i mi in { они СлциЩптш Models

Figure 8.3, whose format follows Hansen and Jagannathan (1991), also illustrates die equity premium puzzle. The figure shows the feasible region for the slot hasiir discount factor implied by equation (8.1.12) and the annual data on real slock and commercial paper returns over the period 1891 to 1991. Ihe figure does not use the noniicgativily restriction discussed in the previous section. The global minimum standard deviation for the stochastic discount lactor is about 0.33, corresponding to a mean stochastic discount factor of about 0.98 and an unconditional riskless return of about 2%. As Ihe mean moves away from 0.98, the standard-deviation bound rapidly increases. The difference between the feasible region in Figure 8.3 and the region above a ray from the origin with slope 0.33 is caused by the fad thai Figure 8.3 uses bond and stock returns separately rather than merely Ihe excess relurn on slocks over bonds. The figure also shows mean-standard deviation points corresponding to various degrees of risk aversion and a fixed lime discount rale in a consumption-based representative agent asset pricing model ol die type discussed in the next section. The first point above the horizontal axis has relative risk aversion of one; successive points have risk aversion of two, three, and so on. The points do not enter the feasible region until relative risk aversion reaches a value of 2Г>.

In interpreting figure 8.3 and similar figures, one should keep in mind thai both the volatility bound for the stochastic discount factor and the points implied by particular asset pricing models are estimated with error. Statistical methods are available to lest whether a particular model satisfies the volatility bound (see for example liuriiside (1994], (дч (belli, l.am, and Mark I 199l, and Hansen, Ilealon, and I.utlmer [1995]). These methods use the Generalized Method ol Moments of Hansen (1982), discussed in the Appendix.

8.2 Consumption-Rased Asset Pricing with Power Utility

In Section 8.1 wc showed how an equation relating asset returns to the stochastic discount factor, (8.1.3), could be derived from the first-order rondilioti ol a single investors intertemporal consumption and porllolio choice problem. This equation is restated here for convenience:

I = K,(l + /, , )Л/,+ . (8.2.1)

It is common in empirical research lo assume that individuals can be aggregated into a single representative investor, so thai aggregate consumption can be used in place of the consumption of any particular individual. Equation (8.2.1) with Л1 , = bl( (. ,)/ V ((.,), where c, is aggregate consumption, is known as the consumption caim, or (ICAPM.

8.2. Consumption-Based Asset li и mgwilh Power Utility

In this section we examine the empirical implications of the CCAPM. We begin by assuming that there is a representative agent who maximizes a time-separable power utility function, so that

г-у 1

(/(С,) = -L-, (8.2.2)

1 -У i

where у is the coefficient of relative risk aversion. As у approaches one, the utility function in (8.2.2) approaches the log utility function U(C,) *= log(C,).

The power utility function has several important properties. First, t is scale-invariant: With constant return distributions, risk premia do not change over time as aggregate wealth and the scale of the economy increase. A related property is that if different investors in the economy have the same power utility function and can freely trade all the risks they face, then even if they have different wealth levels they can be aggregated into a single representative investor with the same utility function as the individual investors.4 This provides some justification for the use of aggregate consumption, rather than individual consumption, in the CCAPM. \

A property of power utility that may be less desirable is that it rigidly links two important concepts. When utility has the power form the elasticity of intertemporal substitution (the derivative of planned log consumption growth with respect to the log interest rate), which we write as ip, is the reciprocal of the coefficient of relative risk aversion/. Hall (1988) has argued that thi linkage is inappropriate because the elasticity of intertemporal subsdtudon concerns the willingness of an investor to move consumpdon between time periods-it is well-defined even if there is no uncertainty-whereas the coefficient of relative risk aversion concerns the willingness of an investor to move consumption between states of the world-it is well-defined even in a one-period model with no time dimension. In Section 8.3.2 below we discuss a more general utility specification, due to Epstein and Zin (1991) and Weil (1989), that preserves the scale-invariance of power utility but breaks the tight link between the coefficient of relative risk aversion and the elasticity of intertemporal substitution.

Taking the derivative of (8.2.2) with respect to consumption, we find that marginal utility U(C,) = C]~Y. Substituting into (8.2.1) we get

1 = E,

(1 +Ri.m)S

(8.2.3)

(.nissiii.m and Shillcr (P.ik2) show that this result generalizes lo a model with nonlraded assets (uninsurable idiosyncratic risks) if consumption and asset prices follow diffusion processes in a continuous-time model.

which was first derived by Grossman and Shiller (1081), following the closely related continuous-lime model of Brccdcn (1979). A typical objective of empirical research is to estimate the coefficient of relative risk aversion у (or its reciprocal ф) and to test the restrictions imposed by (8.2.3). It is easiest to do this if one assumes lhat asset returns and aggregate consumption H** are jointly homoskedastic and lognormal. Although this implies constant

Ж expected excess log returns, and thus cannot fit the dala, il is useful for

*§§** building intuition and understanding methods lhat can be applied to more

Ш realistic models. Accordingly wc make this assumption in Section 8.2.1

<* and relax it in Section 8.2.2, where we discuss the use of Hansens (1982)

Щ Generalized Method of Moments (GMM) to test the power utility model

*► without making distributional assumptions on returns and consumption.

Jfc Our discussion follows closely the seminal work of Hansen and Singleton

(1982,1983).

8.2.1 Power Utility in a Lognormal Model

When a random variable X is conditionally lognormally distributed, it has the convenient property (mentioned in Chapter 1) that.

logE,[X] = E,[logX] + IVar,[logXJ.

(8.2.4)

where Var,[log X] = E,[(log X - E,[log X])2}. If in addition X is conditionally homoskedastic, then Var, [log X] = E[(logX - E,[logX])2] = Var[logX - E,[logX]). Thus with joint conditional lognonnality and ho-moskedasticity of asset returns and consumption, we can take logs of (8.2.3), use ihe notational convention that lowercase letters denote logs, and obtain

0 = Е,[ц,+,] + log<5 - уЕ,[До+] + (I) [of + ГЧ - 2уаЛ (8.2.5)

Here the notation axy denotes the unconditional covariance of innovations Wov[x,+ i - E,[*i+], yi+l ~ E,[y,+ ]], and a2 = axx.

Equation (8.2.5), which was first derived by Hansen and Singleton (1983), has both time-series and cross-sectional implications. In the lime scries, the riskless real interest rale obeys

y2a2

log 5- L~ +yEJAc+i]

(8.2.6)

Tjhe riskless real rate is linear in expected consumption growth, with slope coefficient equal to the coefficient of relative risk aversion. The equation ckn be reversed to express expected consumption growth as a linear function of the riskless real interest rate, with slope coefficient \p = l/y; in fact this relation between expected consumption growth and the interest rate is what (hfines the elasticity of intertemporal substitution.

The assumption of homoskedaslicily makes the log risk premium on any asset over the riskless real rate constant, so expected real relurns on other assets are also linear in expected consumption growth with slope coefficient у. Wc have

Е/1Л/+1 - /.<+]] + у = ya . (8.2.7)

The variance term on the left-hand side of (8.2.7) is a Jensens Inequality adjustment arising from the fact that we are describing expectations of log returns. We can eliminate the need for this adjustment by rewriting the equation in terms of the log of the expected ratio of gross returns:

logE,[(l + /. ,+ ,)/(! +/./.,+ ,)! = yoic.

Equation (8.2.7) shows that risk premia are determined by the coefficient of relative risk aversion times covariance with consumption growth. Of course, we have already presented evidence against the implication of this model thai risk premia arc constant. Nevertheless we explore the model as a useful way to develop intuition and understand econometric techniques used in more general models.

A Second Look al the Equity Premium Puzzle

Equation (8.2.7) clarifies the argument of Mehra and lrcscott (1985) that the equity premium is too high to be consistent with observed consumption behavior unless investors are extremely risk averse. Mehra and Prcscolts analysis is complicated by the lad that they do not use observed stock returns, but instead calculate stock returns implied by the (cotinterlactual) assumption that slock dividends equal consumption. Problem 8.2 carries out a loglinear version of this calculation.

One can appreciate the equity premium puz/.le more directly by examining the moments of log real stock and commercial paper returns and consumption growth shown in Table 8.1. The asset returns are measured annually over the period 1889 to 1994. The mean excess log return of stocks over commercial paper is 4.2% with a standard deviation of 17.7%; using the formula for the mean ofa lognormal random variable, this implies that the mean excess simple return is 6% as stated earlier.

As is conventional in the literature, the consumption measure used in Table 8.1 is consumption of nondurables and services.5 The covariance of the excess log stock return with log consumption growth is the correlation of the two series, limes the standard deviation of the log slock return, times the standard deviation of log consumption growth. Because consumption of nondurables and services is a smooth series, log consumption growth has

This implicitly assumes that utility is sepaiahlc across this lotiti ol t onsuinptinn and other sources оГ utility. In Section 8/1 we discuss ways in which this assumption can he telaxed.

Moments о/ consumption growth and asset returns.

Consuinpiinn growl h is iln-1 hinge in log real 4>nsuni>lion of nondurable), and services. The

stork return is ihe log real leliir.......he SX.I WW index since H!2(i, anil ihe return on a

comparable index trout Cn.ssinan and Sliilli-r (НЖ1) Wore l.l2b. CI is the real return on 0-iiioiiih (otnnieti ial paper, bought in anuarv anil tolled over in luly. All data are annual IKKI to W.l 1.

a small standard deviation of only .4.3%; hence the excess stock return has a invariant e with log consumption growlh of only 0.003 despile die fact that Ihe correlation ol die two series is about 0.5. Substituting the moments in Table 8.1 into (8.2.7) shows that a risk-aversion coefficient of 19 is required to fil Ihe equity premium.1 This is much greater than 10, the maximum value considered plausible by Mehra and lrescott.

Il is worth noting that the implied risk-aversion coefficient is sensitive lo ihe liming convention used for consumption. While asset prices are measured al the end of each period, consumption is measured as a (low during a period. In correlating asset returns with consumption growth one can assume that measured consumption represents beginning-of-pcriod consumption or end-ol-pei iod consumption. The former assumption leads one lo correlate the return over a period with the ratio of the next periods consumption lo this periods consumption, while the latter assumption leads one to correlate the return over a period with the ratio of this periods consumption lo last periods consumption. The former assumption, which we use here, gives a much higher correlation between asset returns and consumption growlh and hence a lower risk-aversion coefficient than the latter assumption.7

table K.I icpoitsilir nioiiieiilsof asset teiiii iisanil rnnsiunptinn growlh whereas equation (H.2.7) requires the inouiellls ol innovations in these seiies. However the variation in condi-

iional espet led iciiu us and consumption growth seems lo be small e......gh that it.....tonieiits

ol innovations are similar lo the iuoiiicills ol the raw seiies.

(rtossniaii. Melino. ami Sliiller (l.IH7) handle this problem more i.uelully by assuming .......del lying loiiiimtoiis-iiuie model and deiiving its implications lor lime-averaged data.

These calculations can be related to the work of Hansen and Jagan- Гк. nathan (1991) in the following way. In the representative-agent model % with power utility, the stochastic discount factor Af,+i = &(C,+\/C,)~y, and the log stochastic discount factor m,+ \ = log(tS) - уДс,+1. If we are willing to make the approximation 7iil+\ M,+\ - 1, which will be accurate if M,+ has a mean close to one and is not too variable, then we have Var[M,+ l] Var[m,+] = y2 Var[Ac,+i]. Equivalently, the standard deviation of the stochastic discount factor must be approximately the coefficient of relative risk aversion times the standard deviation of consumption growth. Using the Hansen-Jagannathan methodology we found that the standard deviation of the stochastic discount factor must be at least 0.33 to fit our annual stock market data. Since the standard deviation of consumption growth is 0.033, this by itself implies a coefficient of risk aversion of at least 0.33/0.033 = 10. But a coefficient of risk aversion this low is consistent with the data j only if stock returns and consumption are perfectly correlated. If we usej the fact that the empirical correlation is about 0.5, we can use the tighter j volatility bound in equation (8.1.15) to double the required standard deviation of the stochastic discount factor and hence double the risk-aversion coefficient to about 20.

The Riskfree Rate Puzzle

One response to the equity premium puzzle is to consider larger values for the coefficient of relative risk aversion y. Kandel and Stambaugh (1991) have advocated this.8 However this leads to a second puzzle. Equation (8.2.5) implies that the unconditional mean riskless interest rate is

Щ,] = - log 5 + Yg- (8.2.8) j

where g is the mean growth rate of consumption. The average riskless interest rate is determined by three factors. First, the riskless rate is high if the time preference rate - logcS is high. Second, the riskless rate is high if ( the average consumption growth rate g is high, for then the representative agent has an incentive to try to borrow lo reduce the discrepancy between consumption today and in the future. The strength of this effect is inversely proportional to the elasticity of intertemporal substitution in consumption; in a power utility model where risk aversion equals the reciprocal of intertemporal substitution, the strength of the effect is directly proportional to у. Finally, the riskless rate is low if the variance of consumption growth is high, Tor then die representative agent has a precautionary motive for

One might think that introspection would be sullicient lo rule out very large values of y. I lowever Kandel and Stambaugh (111*11) point out that introspection can deliver very different estimates of risk aversion depending on ihe size of the gamble considered. This suggests that introspection can be misleading or that some more general model of utility is needed.