Multifactor models and their consistency with the ICAPM

Transcription

1 Multifactor models and their consistency with the ICAPM Paulo Maio 1 Pedro Santa-Clara 2 This version: February Hanken School of Economics. paulofmaio@gmail.com. 2 Nova School of Business and Economics, NBER, and CEPR. psc@novasbe.pt. 3 We thank an anonymous referee, António Antunes, Matti Keloharju, Timo Korkeamaki, Anders Löflund, Peter Nyberg, Bill Schwert (the editor), and seminar participants at the Helsinki Finance Seminar and the Bank of Portugal for helpful comments. We are grateful to Kenneth French, Amit Goyal, Luboš Pástor, and Robert Shiller for making data available on their webpages. A previous version of this paper was entitled The time-series and cross-sectional consistency of the ICAPM. Maio acknowledges financial support from the Hanken Foundation. Santa-Clara is supported by a grant from the Fundação para a Ciência e Tecnologia (PTDC/EGE-GES/101414/2008). All errors are ours.

2 Abstract Can any multifactor model be interpreted as a variant of the ICAPM? The ICAPM places restrictions on time series and cross-sectional behavior of state variables and factors. If a state variable forecasts positive (negative) changes in investment opportunities in time-series regressions, its innovation should earn a positive (negative) risk price in the cross-sectional test of the respective multifactor model. Second, the market (covariance) price of risk estimated from the cross-sectional tests must be economically plausible as an estimate of the coefficient of relative risk aversion (RRA). We apply our ICAPM criteria to eight popular multifactor models tested over 25 portfolios sorted on size and book-to-market (SBM25), and 25 portfolios sorted on size and momentum (SM25). Our results show that most factor models do not satisfy the ICAPM restrictions. Specifically, the hedging risk prices have the wrong sign and the estimates of RRA are not economically plausible. Overall, the Fama and French (1993), and Carhart (1997) models perform the best in consistently meeting the ICAPM restrictions. The remaining models, which represent some of the most relevant examples presented in the empirical asset pricing literature, can still empirically explain the size, value, and momentum anomalies, but they are generally inconsistent with the ICAPM. Keywords: Asset pricing models; Intertemporal CAPM; Predictability of returns; Linear multifactor models; Cross-section of stock returns; Size and value anomalies; Momentum; Timevarying investment opportunities; Fama-French factors JEL classification: G12; G14.

3 1 Introduction Explaining the dispersion in average excess returns in the cross-section of stocks has been one of the most important topics in the asset pricing literature. The inability of the Sharpe (1964) Lintner (1965) CAPM to price portfolios sorted on size, book-to-market, momentum, and other stock characteristics has led to so-called size, value, and momentum anomalies [Fama and French (1992, 1993, 1996), among others]. In response, several multifactor models seeking to explain these various anomalies have emerged in the literature. Typically, these models include factors in addition to the market return whose betas help match the dispersion in excess portfolio returns observed in the cross-section. Many of these multifactor models have been justified as empirical applications of the Intertemporal CAPM (ICAPM, Merton, 1973), leading Fama (1991) to interpret the ICAPM as a fishing license to the extent that authors claim it provides a theoretical background for relatively ad hoc risk factors in their models. However, Cochrane (2005, Chapter 9) notes that although the ICAPM does not directly identify the state variables underlying the risk factors, there are some restrictions that these state variables must satisfy. According to Merton, the state variables relate to changes in the investment opportunity set, which implies that they should forecast the distribution of future aggregate stock returns. Moreover, the innovations in these state variables should be priced factors in the cross-section. We examine the restrictions associated with the ICAPM that prevent it from being a fishing license for any multifactor model that seeks to explain the cross-section of stock returns. We identify three main conditions that a multifactor model must meet to be justifiable by the ICAPM and find that most multifactor models in the literature do not satisfy these restrictions. First, the candidates for ICAPM state variables must forecast the first or second moments of aggregate stock returns. We assess the forecasting power of each variable by conducting time-series long-horizon regressions. Second, if a given state variable forecasts positive expected aggregate returns, its innovation (the risk factor) should earn a positive risk price in cross-sectional tests, while state variables that forecast negatively expected aggregate returns should earn a negative risk price. Risk premiums with opposite signs should accrue to innovations to state variables that forecast market volatility. Thus, it is not enough that the candidate state variables forecast future aggregate expected returns or the volatility of returns, the corresponding factors should also be priced in the cross-section. The intuition for this result is simple. An asset that covaries 1

4 positively with innovations to the state variable also covaries positively with future expected returns. It does not provide a hedge for reinvestment risk because it offers lower returns when aggregate returns are expected to be lower. Hence, a risk-averse rational investor will require a positive risk premium to invest in such an asset, implying a positive price of risk for the factor. A similar argument applies to assets that covary with innovations to market volatility. The third restriction associated with the ICAPM is that the market (covariance) price of risk estimated from the cross-sectional tests must be economically plausible as an estimate of the coefficient of relative risk aversion (RRA) of the representative investor. Most of the empirical literature on the ICAPM uses state variables from the predictability literature (short-term interest rates, bond yields, and aggregate financial ratios) in order to meet the first ICAPM restriction that the state variables should forecast expected market returns. Yet authors largely neglect the other constraints of the ICAPM: that the market price of risk corresponds to the risk aversion of the representative investor and especially that there must be consistency between the hedging factor risk prices and the corresponding slopes from the predictive regressions. In Campbell (1996), the risk prices associated with the VAR state variables that forecast market returns are constrained in the sense that they are linked with the estimated slopes from the VAR. However, Campbell only tests a specific parametrization with Epstein-Zin preferences and a VAR to estimate market discount rate news. This paper extends this work, focusing on whether commonly used empirical factor models satisfy the consistency between time series slopes and cross-sectional risk prices to be justifiable as ICAPM applications. Our work is also related to Lewellen, Nagel, and Shanken (2010), and Lewellen and Nagel (2006), who advocate that cross-sectional tests of asset pricing models in general, and the conditional CAPM in particular, should impose the models theoretical restrictions on the factor risk prices. We apply our ICAPM criteria to eight multifactor models, tested over 25 portfolios sorted on size and book-to-market (SBM25) and 25 portfolios sorted on size and momentum (SM25). We include the market return in the set of testing assets, which enables us to merge the crosssectional literature on the ICAPM with the literature on the time-series aggregate risk-return trade-off. Hence, we have a total of 16 empirical tests in the cross-section: eight models and two sets of portfolios. Table 1 summarizes the main results regarding the multifactor models satisfying the ICAPM criteria. When investment opportunities are driven by changing expected market returns, our 2

5 results show that only two models the Fama and French (1993) three-factor model tested over SBM25, and the Carhart (1997) model tested over SBM25 and SM25 meet the ICAPM consistency criteria. When we consider changes in the investment opportunity set driven by the second moment of aggregate returns, only the Fama and French (1993) model satisfies the ICAPM criteria when tested with the SBM25 portfolios. In most other models the hedging risk prices have the wrong sign and the estimates of RRA are not economically plausible. The Koijen, Lustig, and Van Nieuwerburgh (2010) and Pástor and Stambaugh (2003) models in tests with SBM25 meet the sign restriction on the hedging risk prices but do not produce a reasonable estimate for the risk aversion coefficient. The Hahn and Lee (2006) model in the test with SM25 produces a plausible estimate for RRA, but fails the sign restriction on the risk prices for the state variable factors. These rejections show that the ICAPM is not really a fishing license after all. Our findings are robust to estimating the multifactor models with an intercept, estimating each model by second-stage GMM, using alternative measures of the innovations in the state variables, using alternative test equity portfolios, adding bond risk premia to the menu of test assets, estimating the models in expected return-beta form, conducting a bootstrap simulation for the slopes in the predictive regressions, and using alternative proxies for the expected market return. Our paper is related to the growing empirical literature on the ICAPM. An incomplete list of empirical tests of the ICAPM over the cross-section of stock returns includes Shanken (1990), Brennan, Wang, and Xia (2004), Ang, Hodrick, Xing, and Zhang (2006), Gerard and Wu (2006), Hahn and Lee (2006), Lo and Wang (2006), Petkova (2006), Bali (2008), Guo and Savickas (2008), Ozoguz (2009), and Bali and Engle (2010). In related work, Campbell (1993) develops a theoretical model based on a representative agent with Epstein and Zin (1991) preferences that leads to two risk factors the excess market return (as in the standard CAPM) and expectations about future market returns (discount rate news). Campbell (1996), Chen (2003), Guo (2006a), Campbell and Vuolteenaho (2004), Chen and Zhao (2009), and Maio (2012b) represent variants or extensions of the two-factor model developed in Campbell (1993). A related literature focuses on the time-series aggregate risk-return trade-off. An incomplete list of recent papers includes Scruggs (1998), Whitelaw (2000), Brandt and Kang (2004), Ghysels, Santa-Clara, and Valkanov (2005), Guo and Whitelaw (2006), Lundblad (2007), Pástor, Sinha, and Swaminathan (2008), 3

6 Bali, Demirtas, and Levy (2009), and Guo, Savickas, Wang, and Yang (2009). This paper is organized as follows. In Section 2 we discuss the theoretical restrictions associated with the ICAPM. In Section 3 we analyze the forecasting power of ICAPM state variable candidates with regards to the expected market return. In Section 4 we analyze whether the factor risk prices from cross-sectional asset pricing tests are consistent with the time-series predictability of market returns. In Section 5 we conduct a sensitivity analysis. In Section 6 we evaluate the predictive ability of the ICAPM state variables with regards to the volatility of market return. In Section 7, we conduct a Monte Carlo simulation experiment to assess the plausibility of our results. [Table 1 about here.] 2 Time-series and cross-sectional implications of the ICAPM A simplified version of the Merton (1973) Intertemporal CAPM (ICAPM) is based on the consumption/portfolio choice of a representative investor in continuous time. 1 We discuss the restrictions this model imposes on multifactor asset pricing models. There are N risky assets, and asset i has an instantaneous rate of return given by ds i S i = µ i (z, t)dt + σ i (z, t)dξ i, i = 1,..., N, (1) where S i denotes the price of asset i; dξ i is a Wiener process; and the covariance between two arbitrary risky assets is equal to σ ij dt. In this model, investment opportunities are time-varying since both the mean (µ i ) and volatility (σ i ) of asset returns are functions of a single state variable, z, which also evolves as a diffusion process: dz = a(z, t)dt + b(z, t)dζ, (2) where dζ denotes another Wiener process, and the covariance with the return on risky asset i is equal to σ iz dt. 2 The N + 1th asset is a risk-free asset with instantaneous rate of return equal to r: db B = rdt. (3) 1 For a textbook treatment see Pennacchi (2008), Chapter The Merton (1973) ICAPM does not directly identify the state variables. The main restriction in this simple model is that the state variables forecast the first two moments of stock returns. 4

7 To simplify the exposition, we assume that the risk-free rate is constant. The dynamics of wealth (W ) are given by dw = N N ω i (µ i r)w dt + (rw C)dt + ω i W σ i dξ i, (4) i=1 i=1 where ω i denotes the portfolio weight for asset i, and C stands for consumption. The investor maximizes lifetime utility: [ ] J(W, z, t) = max E t U(C, s)ds, (5) C,ω i s=t subject to the intertemporal budget constraint (4), where J(W, z, t) denotes the value function. It can be shown that the ICAPM equilibrium relation between expected return and risk is given by µ i r = γσ im + γ z σ iz, (6) where γ W J W W (W,z,t) J W (W,z,t) denotes the parameter of relative risk aversion; σ im and σ iz denote the covariances between the return on asset i and the market return and state variable, respectively; and γ z denotes the (covariance) risk price associated with the state variable, which is given by γ z J W z(w, z, t) J W (W, z, t), (7) where J W ( ) denotes the marginal value of wealth; J W W ( ) is the growth in the marginal value of wealth; and J W z ( ) represents a second-order cross-derivative relative to wealth and the state variable. As in Cochrane (2005), Chapter 9, we can approximate Eq. (6) in discrete time, leading to the pricing equation: E t (R i,t+1 ) R f,t+1 = γ Cov t (R i,t+1, R m,t+1 ) + γ z Cov t (R i,t+1, z t+1 ), (8) where R i,t+1 is the return on asset i between t and t + 1; R f,t+1 denotes the risk-free rate, which is known at t; R m,t+1 is the market return; and z t+1 denotes the innovation or change in the state variable. In Eq. (8), the novelty relative to the standard CAPM (Sharpe, 1964; and Lintner, 1965) is the second term on the right-hand side, γ z Cov t (R i,t+1, z t+1 ). This means that if the risk price associated with the state variable is zero, γ z = J W z ( ) = 0, we 5

8 are back to the standard static CAPM. This pricing equation represents the theory behind many multifactor models in the empirical asset pricing literature, leading Fama (1991) to call the ICAPM a fishing license. Yet, as should be clear from the derivation of (8), the non-market factors in such models should proxy for the innovation in some state variable, z t+1, so they cannot be just anything. By using the law of iterated expectations, we can obtain the ICAPM in unconditional form: E(R i,t+1 R f,t+1 ) = γ Cov(R i,t+1, R m,t+1 ) + γ z Cov(R i,t+1, z t+1 ). (9) In Eq. (9) there are two sources of risk that explain average risk premiums. 3 The first is captured by the static market risk premium associated with the CAPM, γ Cov(R i,t+1, R m,t+1 ), which postulates that an asset that covaries positively with the market return earns a positive risk premium over the risk-free rate. The intuition is that such an asset does not provide a hedge against changes in current aggregate wealth, as it pays in good times (periods with high returns on wealth), so a risk averse investor is willing to hold such an asset only if it offers a premium over the risk-free rate. The estimate for the relative risk aversion (RRA) coefficient should be between one and ten (see Mehra and Prescott, 1985, for example). To understand that the second source of risk in Eq. (9) is captured by the term, γ z Cov(R i,t+1, z t+1 ), consider a state variable that predicts future market returns. If the risk price for intertemporal risk (γ z ) is positive, an asset that covaries positively with changes in the state variable (and is thus positively correlated with future market expected return) earns a risk premium. The intuition is that the asset does not provide a hedge against future negative shocks in the returns of aggregate wealth (reinvestment risk), as it offers low returns when future aggregate returns are also expected to be low. Therefore, a rational investor is willing to hold such asset only if it offers an expected return in excess of the risk-free rate. This central economic intuition of the ICAPM puts a constraint on the sign of γ z, when the model is forced to price a set of assets in the cross-section. Specifically, if the state variable is 3 If the factor risk prices were time-varying (as a function of state variables), there would be additional (scaled) risk factors in the pricing equation from the interaction between the original factor and the state variables (see Jagannathan and Wang, 1996; Lettau and Ludvigson, 2001; and Cochrane, 2005, Chapter 8, among others). In our case, and following most of the empirical literature on the ICAPM, we assume that the risk prices are constant through time. In the next sections, we test only the unconditional versions of the empirical multifactor models that are candidates to ICAPM applications. 6

9 positively correlated with future aggregate returns, Cov t (R m,t+2, z t+1 ) = Cov t [E t+1 (R m,t+2 ), z t+1 ] = Cov t [E t+1 (R m,t+2 ), z t+1 ] > 0, (10) then the intertemporal risk price must be positive. To see this point, assume without loss of generality that the return on asset i is positively correlated with the (innovation in the) state variable: Cov t (R i,t+1, z t+1 ) = Cov t (R i,t+1, z t+1 ) > 0. (11) These two conditions imply that the return on asset i is also positively correlated with the future expected market return: Cov t (R i,t+1, R m,t+2 ) = Cov t [R i,t+1, E t+1 (R m,t+2 )] > 0. (12) This last condition has an important economic content: This asset does not provide a hedge for reinvestment risk, and should earn a higher risk premium than an asset with Cov(R i,t+1, R m,t+2 ) = 0, that is, γ z Cov t (R i,t+1, z t+1 ) > 0. This in turn implies that γ z > 0, given the assumption that Cov t (R i,t+1, z t+1 ) > 0. If we assume instead that the state variable forecasts negative expected market returns, then the intertemporal risk price must be negative, and the argument is just symmetric. 4 The main practical implication of this result is that if a state variable positively forecasts expected returns, the asset s covariance with its innovation should earn a risk premium in the cross-section. Thus, it is not enough that the candidate state variables forecast future aggregate returns. It must be the case that the (covariance) risk prices with (the innovations in) those state variables have the correct sign. Otherwise, the risk based explanation associated with those factors is inconsistent with the rational explanation underlying the ICAPM. 4 This positive correlation between the factor risk price and the forecasting slope is also valid under the (more restrictive) Campbell (1993) version of the ICAPM, as long as the coefficient of relative risk aversion is greater than one, γ > 1. In the (implausible) case of an investor less risk averse than the log investor the risk price of discount rate news would be negative, that is, an asset that is positively correlated with good news about future market returns (or alternatively, an asset that is positively correlated with a state variable that forecasts positive market returns) would earn a lower risk premium than an asset that is uncorrelated with future market discount rates. The intuition is that for an investor with very low risk aversion, the upside effect of a positive correlation between a given asset and future market returns (in the sense that it allows the investor to profit from the improvement in future investment opportunities) outweighs the downside effect (a reduced ability to hedge changes in future investment opportunities). However, we follow the extant literature and assume that the representative investor is more risk averse than the log investor, thus ruling out this perverse effect. 7

10 In the predictability literature, most predictive variables forecast positive expected equity market returns. This is the case for aggregate financial ratios (dividend-to-price ratio; earningsto-price ratio; book-to-market ratio), or bond yield spreads like the slope of the yield curve or the default spread. Next, we consider a state variable that forecasts the future variance of market return, and reexamine the corresponding implications for the sign of γ z in the cross-sectional asset pricing tests. The main result in this case is the following: If the state variable is positively correlated with the future volatility of aggregate returns, Cov t (R 2 m,t+2, z t+1 ) = Cov t [E t+1 (R 2 m,t+2), z t+1 ] = Cov t [E t+1 (R 2 m,t+2), z t+1 ] > 0, (13) then the risk price of intertemporal risk must be negative. 5 To see this point, assume again without loss of generality that the return on asset i is positively correlated with the (innovation in the) state variable, which implies that the return on asset i is also positively correlated with the future volatility of market return: Cov t (R i,t+1, R 2 m,t+2) = Cov t [R i,t+1, E t+1 (R 2 m,t+2)] > 0. (14) The economic implication of this last condition is that such an asset provides a hedge for reinvestment risk, as it pays high returns when future aggregate volatility is also high. Thus, such an asset should earn a lower risk premium than an asset with Cov t (R i,t+1, R 2 m,t+2 ) = 0, that is, γ z Cov t (R i,t+1, z t+1 ) < 0, which implies that γ z < 0, given the assumption that Cov t (R i,t+1, z t+1 ) > 0. If we assume instead that the state variable forecasts negative market volatility, then the intertemporal risk price must be positive, and the argument is just symmetric. Thus we have an opposite result relative to the case of expected returns: If a state variable negatively forecasts the volatility of returns, the asset s covariance with its innovation should earn a risk premium in the cross-section. The intuition is that an asset that covaries negatively with future aggregate volatility does not provide an hedge for risk averse investors (who dislike increased uncertainty in their future wealth), who in turn are willing to invest in such an asset if it offers a premium. Because R f,t+1 is known at the beginning of the period, we have Cov t (R f,t+1, R m,t+1 ) = 5 This proposition is consistent with the model developed by Chen (2003). 8

11 Cov t (R f,t+1, z t+1 ) = 0, leading to the pricing equation that we test on our data: E(R i,t+1 R f,t+1 ) = γ Cov(R i,t+1 R f,t+1, R m,t+1 ) + γ z Cov(R i,t+1 R f,t+1, z t+1 ). (15) 3 Do state variables forecast future investment opportunities? In this section, we test the first restriction associated with the Intertemporal CAPM (ICAPM), that is, do the candidates for state variables forecast future investment opportunities? Second, and most important, we want to assess the sign of the correlation between the state variables and future aggregate returns, for comparison with the factor risk prices we estimate, which represent our main criteria to evaluate the ICAPM. Our proxy for the investment opportunity set is the aggregate equity market, which is captured by the monthly return on the value-weighted stock market index available from the Chicago Center for Research in Security Prices (CRSP). We conduct long-horizon predictive regressions, which are commonly used in the predictability literature to assess the forecasting power of the state variables over future expected market returns [Keim and Stambaugh, 1986; Campbell, 1987; Fama and French (1988, 1989), among others]: r t,t+q = a q + b q z t + u t,t+q, (16) where r t,t+q r t r t+q is the continuously compounded return over q periods (from t + 1 to t + q), and u t,t+q denotes a forecasting error with zero conditional mean, E t (u t,t+q ) = 0. It follows that the conditional expected return at time t is given by E t (r t,t+q ) = a q + b q z t. The sign of the slope coefficient, b q, indicates whether a given state variable forecasts positive or negative changes in future expected aggregate stock returns, and the associated t-statistic indicates whether this effect is statistically significant. We use forecasting horizons of 1, 3, 12, 24, 36, 48, and 60 months ahead. 6 The original sample is 1963: :12, which corresponds to the time span used in most empirical asset pricing studies of the cross-section. We evaluate the statistical significance of the regression coefficients by using both Newey and West (1987), and Hansen and Hodrick (1980) asymptotic standard errors with q lags. 7 The first set of state variables we use in our empirical test are variables from the predictability 6 There is a recent debate between in-sample versus out-of-sample predictability of stock market returns (see Campbell and Thompson, 2008; Cochrane, 2008; and Goyal and Welch, 2008, among others). In our case, it makes sense to use in-sample regressions since we are interested in the long run predictive power of the state variables, and also to use the same time series that is used in the estimation of the factor covariances (betas) and average excess returns, which are employed in the cross-sectional tests. 7 We use q lags to correct for the serial correlation in the residuals caused by the overlapping returns. 9

12 literature. The first two variables are bond yield spreads: The slope of the Treasury yield curve (T ERM, Campbell, 1987; Fama and French, 1989) and the corporate bond default spread (DEF, Keim and Stambaugh, 1986; Fama and French, 1989). We measure T ERM as the yield spread between the ten-year and the one-year Treasury bond, while DEF represents the yield spread between BAA and AAA corporate bonds from Moody s. The yield data are available from the FRED (St. Louis Fed) database. We also use the market dividend-to-price ratio [DY, Fama and French (1988, 1989); Campbell and Shiller (1988a)], and the aggregate price-earnings ratio [P E, Campbell and Shiller (1988b); Campbell and Vuolteenaho, 2004], both for the S&P 500 index. DY is computed as the log ratio of the sum of annual dividends to the level of the S&P 500 index, and P E corresponds to the log ratio of the price of the S&P 500 index to a ten-year moving average of earnings. The price, dividend, and earnings data for the S&P 500 index are available from Robert Shiller s website. The next two state variables are the one-month Treasury bill rate (RF, Fama and Schwert, 1977; Campbell, 1991; Hodrick, 1992), which is available from Kenneth French s website, and the value spread (V S, Campbell and Vuolteenaho, 2004). The value spread is computed as the difference between the monthly log book-to-market ratios of small-value and small-growth stocks. 8 The last state variable is the Cochrane and Piazzesi (2005) factor (CP ), which is related to bond risk premia. 9 Hahn and Lee (2006) use T ERM and DEF in their ICAPM application. Petkova (2006) uses DY and RF, in addition to T ERM and DEF, to proxy for changes in future investment opportunities. Campbell and Vuolteenaho (2004) use P E rather than DY and the value spread in addition to T ERM to forecast aggregate returns in their ICAPM application. In Koijen, Lustig, and Van Nieuwerburgh (2010), the two state variables are T ERM and CP. 10 Therefore, in addition to single-variable forecasting regressions, we conduct multiple-variable regressions 8 The value spread is calculated from six portfolios sorted on both size and book-to-market, available from Kenneth French s website. For details on the construction of V S, see the appendix in Campbell and Vuolteenaho (2004). 9 CP is the fitted value from a regression of an average of excess bond returns on forward rates. For details on the construction of CP, see Cochrane and Piazzesi (2005). We thank an anonymous referee for suggesting the inclusion of CP in the empirical analysis conducted in the paper. 10 The proxy for the level of the yield curve in Koijen, Lustig, and Van Nieuwerburgh (2010) is constructed differently than T ERM, however, both proxies are highly correlated. 10

13 to assess the joint forecasting power of the state variables in four ICAPM applications: 11 r t,t+q = a q + b q T ERM t + c q DEF t + u t,t+q, (17) r t,t+q = a q + b q T ERM t + c q DEF t + d q DY t + e q RF t + u t,t+q, (18) r t,t+q = a q + b q T ERM t + c q P E t + d q V S t + u t,t+q, (19) r t,t+q = a q + b q T ERM t + c q CP t + u t,t+q. (20) The second set of state variables is based on risk factors widely used in the empirical asset pricing literature. Although several of these multifactor models are justified as applications of the ICAPM, it is not clear whether the associated state variables do actually forecast future stock returns. The first two variables are the size (SMB) and value (HML) factors used by Fama and French (1993, 1996). 12 We also use the momentum factor (UMD) from Carhart (1997). These three factors are obtained from Kenneth French s website. The fourth empirical factor we use is the liquidity factor from Pástor and Stambaugh (2003) (L), which is obtained from Luboš Pástor s website. 13 To obtain the associated state variables we use the cumulative sums on the factors for UMD and L. For example, in the case of L, the cumulative sum is obtained as: t CL t = L s. (21) s=t 59 We use the cumulative sum over the last 60 months since the total cumulative sum is close to being non-stationary (auto-regressive coefficients around one). In the case of SMB, the corresponding state variable, SMB, is constructed as the difference between the monthly market-to-book ratios of small and big stocks, using the six portfolios sorted on both size and book-to-market, available from Kenneth French s website: SMB = MB SL + MB SM + MB SH 3 MB BL + MB BM + MB BH, (22) 3 where MB SL, MB SM, MB SH, MB BL, MB BM, and MB BH denote the monthly market-to- 11 When there are multiple state variables, we should focus on the marginal predictive role of each variable for changes in future investment opportunities, conditional on all other variables. This is why we use multivariate forecasting regression, instead of univariate regressions for each variable separately. 12 Vassalou (2003) provides evidence that both SMB and HML convey information about future GDP growth while Da and Schaumburg (2011) find that these factors are related to market volatility. 13 We use the non-traded liquidity factor (Eq. (8) in Pástor and Stambaugh, 2003). 11

14 book ratios of small-growth, small-middle BM, small-value, big-growth, big-middle BM, and big-value portfolios, respectively. Similarly, HML corresponds to the difference between the monthly market-to-book ratios of value and growth stocks: HML = MB SH + MB BH 2 MB SL + MB BL. (23) 2 Since SMB SMB and HML HML, this procedure enables us to interpret the original factors as innovations in the state variables, which we use in the cross-sectional regressions conducted later. 14 Thus, we conduct multiple regressions corresponding to three different multifactor models: r t,t+q = a q + b q SMB t + c q HML t + u t,t+q, (24) r t,t+q = a q + b q SMB t + c q HML t + d q CUMD t + u t,t+q, (25) r t,t+q = a q + b q SMB t + c q HML t + d q CL t + u t,t+q. (26) Also we estimate a predictive regression associated with the augmented model estimated in Fama and French (1993): r t,t+q = a q + b q SMB t + c q HML t + d q T ERM t + e q DEF t + u t,t+q. (27) Table 2 presents summary statistics for the state variables described above. Most state variables are highly persistent, with autoregressive coefficients above The least persistent variable is CP with an autoregressive coefficient of [Table 2 about here.] Table 3 presents the results for the single long-horizon regressions associated with T ERM, DEF, DY, RF, P E, V S, and CP. We can see that T ERM, DEF, DY, RF, and CP forecast positive market returns, although only in the cases of DY, DEF, RF, and CP do the asymptotic t-stats indicate statistical significance (and for DEF and RF only at longer horizons). On the other hand, P E consistently forecasts negative market returns at all horizons, and this effect is statistically significant at horizons of and beyond three months. The value spread also forecasts 14 The relation is only approximate, since we are ignoring the dividend component of returns. The construction of SMB and HML is similar in spirit to the value spread computed by Campbell and Vuolteenaho (2004). 12

15 negative market returns at all horizons, and the slopes are significant at the 5% level for horizons between three and 48 months. 15 Basically, all the variables forecast positive market returns with the exception of P E and V S and the forecasting power of most variables increases with the horizon as indicated by the approximately monotonic pattern in the R 2 estimates. [Table 3 about here.] The results for the multiple long-horizon regressions (17)-(20) are displayed in Table 4. To save space, we report results only for horizons of 1, 12, and 60 months. The slope estimates indicate that both T ERM and DEF forecast positive market returns, but only the slope associated with DEF is statistically significant at horizons of 12 and 60 months. In the case of regression (18), at q = 60, for which there is greater evidence of predictability as indicated by the adjusted R 2 estimates, T ERM, DY, and RF are statistically significant and forecast positive market returns but the slope associated with DEF is negatively estimated, although not significant. In the case of regression (19), T ERM forecasts positive market returns, while P E is negatively correlated with future expected returns at all three horizons. V S forecasts negative market returns at q = 1, 12, but the predictive slope is positive at the 60-month horizon. The three forecasting coefficients are statistically significant at q = 60. Regarding the state variables in regression (20), CP forecasts positive market returns, conditional on T ERM, and the slopes are significant at the 5% or 1% levels for q = 1 and q = 60. T ERM forecasts negative market returns for q = 1 and positive returns thereafter, but none of the slopes is significant at the 10% level. [Table 4 about here.] The results for the multiple long-horizon regressions (24)-(27) are displayed in Table 5. Both SMB and HML forecast positive market returns at all horizons; the slope associated with HML is statistically significant at all horizons, while SMB is only marginally significant at q = 60. CUMD forecasts positive market returns at horizons of 12 and 60 months, conditional on both SMB and HML, but the slopes are statistically significant only at very long horizons (q = 60). The liquidity factor is positively correlated with future expected returns at all horizons, but the coefficients are statistically significant only at q = 12. In the fourth multiple regression 15 The negative slopes associated with V S are in line with the results obtained in Campbell and Vuolteenaho (2004). 13

16 (27), we can see that the slopes associated with T ERM and DEF, conditional on SMB and HML, are positive, although only T ERM is marginally significant in predicting the market return for q = 60 (based on the Newey-West standard errors). 16 Overall, these results show that most candidates for ICAPM state variables can forecast market returns, although the evidence of predictability is much stronger at longer horizons. [Table 5 about here.] 4 Are factor risk prices consistent with the ICAPM? To assess the ICAPM restrictions regarding factor risk prices, we test different multifactor models in the cross-section of stock returns. In each model, the first factor is the market equity premium, RM, computed as the monthly return of the value-weighted index in excess of the one-month Treasury bill rate, available from Kenneth French s website. The first group of multifactor models corresponds to models explicitly justified as ICAPM applications in which the risk factors represent innovations to state variables used in the predictability of returns literature to forecast aggregate equity returns. The first model we estimate is the ICAPM version of Hahn and Lee (2006), in which the additional risk factors relative to the market factor are the innovations in T ERM and DEF : E(R i,t+1 R f,t+1 ) = γ Cov(R i,t+1 R f,t+1, RM t+1 ) +γ T ERM Cov(R i,t+1 R f,t+1, T ERM t+1 ) + γ DEF Cov(R i,t+1 R f,t+1, DEF t+1 ). (28) The second model is the ICAPM of Petkova (2006) that consists of the innovations to RF and DY, in addition to T ERM and DEF : E(R i,t+1 R f,t+1 ) = γ Cov(R i,t+1 R f,t+1, RM t+1 ) + γ T ERM Cov(R i,t+1 R f,t+1, T ERM t+1 ) +γ DEF Cov(R i,t+1 R f,t+1, DEF t+1 ) + γ DY Cov(R i,t+1 R f,t+1, DY t+1 ) +γ RF Cov(R i,t+1 R f,t+1, RF t+1 ). (29) The third model is an unrestricted version of the Campbell and Vuolteenaho (2004) ICAPM, 16 We also conduct multiple predictive regressions for the equity premium. In most cases, the signs of the slopes are the same as in the regressions for the market return. In the few cases in which the sign flips the point estimates of the slopes are not statistically significant. 14

17 in which investment opportunities are described by P E, T ERM, and V S: 17 E(R i,t+1 R f,t+1 ) = γ Cov(R i,t+1 R f,t+1, RM t+1 ) + γ T ERM Cov(R i,t+1 R f,t+1, T ERM t+1 ) +γ P E Cov(R i,t+1 R f,t+1, P E t+1 ) + γ V S Cov(R i,t+1 R f,t+1, V S t+1 ). (30) The fourth model is the three-factor model from Koijen, Lustig, and Van Nieuwerburgh (2010), in which the state variables are T ERM and CP : 18 E(R i,t+1 R f,t+1 ) = γ Cov(R i,t+1 R f,t+1, RM t+1 ) +γ T ERM Cov(R i,t+1 R f,t+1, T ERM t+1 ) + γ CP Cov(R i,t+1 R f,t+1, CP t+1 ). (31) Next, we use multifactor models with less theoretical justification, but that some authors consider as possible applications of the ICAPM. The fifth model is the Fama and French (1993, 1996) three-factor model (FF3), in which the factors are SMB and HML in addition to the market risk premium: E(R i,t+1 R f,t+1 ) = γ Cov(R i,t+1 R f,t+1, RM t+1 ) + γ SMB Cov(R i,t+1 R f,t+1, SMB t+1 ) +γ HML Cov(R i,t+1 R f,t+1, HML t+1 ). (32) SM B is used to explain the size premium, the positive spread in average returns between small and big stocks. HM L seeks to explain the value premium, that value stocks (stocks with high book-to-market) have larger average returns than growth stocks (stocks with low book-to-market). The sixth model is the Fama-French three-factor model augmented by T ERM and DEF 17 Campbell and Vuolteenaho estimate a model with two factors, cash flow news (N CF ) and discount rate news (N DR ). Both N CF and N DR are linear functions of the innovations in the state variables, so that the two specifications are equivalent. For further details see Maio (2012a). 18 The Koijen, Lustig, and Van Nieuwerburgh (2010) model is not motivated by the authors as an ICAPM application. However, since the respective factors (other than the market factor) are related to state variables widely used in the predictability literature (T ERM and CP ), it makes sense to test whether this model satisfies the ICAPM restrictions. 15

18 (FF5): E(R i,t+1 R f,t+1 ) = γ Cov(R i,t+1 R f,t+1, RM t+1 ) + γ SMB Cov(R i,t+1 R f,t+1, SMB t+1 ) +γ HML Cov(R i,t+1 R f,t+1, HML t+1 ) + γ T ERM Cov(R i,t+1 R f,t+1, T ERM t+1 ) +γ DEF Cov(R i,t+1 R f,t+1, DEF t+1 ). (33) Fama and French (1993) use this model to explain both equity and bond risk premiums. The seventh model is Carhart s (1997) four-factor model (C): E(R i,t+1 R f,t+1 ) = γ Cov(R i,t+1 R f,t+1, RM t+1 ) + γ SMB Cov(R i,t+1 R f,t+1, SMB t+1 ) +γ HML Cov(R i,t+1 R f,t+1, HML t+1 ) + γ UMD Cov(R i,t+1 R f,t+1, UMD t+1 ). (34) Added to the Fama and French (1993) model is the momentum (UMD) factor. The role of UMD is to explain the momentum anomaly; that is, past short-term winners tend to have higher average returns than past losers (Jegadeesh and Titman, 1993). Finally, we use the four-factor model employed by Pástor and Stambaugh (2003), which incorporates a liquidity related risk factor (L): E(R i,t+1 R f,t+1 ) = γ Cov(R i,t+1 R f,t+1, RM t+1 ) + γ SMB Cov(R i,t+1 R f,t+1, SMB t+1 ) +γ HML Cov(R i,t+1 R f,t+1, HML t+1 ) + γ L Cov(R i,t+1 R f,t+1, L t+1 ), (35) which is denoted as PS. We use two sets of portfolio returns for testing assets. The first group is the 25 portfolios sorted on size and book-to-market (SBM25). The second group is 25 portfolios sorted on size and momentum (SM25). The portfolio return data are obtained from Kenneth French s website. We add the market return to each group of portfolios. Adding the market return enables a more powerful empirical test, and allows us to combine the literature on cross-sectional asset pricing with the literature on the market risk-return trade-off. Thus, the estimates of risk aversion and the intertemporal risk prices incorporate information from both the cross-section of equity premiums and the aggregate equity premium. 19 Moreover, adding factors to the menu 19 Note that it is common practice in the literature on the aggregate risk-return trade-off to estimate only the risk aversion parameter by assuming (in opposition with the underlying theory of the ICAPM) that the risk prices associated with time-varying investment opportunities are negligible, i.e., that hedging motives are marginal. In our case, the hedging factor risk prices are the core of the analysis. 16

19 of testing assets when the factors are returns themselves represents a more appropriate test of asset pricing models, as suggested by Lewellen, Nagel, and Shanken (2010). We estimate each multifactor model in expected return-covariance form by a one-stage generalized method of moments procedure (GMM, Hansen, 1982). This method uses equally weighted moments, which is conceptually equivalent to running an ordinary least squares (OLS) crosssectional regression of average excess returns on factor covariances (right-hand side variables). The advantage of the GMM procedure is that we don t need to have previous estimates of the individual covariances, as these are implied in the GMM moment conditions. This estimation allows us to assess whether each model can explain the returns of a set of economically interesting portfolios (e.g., SBM25). 20 The GMM system includes N + K + 1 moment conditions. The first N sample moments correspond to the pricing errors for each of the N testing returns: g T (b) (R i,t+1 R f,t+1 ) γ(r i,t+1 R f,t+1 ) (RM t+1 µ m ) 1 T T 1 t=0 γ 1 (R i,t+1 R f,t+1 ) (f 1,t+1 µ 1 ) γ 2 (R i,t+1 R f,t+1 ) (f 2,t+1 µ 2 )... γ K (R i,t+1 R f,t+1 ) (f K,t+1 µ K ) RM t+1 µ m f 1,t+1 µ 1. f K,t+1 µ K = 0, i = 1,..., N. (36) K denotes the number of factors in addition to the market factor (that is, each model has K +1 factors). In this system, the market return (RM t+1 ) is the first factor, with unconditional mean given by µ m. The remaining factors are given by (f 1,t+1,..., f K,t+1 ), and the respective means are denoted by (µ 1,..., µ K ). For example, in the case of the Fama-French model, we have K = 2 with f 1,t+1 SMB t+1, f 2,t+1 HML t+1. (γ 1,..., γ K ) denote the (covariance) risk prices associated with the hedging factors. The last K + 1 moment conditions in system (36) 20 The procedure is also more convenient than the two-pass time-series/cross-sectional regressions approach (Cochrane, 2005; Brennan, Wang, and Xia, 2004), since we want to estimate the model in expected returncovariance form rather than in expected return-beta form to obtain the covariance risk prices for each factor, and specifically the market (covariance) risk price, which represents an estimate of the relative risk aversion. 17

20 enable us to estimate the factor means. Thus, the estimated covariance risk prices from the first N moment conditions do account for the estimation error in the factor means, as in Cochrane (2005) (Chapter 13), and Yogo (2006). In this system, there are N K 1 overidentifying conditions (N + K + 1 moments and 2(K + 1) parameters to estimate). The standard errors for the parameter estimates, and the remaining GMM formulas are presented in Appendix 8. To assess the robustness of the asymptotic standard errors, we conduct a bootstrap simulation to produce empirical p-values for the tests of individual significance of the factor risk prices. The bootstrap simulation consists of 10,000 replications in which the portfolio return data and the factors are simulated independently, that is, the data are simulated under the hypothesis that the model is not true. Full details of this bootstrap simulation are available in Appendix 8. Although our focus is on the analysis of the factor risk prices, for completeness we also present some measures for the overall explanatory power of each model. The idea is to assess whether each model satisfies the economic restrictions associated with the ICAPM, in addition to explaining the dispersion in equity premiums over the cross-section. The first measure is the mean absolute pricing error: MAE = 1 N N α i, (37) i=1 where α i, i = 1,..., N represents the first N moments, i.e., the pricing errors associated with the N testing assets. The second goodness-of-fit measure is the cross-sectional OLS coefficient of determination: where R i = 1 T R 2 OLS = 1 Var N(ˆα i ) Var N (R i ), (38) T 1 t=0 (R i,t+1 R f,t+1 ) denotes the average excess return for asset i, and Var N ( ) stands for the cross-sectional variance. R 2 OLS measures the fraction of the cross-sectional variance in average excess returns explained by the model. 21 [Table 6 about here.] We start the empirical analysis by estimating two-factor models that include a hedging 21 We do not present the values for the asymptotic χ 2 test of overidentifying restrictions, since both the mean absolute error (MAE) and R 2 OLS represent more robust measures of the models global fit. 18

21 risk factor in addition to the market return: E(R i,t+1 R f,t+1 ) = γ Cov(R i,t+1 R f,t+1, RM t+1 ) + γ z Cov(R i,t+1 R f,t+1, z t+1 ), (39) where z = T ERM, DEF, DY, RF, P E, V S, and CP. The objective is to analyze whether some of the most relevant predictors of market returns proposed in the predictability literature can be justified in two-factor ICAPM specifications. The factor risk price estimates are displayed in Table 6. In the tests with SBM25 (Panel A), the point estimates for the relative risk aversion (RRA) parameter are negative in most specifications; the exceptions are the models with DEF, V S, and CP. Moreover, the risk price estimates associated with the intertemporal factor are negative in the specifications with DEF, DY, and RF, which are inconsistent with the positive forecasting slopes over the market return associated with the level of these variables, as shown in the last section. In the model with P E, the positive risk price estimate is also inconsistent with the corresponding negative slope estimated in the single predictive regression (16). Only in the specifications with T ERM, V S, and CP is there consistency in sign between the factor risk price estimates from the cross-sectional regressions and the slopes from the forecasting regressions over the market return. Overall, only two two-factor models (including V S or CP ) satisfy the restrictions on the market (covariance) risk price (RRA parameter) and on the hedging factor risk price. In the tests with SM25 (Panel B), only in the specifications with RF and V S is there an implausible negative estimate for γ. The factor risk prices associated with T ERM, DEF, RF, and CP are negative in all cases, which is inconsistent with the corresponding positive slopes estimated in the single predictive regressions. Moreover, the risk price estimate for V S is estimated positively, which is inconsistent with the negative slopes from the single forecasting regressions. On the other hand, the signs of the risk price estimates associated with DY and P E are consistent with the forecasting ability of these variables over market returns, although in both cases the risk price estimates are largely insignificant. Furthermore, in both cases there is no explanatory power over the cross-section of returns, as illustrated by the negative estimates for the cross-sectional R 2, indicating that these two models perform more poorly than a model that predicts constant expected risk premiums in the cross-section of equities. 19

22 The results in Table 6 show overall that most of the state variables in the literature are not valid risk factors under the ICAPM. Specifically, only two two-factor models satisfy the economic restrictions underlying the ICAPM, jointly with a reasonable explanatory power over the crosssection of excess stock returns ( V S and CP in the test with SBM25, with R 2 = 38% and 42%, respectively). We next estimate factor models presented in the literature as empirical applications of the ICAPM, which include combinations of the state variables analyzed above, specifically the threefactor Hahn and Lee (2006) model (henceforth, HL); the five-factor Petkova (2006) model (P); a four-factor model that corresponds to an unrestricted version of Campbell and Vuolteenaho (2004) (CV); and the three-factor model from Koijen, Lustig, and Van Nieuwerburgh (2010) (KLVN). The results are displayed in Table 7. In the tests with SBM25 (Panel A), we have the usual result that the CAPM cannot price the SBM25 portfolios, as shown by the negative cross-sectional R 2 of -42%, while the four multifactor models show a considerable explanatory power over the dispersion in risk premiums within these portfolios, with R 2 estimates above 70%. The point estimates for the RRA parameter, however, are negative and statistically insignificant in all four factor models, compared to a positive estimate of 2.79 in the CAPM, which is statistically significant. In the case of HL, the estimates for γ T ERM and γ DEF are positive and negative, respectively, although only the risk price associated with T ERM is statistically significant. Thus, the sign of γ DEF is inconsistent with evidence above that DEF (conditional on T ERM) forecasts positive market returns. In the case of P, both γ DY and γ RF are negative, while both γ T ERM and γ DEF are positive. All four risk price estimates are not statistically significant, however, which indicates evidence of multicollinearity. Thus, the signs of γ DEF, γ DY, and γ RF are inconsistent with the evidence from the multiple regressions (18) at a horizon of 60 months (for which there is stronger evidence of predictability) that both DY and RF (conditional on the other variables) forecast positive market returns, while DEF forecasts negative market returns (although with no significance). In the case of CV, the estimate for γ P E is positive, and thus inconsistent with the corresponding negative slopes found in the predictive regressions (19), while γ V S is estimated negatively, which also goes against the sign of the predictive slope at q = 60. Both γ P E and γ V S are not statistically significant at the 10% level, suggesting the presence of multicollinearity. Regarding the KLVN model, the point estimates for γ T ERM and γ CP are both positive, 20