7 The budget constraint is:W1=(Wt-C)x'R+1,where x is the portfolio weight vector,subject to x'1= 1. If the allocation of resources to consumption and investment assets is optimal,it is not possible to obtain higher utility by changing the allocation.Suppose an investor considers reducing consumption at time t to purchase more of (any)asset.The expected utility cost at time t of the foregone consumption is the expected marginal utility of consumption expenditures,Uc(Ct,.)>0 (where a subscript denotes partial derivative),multiplied by the price Pit of the asset,measured in the numeraire unit.The expected utility gain of selling the investment asset and consuming the proceeds at time t+1 is Ef(Pit++Di+)Jw(W+1,s)).If the allocation maximizes expected utility,the following must hold:Pit E(U(Ct.))=E(Pi++Di+)Jw(W,s)),which is equivalent to equation (1),with m+1=Jw(W+1,St+1)/E{Uc(Ct,)}. (4) The mu in equation (4)is the intertemporal marginal rate of substitution (IMRS)of the consumer- investor,and equations(2)and(4)combined are the intertemporal Euler equation. Asset pricing models typically focus on the relation of security returns to aggregate quantities. To get there,it is necessary to aggregate the Euler equations of individuals to obtain equilibrium expressions in terms of aggregate quantities.Theoretical conditions which justify the use of aggregate quantities are discussed by Wilson(1968),Rubinstein(1974)and Constantinides(1982),among others. Some recent empirical work does not assume aggregation,but relies on panels of disaggregated data. Examples include Zeldes (1989),Brav,Constantinides and Geczy (2002)and Balduzzi and Yao (2001). Multiple factor models for asset pricing follow when mt can be written as a function of
7 The budget constraint is: Wt+1 = (Wt - Ct) x'Rt+1, where x is the portfolio weight vector, subject to x'1 = 1. If the allocation of resources to consumption and investment assets is optimal, it is not possible to obtain higher utility by changing the allocation. Suppose an investor considers reducing consumption at time t to purchase more of (any) asset. The expected utility cost at time t of the foregone consumption is the expected marginal utility of consumption expenditures, Uc(Ct,.) > 0 (where a subscript denotes partial derivative), multiplied by the price Pi,t of the asset, measured in the numeraire unit. The expected utility gain of selling the investment asset and consuming the proceeds at time t+1 is Et{(Pi,t+1+Di,t+1) Jw(Wt+1,st+1)}. If the allocation maximizes expected utility, the following must hold: Pi,t Et{Uc(Ct,.)} = Et{(Pi,t+1+Di,t+1) Jw(Wt+1,st+1)}, which is equivalent to equation (1), with mt+1 = Jw(Wt+1,st+1)/Et{Uc(Ct,.)}. (4) The mt+1 in equation (4) is the intertemporal marginal rate of substitution (IMRS) of the consumerinvestor, and equations (2) and (4) combined are the intertemporal Euler equation. Asset pricing models typically focus on the relation of security returns to aggregate quantities. To get there, it is necessary to aggregate the Euler equations of individuals to obtain equilibrium expressions in terms of aggregate quantities. Theoretical conditions which justify the use of aggregate quantities are discussed by Wilson (1968), Rubinstein (1974) and Constantinides (1982), among others. Some recent empirical work does not assume aggregation, but relies on panels of disaggregated data. Examples include Zeldes (1989), Brav, Constantinides and Geczy (2002) and Balduzzi and Yao (2001). Multiple factor models for asset pricing follow when mt+1 can be written as a function of
8 several factors.Equation(4)suggests that likely candidates for the factors are variables that proxy for consumer wealth,consumption expenditures or the state variables--the sufficient statistics for the marginal utility of future wealth in an optimal consumption-investment plan. Expected Risk Premiums Typically,empirical work focuses on expressions for expected returns and excess rates of return.Expected excess returns are related to the risk factors that create variation in m1.Consider any asset return Rit+i and a reference asset return,Ro.+1.Define the excess return of asset i,relative to the reference asset as rit1=Rit+i-Ro.t+1.If equation(2)holds for both assets it implies: Efmtti rit)=0 for all i. (5) Use the definition of covariance to expand equation (5)into the product of expectations plus the covariance,obtaining: Efrit+)=Covt(rit+i;-mt+)/Emt+i),for all i, (6) where Cov(.;)is the conditional covariance.Equation (6)is a general expression for the expected excess return,from which most of the expressions in the literature can be derived.The conditional covariance of return with the SDF,m,is a very general measure of systematic risk.Asset pricing models say that assets earn expected return premiums for their systematic risk,not their total risk (i.e., variance of return).The covariance with-mti is systematic risk because it measures the component of the return that contributes to fluctuations in the marginal utility of wealth.If we regressed the asset
8 several factors. Equation (4) suggests that likely candidates for the factors are variables that proxy for consumer wealth, consumption expenditures or the state variables -- the sufficient statistics for the marginal utility of future wealth in an optimal consumption-investment plan. Expected Risk Premiums Typically, empirical work focuses on expressions for expected returns and excess rates of return. Expected excess returns are related to the risk factors that create variation in mt+1. Consider any asset return Ri,t+1 and a reference asset return, R0,t+1. Define the excess return of asset i, relative to the reference asset as ri,t+1 = Ri,t+1 - R0,t+1. If equation (2) holds for both assets it implies: Et{mt+1 ri,t+1} = 0 for all i. (5) Use the definition of covariance to expand equation (5) into the product of expectations plus the covariance, obtaining: Et{ri,t+1} = Covt(ri,t+1; -mt+1) / Et{mt+1}, for all i, (6) where Covt(.;.) is the conditional covariance. Equation (6) is a general expression for the expected excess return, from which most of the expressions in the literature can be derived. The conditional covariance of return with the SDF, mt+1, is a very general measure of systematic risk. Asset pricing models say that assets earn expected return premiums for their systematic risk, not their total risk (i.e., variance of return). The covariance with -mt+1 is systematic risk because it measures the component of the return that contributes to fluctuations in the marginal utility of wealth. If we regressed the asset
9 return on the SDF,the residual in the regression would capture the "unsystematic"risk and would not be "priced,"or command a risk premium. If the conditional covariance with the SDF is zero for a particular asset,the expected excess return of that asset should be zero.The more negative is the covariance with mthe less desireable is the distribution of the random return,as the larger payoffs tend to occur when the marginal utility is low.The expected compensation for holding assets with this feature must be higher than for those with a more desireable distribution.Expected risk premiums should therefore differ across assets in proportion to their conditional covariances with-m+1. Return Predictability Rational expectations implies that the difference between return realizations and the expectations in the model should be unrelated to the information that the expectations in the model are conditioned on.For example,equation(2)says that the conditional expectation of the product of m and Rit+i is the constant,1.0.Therefore,1-mRit+i should not be predictably different from zero using any information available at time t.If we run a regression of 1-m+iRitti on any lagged variable,Zt,the regression coefficients should be zero.If there is predictability in a return Riti using instruments Zt,the model implies that the predictability is removed when Rit+i is multiplied by the correct mu1.This is the sense in which conditional asset pricing models are asked to "explain"predictable variation in asset returns.This view generalizes the older "random walk"model of stock values,which states that stock returns should be completely unpredictable.That model is a special case which can be motivated by 3 Equation (6)is weaker than equation(2),since equation(6)is equivalent to EmR=Aalli, where At is a constant across assets,while equation(2)restricts A=1.Therefore,empirical tests based on equation(6)do not exploit all of the restrictions implied by a model that may be stated in the form of equation(2)
9 return on the SDF, the residual in the regression would capture the "unsystematic" risk and would not be "priced," or command a risk premium. If the conditional covariance with the SDF is zero for a particular asset, the expected excess return of that asset should be zero.3 The more negative is the covariance with mt+1 the less desireable is the distribution of the random return, as the larger payoffs tend to occur when the marginal utility is low. The expected compensation for holding assets with this feature must be higher than for those with a more desireable distribution. Expected risk premiums should therefore differ across assets in proportion to their conditional covariances with -mt+1. Return Predictability Rational expectations implies that the difference between return realizations and the expectations in the model should be unrelated to the information that the expectations in the model are conditioned on. For example, equation (2) says that the conditional expectation of the product of mt+1 and Ri,t+1 is the constant, 1.0. Therefore, 1-mt+1Ri,t+1 should not be predictably different from zero using any information available at time t. If we run a regression of 1-mt+1Ri,t+1 on any lagged variable, Zt, the regression coefficients should be zero. If there is predictability in a return Ri,t+1 using instruments Zt, the model implies that the predictability is removed when Ri,t+1 is multiplied by the correct mt+1. This is the sense in which conditional asset pricing models are asked to "explain" predictable variation in asset returns. This view generalizes the older "random walk" model of stock values, which states that stock returns should be completely unpredictable. That model is a special case which can be motivated by 3 Equation (6) is weaker than equation (2), since equation (6) is equivalent to Et{mt+1Ri,t+1} = ∆t, all i, where ∆t is a constant across assets, while equation (2) restricts ∆t=1. Therefore, empirical tests based on equation (6) do not exploit all of the restrictions implied by a model that may be stated in the form of equation (2)
10 risk neutrality.Under risk neutrality the IMRS,m+1,is a constant.Therefore,in this case the model implies that the return Ritti should not differ predictably from a constant. Conditional asset pricing presumes the existence of some return predictability.There should be instruments Zt for which E(RZt)or E(m+Z)vary over time,in order for the equation E(mt+iR+i- 0 to have empirical bite.Interest in predicting security market returns is about as old as the security markets themselves.Fama(1970)reviews the early evidence. One body of literature uses lagged returns to predict future stock returns,attempting to exploit serial dependence.High frequency serial dependence,such as daily or intra-day patterns,are often considered to represent the effects of market microstructure,such as bid-ask spreads(e.g.Roll,1984) and nonsynchronous trading of the stocks in an index (e.g.Scholes and Williams,1977).Serial dependence at longer horizons may represent predictable changes in the expected returns. Conrad and Kaul (1989)report serial dependence in weekly returns.Jegadeesh and Titman (1993)find that relatively high return,"winner"stocks tend to repeat their performance over three to nine-month horizons.DeBondt and Thaler(1985)find that past high-return stocks perform poorly over the next five years,and Fama and French(1988)find negative serial dependence over two to five-year horizons.These serial dependence patterns motivate a large number of studies which attempt to assess the economic magnitude and statistical robustness of the implied predictability,or to explain the predictability as an economic phenomenon.For more comprehensive reviews,see Campbell,Lo and MacKinlay (1997)or Kaul (1996).Research in this area continues,and its fair to say that the jury is still out on the issue of predictability using lagged returns. 4 At one level this is easy.Since E(m)should be the inverse of a risk-free return,all we need is observable risk free rates that vary over time.Ferson(1989)shows that the behavior of stock returns and short term interest rates imply that conditional covariances of returns with mti must also vary over time
10 risk neutrality. Under risk neutrality the IMRS, mt+1, is a constant. Therefore, in this case the model implies that the return Ri,t+1 should not differ predictably from a constant. Conditional asset pricing presumes the existence of some return predictability. There should be instruments Zt for which E(Rt+1|Zt) or E(mt+1|Zt) vary over time, in order for the equation E(mt+1Rt+1- 1|Zt)=0 to have empirical bite.4 Interest in predicting security market returns is about as old as the security markets themselves. Fama (1970) reviews the early evidence. One body of literature uses lagged returns to predict future stock returns, attempting to exploit serial dependence. High frequency serial dependence, such as daily or intra-day patterns, are often considered to represent the effects of market microstructure, such as bid-ask spreads (e.g. Roll, 1984) and nonsynchronous trading of the stocks in an index (e.g. Scholes and Williams, 1977). Serial dependence at longer horizons may represent predictable changes in the expected returns. Conrad and Kaul (1989) report serial dependence in weekly returns. Jegadeesh and Titman (1993) find that relatively high return, "winner" stocks tend to repeat their performance over three to nine-month horizons. DeBondt and Thaler (1985) find that past high-return stocks perform poorly over the next five years, and Fama and French (1988) find negative serial dependence over two to five-year horizons. These serial dependence patterns motivate a large number of studies which attempt to assess the economic magnitude and statistical robustness of the implied predictability, or to explain the predictability as an economic phenomenon. For more comprehensive reviews, see Campbell, Lo and MacKinlay (1997) or Kaul (1996). Research in this area continues, and its fair to say that the jury is still out on the issue of predictability using lagged returns. 4 At one level this is easy. Since E(mt+1|Zt) should be the inverse of a risk-free return, all we need is observable risk free rates that vary over time. Ferson (1989) shows that the behavior of stock returns and short term interest rates imply that conditional covariances of returns with mt+1 must also vary over time
11 A second body of literature studies predictability using other lagged variables as instruments. Fama and French (1989)assemble a list of variables from studies in the early 1980's,that as of this writing remain the workhorse instruments for conditional asset pricing models.These variables include the lagged dividend yield of a stock market index,a yield spread of long-term government bonds relative to short term bonds,and a yield spread of low-grade (high default risk)corporate bonds over high-grade bonds.In addition,studies often include the level of a short term interest rate(Fama and Schwert (1977),Ferson,1989)and the lagged excess return of a medium-term over a short-term Treasury bill(Campbell(1987),Ferson and Harvey,1991).Recently proposed instruments include an aggregate book-to-market ratio (Pontiff and Schall,1998)and lagged consumption-to-wealth ratios (Lettau and Ludvigson,2000).Of course,many other predictor variables have been proposed and more will doubtless be proposed in the future. Predictability using lagged instruments remains controversial,and there are some good reasons the question the predictability.Studies have identified various statistical biases in predictive regressions(e.g.Hansen and Hodrick(1980),Stambaugh(1999),Ferson,Sarkissian and Simin,2002), questioned the stability of the predictive relations across economic regimes(e.g.Kim,Nelson and Startz,1991)and raised the possibility that the lagged instruments arise solely through data mining(e.g. Campbell,Lo and MacKinlay(1990),Foster,Smith and Whaley,1997). A reasonable response to these concerns is to see if the predictive relations hold out-of- sample.This kind of evidence is also mixed.Some studies find support for predictability in step-ahead or out-of-sample exercises (e.g.Fama and French (1989),Pesaran and Timmerman,1995).Similar instruments show some ability to predict returns outside the U.S.context,where they arose (e.g.Harvey (1991),Solnik (1993),Ferson and Harvey,1993,1999).However,other studies conclude that predictability using the standard lagged instruments does not hold (e.g.Goyal and Welch(1999)
11 A second body of literature studies predictability using other lagged variables as instruments. Fama and French (1989) assemble a list of variables from studies in the early 1980's, that as of this writing remain the workhorse instruments for conditional asset pricing models. These variables include the lagged dividend yield of a stock market index, a yield spread of long-term government bonds relative to short term bonds, and a yield spread of low-grade (high default risk) corporate bonds over high-grade bonds. In addition, studies often include the level of a short term interest rate (Fama and Schwert (1977), Ferson, 1989) and the lagged excess return of a medium-term over a short-term Treasury bill (Campbell (1987), Ferson and Harvey, 1991). Recently proposed instruments include an aggregate book-to-market ratio (Pontiff and Schall, 1998) and lagged consumption-to-wealth ratios (Lettau and Ludvigson, 2000). Of course, many other predictor variables have been proposed and more will doubtless be proposed in the future. Predictability using lagged instruments remains controversial, and there are some good reasons the question the predictability. Studies have identified various statistical biases in predictive regressions (e.g. Hansen and Hodrick (1980), Stambaugh (1999), Ferson, Sarkissian and Simin, 2002), questioned the stability of the predictive relations across economic regimes (e.g. Kim, Nelson and Startz, 1991) and raised the possibility that the lagged instruments arise solely through data mining (e.g. Campbell, Lo and MacKinlay (1990), Foster, Smith and Whaley, 1997). A reasonable response to these concerns is to see if the predictive relations hold out-ofsample. This kind of evidence is also mixed. Some studies find support for predictability in step-ahead or out-of-sample exercises (e.g. Fama and French (1989), Pesaran and Timmerman, 1995). Similar instruments show some ability to predict returns outside the U.S. context, where they arose (e.g. Harvey (1991), Solnik (1993), Ferson and Harvey, 1993, 1999). However, other studies conclude that predictability using the standard lagged instruments does not hold (e.g. Goyal and Welch (1999)