destroyed stockholder value by retaining and reinvesting earnings rather than paying them out. Unusually high stock returns in the late 20th Century could have resulted from unex- pectedly favorable conditions for economic growth.But they could also have resulted from a structural decline in the equity premium.Several economists have recently argued that the equity premium is now far lower than it was in the early 20th Century (Heaton and Lucas 1999,Jagannathan,McGrattan,and Scherbina 2001). 3.3 The Riskfree Rate Puzzle One response to the equity premium puzzle is to consider larger values for the coefficient of relative risk aversion Kandel and Stambaugh(1991)have advocated this.12 However this leads to a second puzzle.Equation(15)implies that the unconditional mean riskless interest rate is Erj+1=-1og6+9-2, (17) where g is the mean growth rate of consumption.Since g is positive,as shown in Table 3, high values of y imply high values of yg.Ignoring the term-202/2 for the moment,this can be reconciled with low average short-term real interest rates,shown in Table 1,only if the discount factor 6 is close to or even greater than one,corresponding to a low or even negative rate of time preference.This is the riskfree rate puzzle emphasized by Weil (1989). Intuitively,the riskfree rate puzzle is that if investors are risk-averse then with power utility they must also be extremely unwilling to substitute intertemporally.Given positive average consumption growth,a low riskless interest rate and a high rate of time preference, such investors would have a strong desire to borrow from the future to reduce their average consumption growth rate.A low riskless interest rate is possible in equilibrium only if investors have a low or negative rate of time preference that reduces their desire to borrow.13 12One might think that introspection would be sufficient to rule out very large values of,but Kandel and Stambaugh(1991)point out that introspection can deliver very different estimates of risk aversion depending on the size of the gamble considered.This suggests that introspection can be misleading or that some more general model of utility is needed. 13As Abel (1996)and Kocherlakota(1996)point out,negative time preference is consistent with finite 24
( % % % % ! % ( ') % = $ " % $ A? : 111# %# * +# " B % , , # $$ I $ % E ( F " % A11B ? .$ A8B ( . P % O A3B % " # 6# % % -%% # % ) # # % # % % ( G A141B - # ( () ' % + % % # ( % # % % % ; ( $ % J - -< - =+ K + + + B + & - C#**#D - + H < - 4 - + 2- - + - + + + + C#**'D B- +< C#**'D = - @ 2
Of course,if the risk aversion coefficient y is high enough then the negative quadratic term-202/2 in equation(17)dominates the linear term and pushes the riskless interest rate down again.The quadratic term reflects precautionary savings;risk-averse agents with uncertain consumption streams have a precautionary desire to save,which can work against their desire to borrow.But a reasonable rate of time preference is obtained only as a knife- edge case. Table 5 illustrates the riskfree rate puzzle in international data.The table first shows the average riskfree rate from Table 1 and the mean consumption growth rate and standard devi- ation of consumption growth from Table 2.These moments and the risk aversion coefficients calculated in Table 4 are substituted into equation (17),and the equation is solved for an implied time preference rate.The time preference rate is reported in percentage points per year;it can be interpreted as the riskless real interest rate that would prevail if consumption were known to be constant forever at its current level,with no growth and no volatility. Risk aversion coefficients in the RRA(2)range imply negative time preference rates in every country except Switzerland,whereas larger risk aversion coefficients in the RRA(1)range imply time preference rates that are often positive but always implausible and vary wildly across countries. An interesting issue is how mismeasurement of average inflation might affect these cal- culations.There is a growing consensus that in recent years conventional price indices have overstated true inflation by failing to fully capture the effects of quality improvements,con- sumer substitution to cheaper retail outlets,and price declines in newly introduced goods. If inflation is overstated by,say,1%,the real interest rate is understated by 1%,which by itself might help to explain the riskfree rate puzzle.Unfortunately the real growth rate of consumption is also understated by 1%,which worsens the riskfree rate puzzle.When y>1, this second effect dominates and understated inflation makes the riskfree rate puzzle even harder to explain. utility in a time-separable model provided that consumption is growing,and marginal utility shrinking, sufficiently rapidly.The question is whether negative time preference is plausible. 25
I # ( E % % % $ $ A3B ( % $ 7 %> () % # ( % = ( ) % 8 ( % ( % ) % ( E 2 $ A3B# $ % > ( ( # % 0( E 00;AB % % ' " # % ( E 00;AB % ; % % 7 % J ) % % 7 % J $ # ) # % - 7 # # 5# 5# % ' ( ! % 5# ( G # J 7 ( ( ' + + + - = + + -< K + + 2- F =- - + + 8
3.4 Bond Returns and the Equity Premium and Riskfree Rate Puzzles Some authors have argued that the riskfree interest rate is low because short-term government debt is more liquid than long-term financial assets.Short-term debt is "moneylike"in that it facilitates transactions and can be traded at minimal cost.The liquidity advantage of debt reduces its equilibrium return and increases the equity premium(Bansal and Coleman 1996,Heaton and Lucas 1996). The difficulty with this argument is that it implies that all long-term assets should have large excess returns over short-term debt.Long-term government bonds,for example,are not moneylike and so the liquidity argument implies that they should offer a large term premium.But historically,the term premium has been many times smaller than the equity premium.This point is illustrated in Table 6,which reports two alternative measures of the term premium.The first measure is the average log yield spread on long-term bonds over the short-term interest rate,while the second is the average quarterly excess log return on long bonds.In a long enough sample these two averages should coincide if there is no upward or downward drift in interest rates. The average yield spread is typically between 0.5%and 1.5%.A notable outlier is Italy, which has a negative average yield spread in this period.Average long bond returns are quite variable across countries,reflecting differences in inflationary experiences;however the average excess bond return rarely exceeds 2%per year.Thus both measures suggest that term premia are far smaller than equity premia. Table 7 develops this point further by repeating the calculations of Table 5,using bond returns rather than equity returns.The average excess log return on bonds over short debt, adjusted for Jensen's Inequality,is divided by the standard deviation of the excess bond return to calculate a bond Sharpe ratio which is a lower bound on the standard deviation of the stochastic discount factor.The Sharpe ratio for bonds is several times smaller than the Sharpe ratio for equities,indicating that term premia are small even after taking account of the lower volatility of bond returns. 26
%- ) , !" # , , # $$ " % ( ) % $ %) ") C ( D $ % $ $ A= 119# ? : 119B E % %) % ' ) :%) % # ' # ( $ % J % = # $ 9# % % %) ) # % $ ' % % - % % % % 85 85 ; - # % % ; % % $ # 7 % J 7 ' > % ' ' 5 %% $ 3 % 8# % $ % ' % # , H - $ # ' " " " $ # % (% 9
This finding is not consistent with a strong liquidity effect at the short end of the term structure,but it is consistent with a consumption-based asset pricing model if bond returns have a low correlation with consumption growth.Table 7 shows that sample consumption correlations often are lower for bonds,so that RRA(1)risk aversion estimates for bonds, which use these correlations,are often comparable to those for equities. A direct test of the liquidity story is to measure excess returns on stocks over long bonds rather than over short debt.If the equity premium is due to a liquidity effect on short-term interest rates,then there should be no "equity-bond premium"puzzle.Table 8 carries out this exercise and finds that the equity-bond premium puzzle is just as severe as the standard equity premium puzzle.14 3.5 Separating Risk Aversion and Intertemporal Substitution Epstein and Zin (1989,1991)and Weil (1989)use the theoretical framework of Kreps and Porteus(1978)to develop a more flexible version of the basic power utility model.That model is restrictive in that it makes the elasticity of intertemporal substitution,the reciprocal of the coefficient of relative risk aversion,y.Yet it is not clear that these two concepts should be linked so tightly.Risk aversion describes the consumer's reluctance to substitute consumption across states of the world and is meaningful even in an atemporal setting, whereas the elasticity of intertemporal substitution describes the consumer's willingness to substitute consumption over time and is meaningful even in a deterministic setting.The Epstein-Zin-Weil model retains many of the attractive features of power utility but breaks the link between the parameters y and v. The Epstein-Zin-Weil objective function is defined recursively by -{a-c宁+6e,）六。 (18) where=(1-7)/(1-1/).When y=1/,0=1 and the recursion (18)becomes linear; 14The excess return of equities over bonds must be measured with the appropriate correction for Jensen's Inequality to adjust from a geometric to an arithmetic mean.From equation(16),the appropriate measure is the log excess return on equities over short-term debt,less the log excess return on bonds over short-term debt,plus one-half the variance of the log equity return,less one-half the variance of the log bond return. 27
% % $ J # ) % % 3 # 00;AB ( # # $ ; $ ' ( % # - $ $ J ) # C $) D 4 ' $) , $ %. ( , ( . K A141# 11B G A141B ( F @ A134B 7 ' ( # # E ( # ( % 0( H % %# H % % % . )K)G ( ( . )K)G , P A B O . A4B A B A B G P # P A4B > 2- A F = - - 5 G > F+ , - ? F C#'D - - + A F - + - + A - + -+ - - + F + -+ - - + 3
it can then be solved forward to yield the familiar time-separable power utility model. The intertemporal budget constraint for a representative agent can be written as W+1=(1+Ru.t+1)(W-Ce), (19) where W+is the representative agent's wealth,and(1+R+)is the gross simple return on the portfolio of all invested wealth.15 This form of the budget constraint is appropriate for a complete-markets model in which wealth includes human capital as well as financial assets.Epstein and Zin use dynamic programming arguments to show that(18)and(19)together imply an Euler equation of the form [《(尝y{a+}ad (20) If I assume that asset returns and consumption are homoskedastic and jointly lognormal, then this implies that the riskless real interest rate is r7m=-g5+aan+”2-品 (21) The riskless interest rate is a constant,plus 1/times expected consumption growth.In the power utility model,1/=y and =1,so(21)reduces to (15) The premium on risky assets,including the wealth portfolio itself,is E小-m1+号-0告+a-a (22) b This says that the risk premium on asset i is a weighted combination of asset i's covariance with consumption growth(divided by the elasticity of intertemporal substitution and asset i's covariance with the return on wealth.The weights are 0 and 1-0 respectively.The Epstein-Zin-Weil model thus nests the consumption CAPM with power utility (=1)and the traditional static CAPM(=0). 15This is often called the "market"return and written Rm.t+1,but I have already used m to denote the stochastic discount factor so I write R.t+1 to avoid confusion. 28
) % % P A O BA B A1B % H # A O B % % ) ( . K % % % A4B A1B % . $ P. A O B A O B AB - - ( , % # ( P % O .QT R O AB ( # ' % - # P P # AB A8B ( # % # .Q R O P O A B AB ( % H % A B H % . )K)G ;@* A P B ;@* A P B 2- ++ - 7< 9 = > - + - - > = 4