scribed in Chapter 7? We shall use intertemporal cquilihiiuin models lo explore these questions.

Section 8.1 begins bv staling tin1 proposition that there exists a stochastic discount factor such that the expected product of any asset return with the stochastic discount factor equals one. This proposition holds very generally in models thai rule out arbitrage opportunities in financial markets. Equilibrium models with optimizing investors imply light links bclween the stochastic discount factor and the marginal utilities of investors consumption. Thus by studying ihe stochastic discount factor one can relate asset prices to the undcrlving preferences of investors.

In Section 8.1 we show how the behavior of asscl prices can be used lo reach conclusions about the behavior of the stochastic discount factor. In particular wc describe Hansen and Jagannathans (1991) procedure for calculating a lower bound on the volatility of the stochastic discount factor, given any set of asset relurns. Using long-run annual dala on US short-term interest rates and slock relurns over die period 1889 to 1994, we estimate the standard deviation of the stochastic discount factor to be 30% per year or more.

Consumption-based asset pricing models aggregate investors into a single representative agent, who is assumed lo derive utility from the aggregate consumption ol the economy. In these models the stochastic discount laclor is the intertemporal marginal rale of substitution-the discounted ratio of marginal iiiiliiies in two successive periods-for the representative agent. The Enterequations-the first-order conditions for optimal consumption and portfolio choices of the representative agent-can be used to link asset returns and consumption.

Section 8.2 discusses a commonly used consumption-based model in which the representative agent has lime-separable power utility. In this model a single parameter governs both risk aversion and the elasticity of intertemporal substitution-the willingness of the representative agent to adjust planned consumption growth in response to investment opportunities. In fact, the elasticity of iiileileinporal substitution is the reciprocal of risk aversion, so in this model risk-averse investors must also be unwilling to adjust their consumption growth rates lo changes in interest rales. The model explains ihe risk premia on assets by their covariances with aggregate consumption growth, multiplied by the risk-aversion coefficient for the representative investor.

Using long-run annual US dala, we emphasize four stylized facts. First, the average excess return on US slocks over shorl-ienn debt-the equity premium-is about f>% per year. Second, aggregate consumption is very smooth, so covariances with consumption growth are small. Cutting these facts together, the power utility model can only fit the equity premium if ihe coefficient of relative risk aversion is very large. This is the equity premium

8.1 The Stochastic Discount Factor

We begin our analysis of the stochastic discount factor in the simplest possible way, by considering the intertemporal choice problem of an investor who can trade freely in asset i and who maximizes the expectation of a time-separable utility function:

Max E, y&UC+f) , (8.1.1)

where S is the lime discount factor, C/+j is the investors consumption in period / + j, and U(C,+j) is the period utility of consumption al t 4- j. One of the first-order conditions or Euler equations describing the investors

: - V.s

puzzle of Mehra and Prescott (1985). Third, there are some predictable movements in short-term real interest rates, but there is little evidence of accompanying predictable movements in consumption growth. This suggests that the elasticity of intertemporal substitution is small, which in the power utility model again implies a large coefficient of relative risk aversion. Finally, there are predictable variations in excess returns on stocks over short-term debt which do not seem to be related to changing covariances of slock returns with consumption growth. These lead formal statistical tests to reject the power-utility model.

In Sections 8.3 and 8.4 we explore some ways in which the basic model can be modified to fit these facts. In Section 8.3 we discuss the effects of markei frictions such as transactions costs, limits on investors ability to borrow or sell assets short, exogenous variation in the asset demands of some investors, and income risks that investors are unable to insure. We argue that many plausible frictions make aggregate consumption an inadequate proxy for the consumption of stock market investors, and we discuss ways to get testable restrictions on asset prices even when consumption is not measured. Wc also discuss a generalization of power utility that breaks the tight link between risk aversion and the elasticity of intertemporal substitution.

In Section 8.4 we explore the possibility that investors have more complicated preferences than generalized power utility. For example, the utility function of the representative agent may be nonseparable between consumption and some other good such as leisure. We emphasize models in which utility is nonseparable over time because investors derive utility from the level of consumption relative to a time-varying habit or subsistence level. Finally, we consider some unorthodox models that draw inspiration from experimental and psychological research.

8. Intertemporal Equilibrium Models

8.1. ihe Storliaslir Discount Factor

optimal consumption and portfolio plan is

U(C,) = <5E,[(1 +/? ,+ , )U\Q+l)]. (8.1.2)

The left-hand side of (8.1.2) is the marginal utility cost of consuming one real dollar less at time t; the right-hand side is the expected marginal utility benefit from investing the dollar in asset i at time (, selling it at time t+1 for (1 + Рч,1+\) dollars, and consuming the proceeds. The investor equates marginal cost and marginal benefit, so (8.1.2) describes the optimum. If we divide both the left- and right-hand sides of (8.1.2) by U(C,), wc

1 = E,[(l + Ru+l)Ml+i), (8.1.3)

where = 8U(Cl+l)/U{Q). The variable M,+i in (8.1.3) is known as the stochastic discount factor, or pricing kernel In the present model it is equivalent

j to the discounted ratio of marginal utilities S (7(C,+ )/ U{C,), which is called the intertemporal marginal rale of substitution. Note that the intertemporal

marginal rate of substitution, and hence the stochastic discount factor, arc

I always positive since marginal utilities are positive.

I Expectations in (8.1.3) are taken conditional on information available al time (; however, by taking unconditional expectations of the left- and right-hand sides of (8.1.3) and lagging one period to simplify notation, wc obtain an unconditional version:

1 = E((l + Н )М,]. (8.1.4)

These relationships can be rearranged so that they explicitly determine jcxpected asset returns. Working with the unconditional form for convenience, wc have E[(l 4- R )M,] = E[l + /< ]ЕЩ] + Cov[ ,- M,J, so

E[l +11 } = - Cov{R, M,}). (8.1.5)

If there is an asset whose unconditional covariance with the stochastic discount factor is zero-an unconditional zero-beta asset-then (8.1.5) implies that this assets expected gross return K[ 1 + It ,] = 1/E[M,]. This can be substituted into (8.1.5) to obtain an expression for the excess return Z on asset i over the zero-beta return:

E[Z ] ее E[/< -/< ] = -E[l + /4 ]Cov[/l, M,]. ( .!.( ,)

This shows that an assets expected return is greater, the smaller its covariance with the stochastic discount factor. The intuition behind this result is that an asset whose covariance with M,+ i is small tends to have low returns when the investors marginal utility of consumption is high-that is, when consumption itself is low. Such an asset is risky in that il fails to deliver

wealth precisely when wealth is most valuable to the investor. The investor therefore demands a large risk premium lo hold it.

Although it is easiest to understand (8.1.3) by reference lo the intertemporal choice problem of an investor, the equation can be derived merely from the absence of arbitrage, without assuming that investors maximize well-behaved utility functions.1 We show this in a discrete-stale setting with states s= I...S and assets i = 1 ... N. Define </,- as the price of asset i and q as the (Лх 1) vector of asset prices, and define X as die payoff of asset i in slate л and X as an (SxN) matrix giving the payolfs of each asset in each state. Provided that all asscl prices are nonzero, we can further define G as an (.SxN) matrix giving the gross return on each asset in each stale. That is, the typical element of G is G = 1 + R = XJq,.

Now define an (.Sxl) vector p, with typical element p to be a slate price vector if it satisfies Xp = q. An asset can be thought of as a bundle of.state-contingent payoffs, one for each stale; the Mb element of the state price vector, p gives the price of one dollar to be paid in stale s, and wc represent each asset price as the sum of its state-contingent payoffs times the appropriate state prices: q, = £, ptX . Equivalent, if we divide through by % wc Kct 1 = E. / (! + ) or Gp = i, where i is an (.Sx 1) vector of ones.

An important result is dial there exists a positive stale price vector if and only if there are no arbitrage opportunities (that is, no available assets or combinations of assets with nonpositive cost today, noiinegative payolfs tomorrow, and a strictly positive payoff in al leasl one stale). Iurlhcriiiore, il there exists a positive stale price vector, then (8.1.3) is satisfied for some positive random variable M. To see this, define Л{, = pjn where я, is the probability of state .v. for any asset the relationship Gp = /. implies

1 = ypl+RJ = JnM{\ + R ) = Y.[(\ + U,)M), (8.1.7) >=i i=t

which is the static discrete-state equivalent of (8.1.3). M, is the ratio ofthe stale price of stale s to the probability of state s; hence it is positive because stale prices and probabilities arc both positive.

If M, is small, then state s is cheap in the sense that investors are unwilling lo pay a high price to receive wealth in state v. An asset that tends to deliver wealth in cheap states has a return thai covaries negatively with M. Such an asset is itself cheap and has a high return on average. This is the intuition for (8.1.0) within a discrete-state framework.

In the discrete-state model, asset markets are complete if for each slate s, one can combine available assets lo gel a nonzero payoff in s and zero

The theory mulei lying cill.itii)ii (H.l.:t) istlisi tisMil.it length iii textbooks sm lias Ingctsoll (1987). The rule nl (ondilioning inlorinaiioii has heell exiiliueit hv Hansen anil Kichanl (1987).

.v. Intertemporal Equilibrium Models

payoffs in all other siairs. Л further important result is that the state price vector is mii(ue if and only if asset markets are complete. In tins case M is unique, Inn with incomplete markets lliere may exist many Ms satisfying equation (К. I.3). This result can he understood by considering an economy with several iiiiliiy-inaxiini/ing investors. The fust-order condition (8.1.2) holds for each investoi, so each investors marginal utilities can he used to construct a stochastic discount factor that prices the assets in the economy. Willi complete markets, the investors marginal utilities are perfectly correlated so lltev all yield the same, unique stochastic discount factor; with incomplete markets then- may he idiosyncratic variation in marginal utilities and hence multiple stochastic discount factors that satisfy (8.1.3).

.S. /. / Volatility Bounds

Any model ol expected asset returns may he viewed as a model of the stochastic discount factor, Before we discuss methods of testing particular models, we ask more generally what asset relurn daia may be able lo tell us about the behavior of the stochastic discount lac lor. I lansen and Jagannathan (1991) have developed a lower bound on the volatility of stochastic discount factors that could be consistent with a given set of asset return data. They begin with the unconditional equation (8.1/1) and rewrite it in vector form as

/. = K(t + R,)cVf,]. (8.1.8)

where i is an A/-vei tor of ones and R, is the N-vector of tiinc-f asset returns, with typical element Л .

Hansen and Jagannathan assume that R, has a nonsiiigtilar variance-covariance matrix ii, in other words, that no asset or combination of assets is unconditionally riskless. There may still exist an unconditional zero-beta asset with gross mean return equal lo die reciprocal of die unconditional mean olthc stochastic discount lactor, but I lansen and Jagannathan assume that il there is such an asset, its identity is not known a priori. Hence they treat the unconditional mean of the stochastic discount factor as an unknown parameter Л/. For each possible Л/, Hansen and Jagannathan form a candidate stochastic discount factor M(M) as a linear combination of asset returns. Thcv show dial die variance of M*(M) places a lower bound on the variance of any stochastic disc omit factor that has mean Л7 and satisfies (8.1.8).

I lansen and ]agaiinathan first show how asset pricing theory determines the coefficients ft- in

Л/;(Л7) = Л7 I (R, - KR,I)ft-. (S.I.9)

If Л1,(Л() is lo be a stochastic discount factor it must satisfy (8.1.8),

/. = кi(;. i к,)Л/(*(л7)1.

8.1. The Stochastic Discount Factor

Expanding the expectation of the product E[(t + Rt)Af*(Af)], we

i = ME[t + R,]+Cov[R M,*(M)]

= ME[c + R,] + E[(R, - E[R,])(m;(M) - Щ

= ME[t + R,l + E[(R, - E[R,])(R, - E[R,])0jj]

= ME[i + R/] + П/37, (8.1.10)

where fl is the unconditional variance-covariance matrix of asset returns. It follows then that j

ftTl = n-d-MEU + R,]). (8.1.llj

and the variance of the implied stochastic discount factor is

Var[M;<A()] = ft-Slftu

= (t- ME[i + R,])fi-U-ME[c.-fR,]). (8.1.12)

The right-hand side of (8.1.12) is a lower bound on the volatility of an stochastic discount factor with mean M. To see this, note that any other M,(M) satisfying (8.1.8) must have the property

E[(l + R,)(M,M)-M;(M))] = Coy[R M,(M)-M*(M)] = 0. (8.1.13)

Since M*(Ai) is just a linear combination of asset returns, it follows that Cov[/Vf,*(A7), M,(M) - M*(M)] = 0. Thus

Var[M,(7Vf)] = Var[M;(M)]+Var[M,(M)-M;(M)]

+ Cov[m;(M), m,(m) - m,(m)]

= Var[/Vf*(Л?)] + Var[/Vf,(Л?) - Af*(m)]

> Var[M*(M)]. (8.1.14)

In fact, we can go beyond this inequality to show that

r n Var[M,*(M)l

VarfM,(M)l =--J - ,., (8.1.15)

(Corr[M,(M), M,(M)])

so a stochastic discount factor can only have a variance close to the lower bound il it is highly correlated with the combination of asset returns M*(M).