Risk Aversion and the Variance Decomposition of the Price-Dividend Ratio

Transcription

1 Risk Aversion and the Variance Decomposition of the Price-Dividend Ratio Kevin J. Lansing Federal Reserve Bank of San Francisco Stephen F. LeRoy y UC Santa Barbara and Federal Reserve Bank of San Francisco February 27, 204 Abstract This paper employs a standard asset pricing model with power utility to derive an analytical variance decomposition for the price-dividend ratio. We show that the fraction of the variance coming from future dividend growth rates and future risk-free rates depends exclusively on the value of the risk aversion coe cient. Higher risk aversion lowers the contribution from future dividend growth rates but raises the contribution from future risk free rates. The variance contribution from future excess returns on equity is exactly zero, in stark contrast to empirical ndings which attribute nearly all of the variance in the pricedividend ratio to this source. Keywords: Asset Pricing, Excess Volatility, Risk Aversion, Variance Decomposition. JEL Classi cation: E44, G2. Corresponding author. Research Department, Federal Reserve Bank of San Francisco, P.O. Box 7702, San Francisco, CA , (45) , FAX: (45) , kevin.j.lansing@sf.frb.org, homepage: y Department of Economics, University of California, Santa Barbara, CA 9306, (805) , sleroy@econ.ucsb.edu, homepage:

2 Introduction Empirical studies starting with Shiller (98) and LeRoy and Porter (98) have shown that stock prices exhibit excess volatility when compared to the value of ex post realized dividends discounted at a constant rate. Campbell and Shiller (988) addressed the issue using a log-linear decomposition of the equity return identity. They showed that the observed volatility of the log stock price-dividend ratio can be attributed mainly to the fact that future equity returns contain a predictable component, contrary to the Shiller-LeRoy-Porter speci cation. This result suggests that the assumption underlying constant discount rates risk neutrality is a misspeci - cation. To investigate this one would like to have in hand a variance decomposition based on a theoretical asset-pricing model that incorporates risk aversion. The Campbell- Shiller variance decomposition of the price-dividend ratio, being based on the identity de ning the equity return and on that alone, cannot deliver results that depend on such model speci cs. However, if one conducts a variance decomposition based on linearization of the Euler equation of an explicit consumption-based asset pricing model incorporating risk aversion, one can see how risk aversion ts in. We conduct this analysis in the present paper. Speci cally, we use a log-linearization of the representative agent s Euler equation to derive expressions for () log dividend growth, (2) discounted risk-free returns, and (3) discounted excess returns as functions of the measure of relative risk aversion. Then we use the Campbell-Shiller log-linearization of the return identity to break down the variance of the log price-dividend ratio into the sum of the covariances of the log price-dividend ratio with each of these terms. Our results are striking. First, the variance decomposition of the theoretical assetpricing model shows that future excess returns do not covary with the current pricedividend ratio, implying that they contribute nothing to the volatility of the pricedividend ratio. The result that excess stock returns are unforecastable is familiar from the case of risk neutrality. One might have anticipated that the absence of correlation between the price-dividend ratio and future excess returns would not carry over when agents are interested in the higher moments of future payo s, as is true under strict risk aversion, rather than just expected future payo s, as under risk neutrality. However, our result is that, contrary to this intuition, the zero-covariance For summaries of this extensive literature, see West (988), Gilles and LeRoy (99), Shiller (2003), and LeRoy (200).

3 result does carry over to the case of risk aversion. Thus regardless of the degree of risk aversion future excess returns are unforecastable, and therefore they make zero contribution to the volatility of the price-dividend ratio. 2 The empirical results of Campbell-Shiller and Cochrane (992), (2005), (20) sharply contradict the nding from the theoretical model that future excess returns do not contribute to the volatility of the current price-dividend ratio. On the contrary, they conclude that more than 00% of the variation in price-dividend ratios comes from this source (re ecting the fact that future dividend growth covaries positively with the current price-dividend ratio, whereas future returns covary negatively). Thus we con rm in a particularly simple setting the ndings of others that the theory of consumption-based asset pricing, at least in its basic versions, does not come anywhere near tting the data. Given the model s result that future excess returns do not covary with the log price-dividend ratio, the variance decomposition based on the return identity implies that the variance of the log price-dividend ratio can be decomposed into the sum of a term that depends on the predictability of future dividend growth rates and a term that depends on the predictability of future risk-free returns. Remarkably, the model implies that the proportion of total variance accounted for by each of these terms depends only on the agents risk aversion it does not depend on the discount factor or the degree of persistence of dividend growth. Low risk aversion results in a small contribution to variance from future risk-free rates and a large contribution from future dividend growth rates, and vice-versa for high risk aversion. The fact that the weights in this decomposition depend only on relative risk aversion implies that empirical estimates of these weights can be converted into measures of risk aversion. We nd that this conversion results in estimated levels of relative risk aversion around 5. Of course, this result cannot be taken as providing a serious estimator for the risk aversion parameter: our main result is that the model must be rejected in its entirety due to its misdiagnosis of the sources of volatility of price-dividend ratio. 2 The result is not completely general. In the model of LeRoy (973) future returns have a forecastable component under strictly positive risk aversion. Also, Lansing and LeRoy (203) nd that, when agents have partial information about future dividend payo s, future returns contain a forecastable component. Even in this case, however, in the example they studied the contribution of future returns to the volatility of the price-dividend ratio is not large. Thus the nding that empirically return forecastability is large remains a puzzle. 2

4 2 Asset Pricing Model We employ a frictionless pure exchange model along the lines of Lucas (978). A representative investor can purchase shares to transfer wealth from one date to another. Each share pays an exogenous stream of stochastic dividends in perpetuity. The representative investor s problem is to maximize ( ) X E 0 t c t subject to the budget constraint t=0 c t + p t s t = (p t + d t ) s t ; c t ; s t > 0; (2) where c t is the investor s consumption in period t; is the coe cient of relative risk aversion, and s t is the number of shares held in period t: The rst-order condition that governs the investor s share holdings is p t = E t ct+ c t (p t+ + d t+)# () ; (3) where E t is the expectation operator conditional on information available at time t. Equity shares are assumed to exist in unit net supply such that s t = for all t: Market clearing therefore implies c t = d t for all t: We assume that the investor s information set consists of current and past values of consumption/dividends. 3 The rst-order condition can be iterated forward to substitute out p t+j for j = ; 2; ::. Applying the law of iterated expectations and imposing the condition that the limiting discounted terminal value of any optimal portfolio strategy is zero, we can exclude bubble solutions. 4 expression for the equilibrium equity price: Exclusion of bubble solutions yields the following X p t = E t M t; t+j d t+j ; (4) j= 3 Lansing and LeRoy (203) consider alternative information sets that allow for investor knowledge about future dividend realizations or imperfect information about the growth rate of dividends. 4 This condition is often called a transversality condition, but this is at best a very loose usage of terminology. More precisely stated, the result is that under appropriate boundedness and nonnegativity restrictions on consumption and portfolio strategies, maximization of the assumed utility function implies a necessary transversality condition (see Kamihigashi [?]). This transversality condition in turn implies the statement in the text. Thus the statement in the text is an implication of a transversality condition, but is not itself a transversality condition. 3

5 where M t;t+j j (c t+j =c t ) is the stochastic discount factor. We assume that the growth rate of dividends x t log (d t =d t ) is governed by the following AR() process: x t+ = x t + ( ) x + t+j N 0; 2 ; iid; t+; jj < : (5) By substituting the equilibrium condition c t = d t into the rst-order condition (3) and de ning the price-dividend ratio as y t p t =d t ; we obtain the following transformed version of the rst-order condition in terms of stationary variables y t = E t f exp [( ) x t+ ] (y t+ + )g : (6) 2. Variance of the Price-Dividend Ratio We now derive an approximate analytical solution for the variance of log(y t ) : This involves solving the transformed rst-order condition (6) subject to the dividend growth process (5). To do so, it is convenient to de ne the following nonlinear change of variables: z t exp [( ) x t ] (y t + ) ; (7) where z t represents a composite variable that depends on both the growth rate of dividends and the price-dividend ratio. The rst-order condition (6) becomes y t = E t z t+ ; (8) which shows that y t is simply the rational forecast of the composite variable z t+ : Combining (7) and (8) yields the following transformed version of the equilibrium condition: z t = exp [( ) x t ] (E t z t+ + ) : (9) Proposition. An approximate analytical solution for the equilibrium value of the composite variable z t is given by z t = a 0 exp [a (x t x)] ; where a solves a = h i exp ( )x + 2 (a ) 2 2 and a 0 exp fe [log (z t )]g is given by a 0 = exp [( )x] h i; exp ( )x + 2 (a ) 2 2 4

6 h i provided that exp ( )x + 2 (a ) 2 2 < : Proof : See Appendix A. Two values of a satisfy the nonlinear equation in Proposition. The inequality restriction selects the value of a with lower magnitude to ensure that a 0 is positive. 5 Given the approximate solution for the composite variable z t, we can recover the solution for the equilibrium price-dividend ratio y t as follows: y t = E t z t+ = a 0 exp a (x t x) + 2 (a ) 2 2 : (0) As shown in Appendix B, the above solution can be used to derive the following unconditional variance of the log price-dividend ratio: 2.2 Variance of Returns V ar [log (y t )] = (a ) 2 V ar (x t ) : () The gross rate of return on equity can be written as R t+ = p t+ + d t+ yt+ + = exp (x t+ ) p t y t (2) = zt+ exp ( x t+ ) ; (3) E t z t+ where we have eliminated y t using the equilibrium condition (8) and eliminated y t+ + using the de nitional relationship which follows directly from (7). y t+ + = exp [ ( )x t+ ] z t+ ; (4) In Appendix B, we show that the approximate law of motion for log(r t+ ) is log(r t+ ) E[log(R t+ )] = (x t+ x) + a t+ (5) where a is given by Proposition. Given the above law of motion for the log equity return, it is straightforward to compute the following unconditional variance: V ar[log(r t+ )] = 2 V ar (x t ) + a [a + 2] 2 ; (6) 5 Lansing (200) compares the approximate solution from Proposition to the exact theoretical solution derived by Burnside (998). The approximate solution is extremely accurate for low and moderate levels of risk aversion ( ' 2) : But even for high levels of risk aversion ( ' 0) ; the approximation error for the equilibrium price-dividend ratio remains below 5 percent. 5

7 where V ar (x t ) = 2 = 2 from (5). by In Appendix B, we show that the law of motion for the log risk-free rate is given log(r f t+ ) E[log(Rf t+ )] = (x t x) ; = (x t+ x) t+; (7) Subtracting the risk-free rate equation (7) from the equity return equation (5) yields the following law of motion for the excess return on equity, i.e., the equity premium: log(r t+ ) log(r f t+ ) = ( + a ) t+ + 2 h 2 (a ) 2i 2 ; (8) where we have substituted in E[log(R t+ )] E[log(R f t+ )] as derived in Appendix B. 3 Mapping to the Campbell-Shiller Framework Campbell and Shiller (988), Campbell (99), and Cochrane (992, 2005) show that a log-linear approximation of the equity return identity implies that the variance of the log price-dividend ratio must equal the sum of the ratio s covariances with: () future dividend growth rates, (2) future risk-free rates, and (3) future excess returns on equity. This variance decomposition, being derived from an identity rather than a theoretical model, cannot be used to ascertain the theoretical connections between risk aversion and asset price volatility. Its use up to now in the nance literature has been to determine the relative empirical importance of the three separate components in explaining the volatility of observed stock prices relative to dividends. However, since the return identity is valid in fully-speci ed theoretical models, it is possible to evaluate the model by performing the variance decomposition analytically and then using a calibrated version of the model to compute the contributions from each of the three components noted above for comparison with the results obtained from U.S. data. This is what we do next. Following the methodology of Campbell and Shiller (988), the de nition of the log equity return (2) can be approximated as follows: log(r t+ ) log (y t+ + ) + x t+ log (y t ) ; ' 0 + log (y t+ ) + x t+ log (y t ) ; (9) 6

8 where 0 and = exp [E log (y t )] = f + exp [E log (y t )]g are Taylor-series coe cients. Solving equation (9) for log (y t ) and then successively iterating the resulting expression forward to eliminate log(y t++j ) for j = 0; ; 2::: yields the following approximate identity: log (y t ) ' 0 P + j=0 ( ) j [x t++j log(r t++j )] ; (20) which shows that movements in the log price-dividend ratio must be accounted for by movements in either future dividend growth rates or future log equity returns. The variables in the approximate identity (20) can be expressed as deviations from their unconditional means while the means are consolidated into the constant term. Multiplying both sides of the resulting expression by log (y t ) then taking the unconditional expectation of both sides yields # P V ar [log (y t )] = Cov log (y t ) ; ( ) j x t++j Cov log (y t ) ; = Cov Cov log (y t ) ; log (y t ) ; j=0 # P ( ) j x t++j j=0 Cov log (y t ) ; E [log (y t )] and # P ( ) j log(r t++j ) ; j=0 # P ( ) j log(r f t++j ) j=0 # P ( ) j log(r t++j =R f t++j ) : (2) j=0 Here the second version of the expression breaks up future log equity returns into two parts: future log risk free rates, denoted by log(r f t++j ); and future excess returns on equity, as given by log(r t++j =R f t++j ). The above equation states that the variance of the log price-dividend ratio must be accounted for by the covariance of the ratio with future dividend growth rates, future risk free rates, or future excess returns on equity. The magnitude of each covariance term is a measure of the predictability of each component when the current price-dividend ratio is employed as the sole regressor in a forecasting equation. The approximate law of motion for the log equity return is given by equation (5). The corresponding law of motion for the log risk-free rate is given by equation (7). Using the approximate laws of motion for the relevant variables, we can analytically compute the three covariance terms in equation (2). Details are provided in Appendix C. The results of the theoretical variance decomposition are as follows: V ar [log (y t )] = a 2 V ar (x t ) a 2 V ar (x t ) 0; (22) where the three terms in the above equation correspond to the three possible sources 7

9 of variation: () future dividend growth rates, (2) future risk free rates, and (3) future excess returns on equity. Equation (22) shows that the variance contribution from excess returns (the third term in the equation) is exactly zero. This result can be understood by examining the law of motion for excess returns on equity as given by equation (8). Equation (8) shows that excess returns are iid in the power-utility model. Hence, the covariance between the log price-dividend ratio at time t and future excess returns must be zero. The theoretical variance decomposition in equation (22) can be expressed more concisely by dividing both sides of the decomposition by the variance of the log pricedividend ratio, as given by equation ().To avoid division by zero, we rule out = 0 and = ; since these result in zero volatility for the price-dividend ratio. Doing this implies that the contributions to variance from each source can be expressed as fractions that sum to unity. Details of this computation are provided in Appendix C. The results of the fractional decomposition are summarized in Table. Table : Analytical variance decomposition for the log price-dividend ratio Fraction of variance attributable to: Future Dividend Growth Future Risk-Free Rates Future Excess Returns 0 Table shows that the fractional variance decomposition depends only on the risk aversion coe cient ; and does not depend on or : When the representative investor is risk neutral, we have = 0: In this case, future dividend growth rates account for 00% of the variance of the log price-dividend ratio, while the fraction of the variance attributable to other sources is zero. For > 0 dividend growth explains more than 00% of the variation in the price-dividend ratio. Accordingly, the contribution of the stochastic discount factor is negative, so as to reduce the sum of the two to 00%. At higher levels of (but maintaining < ) uctuations in future dividends are nearly o set by inverse uctuations in their marginal utilities, implying a low variance of the price-dividend ratio. The weights on dividend growth and stochastic discount factors approach and as approaches. With = ; the log price-dividend ratio is constant and hence there is no variance to be decomposed. 8

10 For > uctuations in dividends are more than o set by inverse uctuations in marginal utilities. As a result the weights on the dividend component and the returns component reverse signs: the former becomes negative and the latter positive. When = 5; we have = ( ) = 0:25 and = ( ) = :25. In this case, 25% of the variance of the log price dividend ratio is attributable to movements in future dividend growth, 25% is attributable to movements in future risk free rates, and 0% is attributable to movements in future excess returns. 6 The variance decomposition computed analytically from the model can be compared to the decomposition obtained from the data without imposing any particular model. Cochrane (2005, p. 400) presents an empirical variance decomposition of the log price-dividend ratio for the value-weighted basket of stocks traded on the New York Stock Exchange using annual data for the period 928 to 988. The data show that 34% of the variance of the log price-dividend ratio is attributable to movements in future dividend growth, while 38% is attributable to movements in future equity returns (i.e., the sum of future risk free rates and future excess returns). WHY DON T THESE SUM TO? The corresponding percentages from our model are 25% and 25% when = 5, which are not too di erent from Cochrane s results. The model can match the 34% gure from Cochrane s empirical decomposition by setting = 3:9; and can match the 38% gure from Cochrane s empirical decomposition by setting = 3:6. However, if the variance contribution from future equity returns is broken down into separate contributions from future risk-free rates and future excess returns, then the results obtained from the model are very di erent from the data. As we have noted, in the data more than 00% of the variation in the log price-dividend ratio is attributable to movements in future excess returns, i.e., the third term in equation (2), while almost nothing can be attributed to movements in future risk-free rates. The power-utility model generates the opposite result: more than 00% of the variation in the log price-dividend ratio is attributable to movements in future risk-free rates, while exactly nothing is attributable to movements in future excess returns. One way in which empirical decomposition manifests itself in the data is the fact that the dividend yield (the inverse of the price-dividend ratio) forecasts excess returns on equity over long horizons, whereas empirical proxies for the risk-free rate do not predict future excess returns (Campbell and Shiller 988). More recently, Cochrane (2008, p. 545) obtains a statistically signi cant long-horizon regression 6 Since the variance decomposition does not require the sources of variation to be orthogonal to one another, the percentage from each source may fall outside the range of 0% to 00%. 9

11 coe cient of.23 when forecasting excess returns using the current dividend yield. His result implies that 23% of the dividend yield variance in the data is coming from future excess returns. The power utility model attributes zero percent of the variance in the dividend yield to this source. 4 Alternative Models Some recent contributions that allow for bubbles or employ models with time-varying risk aversion or time-varying volatility of consumption growth have achieved more success in matching the empirical variance decomposition than we report here. These features introduce persistence (i.e., predictability) in the law of motion for excess returns. Models by Campbell and Cochrane (999) and Bansal and Yaron (2004) are the most notable examples that employ time-varying risk aversion or time-varying volatility of consumption growth, respectively. However, these models must still rely on the assumption of extremely high risk aversion to match observed features of the data. From equation (8), it is straightforward to see how the inclusion of time-varying risk aversion or time-varying volatility of consumption growth could be used to introduce persistence (or predictability) into excess returns. Allowing the risk aversion coe cient to depend on lagged values of consumption growth could be interpreted as a reduced-form version of the habit formation model of Campbell and Cochrane (999). Introducing persistent stochastic variation in the innovation variance 2 (which also appears in the expression for a in Proposition ) would mimic the stochastic volatility setup employed by Bansal and Yaron (2004). It is important to note that our analytical variance decomposition for the powerutility model ruled out the presence of bubbles. Engsted et al. (202) introduce a periodically-collapsing rational bubble into a simple asset pricing model with riskneutral investors. They show that predictability regressions using simulated data from the model yield results that are very similar to those obtained by Cochrane (2008) using U.S. data. Speci cally, in deriving the approximate return identity (20), we applied the transversality condition lim j! ( ) j log (y t+j ) = 0. In the presence of an explosive rational bubble, the transversality condition would not hold. Recent work by Engsted, et al. (202) shows that the introduction of a periodically-collapsing rational bubble is successful in allowing their theoretical asset pricing model to generate predictability regressions that are very similar to those obtained by Cochrane (2008) using U.S. data. The model employed by Engsted et al. is a special case of the 0

12 power-utility model considered here with = = 0: Hence, while Cochrane s empirical results may be interpreted as showing that expected excess returns in the data are predictable, this interpretation is conditioned on the assumption of no bubbles. If one allows for the presence of bubbles, a risk neutral version of the power-utility model can produce data in which the price-dividend ratio has substantial predictive power for future excess returns. I DON T UNDERSTAND THE DISCUSSION OF THIS PARA. YOU DON T GET A CHOICE ABOUT WHETHER OR NOT TO IMPOSE THE TVC. IF THERE IS A NECESSARY TVC THEN IF YOU TRY TO RELAX IT, WHAT YOU GET IS NOT AN OPTIMUM, THEREFORE NOT AN EQUILIBRIUM. NOT HAVING READ ENGSTED, I DON T KNOW HOW TO HANDLE THIS DISCUS- SION. MAYBE WE SHOULD JUST DELETE IT. 5 Conclusion We mapped a standard asset pricing model with power utility into the Campbell- Shiller (988) variance decomposition framework. We showed that the fraction of the variance of the log price dividend ratio coming from future dividend growth rates and future risk free rates depends on a single parameter: the coe cient of relative risk aversion. Moreover, we showed that the fraction of the variance coming from future excess returns on equity is identically zero, in stark contrast to the large contribution to variance from this source found in the data. Our results help to shed light on some elements of a theoretical model that could be used to generate a variance decomposition similar to that in the data. It would appear that a successful asset pricing model must incorporate either time-varying risk aversion and/or time-varying volatility of consumption growth. In this way, excess returns on equity can be made to exhibit signi cant persistence and volatility, in contrast to the basic power-utility model considered here which implies that excess returns are iid; as shown by equation (8). Allowing for the possibility of bubbles is another way in which the basic power-utility model can produce an empiricallyplausible variance decomposition, as demonstrated recently by Engsted et al. (202). Campbell and Cochrane (999) achieve persistent and volatile excess returns on equity by introducing time-varying risk aversion via habit formation. The law of motion for the habit stock in their model is reverse-engineered to deliver a constant risk-free rate, thereby allowing excess returns to make a large contribution to the variance of the log price-dividend ratio, as in the data. However, their calibrated model

13 requires an extremely high coe cient of relative risk aversion to match the various empirical facts around 80 in the model steady state. Moreover, the habit formation model predicts that rational investors expected returns on equity will be low when risk aversion is low a situation that should occur near market peaks when the pricedividend ratio is high. But survey evidence reveals the opposite: investor expectations about future returns on risky assets tend to be high near market peaks when the price-dividend ratio is high suggestive of extrapolative expectations. 7 Bansal and Yaron (2004) introduce exogenous time-varying volatility in the stochastic processes for consumption and dividend growth which also share a persistent component. In addition, they consider Epstein-Zin (989) preferences which allow the elasticity of intertemporal substitution to be varied independently of the risk aversion coe cient. Nevertheless, their model continues to underpredict the volatility of the log pricedividend ratio in the data even when the risk aversion coe cient is raised to a value of 0. Overall, it remains a challenge for fundamentals-based asset pricing models to fully explain observed stock market behavior. 7 For a review of the evidence, see Greenwood and Shleifer (203), Jurgilas and Lansing (203), and Williams (203). 2

14 Appendix A Proof of Proposition Iterating ahead the law of motion for z t speci ed in Proposition and taking the conditional expectation yields h i E t z t+ = y t = a 0 exp a (x t x) + 2 (a ) 2 2 : (A.) Substituting the above expression into the rst-order condition (9) and then taking logarithms yields log (z t ) = F (x t ) = log () + ( )x t n + log a 0 exp h a (x t x) + 2 (a ) 2 2 i o + ; ' log (a 0 ) + a (x t x) ; (A.2) where the Taylor-series coe cients a 0 and a are given by n h i o log (a 0 ) = F (x) = log () + ( ) x + log a 0 exp 2 (a ) ;(A.3) t = + x h i a a 0 exp 2 (a ) 2 2 h i : (A.4) a 0 exp 2 (a ) Solving equation (A.3) for a 0 yields a 0 = exp fe [log (z t )]g = exp [( ) x] h i; (A.5) exp ( ) x + 2 (a ) 2 2 which can be substituted into equation (A.4) to yield the following nonlinear equation that determines a : a = + a exp ( ) x + 2 a2 2 : (A.6) Rearranging equation (A.6) yields the expression shown in Proposition h. There are i two solutions, but only one solution satis es the condition exp ( ) x + 2 (a ) 2 2 < ; which is veri ed after solving (A.6) using a nonlinear equation solver. 3

15 B Asset Pricing Moments This section brie y outlines the derivation of equations (), (6), and (7). Taking the unconditional expectation of log (y t ) in equation (0) yields E [log (y t )] = log (a 0 ) + 2 (a ) 2 2 : (B.) We then have log (y t ) E [log (y t )] = a (x t x) ; (B.2) which in turn implies V ar [log (y t )] = (a ) 2 V ar (x t ) : (B.3) As described in the text, the equity return (3) can be rewritten as R t+ = zt+ exp ( x t+ ) : (B.4) E t z t+ Substituting in E t z t+ from equation (A.) and z t+ = a 0 exp [a (x t+ Proposition and then taking the unconditional mean of log(r t+ ) yields We then have which in turns implies x)] from E[log(R t+ )] = log () + x 2 (a ) 2 2 : (B.5) log(r t+ ) E[log(R t+ )] = (x t+ x) + a t+ ; (B.6) V ar[log(r t+ )] = 2 V ar (x t ) + (a ) a Cov (x t+ ; t+ ) : {z } (B.7) = 2 The log risk free rate is determined by the following rst-order condition ( #) log(r f t+ ) = log ct+ E t ; c t = log fe t [ exp ( x t+ )]g ; = log () + [x t + ( ) x] ; (B.8) where we have made the substitution c t+ =c t = d t+ =d t = exp (x t+ ) and then inserted the law of motion for x t+ from equation (5) before taking the conditional expectation. Taking the unconditional mean of log(r f t+ ) and then subtracting the unconditional mean from equation (B.8) yields the law of motion (7). 4

16 C Variance Decomposition The Taylor series coe cient in the approximate return identify (9) is given by h = exp [E log (y a t)] 0 exp + exp [E log (y t )] = 2 (a ) 2 i 2 h i h i = exp ( ) x + + a 0 exp 2 (a ) (a ) 2 2 ; (C.) where we have made use of equations (A.5) and (B.). Using the law of motion for log (y t ) given by (B.2) and the law of motion for x t+ given by (5), we can compute the following covariance which is the rst term in equation (2): Cov log (y t ) ; # P ( ) j x t++j j=0 = E fa (x t x) (x t+ x) + a (x t x) (x t+2 x) o + ( ) 2 a (x t x) (x t+3 x) + ::: ; n o = a 2 V ar (x t ) + + ( ) 2 + ( ) 3 + ::: ; = a 2 V ar (x t ) : (C.2) Similarly, using the law of motion for log(r f t+ ) given by (7) we can compute the following covariance which is the second term in equation (2): # P Cov log (y t ) ; ( ) j log(r f t++j ) = E fa (x t x) (x t x) j=0 + a (x t x) (x t+2 x) o + ( ) 2 a (x t x) (x t+3 x) + ::: ; n o = a 2 V ar (x t ) + + ( ) 2 + ( ) 3 + ::: ; = a 2 V ar (x t ) : (C.3) The law of motion for excess returns (8) shows that excess returns are iid: Hence, the third covariance term in equation (2) is identically zero. Dividing both sides of equation (2) by V ar [log (y t )] from (B.3) and then substituting in the appropriate 5

17 moments yields = a ( ) a ( ) 0; = 0; (C.4) where we make use of ( ) = ( ) =a from equations (A.6) and (C.). 6

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