quarter's consumption.Table 4 uses the former,"beginning-of-quarter"timing convention because this produces a higher contemporaneous correlation between consumption growth and stock returns. The timing convention has less effect on correlations when the data are measured at longer horizons.Table 4 also shows how the correlations among real consumption growth, real dividend growth,and real stock returns vary with the horizon.Each pairwise correlation among these series is calculated for horizons of 1,4,8,and 16 quarters in the quarterly data and for horizons of 1,2,4,and 8 years in the long-term annual data.The table illustrates three more stylized facts from the introduction. 7.Real consumption growth and dividend growth are generally weakly positively corre- lated in the quarterly data.In many countries the correlation increases strongly with the measurement horizon.However long-horizon correlations remain close to zero for Australia and Canada,and are substantially negative for Italy (with a very small stock market)and Japan (with anomalous dividend behavior).The correlations of consumption and dividend growth are positive and often quite large in the longer-term annual data sets 8.The correlations between real consumption growth rates and stock returns are quite variable across countries.They tend to be somewhat higher in high-capitalization countries (with the notable exception of Switzerland),which is consistent with the view that stock returns proxy more accurately for wealth returns in these countries.Correlations typically increase with the measurement horizon out to 1 or 2 years,and are moderately positive in the longer-term annual data sets. 9.The correlations between real dividend growth rates and stock returns are small at a quarterly horizon but increase dramatically with the horizon.This pattern holds in every country.The correlations also increase strongly with the horizon in the longer-term annual data. After this preliminary look at the data,I now use some simple finance theory to interpret the stylized facts. 14
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3 The Equity Premium Puzzle 3.1 The Stochastic Discount Factor To understand the equity premium puzzle,consider the intertemporal choice problem of an investor,indexed by k,who can trade freely in some asset i and can obtain a gross simple rate of return (1+Rit+1)on the asset held from time t to time t+1.If the investor consumes Cet at time t and has time-separable utility with discount factor 6 and period utility U(Ct), then her first-order condition is U'(Ckt)=6E:[(1+R,t+1)U'(Ckt+i】. (1) The left hand side of(1)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 t,selling it at time t+1,and consuming the proceeds.The investor equates marginal cost and marginal benefit,so(1)must describe the optimum. Dividing (1)by U'(Ckt)yields 1-B+-E+RM (2) where M.t+1=6U'(C.t+)/U(C)is the intertemporal marginal rate of substitution of the investor,also known as the stochastic discount factor.This way of writing the model in discrete time is due originally to Rubinstein(1976),while the continuous-time version of the model is due to Breeden (1979).Grossman and Shiller (1981),Shiller (1982),Hansen and Jagannathan (1991),and Cochrane and Hansen (1992)have developed the implications of the discrete-time model in detail.Cochrane (2001)gives a textbook exposition of finance using this framework. The derivation just given for equation(2)assumes the existence of an investor maximizing a time-separable utility function,but in fact the equation holds more generally.The existence of a positive stochastic discount factor is guaranteed by the absence of arbitrage in markets in which non-satiated investors can trade freely without transactions costs.In general there can be many such stochastic discount factors-for example,different investors k whose marginal 15
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utilities follow different stochastic processes will have different Mk.t+-but each stochastic discount factor must satisfy equation(2).It is common practice to drop the subscript k from this equation and simply write 1=Et[(1+R,t+1)M+i]. (3) In complete markets the stochastic discount factor M+is unique because investors can trade with one another to eliminate any idiosyncratic variation in their marginal utilities. To understand the implications of(3)it is helpful to write the expectation of the product as the product of expectations plus the covariance, E:[(1+Rit+)M+1]Et[(1 Rit+1)]E:[M+]+Cov:[Rit+1,M++]. (4) Substituting into (3)and rearranging gives 1+Er1-Cov,R.M E:M+1 (5) An asset with a high expected simple return must have a low covariance with the stochastic discount factor.Such an asset tends to have low returns when investors have high marginal utility.It is risky in that it fails to deliver wealth precisely when wealth is most valuable to investors.Investors therefore demand a large risk premium to hold it. Equation(5)must hold for any asset,including a riskless asset whose gross simple return is 1+R+.Since the simple riskless return has zero covariance with the stochastic discount factor (or any other random variable),it is just the reciprocal of the expectation of the stochastic discount factor: 1 1+Bj,t+1= EM+i] (6) This can be used to rewrite (5)as 1+E[R,t+]=(1+Rft+1)(1-Covt[R.t+1,M+]) (7) For simplicity I now follow Hansen and Singleton (1983)and assume that the joint con- ditional distribution of asset returns and the stochastic discount factor is lognormal and 16
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homoskedastic.While these assumptions are not literally realistic-stock returns in particu- lar have fat-tailed distributions with variances that change over time-they do make it easier to discuss the main forces that should determine the equity premium. When a random variable X is conditionally lognormally distributed,it has the convenient property that log EX=E:logX+Var:log X, (8) where Vart log X =E[(log X-E:log X)2].If in addition X is conditionally homoskedastic, then Var:log X=E[(log X-E log X)2]=Var(log X-E log X).Thus with joint conditional lognormality and homoskedasticity of asset returns and consumption,I can take logs of(3) and obtain 0-E,r,t+1+Eem+1+(②)[o+o品+2oml (9) Here mt log(M)and ra log(1+Rit),while o?denotes the unconditional variance of log return innovations Var(.t+-Et+),denotes the unconditional variance of innovations to the stochastic discount factor Var(mt+1-E:mt+1),and oim denotes the unconditional covariance of innovations Cov(rit+1-Eri,t+1,mt+1-Erm+1). Equation(9)has both time-series and cross-sectional implications.Consider first an asset with a riskless real return rf.+1.For this asset the return innovation variance o and the covariance ofm are both zero,so the riskless real interest rate obeys Tj.+1 =-Em+1- 2 (10) This equation is the log counterpart of(6). Subtracting (10)from (9)yields an expression for the expected excess return on risky assets over the riskless rate: Elrit+1-rft+i]+ 2 (11) The variance term on the left hand side of(11)is a Jensen's Inequality adjustment arising from the fact that we are describing expectations of log returns.In effect this term converts the expected excess return from a geometric average to an arithmetic average.It would 17
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disappear if we rewrote the equation in terms of the log expectation of the ratio of gross simple returns:log Et[(1+Rit+)/(1+Rj.t+1)]=-0im.The right hand side of (11)says that the risk premium is determined by the negative of the covariance of the asset with the stochastic discount factor.This equation is the log counterpart of(7). The covariance oim can be written as the product of the standard deviation of the asset return oi,the standard deviation of the stochastic discount factor om,and the correlation between the asset return and the stochastic discount factor Pim.Since Pim >-1,-Oim< o;om.Substituting into (11), om≥Er+1-r4l+22 (12) Oi This inequality was first derived by Shiller (1982);a multi-asset version was derived by Hansen and Jagannathan (1991)and developed further by Cochrane and Hansen (1992). The right hand side of(12)is the excess return on an asset,adjusted for Jensen's Inequality, divided by the standard deviation of the asset's return-a logarithmic Sharpe ratio for the asset.(12)says that the standard deviation of the log stochastic discount factor must be greater than this Sharpe ratio for all assets i,that is,it must be greater than the maximum possible Sharpe ratio obtainable in asset markets. Table 4 uses (12)to illustrate the equity premium puzzle.For each data set the first column of the table reports the average excess return on stock over short-term debt,adjusted for Jensen's Inequality by adding one-half the sample variance of the excess log return to get a sample estimate of the numerator in (12).This adjusted or arithmetic average excess return is multiplied by 400 to express it in annualized percentage points.The second column of the table gives the annualized standard deviation of the excess log stock return,a sample estimate of the denominator in (12).This standard deviation was reported earlier in Table 1.The third column gives the ratio of the first two columns,multiplied by 100;this is a sample estimate of the lower bound on the standard deviation of the log stochastic discount factor,expressed in annualized percentage points.In the postwar US data the estimated lower bound is a standard deviation greater than 50%a year;in the other quarterly data sets it is between 15%and 20%for Australia and Italy,between 20%and 30%for Canada 18
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