turn on ttic market and revisions in expectations of future returns on die market:

Gov, [r,.,+ i, До+il

where

7,h = Cov,

r;.i+i. Ki+t

.7=1

rw,l+l+)

о in, + (1 - V) .A.

Lv=i

(8.3.12)

(8.3.13)

ст./, is defined to be the covariance of the return on asset i with news about future returns on the market, i.e., revisions in expected future re-

turns.

I Substituting (8.3.12) into (8.3.7) and using the definition off/ in terms pf the underlying parameters a and y, wc obtain a cross-scetiona! asset pricing formula that makes no reference lo consumption:

E([r,-,i+i] - 1 + у = yo,m + (y-

l)0 l7,-

(8.3.14)

Equation (8.3.14) has several striking features. First, assets can he priced without direct reference lo their covariance with consumption growth, using instead their covariances witli the return on invested wealth and with news about future returns on invested wealth. This is a discrete-lime analogue of Merlons (1973a) continuous-time model in which assets are priced using their covariances with certain hedge portfolios that index changes in die investment opportunity set.

Second, the only parameter of the utility function that enters (8.3.14) is the coefficient of relative risk aversion y. The elasticity of intertemporal substitution i does not appear once consumption has been substituted out of the model. This is in striking contrast with the important role played by ф in the consumption-based Eulcr equation (8.3.7). Intuitively, this result conies from the fact that Ф plays two roles in the theory. A low value of f reduces anticipated fluctuations in consumption, but it also increases the risk premium required to compensate for any contribution lo these fluctuations. These offsetting effects lead \jr lo cancel oul of the asset-based pricing formula (8.3.14).

Third, (8.3.14) expresses the risk premium, net of the Jensens Inequality adjustment, as a weighted sum of two terms. The first term is the assets covariance with the market portfolio; the weight on this term is the coefficient of relative risk aversion y. The second term is the assets covariance with news about future returns on the market; this receives a weight of у - 1. When у is less than one, assets lhat do well when there is good news about future returns on the market have lower mean returns, but when у is greater

than one, such assets have higher mean returns. The intuitive explanation is that such assets arc desirable because they enable the consumer to profit from improved investment opportunities, but undesirable because they reduce the consumers ability to hedge against a deterioration in investment opportunities. When у < 1 the former effect dominates, and consumers are willing to accept a lower return in order to hold assets dial pay off when wealth is most productive. When у > 1 the latter effect dominates, and consumers require a higher return to hold such assets.

There are several possible circumstances under which assets can be priced using only their covariances with the relurn on the market portfolio, as in the logarithmic version of the static CAIM. These cases have been discussed in the literature on intertemporal asset pricing, but (8.3.14) makes it particularly easy lo understand them. First, if ihe coefficient of relative risk aversion у = I, then the opposing effects of covariance with investment opportunities cancel out so thai only covariance with the market return is relevant for asset pricing. Second, if the investment opportunity set is constant, then (7,y, is zero for all assets, so again assets can be priced using only then covariances with ihe market return. Third, if the return on the market follows a univariate stochastic process, then news about future returns is perfectly correlated with the current return; thus, covariance with the current return is a sufficient statistic for covariance with news about future returns and can be used to price all assets. Campbell (I99(>a) argues that the first two cases do not describe US data even approximately, but that the third case is empirically relevant.

A Third Look al the Equity Premium Puzzle

(8.3.14) can be applied to the risk premium on the market itself. When i = m, we get

K/[r ,.,+ i] - r/ + 1 + у = Vol + (y - Do- ,. (8.3.15)

When the market return is unforecastable, there are no revisions of expectations in future returns, so amh - 0. in ibis case the equity premium with llicjensens Inequality adjustment is jusl yafn, and the coefficient of relative risk aversion can be estimated in the manner of Friend and IMrniie (1975) by taking the ratio of die equity premium lo the variance of the market return. Using the numbers from fable 8.1, the estimate of risk aversion is 0.0575/0.0315 = 1.828. This is the risk-aversion coefficient of an investor with power utility whose wealth is entirely invested in a portfolio with an unforecastable return, a risk premium of 5.75% pet year, and a variance of 0.0315 (standard deviation of 17.74% per year). The consumption of such an investor would also have a standard deviation of 17.74% per year. This is far greater than the volatility of measured aggregate consumption in

Table 8.1, which explains why the risk-aversion estimate is much lower than the consumption-based estimates discussed earlier.

The Friend and hlunic (I97f>) procedure can be seriously misleading if the market return is serially correlated. If high stock returns are associated with downward revisions of future returns, for example, then a, tl is negative in (8.3.1Г>). Willi у > 1, ibis reduces the equity risk premium associated with any level of у and increases the risk-aversion coefficient needed to explain a given equity premium. Intuitively, when a ,/, < 0 the long-run risk of slock market investment is less than die short-run risk because the market tends lo inean-ieverl. Investors with high у care about long-run risk rather than short-run risk, so the Friend and Blume calculation overstates risk and correspondingly understates the risk aversion needed to justify the equity premium.

Campbell (IWlia) shows thai the estimated coefficient of relative risk aversion rises by a factor often or more if one allows for the empirically estimated degree of mean-reversion in postwar monthly US dala. In long-run annual US dala the effect is less dramatic but still goes in the same direction. Campbell also shows (hat risk-aversion estimates increase if one allows for human capital as a component of wealth. In ibis sense one can derive die equity premium puzzle without any direct reference to consumption data.

An Equilibrium Multijactor Asset Pricing Model

Willi a few more assumptions, (8.3. Il) can be used to derive an equilibrium multifactor asset pricing model of ihe type discussed in Chapter (i. We write the return on the market as die first element of a Л-eIement state vector x,+ ). The other elements are variables that are known to the market by the end of period /4- I and arc relevant lor forecasting future returns on the market. We assume that the vector х,ц follows a first-order vector autoregression (VAR):

xM, = Ax, + 6,H. (8.3.16)

The assumption that die VAU is fust-order is not restrictive, since a higher-order VAR can always be stacked into fust-order form.

Next we define a Л-elemenl vector 1, whose lirsl element is one and whose other elements are all zero. This vector picks out the real stock return ; , I from the veiloi xM i: >, fi = elx/+i, and r , ,+ - K, r , ,+ = elcH. The lust-order VAU generates simple multiperiod forecasts of future relurns:

F., г ,. , I = cTA+x,. (8.3.17)

It follows that the discounted sum of revisions in forecast returns can be written as

Ei+i

r ..i+i+;

P7n .n-i+, = еГ£рАе,+1 U=- J >=> I

= elM(I-M)-,Ci+i = ve,+ i, (8.3.18)

where уУ is defined to equal elpA(I - pA)~, a nonlinear function of the VAR coefficients. The elements of the vector <p measure the importance of each state variable in forecasting future returns on the market. If a particular element tp* is large and positive, then a shock to variable к is an important piece of good news about future investment opportunities. We now define

aik = Cov,[7V./+i. ?*./+1]. (8.3.19)

where 6* + i is the Ath element of €,+. Since the first element of the state vector is the return on the market, оц = аш. Then (8.3.14) implies that

КДг + 1] - >/.,+, = --у + yan + (y- 1) £wTib (83.20)

where tpk is the Ath element of tp. This is a standard /чГ-factor asset pricing model of the type discussed in Chapter 6. The contribution of the intertemporal optimization problem is a set of restrictions on the risk prices of the factors. The first factor (the innovation in the market return) has a risk price of K\ = у + (у - \)ч>\. The sign of <p\ is the sign of the correlation between markei return innovations and revisions in expected future market returns. As we have already discussed, this sign affects the risk price of the markei factor; with a negative <p\, for example, the market factor risk price is reduced if у is greater than one.

The other factors in this model have risk prices of Л* = (у - \)ipk for A > 1. Factors here are innovations in variables that help to forecast the return on the market, and their risk prices are proportional to their forecasting importance as measured by the elements of the vector <fi. If a particular variable has a positive value of <pk, this means that innovations in that variable are associated with good news about future investment opportunities. Such a variable will have a negative risk price if the coefficient of relative risk aversion у is less than one, and a positive risk price if у is greater than one.

Campbell (1996a) estimates this model on long-term annual and post-World War II monthly US stock markei data. He estimates щ to be negative and large in absolute value, so that the price ofstock market risk Xi is much

smaller than Ihe coefficient of risk aversion y. The other factors in the model have imprecisely estimated risk prices. Although some of these risk prices arc substantial in magnitude, the other factors have minor effects on the mean returns of the assets in the study, because these assets typically have small covariances with the other factors.

8.4 More General Utility Functions

One straightforward response to the difficulties of the standard consumption CAPM is lo generalize the utility function. We have already discussed the Epstcin-Zin-Wcil model, but there are other plausible ways to vary the utility specification while retaining the attractive scale-independence properly of power utility.

For example, ihe utility function may be nonscparable in consumption and some other good. This is easy to handle in a loglincar model if utility is Cobb-Douglas, so thai die marginal utility of consumption can be written as

UcACt.x,) = c;y,x-y (8-4-1)

for some good X, and parameter yj>. The Filler equation now becomes

1 = E,

(8.4.2)

Assuming joint lognormalily and homoskedaslicity, ibis can be written as

E,[ru+I] = m + yiE,[Ac,+i]4-y,E,[Ax,+ iJ. (8.4.3)

Eichcnbaum, Hansen, and Singleton (1988) have considered a model of this form where X, is leisure. Aschauer (1985) andSiartz (1989) have developed models in which X, is government spending and the slock of durable goods respectively. Unfortunately, none of these extra variables greatly improve the ability of the consumption CAPM lo lit die data. The difficulty is that, at least in data since World War II, these variables are not noisy enough to have much effect on the intertemporal marginal rate of substitution.

HA. I Ilahil Formation

A more promising variation of the basic model is to allow for nonscparabil-iily in utility over time. Constantinides (1990) and Sundarcsan (1989) have i

lfAlso, as Campbell and Mankiw (1490) point out, in postwar data there is predictable variation in consumption growth that is uncorrelated with predictable variation in teal interest rales even after one allows lor predictable variation in leisure, government spending, or durable goods.

argued for the importance of habit formation, a positive effect of todayscon-sumption on tomorrows marginal utility of consumption. I lere we discuss some simple ways lo implement this idea.

Several modeling issues arise al the outset. We write (he period utility function as (/(С X,), where X, is the time-varying habit or subsistence level. The first issue is the functional form lor /(). Abel (1990, 1996) has proposed that (7(-) should be a power function of the ralio C,/X while Campbell and Cochrane (1995), Constantinides (1990), and Sundarcsan (1989) have used a power function of the difference C,-X,. The second issue is the effect of an agents own decisions on future levels of habit. In standard internal-habit models such as those in Cousiauliiiides (1990) and Sundarcsan (1989), habit depends on an agents own consumption and the agent lakes account of this when choosing how much lo consume. In external-habit models such as those in Abel (1990, 1996) and Campbell and Cochrane (1995), habit depends on aggregate consumption which is unaffected by any one agents decisions. Abel calls this ratihing up with IheJoneses. The third issue is the speed with which habit reacts to individual or aggregate consumption. Abel (1990, 1996), Dunn and Singleton (1986), and Fcrson and Constantinides (1991) make habit depend on one lag of consumption, whereas Constantinides (1990), Sundarcsan (1989), Campbell and ( .ochrane (1995), and Ilcatun (1995) make habtl react only gradually to changes in consumption.

Ralio Models

Following Abel (1990, 1996), suppose that an agents utility can be written as a power function of the ralio G,/X

U, = E,£s,y-l.

where A, summarizes die influence of past consumption levels on todays utility. X, can be specified as an internal habit or as an external habit. Using one lag of consumption for simplicity, we may have

X, = (8.4.5)

the internal-habit specification where an agents own past consumption matters, or

X, = с.,. (8.4.6)

the external-habit specification where aggregate past consumption C, matters. Since there is a representative agent, in equilibrium the agents consumption must of course equal aggregate consumption, but the (wo formulations yield different Filler equations. In both equations the parameter * governs the degree ol fmie-nonseparabilily.