∫ T

t rs ds WkT ].

(53)

�Proofs of Proposition 9.4, Proposition 9.5 and Theorem 7.1 Propositions 9.4 and 9.5 followdirectly from Lemma 11.5. Theorem 7.1 follows from Theorem 5.2 and Lemma 11.5 since

f (x) = x − Rk(x)

is increasing if and only if

f ′(x) = 1 + (u′′k )

2 − u′k(x) u′′′

k (x)

(u′′k (x))

2 = 1

(u′′k (x))

2 (2 − Pk(x) Rk(x)) ≥ 0 ,

and is decreasing otherwise. �Proof of Proposition 9.8 Item (1) follows from γU ∈ [γ, �]. Items (2) and (3) follow from

0 ≤ −(E Qt [DT ])−1 CovQ

t (γU (DT ) , DT )

= − E Qt [γU (DT )DT ]

E Qt [DT ] + E Q

t [γU (DT )] ≤ −γ + �.(54)

123

106 Math Finan Econ (2014) 8:71–108

Similarly,

0 ≤ −(E Qt [WkT ])−1 CovQ

t (γU (DT ) , Wk T )

= − E Qt [γU (DT )Wk T ]

E Qt [Wk T ] + E Q

t [γU (DT )] ≤ −γ + �(55)

and items (2–3) immediately yield the required inequality for γk > 1. The case γk < 1follows from the directly verifiable identity

πk t = σ e−b (T −t)

σ St

(

γ−1k

E Qt [γU (DT )Wk T ]

E Qt [Wk T ] − CovQ

t (γU (DT ) , Wk T )

)

.

�

Example 11.8 Suppose that the stock price St takes value 1 at time 0, values (u, d) =(1.5, 0.5) with probabilities 0.5 at time 1, and the IR is 0. In the second period, conditionedon u (resp. d), S2/S1 = (u1, d1) (resp. (u2, d2)). Let us denote by π0 γ the optimal stockweight at time 0 of a CRRA investor with risk aversion γ. Then a direct calculation showsthat

π0 γ = −2 + 4

1 + m1−b2 +(2−m2)1−b

m1−b1 +(2−m1)1−b

where m1 = 2(1−d1)u1−d1

, m2 = 2(1−d2)u2−d2

and b = γ−1. Clearly, by choosing different (u1, d1)

and (u2, d2) we can achieve any desirable monotonicity behavior in γ.

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