Macroeconomic Sources of Equity Risk

Transcription

1 Macroeconomic Sources of Equity Risk P.N. Smith, S. Sorenson and M.R. Wickens University of York June 2003 Abstract Empirical studies of equity returns can be classified broadly into two types: those based on a theoretical model of the risk premium that satisfies the no-arbitrage condition, and statistical models constructed to best fit the data with no regard to the no-arbitrage condition. In this paper we show how the stochastic discount factor (SDF) model can be used to measure the equity risk premium for theoretical models. We provide new estimates of the equity risk premium for the US and the UK based on monthly data We show how most of the theoretical models of the equity risk premium that have been proposed in the literature are special cases the SDF model. These include CAPM, consumption CAPM, habit persistence models and the Epstein-Zin model. Several of these models were proposed especially to deal with the equity premium puzzle. We explain why some of them are unable to do this as formulated. Due to the difficulties of estimation, most of them have either been calibrated, or estimated using GMM, neither of which delivers an estimate of the risk premium. We use a new empirical approach that does produce estimates of the risk premium and allows tests ofthetheories. Asaresultweprovidethefirst estimates of the equity risk premium for some of these models. In addition to examining existing theories of the equity risk premium, we use the SDF model to generate new theories. We find that macroeconomic variables not previously considered and not consistent with standard general equilibrium theory, such as production, appear to be priced for the equity risk premium. This suggests that traditional general equilibrium considerations may not be the sole explanation for the equity risk premium; other short-term factors associated with pure price risk may also be involved. We then use our estimates to investigate the importance of different components of the equity risk premium including, amongst others, return volatility. A related, and rapidly growing, literature adopts a more statistical approach. It focusses on the empirical relation between the return on equity (or the Sharpe ratio) and return volatility. We use SDF theory to show that this relation is misconceived. 0

2 1 Introduction Empirical studies of equity returns can be classified broadly into two types: those based on a theoretical model of the risk premium that satisfies the no-arbitrage condition, and statistical models constructed to best fit the data with no regard to the no-arbitrage condition. The theory-based approach typically involves the use of a partial equilibrium model such as CAPM, or a general equilibrium model like consumption-capm (CCAPM). Due to the difficulty of directly estimating the risk premium given by CCAPM either its associated Euler equation is estimated by GMM (one of the first to this was Hansen and Singleton (1983), or the risk premium is calibrated (the best known example is Mehra, R. and E. C. Prescott (1985)). As a result, it has not proved possible to extract estimates of a time-varying equity risk premium for CCAPM. The main finding for CCAPM is that the coefficient of relative risk aversion must be implausibly large in order to match the observed (ex-post) risk premium. This is the equity premium puzzle. The response has been to try to modify the theory so that it possesses other means of capturing the risk premium, and doesn t have to rely solely the coefficient of relative risk aversion to do so. One route is to assume that the utility function is time non-separable. This permits the coefficient of relative risk aversion to be different from the elasticity of inter-temporal substitution, and so introduces additional variables into the expression for the equity risk premium. Again the resulting risk premia are not estimated directly; GMM estimates of the Euler equation (Epstein and Zin (1991)), or calibration of the risk premium is undertaken instead (Campbell, J.Y. (2002)). Another route, also based on time non-separable utility, is to allow the utility function used in general equilibrium models of risk to display habit persistence, (see Abel (1990), Constantinides (1990) and Campbell and Cochrane (1999)). This has the effect of causing the inter-temporal marginal rate of substution to become more variable and so be better able to match the observed volatility of equity returns. One of the main findings to emerge from the statistical approach is that average equity returns are positively related to their volatility. The usual interpretation, informally offered, is that higher volatility implies greater risk, and larger returns are required to compensate for this. In other words, the volatility is capturing a risk premium. This explanation cannot be correct, however, as risk should be should be expressed in terms of the covariance of returns with other factors, and not the volatility of returns. It would seem, therefore, that the empirical models are simply picking up the fact that one component of all of these covariances is in terms of the volatility of returns. A more general theoretical framework of asset pricing than either CCAPM or CAPM is the stochastic discount factor (SDF); CAPM and CCAPM are special cases of this. In this paper we 1

3 show how to obtain direct estimates of a time-varying risk premium derived from the SDF model. We apply the methodology to the following models: CAPM and CCAPM with power utility and with Epstein-Zin time non-separable preferences. We show that as formulated several of the habit persistence models do not admit a time-varying equity risk premium. This only becomes apparent when one attempts to derive the risk premium for the habit persistence model. The greater generality of SDF model compared with general equilibrium models allows us to consider whether there are other priced sources of equity risk. General equilibrium models imply that investors are concerned with future consumption, and in particular, consumption next period. The main holders of equity are financial institutions, especially pension funds. They act on behalf of investors consumption at a much more distant point in the future. In assessing risk, financial institutions focus largely on short-term performance, and on the value of the portfolio. This suggests that the market equity risk premium may be more influenced by short-term price risk than longer-term considerations of consumption. The factors that affect the price of equity in the short term are associated with the business cycle and inflation. We therefore examine whether output and inflation are priced sources of equity risk. In empirical implementations of the SDF model, the factors are latent variables. Equations for the latent factors must be specified. Since data only exist for the returns, estimates of the parameters of these equations must be obtained from the likelihood function for returns. Estimates of the factors and a time-varying risk premium can then be backed out of the returns data. This is the approach commonly followed for bond pricing. It is not, however, suitable for CAPM and CCAPM because the factors are observable macroeconomic variables, namely, the market interest rate in the case of CAPM, and consumption growth (among others) for CCAPM. To obtain direct estimates of a time-varying risk premium for the SDF model when the factors are observable we therefore need a new econometric approach. In the SDF model the risk premium is represented by the conditional covariances of excess equity returns with the factors. This implies that we must model the joint distribution of excess returns and the factors in such a way that the mean of the conditional distribution of excess returns is a linear function of selected conditional covariances of the joint distribution. A convenient model to use is the multi-variate GARCH model with in mean effects. It is important to stress that the in mean effect is not just the conditional variance of returns, but also includes conditional covariance terms. In order for the econometric model to satisfy the no-arbitrage condition, restrictions must be imposed on the conditional mean. A special case of this model that omits the conditional covariances in the mean relates equity returns solely to their volatility. This enables us to compare the statistical approach with the theory-based approach. We apply this new methodology to US 2

4 and UK monthly equity returns The paper is set out as follows. In Section 2 we consider various theoretical models of the equity risk premium. As they are all special cases of the SDF model of asset pricing, first we derive the equity risk premium for a general SDF model. We then show that a number of general equilibrium models that differ due to the specification of the investor s utility function are special cases of the SDF model. Three of these models are versions of CCAPM: with power utility, with habit persistence and with Epstein-Zin utility. The first has a time separable utility function and the other two have time non-separable utility functions. We then consider CAPM and our proposed alternative to general equilibrium models of equity risk. In Section 3 we examine the theoretical plausibility of models that relate equity returns to the volatility of returns. Our new econometric methodology is described in Section 4. Estimates of the various models are reported in Section 5 together with the implied estimates of the equity risk premium. 2 Theoretical models of the equity risk premium 2.1 The SDF model of asset pricing for equity returns TheSDFmodelisbasedonthesimpleideathatP t, the price of an asset at the beginning of period t, is determined by the expected discounted value of its pay-off at the start of period t +1, namely, X t+1 : P t = E t [M t+1 X t+1 ] where M t+1 is the stochastic discount factor, or pricing kernel (see Cochrane (2001) for a survey of SDF theory). For equity, the payoff is X t+1 = P t+1 + D t+1,whered t+1 are dividend payments assumed to be made at the start of period t +1. The pricing equation can also be written X t+1 1=E t [M t+1 ]=E t [M t+1 R t+1 ], P t where R t+1 =1+r t+1 = X t+1 /P t =(P t+1 + D t+1 )/P t is the gross return and r t+1 is the return. Taking logarithms and assuming log-normality (and noting that if ln x is N(µ, σ 2 ) then ln E(x) = µ + σ2 2 )weobtain 0 = lne t [M t+1 R t+1 ] = E t [ln(m t+1 R t+1 )] + V t [ln(m t+1 R t+1 )]/2 = E t (m t+1 )+E t (r t+1 )+V t (m t+1 )/2+V t (r t+1 )/2+cov t (m t+1,r t+1 ) where m t+1 =lnm t+1. Hence the pricing equation can be written E t (r t+1 )+E t (m t+1 )+V t (m t+1 )/2+V t (r t+1 )/2 = cov t (m t+1,r t+1 ) 3

5 If the asset is risk-free then its return is known at the start of period t implying r t+1 r f t, E t (r t+1 )=r f t and V t (r t+1 )=0. The pricing equation for the risk-free asset can therefore be written E t (m t+1 )+r f t V t(m t+1 )=0. Subtracting the two pricing equations gives the expected excess return on the risky asset E t (r t+1 r f t )+ 1 2 V t(r t+1 )= Cov t (m t+1,r t+1 ). This is the key no-arbitrage condition that all correctly priced assets must satisfy when their returns are lognormally distributed. The right-hand side is the risk premium and 1 2 V t(r t+1 ) is the Jensen effect Real versus nominal returns This no-arbitrage condition holds whether we measure returns in nominal or real terms. If real returns are measured ex-ante, i.e. as nominal returns less expected future inflation and the discount factor is defined in real ex-ante terms, then we can use either real or nominal returns in formula as expected future inflation is known at time t. In general equilibrium models it is usual to express the discount factor implicitly in real ex-ante terms. However, most empirical studies use ex-post and not ex-ante returns and discount factors. In this case the real ex-ante discount factor will different from the real ex-post. If we adopt this convention and define M t+1 as the real ex-post discount factor and r t+1 and r f t as real ex-post rates of return, then the no-arbitrage condition for nominal returns will require re-stating. Since only a nominal risk-free rate exists in practice (assuming no default risk this is the Treasury bill rate), it makes more sense to conduct the analysis of equity returns using nominal variables. Let Mt+1 N denote the nominal discount factor, let i t+1 and i f t be the respective nominal rates of return and let P t be the consumer price index at the start of period t. It follows that for ex-post real returns 1+i t+1 =(1+r t+1 ) P t+1 =(1+r t+1 )(1 + π t+1 ) P t where π t+1 is the inflation rate. Hence 1 = E t [M t+1.(1 + r t+1 )] = E t [M t+1. 1+i t+1 1+π t+1 ] M t+1 = E t [.(1 + i t+1 )] 1+π t+1 4

6 and so Mt+1 N = Mt+1 1+π t+1. It can be shown that the no-arbitrage condition for nominal returns and a real discount factor is E t (i t+1 i f t )+ 1 2 V t(i t+1 )= Cov t (m t+1,i t+1 )+Cov t (π t+1,i t+1 ). We now consider a number of models of the discount factor. 2.2 Consumption-based models C-CAPM with power utility The canonical model of the discount factor is the consumption-based CAPM for real asset returns. Asset prices derive their value from the expected consumption streams of investors who choose to subject to the budget constraint max C t {U t = U(C t )+βe t (U t+1 )} C t + W t+1 = Y t + W t R t where C t is real consumption, Y t is real non-asset income and W t is real financial wealth at the start of period t. The solution is the Euler equation E t [ βu 0 (C t+1 ) U 0 R t+1 ]=1. (C t ) This implies that the CCAPM has implicitly defined the real SDF as M t+1 = βu 0 (C t+1 ) U 0 (C t ) 1 For the power utility function U = C1 σ t 1 σ with constant coefficient of relative risk aversion σ, the real discount factor is: µ σ Ct+1 M t+1 = β Since real consumption is usually defined in ex-post terms, the discount factor will also be in real ex-post terms. Taking logarithms, and ignoring all constants, we obtain C t m t+1 = σ ln C t+1 where c t =lnc t. The no-arbitrage condition in real terms becomes E t (r t+1 r f t )+ 1 2 V t(r t+1 )=σcov t (r t+1 c t+1 ) 5

7 The interpretation of the real equity risk premium is that investors lose utility today by not consuming. To compensate investors who defer the utility from an extra unit of consumption today they need additional marginal utility from future consumption. Because marginal utility declines as consumption increases, a higher level of consumption is needed in the future. The return on the investment must be large enough to generate the required consumption in the future. The greater the consumption needed, the larger the return must be, hence the risk premium is larger the greater the predicted covariance between consumption and returns. Another way to think about the equity risk premium is to note that over the business cycle equity returns tend to be positively correlated with consumption growth. In the downturn when consumption growth tends to be low, equity returns are also low, hence risk is associated with the predicted covariance of returns with consumption. In nominal terms the no-arbitrage condition can be written E t (i t+1 i f t )+ 1 2 V t(i t+1 )=σcov t (i t+1 c t+1 )+Cov t (i t+1 π t+1 ) Thus the nominal risk premium involves the nominal return and has a second covariance term associated with consumer price inflation. The greater the covariance between nominal returns and inflation, the larger the risk premium. We have argued that the larger the future consumption needed, the higher real returns must be. This is also true for nominal returns. The extra risk is that nominal returns will be larger solely due to inflation C-CAPM with Epstein-Zin preferences CCAPM with power utility restricts the coefficient of relative risk aversion to be equal to the elasticity of inter-temporal substitution. This restriction can be relaxed using a time non-separable utility function. A general formulation of time non-separable utility proposed by Kreps and Porteus (1978) is U t = U[C t,e t (U t+1 )] Epstein and Zin (1989), Epstein and Zin (1991) have implemented a special case of this based on the constant elasticity of substitution (CES) function: U t = (1 β)c 1 γ 1 t + β[e t (U 1 1 σ 1 1 γ 1 t+1 )]1 1 γ where β is the discount rate, σ is the coefficient of relative risk aversion and γ is the elasticity of inter-temporal substitution. In the separable, power utility case σ =1/γ. Epstein and Zin show that maximising U t subject to a slightly different budget constraint W t+1 = R m t+1(w t C t ) 6

8 where Rt+1 m is the return on the market portfolio of all invested wealth. The asset pricing equation for the portfolio return derived by Epstein and Zin is ( E t β( C 1 σ ) t+1 ) 1 1 γ R m γ 1 C t+1 =1 t and for the return on an individual asset with gross real return R t+1 is ( E t β( C t+1 C t ) 1 γ 1 σ 1 1 γ The real stochastic discount factor is therefore M t+1 = In logarithms it is β( C t+1 ) 1 γ C t ) 1 σ (Rt+1) m 1 γ 1 1 Rt+1 1 σ 1 1 γ 1 σ (Rt+1) m 1 γ 1 1 m t+1 = 1 σ 1 γ ln C t+1 1 γσ 1 γ rm t+1 The resulting no-arbitrage condition first derived by Campbell and MacKinlay (1997) is E t (r t+1 r f t )+ 1 2 V t(r t+1 )= 1 σ 1 γ Cov t(r t+1 c t+1 )+ 1 γσ 1 γ Cov t(r t+1 r m t+1) =1 Using nominal gross market returns R N t+1, the Euler equation becomes ( E t β( C t+1 C t ) 1 γ 1 σ 1 1 γ and the log nominal stochastic discount factor is ) Rt+1 N 1 σ 1 ( ) 1 1 R γ t+1 ( ) =1 P t+1 /P t P t+1 /P t m t+1 = 1 σ 1 γ ln C t+1 1 γσ γ(1 σ) 1 γ im t γ π t+1 where i m t+1 is the nominal market return. The no-arbitrage condition for the nominal return on an individual asset with Epstein-Zin preferences is E t (i t+1 i f t )+ 1 2 V t(i t+1 ) = 1 σ 1 γ Cov t(i t+1 c t+1 ) + 1 γσ 1 γ Cov t(i t+1 i m t+1) γ(1 σ) 1 γ Cov t(i t+1 π t+1 ) Adifficulty in implementing this in practice is choosing a market rate of return. A convenient alternative is to assume that the market portfolio consists of equity and a risk-free asset. The market return is then a weighted average of the two. i m t+1 = θ t i t+1 +(1 θ t )i f t 7

9 Iftheweassumethattheweightsθ t = θ, a constant, the Epstein-Zin no-arbitrage condition then becomes the fixed parameter model E t (i t+1 i f t )+( 1 2 θ 1 γσ 1 γ )V t(i t+1 ) = 1 σ 1 γ Cov t(i t+1 c t+1 ) γ(1 σ) 1 γ Cov t(i t+1 π t+1 ) Comparing the no-arbitrage condition for power utility with this condition, we note that the interpretation of all of the coefficients is changed. The coefficient on Cov t (i t+1 c t+1 ) is no longer the coefficient of relative risk aversion, and the coefficients on V t (i t+1 ) and Cov t (i t+1 π t+1 ) are no longer restricted Habit persistence An alternative example of time non-separable utility is the habit persistence model which assumes that U t = U(C t,x t ) where X t is the habitual level of consumption. Again the aim of this approach is to introduce additional terms in the risk premium. Various specific functional forms for the utility function have been suggested. Constantinides (1990) proposed the form U t = (C t λx t ) 1 σ 1 1 σ The stochastic discount factor implied by this is µ Ct+1 λx t+1 M t+1 = β C t λx t By a suitable choice of λ and X t it is possible to produce a discount factor that displays greater volatility and hence a larger risk premium. Campbell and Cochrane (1999) implemented this with the restriction that λ = 1. They introduced the concept of surplus consumption, defined as S t = C t X t C t. The discount factor can then be written µ σ Ct S t+1 M t+1 = β C t+1 S t Assuming log normality, the discount factor is σ no-arbitrage condition is m t+1 = σ c t+1 σ s t+1 E t (r t+1 r f t )+ 1 2 V t(r t+1 )= σcov t (r t+1, c t+1 )+σcov t (r t+1,s t+1 ) 8

10 where s t+1 =lns t+1. The attraction of this approach is the inclusion of the extra term to account for the risk premium. The problem lies in the way it has been implemented. Campbell and Cochrane assume that s t+1 is generated by an AR(1) process with a disturbance term whose variance depends on s t. They do not estimate the resulting model but, like Constantinides, use calibration methods. By calibrating the unconditional variance of the error term of the AR process suitably, it is possible to force the covariance between r t+1 and s t+1 to be of the necessary size. By this means success is virtually guaranteed. Unfortunately, by making s t+1 an AR(1) process, Cov t (r t+1,s t+1 )=0. As a result, this version of the habit persistence hypothesis is incapable of providing any additional explanation of the risk premium as the implied conditional covariance r t+1 and s t+1 is necessarily zero. A similar analysis applies to the habit persistence utility function proposed by Abel (1990) U t = ³ Ct X t 1 σ 1 1 σ where X t is a function of past consumption, for example X t = Ct 1. δ The stochastic discount à Ct+1! σ factor becomes M t+1 = β and so X t+1 C t X t m t+1 = σ c t+1 + σ x t+1 where x t =lnx t. Assuming log normality, the no-arbitrage condition is now E t (r t+1 r f t )+ 1 2 V t(r t+1 )=σcov t ( c t+1,r t+1 ) σcov t (r t+1, x t+1 ) If, however, it is assumed that x t+1 = δc t then Cov t (r t+1, x t+1 )=δcov t (r t+1, c t )=0 and so once again the habit persistence model is of no help in providing additional terms in the risk premium. Thus, the problem is that if habit is solely related to past consumption - which is its natural meaning - then the conditional covariance of returns with the discount factor are unaffected. As unconditional moments are used in GMM estimation of the Euler equation, measuring habit by past consumption does not prevent the parameters of the habit persistence model from being estimated. It is only when trying to estimate the risk premium itself that the problem with the habit persistence model becomes evident. 9

11 2.3 CAPM CAPM relates the expected excess return to the excess return of the market portfolio: E t (r t+1 r f t )=β t E t (r m t+1 r f t ) where and β t = Cov t(r t+1,r m t+1 ) V t (r m t+1 ) E t (r m t+1 r f t )=σ t V t (r m t+1). It follows that E t (r t+1 r f t )=σ t Cov t (r t+1,r m t+1) We also note that (1 + rt+1) m = W t+1 W t,wherew t is real wealth. And if consumption is proportional to wealth, as in the life cycle model, we obtain E t (r t+1 r f t )=σ t Cov t (r t+1, c t+1 ) Thus we can interpret CAPM as an SDF model in which the discount factor is σ t (1 + r m t+1 ) -or σ t W t+1 W t - and no Jensen effect is needed. The corresponding expressions for nominal returns are E t (i t+1 i f t )=σ t Cov t (i m t+1,i t+1 ) as adjusting real returns by E t π t+1 has no effect on conditional covariances The relation between equity returns and volatility The finding that the return on equity is related to its own variance does not in general satisfy a no-arbitrage condition. Much of the current view of this relation between mean and variance dates from Campbell (1987) who find that the relation is generally insignificantly positive when the conditional variance term is modelled as a GARCH-in-mean term. This result on the monthly CRSP market-wide index has been further reinforced by Baillie and DeGennero (1990) using a similar methodology. However, the results of Glosten, L.R., Jagannathan R. and D.E. Runkle (1993) and Campbell (1987) have subsequently found evidence of a significant negative relationship between mean and conditional variance. In the case of Glosten et al. the inclusion of seasonal effects to the GARCH-M model appears to be of importance whilst Campbell adopts a different approach deriving the unconditional implications of a conditional model using GMM to carry out the estimation. Recently, focus on additional factors has provided more support for a positive 10

12 (partial) relation between mean and variance; see Scruggs (1998) and Campbell, J.Y. and L. Hentschel (1992). A closely related approach is that of Lettau, M. and S.C. Ludvigson (2002) who examine the behaviour of the Sharpe ratio. Instead of using an ad hoc model of this, it is possible to infer the behaviour of the Sharpe ratio from the SDF model by re-writing the SDF no-arbitrage condition as E t (r t+1 r f t ) = 1 SD t (r t+1 ) 2 SD t(r t+1 ) ρ t (m t+1,r t+1 )SD t (m t+1 ) where ρ t (m t+1,r t+1 ) is the conditional correlation and SD t denotes a conditional standard deviation. Lettau and Ludvigson simple relate the Sharpe ratio to the conditional standard deviation of returns. Both the previous model and the Sharpe ratio models can therefore be easily tested within the SDF framework since they imply that conditional covariances can be excluded from the conditional mean and the coefficient on the own conditional variance should not be restricted to satisfy the Jensen effect. 2.4 Other SDF models The SDF model shows how any source of risk can be incorporated into an explanation of the risk premium in a way that satisfies the no-arbitrage conditiond. If z it (i =1,..., n) are n factors which are jointly log normally distributed with equity returns then the discount factor can be written m t+1 = X n β iz i,t+1 i=1 This implies the no-arbitrage condition E t (r t+1 r f t )+ 1 2 V t(r t+1 ) = X i β i Cov t (z i,t+1,r t+1 ) = X i β i f it, where the f it are known as common factors. In the absence of the sort of clear theoretical foundations provided by general equilibrium theories of asset pricing, the problem is to identify potential souces of risk to include in the SDF model. The latent factor literature simply assumes that unobserved processes can be specified for the factors. As noted above, general equilibrium models imply that investors are concerned with future consumption, and in particular, consumption next period. The main holders of equity are financial institutions, especially pension funds. They act on behalf of investors consumption at amuchmoredistantpointinthefuture. Inassessingrisk,financial institutions focus largely on 11

13 short-term performance, and on the value of the portfolio. This suggests that the market equity risk premium may be more influenced by short-term price risk than longer-term considerations of consumption. The sort of factors that are likely to affect the price of equity in the short term are associated with the business cycle and inflation. We therefore examine whether output is an additional source of equity risk to consumption and inflation. 2.5 A taxonomy of SDF models The following is a summary of the models above. 1. C-CAPM (a) based on time-separable power utility model (i) with real returns (ii) with nominal returns (b) based on the Epstein-Zin time non-separable utility model (i) with nominal returns and an explicit market return (ii) with nominal returns but no explicit market return 2. SDF model (a) based on two macroeconomic factors: consumption and inflation (b) based on three macroeconomic factors: consumption, inflation and industrial production 3. General model The SDF model based on three macroeconomic factors but removing the Jensen restriction on V t (i t+1 ) as in the E-Z model These can all be represented as restricted versions of the SDF model E t (i t+1 i f t ) = β 1 V t (i t+1 )+β 2 Cov t (i t+1 c t+1 )+β 3 Cov t (i t+1 π t+1 ) +β 5 Cov t (i t+1 q t+1 )+β 4 Cov t (i t+1 i m t+1) where q t is the logarithm of a measure of output. The SDF model restricts the coefficient on the conditional variance of equity returns to be β 1 = 1/2 (the Jensen effect). In general, the different SDF models can be distinguished by restricting selected β i to be zero. We note, however, that in SDF models which include the market return as a factor, because the market portfolio includes equity, we have found that we can replace i m t+1,setβ 5 =0and remove the restriction on β 1. The various models are summarised in Table 1 below. The statistical approach relating the conditional mean to its conditional variance implies that β 1 is unrestricted and all other β i =0 (i =2,...). Table 1 : Model Taxonomy 12

14 Model β 1 β 2 β 3 β 4 β 5 CAPM σ PU:real returns 1 2 σ PU:nominal returns 1 2 σ 1 EZ-standard θ(1 γσ) 1 γ 1 σ 1 γ SDF EZ-restricted γσ 1 γ 1 σ 1 γ Two factors 1 2 δ 1 δ 2 γ(1 σ) 1 γ γ(1 σ) 1 γ Three factors 1 2 γ 1 γ 2 γ 3 3 The Econometric Framework 3.1 Multivariate conditional heteroskedasticity models We need to model the distribution of the excess return on equity jointly with the macroeconomic factors in such a way that the mean of the conditional distribution of the excess return in period t +1, given information available at time t, satisfies the no-arbitrage condition. The conditional mean of the excess return involves selected time-varying second moments of the joint distribution. We therefore require a specification of the joint distribution that admits a time-varying variancecovariance matrix. A convenient choice is the multi-variate GARCH-in-mean (MGM) model. Let x t+1 =(r t+1 r f t,z 1,t+1,z 2,t+1,...) 0, where the variables z 1,t+1,z 2,t+1,... are the macroeconomic variables in the SDF. The MGM model can then be written x t+1 = α + Γx t + Bg t + ε t+1 where ε t+1 I t D[0, H t+1 ] g t = vech{h t+1 } The vech operator converts the lower triangle of a symmetric matrix into a vector. The distribution is the multivariate t distribution. The first equation of the model is restricted to satisfy the noarbitrage condition. Thus, in general, the first row of Γ is zero and the first row of B is ( 1 2, β 11, β 12, β 13,...). It will be noted that the theory requires that the macroeconomic variables display conditional heteroskedasticity. This is not something traditionally assumed in macro-econometrics, but seems 13

15 to be present in our data. Ideally, we would like to use high frequency data for asset returns, but very little macroeconomic data are published for frequencies higher than one month, and then only a few variables are available. Although more macroeconomic variables are published at lower frequencies, they tend not to display conditional heteroskedasticity. Whilst the MGM model is convenient, it is not ideal. First, it is heavily parameterised which can create problems for the numerical convergence of the maximum likelihood due to the likelihood being surface being relatively flat, and hence uninformative. Second, asset returns tend to be excessively volatile. Assuming a non-normal distribution such as a t distribution can sometimes help in this regard by dealing with thick tails. The main problem, however, is not thick tails, but a small number of extreme values. The coefficients of the variance process of the MGM model have a tendency to produce a near unstable variance process in their attempt to fit these extreme values. In principle, a stochastic volatility model, which includes an extra random term in the variance, could capture these extreme values. Unfortunately, as far as we are aware, no multivariate stochastic model with in mean effects in the conditional covariances has been proposed in the literature. In view of the need to restrict the number of coefficients to estimate, a commonly used specification of H t+1 is the Constant Conditional Correlation model of Bollerslev (1990) where the dynamics of the conditional covariances are driven by individual GARCH processes for the variances of each variable. Given that the SDF approach focusses on the importance of the contribution of covariances, restricting their dynamics in this way, and not allowing the correlations to be time-varying, seems too restrictive. 1 As a result, we specify H t+1 using the BEKK model originally proposed by Engle and Kroner (1995). This takes the form: vech(h t+1 )=Λ + Σ p 1 i=0 Φ ivech(h t i )+Σ q 1 j=0 Θ jvech(ε t j ε 0 t j) where Λ, Φ and Θ may be unrestricted. With n 1 factors z it then Φ and Θ are both square matrices of size n(n +1)/2 and Λ is a size n(n +1)/2 vector. A formulation of this model which can make implementation easier is the error-correction formulation or VECM BEKK: H t+1 = V 0 V + A 0 (H t V 0 V)A + B 0 (ε t ε 0 t V 0 V)B. where the first term on the RHS is the long-run or unconditional covariance matrix. This can be initialised with starting values from sample averages. The remaining terms capture short-run 1 The attraction of the reduction in parameterisation offered by the CCC model has led to an extension to the dynamics in the DCC proposed recently by Engle (2000). 14

16 deviations from this long run. A restricted version of this formulation is to specify V to be lower triangular and A and B to be symmetric matrices which further reduces parameter numbers. A comparison of parameterisation for n =3and p = q =1is then: BEKK= n(n +1)/2+(p + q)n 2 (n +1) 2 /4 = 78;VECM BEKK= 3n 2 =27; Restricted VECM BEKK= 3n(n +1)/2 =18; Constant Correlation= 3n+n(n 1)/2 =12. We require that the covariance function is stationary. This is satisfied if the absolute value of the eigenvalues of (A A)+(B B) lie inside the unit circle where is the Kronecker product. The structure of the VECM BEKK model that we employ is common to all of the models that we estimate. In each case we condition on the same set of variables in the macroeconomic environment even though the terms in the no-arbitrage condition for the excess return differ between models. Thus the vector x t+1 for the models of the nominal equity return is x t+1 =(i t+1 i f t, π t+1, c t+1, q t+1 ) 0 whilstthatfortherealreturnis:x t+1 =(r t+1 r f t, π t+1, c t+1, q t+1 ) 0. A first order vector autoregression for the macroeconomic variables is found to be sufficient to capture the serial dependence in their means, a VECM BEKK(1,1) model is found to be adequate for the multivariate variance-covariance process. 3.2 GMM Estimation It is informative to contrast the above approach with the GMM estimation of the Euler equation of the general equilibrium condition. This may be written E t [(M t+1 (θ)(r t+1 r f t )] = 0 where we denote that the discount factor depends on the parameters θ. GMM estimation exploits the lack of correlation between the discounted pay-off and the information set used in conditioning. The null hypothesis is that E[(M t+1 (θ)(r t+1 r f t )I t ]=0 I t is the information set which by implication contains no information about M t+1 (θ)(r t+1 r f t ), the discounted excess return in period t +1. Two things may be noted. First, unless the information set has time-varying volatility, it will be unlikely to prove a suitable basis for a time varying risk premium. Surprisingly perhaps, in practice, this has not usually been a consideration. Second, the risk premium itself cannot be obtained even if we knew θ. 15

17 4 Results and Risk Premia 4.1 The Data The data are monthly for the US ( ) and UK: ( ). The US data consists of the excess return on equity of Fama and French, 2 real non-durable growth consumption from FRED, CPI inflation from Datastream and the volume index of industrial production volume index from Datastream. The UK data are the MSCI total equity return index, from Datastream, totalrealnon-durableconsumptiongrowth specially provided by the NIESR, RPI inflation and the volume index of industrial production both from Datastream. All data are express in equivalent annual percentages. Table 2 reports various descriptive statistics for the data, including sweness, kurtosis and autocorrelations of levels and squares. Table 2. DESCRIPTIVE STATISTICS FOR THE WHOLE SAMPLE, RAW DATA FOR THE UK AND US i us t+1 i uk t+1 π us t+1 π uk t+1 c us t+1 c uk t+1 qt+1 us qt+1 uk Mean Std. Dev Skewness Excess Kurtosis Normality ρ(x t,x t 1 ) ρ(x t,x t 2 ) ρ(x t,x t 3 ) ρ(x t,x t 4 ) ρ(x t,x t 5 ) ρ(x t,x t 6 ) ρ(x 2 t,x 2 t 1) ρ(x 2 t,x 2 t 2) ρ(x 2 t,x 2 t 3 ) ρ(x 2 t,x 2 t 4) ρ(x 2 t,x 2 t 5) ρ(x 2 t,x 2 t 6) Estimates A complete set of estimates for one model To illustrate, a full set of model estimates with their restrictions is reported for the variables x(t +1) 0 = {i t+1 i f t, π t+1, c t+1, q t+1 } based on UK data, Jan Nov 2001 (321 obs). 2 Available from Ken French s web site: 16

18 The model is CCAPM with power utility and nominal (ex-post real) returns. The production variable is included in the joint distribution but not in the excess return equation. In this way output growth is allowed to be jointly distributed with the other macroeconomic variables which will in general affect the estimates of the variance-covariance matrix. Also a dummy variable is included for October 1987 on the grounds that the excess return for this observation is drawn from adifferent distribution. This prevents an extreme observation from contaminating the estimates of the GARCH parameters for the whole sample. The main points to note at this stage are that the multivariate GARCH process is well determined with the macroeconomic variables displaying conditional heteroskedasticity, conditional covariance of returns with consumption growth is highly sugnificant, but the size of the coefficient implies a very large coefficient of relative risk aversion. Thus these results display the equity premium puzzle ( ) (6.50) (0.17) (4.18) (0.48) (0.78) α = Γ = (8.64) (1.27) (1.14) (4.49) (1.14) (2.22) (1.82) (0.30) (1.53) (4.68) (2.54) (1.53) Φ = (1.41) (2.69) (3.70) A = (1.31) (2.24) (3.43) B = (3.51) (2.68) (4.05) (3.34) (3.19) (0.09) (6.46) (1.98) (2.60) (0.59) (0.12) (2.10) (4.77) (7.61) The eigenvalues for GARCH process are: 0.958, 0.919, 0.902, 0.882, 0.802, etc. 17

19 4.2.2 US estimates Estimates of the no-arbitrage condition for the US data are pre- Theno-arbitrageequation sented in Table 3. Table 3. Estimates of the no-arbitrage equation for all US models US General SDF EZ1 PU-N-H1PU-R-H0PU-N-H0 EZ2 CAPM1 CAPM V t (i us t+1) (0.72) t+1) C t (i us t+1, π us C t (i us t+1, c us (1.58) t+1) (2.34) C t (i us t+1, qt+1) us (0.08) Dummy (α) (1.98) Deg of Freedom (3.10) (1.68) (2.89) (0.29) (2.02) (3.15) (0.75) (1.65) (2.38) (1.95) (3.13) (1.69) (2.94) (1.90) (3.21) (0.63) (0.63) (3.10) (1.80) (3.25) (3.10) (1.80) (3.25) (2.38) (1.88) (3.18) (0.63) (2.11) (3.60) (3.60) (1.69) (1.47) (2.67) (3.07) Log-likelihood Mean risk premium 8.83% 9.75% 8.85% 9.99% 12.01% 12.00% 10.79% 7.01% 10.66% λ max ε t V(φ t+1 ) V(φ t+1 ) V(i us t V t(i us t+1 ) bαd) Estimates in parentheses are t statistics In reverse order of the number of restrictions, estimates of CCAPM with power utility assuming all variables are measured in ex-ante real terms are reported in column 5. Column 6 assumes that all variables are measured in nominal (real ex-post) terms and column 4 is the equivalent SDF model where the restriction on coefficient on Cov t (i us t+1, π us t+1) is removed. All of the estimates of the coefficient of relative risk aversion display the equity premium puzzle effect of being far too large. Removing the restriction on the inflation term only reduces the estimate of the CRRA a little. The inflation coefficient suggests that the restrictions are satisfied by the data. Estimates of the restricted Epstein-Zin model are reported in column 7. Column 3 removes the restriction on the inflation covariance term. Neither the restricted nor the unrestricted model is significantly different from the corresponding power utility model. The χ 2 (2) test statistic is 0.62, which is not significant. We conclude that, in practice, the Epstein-Zin model offers only a minor generalisation of power utility. Column 2 reports estimates of the more general SDF model that includes output. Column 1 includes the market return as an extra variable in the SDF model, but assumes that this is 18

20 a weighted average of the return on equity and the risk-free rate and so affects the coefficient on the own conditional variance. Neither variable seems to be significantly priced. The lack of significance of the own variance from -0.5 suggests that the market return may not be priced. The results also show that statistical models that relate excess returns just to volatility can be rejected in favour of an SDF model with consumption and inflation risk. Risk premia Table 3 reports the mean risk premia for the models and the proportion of the variance of excess returns explained by each model. The power utility models give an estimated average risk premium of about 12%, while the other models give much lower values. The lowest is about 9% for the more general models. Correspondingly, the most restricted power utility model explains 2.25% of the variance of excess returns by variations in the risk premium, and the other models explain up 3.6%. Thus, the size of the risk premium is linked to the explanatory power of the model. Figure 1 gives a plot of excess returns together with the estimates risk premia, and a comparison of the risk premia for restricted and unrestricted CCAPM with power utility and the SDF model. The latter graph shows that removing the restriction on the coefficient of the inflation covariance term gives a much more variable risk premium. In the restricted CCAPM model, consumption is in effect the only source of time variation in the risk premium and this reduces the variability of the risk premium considerably. 19

21 i s,t V t ( i s,t + 1 ) - US φ s,t - US Figure 1. Plots of the US risk premia UK estimates The no-arbitrage equation Estimates for the UK data are presented in Table 4. The results for the UK differ various respects. The estimates of the CRRA are smaller, but are still far too large. Inflation seems to be a more significant source of risk than for the US. The Epstein-Zin model is again a minor generalisation of power utility; the χ 2 (2) statistic is Risk premia The mean risk premium for the UK is somewhat higher than for the US, and is higher for the more general models than for power utility. The proportion of the variance of excess returns explained by the variations in the risk premium is about the same size. Figure 2 plots the excess returns and estimates risk premia, and gives a comparison of the risk premia for restricted and unrestricted CCAPM with power utility and the SDF model. The results are similar to those of the US, except that the risk premia are slightly more volatile. 20

22 Table 4. Estimates of the no-arbitrage equation for all UK models UK General SDF EZ1 PU-N-H1 PU-N-H0 PU-R-H0 EZ2 CAPM1 CAPM V t (i uk t+1) (0.74) C t (i uk t+1, πuk t+1 ) (1.72) C t (i uk t+1, c uk t+1) C t (i uk t+1, q uk (1.95) t+1) (1.07) Dummy (α) (1.86) Deg of Freedom (3.32) (2.06) (1.85) (0.60) (1.80) (3.37) (0.45) (2.73) (1.38) (1.65) (3.42) (2.75) (1.57) (1.70) (3.43) (2.03) (2.03) (2.54) (1.53) (3.16) (2.54) (1.53) (3.16) (1.50) (1.54) (3.05) (0.83) (2.67) (1.74) (3.43) (3.24) (3.24) (1.45) (3.06) Loglikelihood Mean risk premium 10.54% 10.41% 10.81% 10.65% 7.23% 7.21% 9.76% 10.84% 9.70% λ max ε t V(φ t+1 ) V(φ t+1 ) V(i uk t V t(i uk t+1 ) bαd) i s, t V t ( i s,t + 1 ) - UK φ s, t - UK Figure 2. Plots of the UK risk premia 21

23 5 Conclusions In this paper, we have undertaken a critical re-examination of models of the equity risk premium and have proposed an alternative approach to testing these theories based on direct estimation of the implied risk premium. The theoretical framework is the SDF model and a new econometric model is suggested. Using this new methodolgy, we have provided some of the first estimates of a time-varying risk premium for the US and UK based on standard models of risk. The estimated risk premia are about 10% for both countries, but the estimates differ for different models. Our results suggest that the Epstein-Zin model adds little in empirical terms to power utility. They also support the prevailing evidence obtained using very different methods, that the coefficient of relative risk aversion seems too large to be plausible. We therefore find strong evidence of the equity premium puzzle. The most successful model is a general SDF model based on consumption and inflation both of which appear to be priced for the US and UK. Output does not appear to be priced. Our results also show that statistical models that relate excess returns just to volatility canberejectedinfavourofansdfmodelwithconsumptionandinflation risk. References Abel, A. (1990): Asset Prices under Habit Formation and Catching Up with the Joneses, American Economic Review Papers and Proceedings, 80, Bollerslev, T. (1990): Modelling the Coherence in Short-Run Nominal Exchange Rates: A Multivariate Generalised ARCH Model, The Review of Economics and Statistics, pp Campbell, J., and J. Cochrane (1999): By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior, Journal of political economy, pp Campbell, J. Y. (1987): Stock Returns and the Term Structure, Journal of Financial Economics, 18, Campbell, J.Y., L. A., and A. MacKinlay (1997): The Econometrics of Financial Markets. Princeton University Press, second printing edn. Campbell, J.Y. (2002): Consumption-Based Asset Pricing, Chapter prepared for the Handbook of the Economics of Finance, George Constantinides, Milton Harris, and Rene Stulz eds., North-Holland, Amsterdam. Campbell, J.Y. and L. Hentschel (1992): No News is Good News, Journal of Financial Economics, pp Cochrane, J. (2001): Asset Pricing. Princeton University Press, 1 edn. Constantinides, G. (1990): Habit Formation: A Resolution of the Equity Premium Puzzle, Journal of Political Economy, 98, Engle, R. (2000): Dynamic Conditional Correlation - A Simple Class of Multivariate GARCH Models, Mimeo, University of California San Diego, USA. 22

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