and Japan,and above 30%for all the other countries.In the long-run annual data sets the lower bound on the standard deviation exceeds 30%for all three countries. 3.2 Consumption-Based Asset Pricing with Power Utility To understand why these numbers are disturbing,I now follow Rubinstein (1976),Lucas (1978),Breeden (1979),Grossman and Shiller (1981),Mehra and Prescott (1985)and other classic papers on the equity premium puzzle and assume that there is a representative agent who maximizes a time-separable power utility function defined over aggregate consumption Ct: U(C)= C4--1 1-y (13) where y is the coefficient of relative risk aversion.This utility function has several important properties. First,it is scale-invariant;with constant return distributions,risk premia do not change over time as aggregate wealth and the scale of the economy increase.This is important because over the past two centuries wealth and consumption have increased manyfold,yet riskless interest rates and risk premia do not seem to have trended up or down.Power utility is one of the few utility specifications that are consistent with this fact.Related to this,if different investors in the economy have different wealth levels but the same power utility function,then they can be aggregated into a single representative investor with the same utility function as the individual investors. A possibly less desirable property of power utility is that the elasticity of intertemporal substitution,which I write as is the reciprocal of the coefficient of relative risk aversion y. Epstein and Zin(1991)and Weil(1989)have proposed a more general utility specification 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.I discuss this form of utility in section 3.4 below. Power utility implies that marginal utility U(C)=C,and the stochastic discount factor M+1=6(C+1/Ct)-7.The assumption made previously that the stochastic discount 19
# 65 - %) ' 65 % ' ( ) # * # * + %# - 0 A139B# : A134B# = A131B# + " A14B# * @ A148B $ % ' ) %% % AB P A6B E ( # ) > # ( % %% % # ( ( @ 0 # J J # %% % % ; # - # E ( . K A11B G A141B % ) ( % ( E ( - 62 @ % AB P # P A B 1
factor is conditionally lognormal will be implied by the assumption that aggregate consump- tion is conditionally lognormal (Hansen and Singleton 1983).Making this assumption for expositional convenience,the log stochastic discount factor is mt+1=log(6)-Ac+1,where c:=log(C:),and (9)becomes 0=Eirit+1+l0g6-YEAc+1+[o2+7202-2Y0icl. (14) Here o2 denotes the unconditional variance of log consumption innovations Var(c+1-Ect+1), and ci denotes the unconditional covariance of innovations Cov(rit+1-Eri+1,c+1-Ec+1). Equation (10)now becomes 141=-log5+7E,Ae1-a 2 (15) This equation says that the riskless real rate is linear in expected consumption growth,with slope coefficient equal to the coefficient of relative risk aversion.The conditional variance of consumption growth has a negative effect on the riskless rate which can be interpreted as a precautionary savings effect. Equation (11)becomes E-r小+竖=7 (16) The log risk premium on any asset is the coefficient of relative risk aversion times the covariance of the asset return with consumption growth.Intuitively,an asset with a high consumption covariance tends to have low returns when consumption is low,that is,when the marginal utility of consumption is high.Such an asset is risky and commands a large risk premium. Table 4 uses(16)to illustrate the equity premium puzzle.As already discussed,the first column of the table reports a sample estimate of the left hand side of(16),multiplied by 400 to express it in annualized percentage points.The second column reports the annual- ized standard deviation of the excess log stock return(given earlier in Table 1),the fourth column reports the annualized standard deviation of consumption growth (given earlier in Table 2),the fifth column reports the correlation between the excess log stock return and 20
% %% % ) % A? "% 146B *(% ' # % P %ABT # P %AB# A1B P. O % .T O Q O R A2B ? % LA . B# A . # . B .$ AB P % O .T A8B $ ( ' % # E $ E ( % % J ( % J .$ AB .Q R O P A9B % ( E ( % - # % # # % % " ( % ( 2 A9B $ ; # A9B# 2 ' % ) ' % ( A% B# % A% B# ' % (
consumption growth,and the sixth column gives the product of these three variables which is the annualized covariance oie between the log stock return and consumption growth. Finally,the table gives two columns with implied risk aversion coefficients.The column headed RRA(1)uses (16)directly,dividing the adjusted average excess return by the es- timated covariance to get estimated risk aversion.9 The column headed RRA(2)sets the correlation of stock returns and consumption growth equal to one before calculating risk aversion.While this is of course a counterfactual exercise,it is a valuable diagnostic because it indicates the extent to which the equity premium puzzle arises from the smoothness of consumption rather than the low correlation between consumption and stock returns.The correlation is hard to measure accurately because it is easily distorted by short-term mea- surement errors in consumption,and Table 4 indicates that the sample correlation is quite sensitive to the measurement horizon.By setting the correlation to one,the RRA(2)column indicates the extent to which the equity premium puzzle is robust to such issues.A corre- lation of one is also implicitly assumed in the volatility bound for the stochastic discount factor,(12),and in many calibration exercises such as Mehra and Prescott (1985),Abel (1999),or Campbell and Cochrane (1999). Table 4 shows that the equity premium puzzle is a robust phenomenon in international data.The coefficients of relative risk aversion in the RRA(1)column are generally extremely large.They are usually many times greater than 10,the maximum level considered plausible by Mehra and Prescott (1985).In a few cases the risk aversion coefficients are negative because the estimated covariance of stock returns with consumption growth is negative, but in these cases the covariance is extremely close to zero.Even when one ignores the low correlation between stock returns and consumption growth and gives the model its best chance by setting the correlation to one,the RRA(2)column still has risk aversion coefficients above 10 in all countries except Australia and Japan.Thus the fact shown in Table 3,that for some countries the correlation of stock returns and consumption increases with the horizon, The calculation is done correctly,in natural units,even though the table reports average excess returns and covariances in percentage point units.Equivalently,the ratio of the quantities given in the table is multiplied by 100. 21
% # ' % % ( % # % ( E 00;AB A9B # % , % ' ) % ( 00;AB ( % $ % ( G ' # % ' $ ( ) ) # 2 $ = % # 00;AB ' $ ; ) # AB# ' * @ A148B# ; A111B# A111B 2 $ E ( 00;AB % ' % % # ' * @ A148B - ( E % ( % % # ' . % ( % % % # 00;AB ( E ' ; 6# ( # 2- ++ + + - - - + A F+ + - - F - + + + #!!
is unable by itself to resolve the equity premium puzzle. Gabaix and Laibson (2001)and Parker (2001)have argued that adjustment costs in consumption artificially reduce the short-run variability of consumption and its correlation with stock returns,biasing upwards the estimated risk aversion coefficients in Table 4.Ad- justment costs should dampen the short-term volatility of consumption growth but not the volatility over longer horizons;equivalently,short-term consumption growth should be pos- itively autocorrelated.The first-order autocorrelation coefficients for consumption growth, shown in Table 2,do not generally support this model since they are typically small and often negative.10 However Gabaix and Laibson(2001)look at higher-order autocorrelations and point out that they tend to be larger in countries with larger stock markets.The re- sults of other studies of US consumption growth are mixed.Campbell and Mankiw (1989), Cochrane (1994),and Lettau and Ludvigson(2001)find that US consumption growth is almost unforecastable,although discrete-state Markov models estimated by Cecchetti,Lam, and Mark (1990,1993),Kandel and Stambaugh (1991),and Mehra and Prescott (1985) imply modest but persistent predictable variation in US consumption growth. The risk aversion estimates in Table 4 are of course point estimates and are subject to sampling error.No standard errors are reported for these estimates.However authors such as Cecchetti,Lam,and Mark (1993)and Kocherlakota (1996),studying the long-run annual US data,have found small enough standard errors that they can reject risk aversion coefficients below about 8 at conventional significance levels. Of course,the validity of these tests depends on the characteristics of the data set in which they are used.Rietz (1988)has argued that there may be a peso problem in these data.A peso problem arises when there is a small positive probability of an important event, and investors take this probability into account when setting market prices.If the event does not occur in a particular sample period,investors will appear irrational in the sample and economists will misestimate their preferences.While it may seem unlikely that this could 10These autocorrelations are biased upwards by the time-averaging of consumption data,but outside the US are biased downwards by the durable component of total consumption expenditure.The absence of positive autocorrelations in consumption growth is also evidence against the Constantinides(1990)model of habit formation,discussed in section 5.1,which has similar implications for consumption growth. 22
$ + ' : AB @( AB % , ) ( # % ( E 2 ;) , ) % % > $ # ) % ) ) E % # # % % ? + ' : AB ( % ) % % ( ( ) !" % ' *( A141B# A112B# : : % AB !" % # % ) *( # : # *( A11# 116B# F " % A11B# * @ A148B !" % ( 2 , % M ? # : # *( A116B F ( A119B# % %) !" # % , ( E 4 % I # 0 A144B % ; # ( % ( - # G ( 2- + = - - & == - + + A 2- + = - + - C#**!D + - E# =-- - + + = -
be an important problem in 100 years of annual data,Rietz(1988)argues that an economic catastrophe that destroys almost all stock-market value can be extremely unlikely and yet have a major depressing effect on stock prices. One difficulty with this argument is that it requires not only a potential catastrophe, but one which affects stock market investors more seriously than investors in short-term debt instruments.Many countries that have experienced catastrophes,such as Russia or Germany,have seen very low returns on short-term government debt as well as on equity. A peso problem that affects both asset returns equally will affect estimates of the average levels of returns but not estimates of the equity premium.The major example of a disaster for stockholders that did not negatively affect bondholders is the Great Depression of the early 1930's,but of course this is included in the long-run annual data for Sweden,the UK, and the US,all of which display an equity premium puzzle. Also,the consistency of the results across countries requires investors in all countries to be concerned about catastrophes.If the potential catastrophes are uncorrelated across countries,then it becomes less likely that the data set includes no catastrophes;thus the argument seems to require a potential international catastrophe that affects all countries simultaneously. Even if the equity premium puzzle is not entirely spurious,there are several reasons to think that stock returns exceeded their true long-run mean in the late 20th Century.Dimson, Marsh,and Staunton(2002)find that international returns were generally higher in the late 20th Century than in the early 20th Century.Siegel(1994)reports similar results for US data going back to the early 19th Century.Fama and French (2000)point out that average US stock returns in the late 20th Century were considerably higher than accountants'estimates of the return on equity for US corporations.Thus if one uses average returns as an estimate of the true cost of capital,one is forced to the implausible conclusion that corporations 1iThis point is relevant for the study of Jorion and Goetzmann(1999).These authors measure average growth rates of real stock prices,as a proxy for real stock returns.They find low real stock-price growth rates in many countries in the early 20th Century,but Dimson,Marsh,and Staunton(2002)show that in many cases these were accompanied by low returns to holders of short-term debt,and also by high dividend yields. 23
# 0 A144B % () ( ' ( , % J ( I E % $ # J ( ( ) * ' # 0 + # ) % $ ; J $ J % $ , ' ( % J + & 16H# %) " # !F# !"# $ ; # $ - # ( > % $ J . $ # ( ( ' %) & # *# " AB % % " % A112B !" %% ( 1 AB % !" ( % H $ !" % # 2- + - 5 3 4 C#***D 2- - = - + < A + < 2- @ += + < = - - + "! - - & C"!!"D -= - - = += -+ - + - - + 6