Value, momentum, and shortterm interest rates


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1 Value, momentum, and shortterm interest rates Paulo Maio 1 Pedro SantaClara 2 First version: July 2011 This version: December Hanken School of Economics. 2 Millennium Chair in Finance. Nova School of Business and Economics, NBER, and CEPR. 3 We thank John Cochrane for comments on a preliminary version of this paper.
2 Abstract This paper offers a simple asset pricing model that goes a long way forward in explaining the value and momentum anomalies. We specify a threefactor conditional intertemporal CAPM, denoted as (C)ICAPM, where the factors (other than the market return) are the market factor scaled by the lagged state variable and the hedging, or intertemporal, risk factor. These two factors are based on the same macroeconomic state variable: the shortterm interest rate. We test our threefactor model with 25 portfolios sorted on size and booktomarket and 25 portfolios sorted on size and momentum. The (C)ICAPM outperforms the Fama and French (1993) threefactor model in pricing both sets of portfolios, and only marginally underperforms the Carhart (1997) fourfactor model. The ICAPM hedging risk factor explains the dispersion in risk premia across the BM portfolios, while the scaled factor prices the dispersion in risk premia across the momentum portfolios. According to our model, value stocks enjoy higher expected returns than growth stocks because they have higher interest rate risk; that is, they have more negative loadings on the hedging factor. Past winners also enjoy higher average returns than past losers, because they have greater conditional market risk; that is, past winners have higher market betas in times of high shortterm interest rates. Keywords: Crosssection of stock returns; Asset pricing; Intertemporal CAPM; Conditional CAPM; Conditioning information; State variables; Linear multifactor models; Predictability of returns; FamaFrench factors; Value premium; Momentum; Longterm reversal in returns JEL classification: G12; G14; E44
3 1 Introduction There is much evidence that the standard Sharpe (1964)Lintner (1965) Capital Asset Pricing Model (CAPM) cannot explain the crosssection of U.S. stock returns in the postwar period. Value stocks (stocks with high booktomarket ratios, (BM)), for example, outperform growth stocks (low BM), which is known as the value premium anomaly [Rosenberg, Reid, and Lanstein (1985), Fama and French (1992)]. Also, stocks with high prior oneyear returns outperform stocks with low prior returns, which is the momentum anomaly [Jegadeesh and Titman (1993)]. We offer a simple asset pricing model that goes a long way forward in explaining these two anomalies. We specify a threefactor conditional intertemporal CAPM, denoted as (C)ICAPM, that merges the conditional CAPM (CCAPM) and the intertemporal CAPM (ICAPM) from Merton (1973). The factors in the model are the market equity premium (as in both the CCAPM and the ICAPM); the market factor scaled by the state variable (as in the CCAPM); and the hedging or intertemporal factor (as in the ICAPM). The first source of systematic risk other than the market factor (the scaled factor) arises from timevarying betas. The second source of systematic risk (the innovation in the state variable) arises because stocks that are more correlated with good future investment opportunities should earn a higher risk premium as they do not provide a hedge for reinvestment risk (unfavorable changes in aggregate wealth for future periods). In the empirical applications of both the ICAPM and CCAPM, the ultimate source for the additional risk factors (relative to the usual market factor) is the same, that is, a timevarying market risk premium in the current (CCAPM) or future (ICAPM) periods, or timevarying betas (CCAPM), where the time variation is driven by common state variables. 1 In our threefactor model, we use shortterm interest rates (proxied either by the Federal funds rate, F F R, or the relative or stochastically detrended Treasurybill rate, RREL) as the single state variable that drives both future aggregate investment opportunities and conditional market betas. There is evidence in the return predictability literature that shortterm interest rates forecast expected (excess) market returns, especially at short forecasting horizons [Campbell (1991), Hodrick (1992), Jensen, Mercer, and Johnson (1996), Patelis (1997), Thorbecke 1 State variables used to proxy for the expected market return or conditional betas are largely borrowed from the fast growing literature on equity premium predictability: the slope of the yield curve or term structure spread [Campbell (1987), Fama and French (1989)]; the spread between higher and lowerrated corporate bond yields (default spread) [Keim and Stambaugh (1986), Fama and French (1989)]; shortterm interest rates [Campbell (1991), Hodrick (1992)]; and aggregate valuation ratios like the dividend yield [Fama and French (1988, 1989)] or the earnings yield [Campbell and Shiller (1988)], among others. 1
4 (1997), Ang and Bekaert (2007), among others]. Thus, both the Fed funds rate and the relative Tbill rate represent valid ICAPM state variables. It is not surprising that a factor model based on the shortterm interest rate would perform well in driving equity risk premia. Specifically, the Fed funds rate represents the major instrument of monetary policy, so changes in it should reflect the privileged information of the monetary authority about the future state of the economy. 2 We test the two versions of our threefactor model with 25 portfolios sorted on size and booktomarket and 25 portfolios sorted on size and momentum. The crosssectional tests show that the (C)ICAPM explains a large percentage of the dispersion in average equity premia of the two portfolio groups, with explanatory ratios around 70%. The (C)ICAPM outperforms the Fama and French (1993) threefactor model when it comes to pricing both sets of portfolios, and is only marginally behind the Carhart (1997) fourfactor model, which has explanatory ratios around 80%. The ultimate sources of systematic risk in our model (other than the market factor), however, are associated with a single variable from outside the equity market, the shortterm interest rate. In contrast, the fourfactor model has three equity financialbased sources of systematic risk (other than the market factor). Thus, our model is more parsimonious in this sense. Moreover, our model represents an application of the ICAPM using a macroeconomic variable, while the foundation for the Carhart (1997) model is less clear. 3 In this sense, our model is a step in the direction of a fundamental model of asset pricing instead of simply explaining equity portfolio returns with the returns of other equity portfolios. In other words, the Fed funds rate or the relative interest rate are not a priori mechanically related to the test portfolios, as is the case with some of the equitybased factors in Fama and French (1993) and Carhart (1997). Interestingly, the (C)ICAPM outperforms both the Fama and French (1993) and Carhart (1997) models in fitting the difficulttoprice smallgrowth portfolio. The hedging risk factor explains the dispersion in risk premia across the sizebm portfolios, while the scaled factor prices the dispersion in risk premia across the sizemomentum portfolios. According to our model, value stocks enjoy higher expected returns than growth stocks because they have more exposure to changes in the state variable; that is, they have more negative 2 Bernanke and Blinder (1992) and Bernanke and Mihov (1998) argue that the Fed funds rate is a good proxy for Fed policy actions. 3 There is some evidence that the FamaFrench size and value factors proxy for future investment opportunities [Petkova (2006) and Maio and SantaClara (2011)] and future GDP growth [Vassalou (2003)]. The justification for the momentum factor that Carhart (1997) uses is more controversial. 2
5 loadings on the hedging factor. One possible explanation for these loadings is that many value firms have a poor financial position and thus are more sensitive to rises in shortterm interest rates that further constrain their access to external finance. As for explaining momentum, in our model past winners enjoy higher average returns than past losers because they have greater conditional market risk; that is, they have higher market betas in times of high shortterm interest rates. The explanation for this is that winner and loser stocks have different characteristics at different points of the business cycle. Specifically, during economic expansions, which are associated with high shortterm interest rates, winners tend to be cyclical firms, which have high market betas. Conversely, in recessions, with low shortterm interest rates, winners tend to be noncyclical firms, with low market betas [see Grundy and Martin (2001) and Daniel (2011) for a related discussion]. The results of the (C)ICAPM hold under a battery of robustness checks: conducting a bootstrap simulation; including bonds in the test assets; testing the model simultaneously on value and momentum portfolios; using an alternative measure of the innovation in the state variable; including the market equity premium in the test assets; estimating the model with an alternative sample; using alternative standard errors for the factor risk prices; and estimating the model in expected returncovariance form. We also test the (C)ICAPM over an alternative group of portfolios, 25 portfolios sorted on size and longterm prior returns (SLTR25), to assess whether the model explains the longterm reversal anomaly [De Bondt and Thaler (1985, 1987)]. The results show that our threefactor model can explain a significant fraction of the dispersion in equity premia of these portfolios, with explanatory ratios above 50%. As in the test with the SBM25 portfolios, it is the hedging factor that drives the explanatory power of the (C)ICAPM over the SLTR25 portfolios. Our work is related to the growing empirical literature on the ICAPM, in which the factors (other than the market return) proxy for future investment opportunities. 4 It is also related to the large conditional CAPM literature, which postulates that the CAPM should hold on a periodbyperiod basis, i.e., conditionally rather than unconditionally. 5 4 An incomplete list of papers that have implemented empirically testable versions of the original ICAPM over the cross section of stock returns includes Shanken (1990), Campbell (1996), and more recently, Chen (2003), Brennan, Wang, and Xia (2004), Campbell and Vuolteenaho (2004), Guo (2006), Hahn and Lee (2006), Petkova (2006), Guo and Savickas (2008), and Bali and Engle (2010). 5 An incomplete list of references includes Ferson, Kandel, and Stambaugh (1987), Harvey (1989), Cochrane (1996, 2005), He, Kan, Ng, and Zhang (1996), Jagannathan and Wang (1996), Ghysels (1998), Ferson and Harvey (1999), Lewellen (1999), Lettau and Ludvigson (2001), Wang (2003), Petkova and Zhang (2005), Avramov and Chordia (2006), and Ferson, Sarkissian, and Simin (2008). 3
6 Our paper is organized as follows. In Section 2, we derive our threefactor model. Section 3 describes the econometric methodology and the data. In Section 4, we present and analyze the main results for the crosssectional tests of the (C)ICAPM. Section 5 provides a number of robustness checks. In Section 6, we analyze the longterm reversal anomaly. 2 A threefactor model We use a simple version of the Merton (1973) intertemporal CAPM (ICAPM) in discrete time (the full derivation is presented in Appendix A). 6 The expected returncovariance equation is given by E t (R i,t+1 ) R f,t+1 = γ Cov t (R i,t+1 R f,t+1, R m,t+1 ) + γ z Cov t (R i,t+1 R f,t+1, z t+1 ), (1) where R i,t+1 denotes the return on asset i; R f,t+1 stands for the riskfree rate; γ denotes the (constant) coefficient of relative risk aversion (RRA); R m,t+1 is the market return; and γ z represents the (covariance) risk price associated with statevariable risk, which is given by γ z J W z(w t, z t ) J W (W t, z t ). In this expression, J W ( ) denotes the marginal value of wealth (W ), and J W z ( ) represents a secondorder crossderivative relative to wealth and the state variable (z). γ z can be interpreted as a measure of aversion to state variable/intertemporal risk, with z t+1 = z t+1 z t representing the innovation in the state variable. We can rewrite the pricing equation (1) in expected returnbeta form: E t (R i,t+1 ) R f,t+1 = γ Var t (R m,t+1 ) Cov t(r i,t+1 R f,t+1, R m,t+1 ) Var t (R m,t+1 ) +γ z Var t ( z t+1 ) Cov t(r i,t+1 R f,t+1, z t+1 ) Var t ( z t+1 ) = λ M,t β i,m,t + λ z,t β i,z,t, (2) where λ M,t and λ z,t represent the conditional (beta) risk prices associated with the market and state variable factors, respectively, and β i,m,t and β i,z,t denote the corresponding conditional betas for asset i. 7 Thus, although the market price of covariance risk is constant over time, the 6 Cochrane (2005) presents a similar covariance pricing equation based on a continuous time pricing kernel. 7 We call the innovation to the state variable a risk factor. 4
7 market price of beta risk is timevarying. We assume that the conditional beta associated with the state variable innovation is constant through time, that is, β i,z,t = β i,z, but, following the conditional CAPM literature [Harvey (1989), Ferson and Harvey (1999), Lettau and Ludvigson (2001), and Petkova and Zhang (2005), among others], we let the conditional market beta for asset i be linear in the lagged state variable: β i,m,t = β i,m + β i,mz z t. (3) We estimate β i,m and β i,mz (and also β i,z ) from the timeseries multiple regression: R i,t+1 R f,t+1 = a i + β i,m,t R m,t+1 + β i,z z t+1 + ε i,t+1 = a i + (β i,m + β i,mz z t )R m,t+1 + β i,z z t+1 + ε i,t+1 = a i + β i,m R m,t+1 + β i,mz R m,t+1 z t + β i,z z t+1 + ε i,t+1, (4) where β i,m and β i,mz represent the unconditional betas associated with the market factor (R m,t+1 ) and scaled factor (R m,t+1 z t ), respectively. 8 By substituting equation (3) in (2), we obtain a threefactor model: E t (R i,t+1 ) R f,t+1 = λ M,t β i,m + λ M,t z t β i,mz + λ z,t β i,z. (5) By applying the law of iterated expectations, we define the model in unconditional form: E(R i,t+1 R f,t+1 ) = E(λ M,t )β i,m + E(λ M,t z t )β i,mz + E(λ z,t )β i,z = λ M β i,m + λ Mz β i,mz + λ z β i,z, (6) where λ M, λ Mz, and λ z represent the unconditional risk prices for the market, scaled, and hedging factors, respectively. This is a conditional intertemporal CAPM, (C)ICAPM. The economic intuition underlying the (C)ICAPM is that an asset that covaries positively with changes in the state variable earns a higher risk premium than an asset that is uncorrelated with the state variable. The reason is that the first asset does not provide a hedge against future negative shocks in the returns of aggregate wealth, since it offers high returns when 8 The interaction variable, R m,t+1z t, is often interpreted as a managed return. See Hansen and Richard (1987), Cochrane (1996, 2005), Bekaert and Liu (2004), Brandt and SantaClara (2006), among others. 5
8 future aggregate returns are also high. 9 Therefore, a rational investor is willing to hold such an asset only if it offers a higher expected return in excess of the riskfree rate. This additional risk premium is captured by the term λ z β i,z. The term λ Mz β i,mz represents an additional risk premium that arises from the fact that the market beta is timevarying and increases with the state variable. When z increases, asset i becomes more correlated with the market return, making this asset riskier. This threefactor model is parsimonious, since a single state variable drives the two sources of systematic risk (other than the market factor). In the empirical tests of the (C)ICAPM we use the Fed funds rate (F F R), and in an alternative version the relative Treasury bill rate (RREL), as the single state variable that drives future aggregate investment opportunities (market returns), and that also drives conditional market betas. There is strong evidence in the return predictability literature that shortterm interest rates forecast expected market returns, especially at shortterm forecasting horizons [Campbell (1991), Hodrick (1992), Jensen, Mercer, and Johnson (1996), Patelis (1997), Thorbecke (1997), and Ang and Bekaert (2007), among others] Econometric methodology and data In this section, we describe the econometric methodology and the data used in the asset pricing tests conducted in the following sections. 3.1 Econometric methodology The empirical methodology is the timeseries/crosssectional regressions approach (TSCS) presented in Cochrane (2005) (Chapter 12), which enables us to obtain direct estimates for factor betas and (beta) prices of risk. This method has been employed by Brennan, Wang, and Xia (2004), and Campbell and Vuolteenaho (2004), among others. The factor betas are estimated from the timeseries multiple regressions for each test asset: 11 R i,t+1 R f,t+1 = δ i + β i,m RM t+1 + β i,m,z RM t+1 z t + β i,z z t+1 + ε i,t. (7) 9 In this reasoning, we are assuming that the state variable covaries positively with future investment opportunities. 10 Under some assumptions, Brennan and Xia (2006) and Nielsen and Vassalou (2006) show that the intercept of the capital market line, which corresponds to the riskfree rate, represents one valid state variable in the ICAPM. 11 The lagged conditioning variable is previously demeaned, which is a common practice in the conditional CAPM literature [see, for example, Lettau and Ludvigson (2001) and Ferson, Sarkissian, and Simin (2003)]. 6
9 We use the monthly excess market return (RM) to compute the betas, rather than the raw market return, as in most applications of linear factor models in the empirical asset pricing literature. RM is based on the valueweighted market return from CRSP and it is available on Kenneth French s website. RM t+1 z t denotes the scaled factor (the interaction between the equity premium and the lagged state variable), and z t+1 z t+1 z t stands for the innovation in the shortterm interest rate, z = F F R or RREL. The expected returnbeta representation from equation (6) is estimated in a second step by the OLS crosssectional regression: R i R f = λ M β i,m + λ M,z β i,m,z + λ z β i,z + α i, (8) which produces estimates for factor risk prices ( λ) and pricing errors (ˆα i ). In this crosssectional regression, R i R f represents the average timeseries excess return for asset i. 12 We do not include an intercept in the crosssectional regression since we want to impose the economic restrictions associated with the model. If the model is correctly specified, the intercept in the crosssectional regression should be equal to zero; that is, assets with zero betas with respect to all the factors should have a zero risk premium relative to the riskfree rate. 13 Other studies use generalized least squares (GLS) or weighted least squares (WLS) crosssectional regressions to estimate factor risk prices in the crosssection of returns [e.g., Ferson and Harvey (1999), Shanken and Zhou (2007), Lewellen, Nagel, and Shanken (2010)]. The OLS crosssectional regression is economically appealing and easy to interpret since it assigns equal weight to all testing returns. Thus, we can assess if some economically interesting group of portfolios (e.g., value or momentum portfolios) is properly priced by each model. Furthermore, the GLS or WLS crosssectional regressions are more difficult to interpret, since the testing returns usually receive large positive and negative weights (the weights come from the inverse of the covariance matrix of the residuals associated with the timeseries regressions). Therefore, it is harder to assess whether a particular model is able to explain the CAPM anomalies. Moreover, use of OLS regressions allows us to directly compare different models, unlike either 12 If the factor loadings are based on the whole sample, the risk price estimates from the TSCS approach are numerically equal to the risk price estimates from Fama and MacBeth (1973) regressions. The standard errors of the risk price estimates in the FamaMacBeth procedure, however, do not take into account the estimation error in the factor loadings from the firstpass timeseries regressions. In the TSCS approach, we use Shanken (1992) standard errors that correct for the errorinvariables bias, as discussed below. 13 Another reason for not including the intercept in the crosssectional regressions is that often the market betas for equity portfolios are very close to 1 (e.g., 25 size/booktomarket portfolios), creating a multicollinearity problem [see Jagannathan and Wang (2007)]. 7
10 GLS or WLS regressions, in which the weights are modelspecific, and thus prevent us from directly comparing the fit of two different models (e.g., (C)ICAPM versus the CAPM). A test for the null hypothesis that the N pricing errors are jointly equal to zero (that is, the model is perfectly specified) is given by ˆα Var ( ˆα) 1 ˆα χ 2 (N K), (9) where K denotes the number of factors (K = 3 in the (C)ICAPM), and ˆα is the (N 1) vector of crosssectional pricing errors. Both the tstatistics for the factor risk prices and the computation of Var( ˆα) are based on Shanken (1992) standard errors, which introduce a correction for the estimation error in the factor betas from the timeseries regressions, thus accounting for the errorinvariables bias in the crosssectional regression [see Cochrane (2005), Chapter 12]. Although the statistic (9) represents a formal test of the validation of a given asset pricing model, it is not particularly robust [Cochrane (1996, 2005), Hodrick and Zhang (2001)]. In some cases, the near singularity of Var( ˆα), and the inherent problems in inverting it, points to rejection of a model with low pricing errors. In other cases, it is possible that the low values for the statistic are a consequence of low values for Var(ˆα) 1 (overestimation of Var(ˆα)), rather than the result of low individual pricing errors. In both cases, this asymptotic statistic provides a misleading picture of the overall fit of the model. A simpler and more robust measure of the global fit of a given model over the crosssection of returns is the crosssectional OLS coefficient of determination: R 2 OLS = 1 Var N (ˆα i ) Var N (R i R f ), where Var N ( ) stands for the crosssectional variance. R 2 OLS represents a proxy for the proportion of the crosssectional variance of average excess returns explained by the factors associated with a given model. A related measure is the mean absolute pricing error, computed as MAE = 1 N N α i, i=1 which represents the average pricing error associated with a given model. 8
11 3.2 Data and variables The data on the Federal funds rate and the threemonth Treasury bill rate (T B) are from the FRED database (St. Louis Fed). The relative Treasurybill rate (RREL) represents the difference between T B and its moving average over the previous twelve months, RREL t = T B t j=1 T B t j. The portfolio return data, the onemonth Treasury bill rate used to construct portfolio excess returns, and the risk factors from alternative models are all obtained from Kenneth French s data library. The sample period we use is 1963: :12, where the starting date coincides with most crosssectional asset pricing tests in the literature. Table 1 presents descriptive statistics for the factors in the (C)ICAPM, RM t+1, RM t+1 F F R t, RM t+1 RREL t, F F R t+1 and RREL t+1. We also present descriptive statistics for the size (SMB), value (HML), and momentum factors (UMD) from the Fama and French (1993) and Carhart (1997) factor models. We can see that the three (C)ICAPM factors are not persistent, with the innovation in the Fed funds rate being the most persistent variable, with an autoregressive coefficient of Moreover, the three factors are not significantly correlated among themselves, with correlation coefficients varying between (RM t+1 F F R t and F F R t+1 ) and 0.15 (RM t+1 and RM t+1 F F R t ), when the state variable is F F R. In the version with RREL the magnitudes of the correlations among the three factors are smaller than Hence the three factors from the (C)ICAPM seem to proxy for different sources of systematic risk. As for the correlation with the other risk factors, RM t+1 F F R t is marginally negatively correlated with HML (0.26) and marginally positively correlated with UMD (0.22), while RM t+1 RREL t is also slightly positively correlated with the momentum factor (0.25). Figure 1 depicts the timeseries of the changes in both the Fed funds rate and RREL. We can see that these two variables present an approximate procyclical pattern, with sharp increases during economic expansions, and some significant declines during recessions. The average Fed funds rate change in expansions (as measured by the National Bureau of Economic Research (NBER)) is 0.06% per month, and in recessions 0.38% per month. In the case of RREL, we have an average of 0.14% in expansions and 0.21% in recessions. 9
12 4 Main empirical results 4.1 Testing the (C)ICAPM We assess whether the threefactor (C)ICAPM explains the value and momentum anomalies. The value premium corresponds to the empirical evidence showing that value stocks (stocks with a high booktomarket ratio) have higher average returns than growth stocks (stocks with a low booktomarket) [see Rosenberg, Reid, and Lanstein (1985), and Fama and French (1992), among others]. This spread in average returns is called an anomaly in the sense that the baseline CAPM [Sharpe (1964) and Lintner (1965)] is not able to explain such a premium [see Fama and French (1992, 1993, 2006)]. We use the standard 25 size/booktomarket portfolios (SBM25) from Fama and French (1993) to test the value premium puzzle. The momentum anomaly is that past winners (stocks with higher returns in the recent past) continue to have subsequent higher returns, while past losers continue to underperform in the near future [Jegadeesh and Titman (1993), and Chan, Jegadeesh, and Lakonishok (1996), among others]. This return premium is not explained by either the baseline CAPM or the Fama and French (1993) threefactor model [see Fama and French (1996)]. In fact, the momentum anomaly represents one of the major challenges for most asset pricing models in the literature (Cochrane (2007)). In order to assess the explanatory power of the (C)ICAPM for the momentum anomaly we use 25 portfolios sorted on both size and prior oneyear returns (SM25). 14 The use of these portfolios allows us to assess whether momentum is persistent across different size groups [see Fama and French (2008)]. 15 The estimation results for the (C)ICAPM are displayed in Table 2. The results for the test with the SBM25 portfolios (Panel A) show that the (C)ICAPM s version with F F R explains a significant fraction of the dispersion in average returns of these portfolios, with an R 2 estimate of 70% and an average pricing error of only 0.10% per month (which compares with a crosssectional average portfolio risk premium of 0.67% per month). Moreover, the model passes the χ 2 test with a pvalue of 6%. The point estimate for the hedging risk price, λ z, is negative and strongly statistically significant (1% level), while the point estimate for the risk price of the scaled factor, λ M,z, is largely insignificant. In the version with RREL, the model s fit is somewhat worse, but still shows a good ex 14 Fama and French (1996), Bansal, Dittmar, and Lundblad (2005), Liu and Zhang (2008), He, Huh, and Lee (2010), and Maio (2011), among others, conduct asset pricing tests over portfolios sorted on momentum. 15 Some authors argue that doublesort portfolios produce a greater dispersion in average returns [see, for example, Lakonishok, Shleifer, and Vishny (1994)]. 10
13 planatory power with an R 2 estimate of 46% and an average pricing error of 0.14% per month. This version also passes the χ 2 test with a pvalue of 9%. As in the version with F F R, the estimate for λ z is negative and strongly significant, while the estimate for λ M,z is now positive, although not significant at the 10% level. Thus, the key factor that drives the fit of the model over the SBM25 portfolios seems to be the innovation in the shortterm interest rate, F F R t+1 or RREL t+1, rather than the scaled factor, RM t+1 F F R t or RM t+1 RREL t. 16 The results for the test with the SM25 portfolios (Panel B) indicate that the (C)ICAPM based on F F R also explains a large fraction of the dispersion in average returns of these portfolios, with an R 2 estimate of 71%, which is very close to the explanatory ratio in the test with the SBM25 portfolios. The average pricing error is 0.16% per month (compared to a crosssectional average portfolio risk premium of 0.60% per month), which is higher than the corresponding mispricing in the test with SBM25, confirming that the sizemomentum portfolios are harder to price than the sizebm portfolios. The (C)ICAPM does not pass the χ 2 test, although this rejection is largely explained by a mismeasured inverse of the covariance matrix of the pricing errors, Var( ˆα), given the good fit associated with the model. The point estimate for the risk price of the scaled factor is positive and strongly significant (1% level), but the risk price estimate associated with the hedging factor is not statistically significant at the 10% level. In the version based on RREL, the model s fit is marginally above the first version, with an explanatory ratio of 74% and an average pricing error of 0.15% per month. This shows that this version of the model performs relatively better in pricing the SM25 portfolios than the SBM25 portfolios. The model is rejected by the χ 2 statistic only marginally (pvalue = 4%). As in the case of F F R, the estimate for λ M,z is positive and strongly significant, while the estimate for λ z is negative and significant at the 5% level. Thus, the scaled factor seems to be the key factor that drives the explanatory power of the (C)ICAPM over the SM25 portfolios. Therefore, the two key factors in the (C)ICAPM seem to measure two different and complementary sources of systematic risk. The hedging factor is able to capture the value anomaly, and the scaled factor prices momentum. 16 Brennan, Wang, and Xia (2004) and Petkova (2006) also price the SBM25 portfolios with multifactor models that contain the innovation in shortterm interest rates as one of the factors. However, it is not clear in their models what is the contribution of the interest rate factor to drive the explanatory power over the size/bm portfolios. 11
14 4.2 Comparison with alternative factor models We compare the performance of the (C)ICAPM with three alternative linear factor models, the baseline unconditional CAPM; the Fama and French (1993, 1996) threefactor model (FF3); and the Carhart (1997) fourfactor model (C4). FF3, the most widely used model in the empirical asset pricing literature, seeks to offer a riskbased explanation for both the size and value premiums. To the excess market return, Fama and French add two factors SMB (small minus big), and HML (high minus low) to account for the size and value premiums. The FF3 model can be represented in expected returnbeta form as E (R i,t+1 R f,t+1 ) = λ M β i,m + λ SMB β i,smb + λ HML β i,hml, (10) where (λ SMB, λ HML ) denote the (beta) risk prices associated with the SMB and HML factors, respectively, and (β i,smb, β i,hml ) stand for the corresponding factor betas for asset i. The fourfactor model is represented as E (R i,t+1 R f,t+1 ) = λ M β i,m + λ SMB β i,smb + λ HML β i,hml + λ UMD β i,umd, (11) where λ UMD denotes the risk price associated with the momentum factor, and β i,umd represents the corresponding beta for asset i. The novelty relative to the FF3 model is the risk premium associated with the momentum (UMD) factor. UMD (up minus down or winner minus loser) refers to the return of a selffinancing portfolio (like SMB and HML), representing the spread in average returns between past shortterm winner stocks and past shortterm loser stocks. The results for these two factor models are displayed in Table 3. We can see that the baseline CAPM cannot price both sets of equity portfolios, with explanatory ratios of 42% and 18% in the tests with SBM25 and SM25, respectively. These negative estimates indicate that the model performs more poorly than a model that predicts constant risk premia in the cross section. The FF3 model, however, explains a significant proportion of the dispersion in average returns of the SBM25 portfolios, with an R 2 estimate of 67% and an average mispricing of 0.10% per month. These results are consistent with the evidence in Fama and French (1993, 1996). The risk price estimate associated with HM L is statistically significant at the 1% level, while the risk price for SMB is not significant. Yet the FF3 model cannot price the SM25 portfolios, as illustrated by the nearly zero estimate of the coefficient of determination (3%), 12
15 which is in line with previous evidence [Fama and French (1996)]. When we compare the performance of FF3 and the (C)ICAPM (version with F F R), we see that both models have similar performance in pricing the SBM25 portfolios, but the (C)ICAPM (both versions) clearly outperforms in pricing the sizemomentum portfolios. In other words, the (C)ICAPM can explain the two anomalies, while the FF3 can price only the value premium. The C4 model explains a large fraction of the crosssectional dispersion in average returns, with explanatory ratios of 78% and 85% in the tests with SBM25 and SM25, respectively, and the UMD factor is priced in both cases (1% level). Thus, the fourfactor model outperforms the (C)ICAPM (version with F F R) by about 8% and 14% (in the explanatory ratios) in explaining the SBM25 and SM25 portfolios, respectively. When we compare against the version based on RREL, the C4 model outperforms by about 0.32% and 0.11% in pricing the SBM25 and SM25 portfolios, respectively. Note that the C4 model includes three independent sources of systematic risk (in addition to the market return), while in the (C)ICAPM the two key sources of systematic risk (scaled factor and innovation in the state variable) are associated with the same state variable, the shortterm interest rate. 4.3 Individual pricing errors Although both R 2 OLS and MAE represent measures of the overall fit of the (C)ICAPM, it is important to assess the relative explanatory power of the model over the different portfolios within a certain group (e.g., value versus growth portfolios, or past winners versus past losers portfolios). Figure 2 plots the pricing errors (and respective tstatistics) associated with the SBM25 portfolios in the version with F F R. For the (C)ICAPM, the biggest negative outlier is the extreme largevalue portfolio (S5BM 5) with a pricing error of 0.21% per month, and the main positive outliers are the smallvalue portfolios (S1BM 4 and S1BM 5), with pricing errors of 0.26% per month. In terms of statistical significance, only the pricing error for portfolio S1BM5 is significant at the 5% level. In comparison, in the test with the FF3 model there are seven portfolios with significant pricing errors. It is interesting to see that the (C)ICAPM outperforms both the FF3 and C4 models in pricing the extreme smallgrowth portfolio (S1BM1) with a pricing error of 0.15% per month, compared to mispricing of 0.41% and 0.31% for FF3 and C4, respectively. This portfolio is particularly hard to price for most models in the empirical asset pricing literature [see, for example, Fama and French (1993) and Campbell and Vuolteenaho 13
16 (2004)]. In untabulated results for the version with RREL, there are four portfolios with significant pricing errors at the 5% level (S1BM1, S1BM4, S1BM5 and S5BM1), which is consistent with the lower explanatory power of this version relative to the version with F F R in pricing the SBM25 portfolios. Figure 3, which is similar to Figure 2, provides a visual representation of the model s fit (version with F F R) in a crosssectional test with the SM25 portfolios. The main negative outliers associated with the (C)ICAPM are the bigintermediate (S5M 3) and bigwinner (S5M 4) portfolios, with pricing errors of 0.37% and 0.40% per month, respectively. The main positive outlier is the smallwinner portfolio (S1M5), with a pricing error of 0.43% per month. We can see that for most portfolios the (C)ICAPM produces significantly lower pricing errors than the FF3 model and similar errors to the C4 model. Regarding the statistical significance, there are five portfolios with significant mispricing (S1BM5, S3BM5, S5BM3, S5BM4 and S5BM5). However, in the case of the alternative factor models, the number of significant pricing errors is greater (nine and eighteen for C4 and FF3, respectively). Untabulated results show that in the version based on RREL, there are only three portfolios with significant mispricing (S1BM 3, S1BM 5 and S2BM 1). 4.4 Which factors explain the value and momentum premiums? The results in Table 2 suggest that the innovation in the state variable drives the fit of the (C)ICAPM for pricing the SBM25 portfolios, while the scaled factor seems to drive the explanatory power of the model for the SM25 portfolios. To see more clearly which factors drive the explanatory power of the (C)ICAPM in pricing each set of portfolios, we conduct an accounting analysis of the contribution of each factor to the overall fit of the model. Specifically, we compute the average factor risk premium (average beta times risk price) for each factor and across every booktomarket(bm)/momentum quintile. For example, the average market risk premium associated with the BM/momentum quintile j is given by λ M β j,m, where the average beta for BM/momentum quintile j = 1,..., 5 is computed as the simple 14
17 average of the market betas for portfolio j across the 5 size quintiles within SBM25 or SM25: β j,m = β i,m, i = 1j, 2j, 3j, 4j, 5j, i=1 where the first number refers to the size quintile, and j refers to the BM/momentum quintile. The results for this accounting decomposition are shown in Table 4. The spread in average excess returns between the first (Q1, growth) and the fifth BM quintile (Q5, value) is 0.53% per month, which corresponds to the (symmetric of the) value premium in our sample. This gap has to be (partially) matched by the risk premium associated with one or more of the factors in the (C)ICAPM, as shown in the respective beta pricing equation (6), if this model is able to price the value premium. In the version with F F R, the spread Q1 Q5 in the market risk premium is 0.13% per month, which moves the model farther from explaining the value premium, and confirms why the baseline CAPM does not price the value anomaly. The spread associated with the scaled factor has the right sign, but the magnitude is quite low (0.03% per month). Thus, it is the innovation in the Fed funds rate that accounts for the value premium, with a spread in the respective risk premium of 0.58% per month, which more than explains the original value premium of 0.53%. Only 0.04%, of the original gap of 0.53%, is left unexplained by the threefactor ICAPM; this is another way to gauge the success of the model in explaining the value anomaly. Thus, value stocks covary negatively with innovations to the Fed funds rate, which has a negative risk price. In the version based on RREL, the accounting decomposition is qualitatively similar. The gaps in risk premiums associated with the market, scaled and hedging factors are 0.15%, 0.17% and 0.69%, respectively, producing a mispricing Q1 Q5 of 0.16% per month. Thus, it is the hedging factor that drives the value premium. With regard to the momentum spread, the gap Q1 Q5 (loser minus winner) in average excess returns is about 1% per month, nearly twice the size of the value premium in our sample. As in the case of the value anomaly, the CAPM cannot explain momentum, as the gap (Q1 Q5) in the market risk premium is positive (0.14% per month). A similar positive spread in risk premium (0.19% per month) is generated by the innovation in the Fed funds rate, showing that this factor does not help to price momentum. Thus, it is the scaled factor that is key in pricing momentum, generating a gap Q1 Q5 in risk premia of 15
18 about 1.16% per month that more than matches the original return spread of 1.01%. Only 0.18% of this last spread is left unexplained by the threefactor ICAPM, thus justifying the large fit of the model in pricing the SM25 portfolios, as documented above. In the case of the version based on RREL, once again we have similar results. The spreads in risk premiums for the market, scaled and hedging factors are 0.08%, 1.43% and 0.49%, respectively, leading to a mispricing of only 0.14% per month. These results confirm that the two nonmarket factors in the (C)ICAPM drive two different sources of systematic risk, one related to the value premium, and another related to momentum. We conduct a similar decomposition for the FF3 and C4 models to assess the factors that drive the explanatory power of these models over the value and momentum quintiles, which is presented in Table 5. In the case of the FF3 model, the gap Q1 Q5 associated with the HML factor is 0.56% per month, which is about the same as the risk premium gap associated with the hedging factor in the (C)ICAPM, and nearly matches the original value spread of 0.53%. When it comes to pricing the momentum quintiles, the spread associated with HML is only 0.29% per month, about onethird of (the size of) the original momentum spread of 1.01%, thus leading to a gap Q1 Q5 in mispricing of 0.84% per month. In other words, the FF3 cannot price the momentum spread. In the case of the C4 model and BM quintiles, the risk premium gap associated with HML is very similar to the corresponding spread of HML in the FF3 model (0.55% per month). For the momentum quintiles, the risk premium spread associated with UMD is 0.94%, which almost matches the original momentum spread. Thus, the key factor that prices the value premium in both models is HML (similarly to the hedging factor in the (C)ICAPM), while the UMD factor drives momentum in the C4 model (just like the scaled factor in the (C)ICAPM). These results also suggest that the innovation in our state variable is correlated (conditional on the other factors of the (C)ICAPM) with the HML factor, while the scaled factor is correlated with the U M D factor. To assess this conjecture, we conduct timeseries multiple 16
19 regressions: 17,18 HML t+1 = ρ 0 + ρ 1 RM t+1 + ρ 2 RM t+1 z t + ρ 3 z t+1, (12) UMD t+1 = ρ 0 + ρ 1 RM t+1 + ρ 2 RM t+1 z t + ρ 3 z t+1. (13) To measure the individual statistical significance of the regressors, we compute heteroskedasticityrobust GMM standard errors [White (1980)]. The results displayed in Table 6 show that in the regression for HML (Panel A), conditional on the market return, both the hedging factor, F F R t+1, and the scaled factor, RM t+1 F F R t, are negatively correlated with HML t+1, and the slopes are strongly significant (1% level). In the regression for UMD, the scaled factor is positively correlated with UMD t+1, and this effect is strongly significant (at the 1% level), while the slope associated with F F R t+1 is not significant at the 10% level. The correlations are far from perfect, however, as indicated by the R 2 estimates of 16% and 8% in the regressions for HML and UMD, respectively. When the state variable is RREL, the results are qualitatively similar. Both RM t+1 RREL t and RREL t+1 are negatively correlated with HML t+1, although the slope associated with the scaled factor is not significant at the 10% level. In the regression of UMD, both factors are positively correlated with UMD t+1, and both coefficients are statistically significant. Hence, F F R t+1 and RREL t+1 both measure some of the risks captured by HML, and the same happens to the scaled factor in relation to UMD, but these effects are only partial. This result for UMD is consistent with other evidence showing that the payoffs of momentum strategies can be, at least partially, accounted for by lagged macroeconomic variables linked to the business cycle as is the case of the Fed funds rate or relative Treasury bill rate [see Chordia and Shivakumar (2002) and Ahn, Conrad, and Dittmar (2003)]. 4.5 Factor betas and intuition Our analysis shows that the innovation in the Fed funds (or in the relative Tbill rate) is the factor in the (C)ICAPM responsible for pricing the value spread, and the scaled factor accounts for the momentum anomaly. Put differently, there is a dispersion in the betas associated with 17 Other evidence shows that some of the risk factors in the ICAPM or conditional CAPM measure approximately the same types of risks associated with HML [e.g., Lettau and Ludvigson (2001), Vassalou (2003), Hahn and Lee (2006), Petkova (2006), among others]. 18 Ferguson and Shockley (2003) and Hahn and Lee (2006) conduct similar timeseries regressions for SMB and HML. 17
20 the hedging factor within the sizebm portfolios that fits the value premium, and there is a similar dispersion in the betas associated with the scaled factor within the sizemomentum portfolios that fits the momentum premium. The multipleregression betas associated with both factors in the case of the SBM25 portfolios are displayed in Figure 4. We can see that value stocks have negative betas associated with F F R t+1, while growth stocks have positive betas for this same factor. This dispersion in betas scaled by the negative risk price for F F R t+1 generates a spread in risk premia. A similar pattern holds for the factor loadings associated with RREL t+1. Why are value stocks more (negatively) sensitive to unexpected rises in shortterm interest rates? One possible explanation is that many of these firms are near financial distress as a result of a sequence of negative shocks to their cash flows [Fama and French (1992)], and are thus more sensitive to rises in shortterm interest rates. According to the credit channel theory of monetary policy [Bernanke and Gertler (1995)], a monetary tightening (increase in the Fed funds rate) represents an increase in financial costs and restricts access to external financing. This effect should be stronger for firms in poorer financial position, as typically those firms have higher costs of external financing, and the value of their assets (which act as collateral for new loans) is relatively depressed. Increases in interest rates would thus constrain access to financial markets and prevent those firms from investing in profitable investment projects. This mechanism is consistent with the analysis of Lettau and Wachter (2007) who show that the prices (and realized returns) of value stocks are more sensitive to shocks in nearterm cash flows, while the prices of growth stocks are more related to shocks to discount rates (longterm expected returns). The analysis of the betas for the SM25 portfolios in Figure 5 shows that past winners have slightly positive betas with the scaled factor, while past losers have large negative betas. This dispersion in betas multiplied by the positive risk price of the scaled factor, RM t+1 F F R t, generates the risk premium necessary to explain the momentum spread. In the case of RM t+1 RREL t the dispersion of betas (negative versus positive) between past losers and winners is even more clear. Thus, past winners are riskier not because they have higher market betas in average times, but because they are more correlated with the market in periods of high shortterm interest rates. 18
21 We can assess this in greater detail by computing the average conditional market betas: 19 β i,m,t = β i,m + β i,m,z z t, where z t represents the average of the scaling variable calculated over periods with high and low interest rates. A period with high interest rates occurs when the Fed funds rate or RREL is 1.5 standard deviations above its mean; similarly, a period with low interest rates occurs when F F R (RREL) is 1.5 standard deviations below its mean. When we consider all the periods, the average conditional market beta corresponds to the unconditional (multipleregression) market beta since the scaling variable has unconditional zero mean, E(F F R t ) = E(RREL t ) = 0. Figure 6 plots the average conditional market betas in the test with the SM25 portfolios for both versions of the model. 20 In Panels A and B, we can see that past winners have lower unconditional market betas than past losers across all size quintiles. That is, past losers are unconditionally riskier than past winners. This shows the inability of the simple CAPM to price the momentum portfolios. In Panels C and D, however, we can see that in periods of high interest rates, past winners have higher market betas than past losers, an effect that is robust across all size deciles. On the other hand, in periods with low interest rates, past losers have higher market betas than past winners, as shown in Panels E and F. Thus, past winners are riskier than past losers because they have greater market risk in times of high shortterm interest rates. Why are past winners riskier than past losers in periods with high interest rates? A possible explanation relies on the different characteristics of winner and loser stocks at different points of the business cycle. That is, during economic expansions (which are associated with high shortterm interest rates) winners tend to be cyclical firms, which have high market betas. Conversely, during recessions (periods with low shortterm interest rates) winners tend to be noncyclical firms, with low market betas. The changing composition of the momentum portfolios leads to the time variation in its market betas. This reasoning is consistent with evidence in the momentum literature that momentum profits are procyclical. 21 The mean of the momentum factor (UMD) in economic expansions 19 Lettau and Ludvigson (2001) perform a similar analysis. 20 The analysis is conducted only for the SM25 portfolios, since the scaled factor is not relevant to price the SBM25 portfolios. 21 See Johnson (2002), Chordia and Shivakumar (2002), Cooper, Gutierrez, and Hameed (2004), Sagi and Seasholes (2007), and Stivers and Sun (2010). Specifically, the theoretical analysis in Johnson suggests that momentum profits might be the result of episodic but persistent shocks in cash flows, which can be related with 19
22 (as classified by the NBER) is 0.85% per month compared to only 0.01% per month in recessions. Thus, the time variation in market betas matches the time variation in momentum returns. This variation in risk premium is justified because under positive business conditions and high shortterm interest rates, the market future risk premium is low. Riskaverse investors are willing to invest in winner stocks, which are cyclical at this point of the cycle and have high betas, only if these stocks sell at a greater discount, that is, offer a higher expected return. Our results are also consistent with the evidence provided in Grundy and Martin (2001) and Daniel (2011) that momentum profits are associated with timevarying market betas of winner and loser portfolios. They find that after a bear equity market, the market beta of the momentum factor is low since past winners have low betas (defensive stocks that performed relatively better in the bear market) and past losers have high betas (aggressive or cyclical stocks that underperformed more in the bear market). At the same time, in a bear market interest rates are usually at low levels, and so it follows that past losers have high betas when interest rates are low while past winners have low betas. On the other hand, in a bull market interest rates are at high levels, and thus past winners (those that have outperformed in the bull market) have high market betas while past losers exhibit low betas. Thus, interest rates represent an instrument that signals time variation in market betas of the winner and loser portfolios as a result of changing market conditions and hence of the changing composition of the momentum portfolios and of their market betas. Figure 7 shows that there is some correlation over time between the momentum factor and the conditional market beta of UMD, computed as β UMD,M,t = β UMD,M + β UMD,M,z z t. Specifically, the momentum crashes that occurred in 2001 and 2009 [as documented by Daniel (2011)] are roughly associated with a sharp decline in the current and lagged market betas of the UMD factor. 5 Additional results In this section, we apply a battery of robustness checks to our main results. shortterm business conditions. 20
23 5.1 Bootstrap simulation Following Lewellen, Nagel, and Shanken (2010), we estimate empirical confidence intervals for the coefficient of determination and average pricing error in the crosssectional regressions. We use a bootstrap simulation with 5,000 replications in which the excess portfolio returns and risk factor realizations are simulated (with replacement from the original sample) independently and without imposing the (C)ICAPM s restrictions. Thus, the data generating process is derived under the assumption that the model is not true. We want to investigate the following question: under the assumption that the (C)ICAPM does not hold, how likely is it that we obtain the fit found in the data. In other words, are our results in the crosssectional tests spurious? In untabulated results and in the test with SBM25, the 95% confidence intervals for R 2 are [ 1.09, 0.19] and [ 1.08, 0.19] when the state variables are F F R and RREL, respectively. The 95% confidence intervals for the average pricing error are [0.48, 0.72] and [0.48, 0.72] for the versions with F F R and RREL, respectively. When we compare these intervals with the actual estimates, it follows that for both versions of the model the estimated coefficients of determination (70% and 46%) are well above the upper bounds on the intervals. Simultaneously, the sample MAE estimates (0.10% and 0.14%) are significantly below the lower bounds on the corresponding empirical intervals. In the test with SM25, the confidence intervals for MAE are quite similar to those in the test with SBM25, which implies that the MAE estimates from the original sample (0.16% and 0.15%) are statistically significant. On the other hand, the 95% interval for R 2 is [ 0.46, 0.27] for both versions of the model, implying that also in this case, the actual R 2 estimates of 71% and 74% are well above the upper limit. Overall, these results suggest that the fit of the model in pricing the BM and momentum portfolios is not spurious. 5.2 Pricing bond returns Adding bond returns to the empirical tests of the (C)ICAPM enables us to assess whether the model can jointly price stocks and bonds. 22 We add to each equity portfolio group the excess returns on seven Treasury bonds with maturities of 1, 2, 5, 7, 10, 20, and 30 years. The data are available from CRSP. This involves a total of 32 test assets in each estimation (SBM25 or 22 Fama and French (1993) and Koijen, Lustig, and Van Nieuwerburgh (2010) also estimate factor models over the joint crosssection of stock and bond returns. 21
24 SM25). The results are presented in Table 7. In the test with SBM25 and version based on F F R, the explanatory ratio increases to 85% from 70% in the benchmark test, while the average pricing error is 0.09% per month. The point estimate for λ F F R is slightly lower (0.48) than in the benchmark test, but remains strongly significant (1% level). In the test with SM25, the R 2 and average pricing error estimates are the same as in the tests for the equity portfolios. The risk price for the scaled factor, λ M,F F R, is close to the corresponding estimate in the benchmark test and is significant at the 1% level. In the tests with either portfolio group, the model is rejected by the χ 2 test, likely mainly because of a poor inversion of Var( ˆα) when the number of test returns is relatively large. In the version based on RREL, the explanatory ratio in the test with SBM25 is 57% (up from 46% in the benchmark test), and the average pricing error is 0.16% per month. The estimate for the hedging risk price is significantly lower in magnitude than in the baseline case (0.18%) but is still significant at the 1% level. In the test containing the SM25 portfolios, the fit of the model is basically the same as in the benchmark case, with a coefficient of determination of 76%. As before, the estimates for λ M,RREL and λ RREL are positive and negative, respectively, but only the scaled factor is priced. We also estimate the alternative linear factor models by including bond risk premiums in the menu of test assets. Untabulated results show explanatory ratios of 31% and 12% for the baseline CAPM in the tests with SBM25 and SM25, respectively. This shows that the CAPM has some explanatory power over bond risk premia. The FF3 model has a fit very similar to the (C)ICAPM (version with F F R) in the test with SBM25 (R 2 = 82%), but it underperforms significantly in the test with SM25 (R 2 = 26%). The explanatory ratio for the C4 model is quite similar to the (C)ICAPM in the test with SBM25 (88%), while it outperforms in the test with SM25, with an explanatory ratio of 89%. Overall, when we price equity and bond risk premia jointly, the results for the (C)ICAPM are quite similar to the benchmark results. 5.3 Pricing alternative equity portfolios We estimate the (C)ICAPM with alternative equity portfolios 10 portfolios sorted on size, 10 portfolios sorted on BM and 10 momentum portfolios, for a total of 30 portfolios. This crosssectional test enables us to check whether our threefactor model prices simultaneously 22
25 the BM and momentum portfolios. The results are displayed in Table 8. We can see that in the version based on F F R the model s fit is smaller than in the tests with either SBM25 or SM25, with an explanatory ratio of 29% and an average pricing error of 0.15% per month. The risk price estimates for the nonmarket factors have the same signs than in the test with SM25, and both estimates are significant at the 1% level. In the version with RREL, the explanatory power is significantly greater than in the first version, with an R 2 of 60% and an average mispricing of 0.12% per month. This fit is halfway the one obtained for the tests with the SBM25 and SM25 portfolios. The risk price estimates for the scaled and hedging factors have the same signs as in the version with F F R, and both estimates are statistically significant. In untabulated results, the explanatory ratios for the FF3 and C4 models are 9% and 85%, respectively. These results show that overall, the (C)ICAPM does a good job in pricing simultaneously the size, BM and momentum portfolios. 5.4 Pricing the market return We next reestimate the (C)ICAPM by including the market equity premium (RM) in the set of test assets. 23 This enables us to assess whether the model can jointly price the equity portfolios (SMB25 or SM25) and the market return. Results not tabulated show that the (C)ICAPM fit is very close to that of the benchmark test including only equity portfolios, with R 2 estimates of 71% and 69% in the tests with SBM25 and SM25, respectively, when the state variable is F F R. In the version based on RREL, the explanatory ratios are 48% and 74% in the tests with SBM25 and SM25, respectively. For both versions of the model, the risk price estimates are also nearly the same as in the benchmark test of the (C)ICAPM. Thus, forcing the model to price the aggregate equity premium does not have an impact on the fit of the (C)ICAPM. 23 Lewellen, Nagel, and Shanken (2010) advocate that when the factors are returns, they should be included in the set of test assets. 23
26 5.5 Alternative ICAPM specification In an alternative ICAPM specification, the innovation in the state variable represents the residual from an AR(1) model: z t+1 ε t+1 = z t+1 φ z ρ z z t. (14) By using this new proxy for z t+1, we want to assess whether the results for the (C)ICAPM are sensitive to the measurement of the innovation in the state variable. Untabulated results are very similar to the benchmark test using the first difference in either F F R or RREL. In the version based on F F R, the explanatory ratios are 72% and 71% in the tests with SBM25 and SM25, respectively, while average pricing errors are the same as in the benchmark test. In the version with RREL, the R 2 estimates are 54% and 76% in the tests with SBM25 and SM25, respectively. The corresponding MAE estimates are 0.13% and 0.14% per month, which are very similar to the corresponding values in the benchmark test. The point estimates for the factor risk prices are also very close to the estimates in the benchmark test, for both versions of the model. Thus, the results of the (C)ICAPM are robust to the way we measure the innovation in the state variable, the hedging risk factor. 5.6 Alternative standard errors We use alternative standard errors for the factor risk prices and pricing errors. These GMMbased standard errors can be interpreted as a generalization of the Shanken (1992) standard errors to the extent that they relax the implicit assumption of independence between the factors and the residuals from the timeseries regressions [see Cochrane (2005) (Chapter 12)]. The full details are provided in Appendix B. Untabulated results show that the tstatistics for the risk price estimates based on the new standard errors lead to the same qualitative decisions as the Shanken tstatistics. Specifically, λ z in the test with SBM25 and λ M,z in the test with SM25 are both significant at the 1% level. The main difference occurs with the χ 2 statistic in the tests with SBM25, which now has pvalues of 2% and 4% in the versions with F F R and RREL, respectively. This values are related to a poor inversion of the covariance matrix of the pricing errors, given their lowness (0.10% or 0.14% per month). 24
27 5.7 Alternative sample We estimate out threefactor model in the test with the SM25 portfolios for the 1963: :12 period. We want to assess whether the fit of the model in pricing the momentum portfolios is robust to removing the momentum crash occurred in 2009, as documented by Daniel (2011). Untabulated results show that the explanatory power of the version based on F F R is only marginally lower than in the test for the full sample, with an R 2 estimate of 60% and an average pricing error of 0.22% per month. On the other hand, in the version based on RREL, the fit of the (C)ICAPM is basically the same as in the benchmark test (R 2 = 73%, MAE = 0.17%). In both versions, the point estimates of the risk price for the scaled factor are strongly significant (1% level). Overall, these results show that the 2009 momentum crash does not have a meaningful impact on the capacity of the model in pricing the momentum portfolios. 5.8 Estimating the (C)ICAPM in expected returncovariance representation We define and test the (C)ICAPM in expected returncovariance representation: E(R i,t+1 R f,t+1 ) = γ M Cov(R i,t+1 R f,t+1, RM t+1 ) +γ M,z Cov(R i,t+1 R f,t+1, RM t+1 z t ) + γ z Cov(R i,t+1 R f,t+1, z t+1 ), (15) where (γ M, γ M,z, γ z ) denote the covariance risk prices associated with the market return, the scaled factor, and the innovation in the state variable, respectively. This version of the model is equivalent to an expected returnsingle beta pricing equation. Thus, the model should fit as well as the version with multipleregression betas, although the risk prices might have different signs, given possible correlation among the factors. Though, as the factors in the (C)ICAPM are not significantly correlated, as shown in Table 1, the factor risk prices should have the same signs in either multiple or singleregression betas (or equivalently, covariances). We estimate specification (15) by firststage GMM [Hansen (1982) and Cochrane (2005)]. This method uses equally weighted moments, which is conceptually equivalent to running an OLS crosssectional regression of average excess returns on factor covariances (righthand side variables). One advantage of using the GMM procedure is that we do not need to have previous estimates of the individual covariances, since these are implied in the GMM moment conditions. 25
28 The GMM system has N + 3 moment conditions, where the first N sample moments correspond to the pricing errors for each of the N testing returns: (R i,t+1 R f,t+1 ) γ M (R i,t+1 R f,t+1 ) (RM t+1 µ M ) γ M,z (R i,t+1 R f,t+1 ) (RM t+1 z t µ M,z ) g T (b) 1 T T 1 t=0 γ z (R i,t+1 R f,t+1 ) ( z t+1 µ z ) RM t+1 µ M = 0. RM t+1 z t µ M,z z t+1 µ z i = 1,..., N, (16) In this system, the last three moment conditions enable us to estimate the factor means. Thus, the estimated covariance risk prices from the first N moment conditions correct for the estimation error in the factor means, as in Cochrane (2005) (Chapter 13) and Yogo (2006). There are N 3 overidentifying conditions (N + 3 moments and 2 3 parameters to estimate). The standard errors for the parameter estimates and the remaining GMM formulas are presented in Appendix C. By defining the first N residuals from the GMM system as the pricing errors associated with the N test assets, α i, i = 1,..., N, the χ 2, ROLS 2, and MAE measures are defined analogously to the formulas presented in Section 3. The GMM estimation results are displayed in Table 9. As expected, the R 2 and MAE estimates are the same as in the benchmark test of the beta pricing equation. Now, however, the (C)ICAPM version based on F F R is rejected by the χ 2 statistic in the estimation with the SBM25 portfolios (pvalue = 1%), which again should be the result of a poor inversion of Var(ˆα). The point estimate for γ F F R is negative and statistically significant (at the 5% level) in the test with SBM25, while the point estimate for γ M,F F R is positive and strongly significant (1% level) in the test with SM25. Thus, the signs of the covariance risk prices of the nonmarket factors are the same as in the test of the beta pricing equation. The market covariance risk prices, γ M, are negative, but these point estimates are largely insignificant. In the version with RREL, the (C)ICAPM continues to pass the χ 2 test when the test portfolios are SBM25 (pvalue = 6%). The estimates for γ M,RREL and γ RREL have the same signs as λ M,RREL and λ RREL, in the benchmark test. The hedging risk price in the test with SBM25 and the scaled factor risk price in the test with SM25 are statistically significant at the 26
29 1% and 5% levels, respectively. In contrast with the version based on F F R, the estimates for the market risk price are now positive, although largely insignificant. Overall, the estimation results for the covariance pricing equation are consistent with those in the benchmark test. We also estimate the expected returncovariance equation by including an intercept that represents a proxy for the excess zerobeta rate: E(R i,t+1 R f,t+1 ) = γ 0 + γ M Cov(R i,t+1 R f,t+1, RM t+1 ) +γ M,z Cov(R i,t+1 R f,t+1, RM t+1 z t ) + γ z Cov(R i,t+1 R f,t+1, z t+1 ). (17) As we note in Section 3, if the (C)ICAPM is correctly specified, the estimate for γ 0 should not be statistically different from zero. Results not tabulated show that the point estimates for γ 0 are nearly zero and largely insignificant in the tests with both the SBM25 and SM25 portfolios, and for both versions of the model. Moreover, the MAE and R 2 estimates are nearly the same as in the benchmark restricted pricing equation without intercept, thus showing that the constant factor plays no relevant role. These results seem to suggest that the (C)ICAPM is not misspecified. That is, there are no relevant missing risk factors, at least when it comes to price the value and momentum portfolios. 5.9 Nested models The (C)ICAPM consists of two important nested models. The standard ICAPM can be obtained as a special case of the (C)ICAPM by imposing β i,mz = 0, i.e., that the conditional market beta is constant over time: E(R i,t+1 R f,t+1 ) = λ M β i,m + λ z β i,z. (18) Similarly, the conditional CAPM in unconditional form can be obtained from (6) by imposing λ z = 0; that is, investment opportunities are constant through time: E(R i,t+1 R f,t+1 ) = λ M β i,m + λ Mz β i,mz. (19) We estimate the two nested models of the (C)ICAPM: the twofactor conditional CAPM in equation (19), and the twofactor (unconditional) ICAPM in equation (18). This analysis 27
30 allows us to evaluate the incremental explanatory power of the benchmark threefactor model against each nested model in pricing both sets of equity portfolios. The estimation results are displayed in Table 10. In the test with SBM25, the conditional CAPM based on F F R has some explanatory power over the sizebm portfolios with an R 2 estimate of 34% and an average mispricing of 0.15% per month. The fit of the twofactor ICAPM is significantly better, with an explanatory ratio of 67% and an average pricing error of 0.11% per month, for almost the same explanatory power as in the benchmark (C)ICAPM. Moreover, the point estimate for λ F F R is negative and strongly significant (1% level). When the state variable is RREL, the CCAPM cannot price the size/bm portfolios, with a coefficient of determination of 18% and a MAE estimate of 0.22% per month. On the other hand, the fit of the twofactor ICAPM (R 2 = 40%) is almost the same as in the (C)ICAPM, and the estimate for λ z is significant at the 1% level. In the test with SM25, the twofactor ICAPM cannot price these portfolios, with a negative R 2 estimate (18%) and an average pricing error as high as 0.32% per month. On the other hand, the fit of the conditional CAPM is nearly the same as that of the (C)ICAPM, with a coefficient of determination of 69% and an MAE estimate of 0.16% per month. Moreover, the risk price of the scaled factor is positive and highly significant. The results for the version based on RREL are qualitatively similar. The twofactor ICAPM performs poorly with an explanatory ratio of just 7%. In contrast, the fit of the CCAPM (66%) is close to that of the benchmark threefactor model and the risk price estimate for the scaled factor is significant at the 1% level. Thus, these results are consistent with the analysis so far. The (C)ICAPM provides the best of both worlds, that is, the best characteristics of the two nested models. It includes the hedging risk factor that prices the BM portfolios (as in the baseline ICAPM), and also the scaled factor that prices the momentum portfolios (as in the conditional CAPM). 6 Longterm reversal Can the (C)ICAPM explain the longterm reversal in returns anomaly [De Bondt and Thaler (1985, 1987)]? The anomaly is that stocks with low returns over the long term (three to five years) have higher subsequent future returns, while past longterm winners have lower future returns. This longterm mean reversion in stock returns is not explained by the CAPM. This 28
31 anomaly should be closely related to the value anomaly, as longterm underperformers end up with high booktomarket ratios. To test the explanatory power of the (C)ICAPM for this anomaly, we use 25 portfolios sorted on both size and longterm past returns (SLTR25). The portfolios come from the intersection of five portfolios formed on size (market equity) and five portfolios formed on past returns (13 to 60 months before the portfolio formation date). The portfolios are obtained from Kenneth French s data library. 24 The results for the (C)ICAPM pricing equation in the test with the SLTR25 portfolios are shown in Table 11. We can see that the (C)ICAPM based on F F R has considerable explanatory power, with a coefficient of determination of 63%, while the corresponding average pricing error is 0.09% per month. This is a relatively similar fit to the test with the SBM25 portfolios. Moreover, the point estimate for λ F F R is negative and strongly significant (1% level), while the risk price of the scaled factor is largely insignificant. In the version based on RREL, the fit is only marginally lower with an explanatory ratio of 51% and an average mispricing of 0.11% per month. Moreover, the hedging risk factor is strongly priced (1% level). Thus, as in the test with SBM25, most of the explanatory power of the model over the SLTR25 portfolios seems to be driven by the hedging factor. We also estimate the alternative factor models with these SLTR25 portfolios. Results not tabulated show that the baseline CAPM cannot price these portfolios, with an R 2 of 9%, and an average pricing error of 0.17% per month. The FF3 model significantly outperforms the CAPM, with an explanatory ratio of 75%, marginally better than the fit of the (C)ICAPM. Moreover, the risk price for HML is strongly priced. The C4 model has the best overall fit, with an R 2 estimate of 92%, indicating that UMD, in addition to HML, helps to price these portfolios. The plot of the individual pricing errors, presented in Figure 8, shows that the main outlier in the test with the (C)ICAPM s version with F F R is the small/past winner portfolio (S1LT R5), with a pricing error of 0.36% per month; the corresponding mispricing in the case of the FF3 model is 0.33% per month. The pricing errors for portfolios S1LT R3 and S1LT R5 are statistically significant at the 5% level, while in the case of the FF3 model there are three portfolios with significant errors. Untabulated results show that, similarly to the version with F F R, 24 Fama and French (1996), Da (2009), and Da and Warachka (2009), among others, also conduct asset pricing tests over portfolios sorted on prior longterm returns. 29
32 when the state variable is RREL only the pricing errors associated with portfolios S1LT R3 and S1LT R5 are statistically significant. Thus, as in the test over the size/bm portfolios, the (C)ICAPM seems to behave much like the FF3 model. We also conduct an accounting decomposition of the longterm reversal spread, similar to the analysis made for the value and momentum spreads in Section 4. In untabulated results, the gap Q1 Q5 in average excess returns (past longterm loser minus past longterm winner) is about 0.45% per month, which corresponds to the longterm reversal spread in our sample. This premium is comparable to the size of the value premium reported above (0.53%). The risk premium (beta times risk price) gap (Q1 Q5) associated with the market factor is 0.03% per month, thus confirming that the baseline CAPM cannot price the longterm reversal (LTR) quintiles. The spreads in risk premium associated with the hedging and scaled factors are 0.26% and 0.09% per month, respectively. Of the original 0.45% spread in returns, 0.14% is not explained by the model, which represents about onethird of the original gap. Thus, the key factor responsible for the explanatory power of the (C)ICAPM over the longterm reversal portfolios is the hedging factor, similar to the results obtained for the value premium. When the state variable is RREL, the results are qualitatively similar: the risk premium gaps for the market, scaled and hedging factors are 0.04%, 0.05% and 0.19%, respectively, producing a spread in pricing errors of 0.25% per month. Thus, as in the case with F F R, the hedging factor drives most of the explanatory power of the model over the longterm reversal spread. An analogous decomposition for the FF3 model shows that the HML factor is the key driver of the LTR spread, with a gap in risk premium of 0.34% per month, while the SMB makes a marginal contribution (gap of 0.02%) leading to a gap in mispricing of 0.11% per month. In the case of the C4 model, the gaps in risk premium associated with the SMB, HML, and UMD factors are 0.12%, 0.26%, and 0.07% per month, respectively, producing a gap Q1 Q5 in average pricing error of only 0.02% per month, consistent with the high explanatory ratio. Analysis of the factor loadings in Figure 9 sheds light on the way the (C)ICAPM, more precisely, the hedging factor, prices the longterm reversal anomaly. We can see that, across all size quintiles, past longterm losers have relatively high negative betas associated with the innovation in the Fed funds rate, while past longterm winners have positive loadings (within the first size quintile, negative betas but with lower magnitudes). This spread in betas scaled 30
33 by the corresponding risk price generates the risk premium necessary to partially explain the longterm reversal return spread. When the state variable is RREL, with the exception of the first size quintile, we also have negative factor loadings for past longterm losers and positive betas for past winners. Why are past longterm losers have greater interest risk than past longterm winners? Past longterm losers are likely to have a long sequence of negative shocks in their cash flows, and hence become more financially constrained through time. Hence, these firms will be more sensitive to additional negative shocks in their earnings, specifically further rises in shortterm interest rates. Hence, past longterm losers act much like value stocks, while pastwinners behave more like growth stocks. 7 Conclusion We offer a simple asset pricing model that goes a long way forward in explaining the value and momentum anomalies. We specify a threefactor conditional intertemporal CAPM, denoted as (C)ICAPM. The factors are the market equity premium, the market factor scaled by the state variable (arising from timevarying market betas), and the hedging or intertemporal factor. These last two factors are based on the same macroeconomic state variable, the Federal funds rate or the relative Tbill rate. We test our threefactor model with 25 portfolios sorted on size and booktomarket and 25 portfolios sorted on size and momentum. The crosssectional tests show that the (C)ICAPM explains a large faction of the dispersion in average equity premia of the two portfolio groups, with explanatory ratios around 70%. The (C)ICAPM outperforms the Fama and French (1993) threefactor model when it comes to pricing both sets of portfolios, and only marginally underperforms the Carhart (1997) fourfactor model. The ultimate nonmarket sources of systematic risk in our model are associated with one single variable, a proxy for shortterm interest rates; in the fourfactor model, there are three unrelated nonmarket sources of systematic risk. Moreover, the factors in Carhart (1997) are selffinancing portfolios related to the test portfolios, while we use a macroeconomic state variable that a priori is not mechanically related to the test portfolios. The ICAPM hedging risk factor explains the dispersion in risk premia across the booktomarket portfolios, and the scaled factor prices the dispersion in risk premia across the momentum 31
34 portfolios. According to our model, the reason that value stocks enjoy higher expected returns than growth stocks is because they have higher interest rate risk; that is, they have more negative factor loadings on the hedging factor. Furthermore, in our model past winners enjoy higher average returns than past losers because they have greater conditional market risk; that is, past winners have higher market risk in times of high shortterm interest rates. 32
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42 A Derivation of the ICAPM in discrete time The problem for the representative investor in the economy is stated as J (W t, z t ) max {C t+j } j=0,{ω i,t+j} j=0 E t δ j U (C t+j ) j=0 W t+1 = R p,t+1 (W t C t ) s.t. R p,t+1 = g (z t ) + ε t+1 R p,t+1 = N i=1 ω i,tr i,t+1 and can be represented in a dynamic programming framework, as follows:, J (W t, z t ) max {U (C t ) + δ E t [J (W t+1, z t+1 )]} C t,ω i,t W t+1 = R p,t+1 (W t C t ) s.t. R p,t+1 = g (z t ) + ε t+1, (A.1) R p,t+1 = N i=1 ω i,tr i,t+1 where J (W t, z t ) denotes the time t value function; U (C t ) denotes the utility over consumption; R p,t+1 is the gross return on the aggregate portfolio; z t is the state variable that forecasts R p,t+1 ; ω i,t is the weight for asset i in the representative investor s portfolio; and δ is a timesubjective discount factor. 25 ε t+1 represents a forecasting error, and g (z t ) denotes a function of the state variable that represents the component of the market return that is predictable by the state variable. The firstorder condition (f.o.c.) with respect to C t is equal to U C (C t ) = δ E t [J W (W t+1, z t+1 ) R p,t+1 ], (A.2) where U C ( ) and J W ( ) denote the firstorder partial derivatives of U( ) relative to C t and J ( ) with respect to W t+1, respectively. The return on aggregate wealth can be rewritten as R p,t+1 = N 1 i=1 ω i,t (R i,t+1 R f,t+1 ) + R f,t+1, (A.3) 25 For notational convenience we assume there is only one state variable, i.e., z t is a scalar. 40
43 where R f,t+1 denotes a benchmark return (for example, the riskfree rate), and where we impose the constraint that the portfolio weights must sum up to 1, N i=1 ω i,t = Then, the f.o.c. with respect to ω i,t is given by E t [J W (W t+1, z t+1 ) (W t C t ) (R i,t+1 R f,t+1 )] = 0. (A.4) By applying the envelope theorem to (A.1), J W ( ) can be represented as J W (W t, z t ) = C t W t {U C (C t ) δ E t [J W (W t+1, z t+1 ) R p,t+1 ]} + δ E t [J W (W t+1, z t+1 ) R p,t+1 ] + ω i,t W t E t [J W (W t+1, z t+1 ) (W t C t ) (R i,t+1 R f,t+1 )]. (A.5) By using Equations (A.2) and (A.4), Equation (A.5) simplifies to J W (W t, z t ) = δ E t [J W (W t+1, z t+1 ) R p,t+1 ], (A.6) and by combining with Equation (A.2), this leads to the usual envelope condition: J W (W t, z t ) = U C (C t ). (A.7) By updating (A.7), substituting the result in (A.2), and rearranging, we obtain the Euler equation: [ 1 = E t δ U ] [ C(C t+1 ) U C (C t ) R p,t+1 = E t δ J ] W (W t+1, z t+1 ) R p,t+1. (A.8) J W (W t, z t ) Given (A.8), we can substitute consumption out of the model, and the resulting stochastic discount factor (SDF) is equal to M t+1 = δ J W (W t+1, z t+1 ). (A.9) J W (W t, z t ) To derive the Euler equation for an arbitrary individual risky return, R i,t+1, by using the law of iterated expectations, the f.o.c. with respect to ω i,t can be rewritten as E t (M t+1 R i,t+1 ) = E t (M t+1 R f,t+1 ). (A.10) 26 The normalization that the benchmark return is the Nth asset does not play any role in the derivation. 41
44 By substituting (A.3) in (A.8), and rearranging, we obtain, 1 = N 1 i=1 ω i,t E t [M t+1 (R i,t+1 R f,t+1 )] + E t (M t+1 R f,t+1 ). (A.11) By using (A.10), we derive the pricing equation for asset i: 1 = E t (M t+1 R f,t+1 ) = E t (M t+1 R i,t+1 ). (A.12) To linearize the model, we use the general expected returncovariance representation: E t (R i,t+1 ) R f,t+1 = Cov t(r i,t+1 R f,t+1, M t+1 ). (A.13) E t (M t+1 ) By using Stein s lemma, we can rewrite the covariance term Cov t (R i,t+1 R f,t+1, M t+1 ) as: 27 [ Cov t (R i,t+1 R f,t+1, M t+1 ) = Cov t R i,t+1 R f,t+1, δ J ] W (W t+1, z t+1 ) J W (W t, z t ) δ = J W (W t, z t ) {E t[j W W (W t+1, z t+1 )] Cov t (R i,t+1 R f,t+1, W t+1 ) = + E t [J W z (W t+1, z t+1 )] Cov t (R i,t+1 R f,t+1, z t+1 )} ( δ {W t E t [J W W (W t+1, z t+1 )] Cov t R i,t+1 R f,t+1, W ) t+1 J W (W t, z t ) W t + E t [J W z (W t+1, z t+1 )] Cov t (R i,t+1 R f,t+1, z t+1 )}. (A.14) The conditional mean SDF is given by E t (M t+1 ) = δ J W (W t, z t ) E t[j W (W t+1, z t+1 )]. (A.15) By substituting Equations (A.14) and (A.15) into (A.13), we obtain: E t (R i,t+1 ) R f,t+1 = W t E t [J W W (W t+1, z t+1 )] E t [J W (W t+1, z t+1 )] ( Cov t R i,t+1 R f,t+1, W ) t+1 W t E t[j W z (W t+1, z t+1 )] E t [J W (W t+1, z t+1 )] Cov t(r i,t+1 R f,t+1, z t+1 ). (A.16) 27 For applications of the Stein (1981) lemma to asset pricing, see, for example, Brandt and Wang (2003), Cochrane (2005), and Balvers and Huang (2009). 42
45 Finally, by assuming the approximations, E t [J W (W t+1, z t+1 )] = J W (W t, z t ), E t [J W W (W t+1, z t+1 )] = J W W (W t, z t ), E t [J W z (W t+1, z t+1 )] = J W z (W t, z t ), we obtain the ICAPM pricing equation: E t (R i,t+1 ) R f,t+1 = γ Cov t (R i,t+1 R f,t+1, R m,t+1 ) J W z(w t, z t ) J W (W t, z t ) Cov t(r i,t+1 R f,t+1, z t+1 ), where γ WtJ W W (W t,z t) J W (W t,z t) (A.17) denotes the parameter of relative risk aversion (assumed to be constant), and we use the result from the intertemporal budget constraint that the return on aggregate wealth is approximately equal to the change in wealth, W t+1 W t R m,t Since Cov t (R i,t+1 R f,t+1, z t ) = 0, we use the innovation in the state variable, which is measured by the first difference in z t+1 : 29 z t+1 = z t+1 z t. (A.18) The resulting pricing equation is given by E t (R i,t+1 ) R f,t+1 = γ Cov t (R i,t+1 R f,t+1, R m,t+1 ) J W z(w t, z t ) J W (W t, z t ) Cov t(r i,t+1 R f,t+1, z t+1 ). (A.19) This specification is also consistent with the original ICAPM in continuous time, which is based on the innovations in the state variables. 28 This is true if consumption is low relative to wealth, C t W t. 29 The simple change corresponds to the innovation if the state variable follows a randomwalk process. 43
46 B Crosssectional regressions with GMM robust standard errors The GMM system equivalent to the time series/crosssectional regressions approach has a set of moment conditions given by g T (Θ) = 1 T T t=1 (r t R f,t 1 N δ βf t ) T t=1 (r t R f,t 1 N δ βf t ) f t = T t=1 (r t R f,t 1 N βλ) 0 (N 1) 0 (NK 1) 0 (N 1), (B.20) where r t (N 1) is a vector of simple returns; 1 N (N 1) is a vector of ones; δ(n 1) is a vector of constants for the time series regressions; β(n K) is a matrix of K factor loadings for the N test assets; f t (K 1) is a vector of common factors used to price assets; λ(k 1) is a vector of beta risk prices; denotes the Kronecker product; and 0 denotes conformable vectors of zeros. The first two sets of moment conditions identify the factor loadings (including the constants or Jensen alphas), and thus are equivalent to the timeseries regressions. These moment conditions are exactly identified with N + NK orthogonality conditions and N + NK parameters to estimate. The third set of moments corresponds to the crosssectional regression, and identifies the beta risk prices, λ. Hence, the third set of moments has N moment conditions and K parameters to estimate, leading to N K overidentifying restrictions, which also corresponds to the number of overidentifying conditions in the entire system. System (B.20) represents a straightforward generalization of the system presented in Cochrane (2005) (Chapter 12), for the case of K > 1 risk factors affecting the crosssection of returns. The vector of parameters to estimate in this GMM system is given by Θ = [δ β λ ], (B.21) where β vec(β ), and vec is the operator that enables us to stack the factor loadings for the N assets into a column vector. The matrix that chooses which moment conditions are set to zero in the GMM firstorder 44
47 condition, ag T ( ˆΘ) = 0, is given by 0 (N(K+1) N) a = I N I K+1 0 (K N(K+1)) β, (B.22) where I m denotes an identity matrix of order m. The matrix of sensitivities of the moment conditions to the parameters is given by d g T (Θ) Θ ( I N I N 1 ) T T t=1 f t ( = I N 1 ) ( T T t=1 f t I N 1 ) T T t=1 f tf t 0 (N N) I N λ β 0 (N K) 0 (NK K). (B.23) The variancecovariance matrix of the moments, S, has the form: r t R f,t 1 N δ βf t r t j R f,t j 1 N δ βf t j S = E (r t R f,t 1 N δ βf t ) f t (r t j R f,t j 1 N δ βf t j ) f t j j= r t R f,t 1 N βλ r t j R f,t j 1 N βλ = j= E ɛ t ɛ t f t β(f t E(f t )) + ɛ t ɛ t j ɛ t j f t j β(f t j E(f t )) + ɛ t j, (B.24) where ɛ t r t R f,t 1 N δ βf t, represents the vector of timeseries residuals. In the last equality, we impose the null that the asset pricing model relation is true, E (r t R f,t 1 N ) = βλ: r t R f,t 1 N βλ = r t R f,t 1 N E (r t R f,t 1 N ) = r t R f,t 1 N δ β E(f t ) = β(f t E(f t )) + ɛ t. (B.25) By using the general GMM formula for the variancecovariance matrix of the parameter estimates, Var( ˆΘ) = 1 T (ad) 1 aŝa (ad) 1, (B.26) the last K elements of the main diagonal give the variances of the estimated factor risk prices, used to calculate the tstatistics. In addition, if we use the formula for the variancecovariance matrix of the GMM moment 45
48 conditions (errors), Var(g T ( ˆΘ)) = 1 T ( ( I N(K+2) d(ad) a) 1 Ŝ I N(K+2) d(ad) a) 1, (B.27) we obtain the covariance matrix of the crosssectional pricing errors ( ˆα) from the bottomright (N N) block of Var(g T ( ˆΘ)), which is used to conduct the test that the pricing errors are jointly equal to zero: ˆα Var( ˆα) 1 ˆα χ 2 (N K). (B.28) The Shanken (1992) standard errors can be derived as a special case of the GMM robust standard errors derived above, as noted by Cochrane (2005) (Chapter 12). If we assume that ɛ t is jointly i.i.d.; ɛ t and f t are independent; and finally f t has no serial correlation, then the spectral density matrix S in (B.24) specializes to ɛ t ɛ t S = E ɛ t f t ɛ t f t β(f t E(f t )) + ɛ t β(f t E(f t )) + ɛ t Σ Σ E(f t) Σ = Σ E(f t ) Σ E(f t f t) Σ E(f t ), Σ Σ E(f t) βσ f β + Σ (B.29) where Σ f E [(f t E(f t ))(f t E(f t )) ] represents the variancecovariance matrix associated with the factors, and Σ E(ɛ t ɛ t) denotes the variancecovariance matrix associated with the residuals from the timeseries regressions. By replacing (B.29) in (B.26) we obtain the Shanken variances for the estimated factor risk premia: Var(ˆλ) = 1 T [ (β β ) 1 β Σβ ( β β ) 1 ( 1 + λ Σ 1 f λ ) + Σ f ]. (B.30) Similarly, the Shanken variances for pricing errors are given by Var(ˆα) = 1 T ( I N β ( β β ) ) ( 1 β Σ I N β ( β β ) ) ( ) 1 β 1 + λ Σ 1 f λ. (B.31) 46
49 C GMM formulas Following Cochrane (2005), the weighting matrix associated with the GMM system (16) is given by W = W 0, 0 I K (C.32) where W = I N is an Ndimensional identity matrix; 0 denotes a conformable matrix of zeros; and I K denotes a Kdimensional identity matrix, with K representing the number of factors in the model. In this specification, W is the weighting matrix for the first N moment conditions (corresponding to the N pricing errors), while I K is the weighting matrix associated with the last K orthogonality conditions that identify the factor means. The risk price estimates ˆb have variance formulas given by Var(ˆb) = 1 T (d Wd) 1 d WŜWd(d Wd) 1, (C.33) where d g T (b) b represents the matrix of moments sensitivities to the parameters; and Ŝ is an estimator for the spectral density matrix S derived under the heteroskedasticityrobust or White (1980) standard errors (that is, no lags of the moment functions are considered in the computation of Ŝ). The variance covariance matrix for the moments from firststage GMM is given by ( ) Var g T (ˆb) = 1 T ( ( I N+K d(d Wd) 1 d W) Ŝ I N+K Wd(d Wd) 1 d ), (C.34) where the first (N, N) block of (C.34) represents the covariance matrix of the N pricing errors. 47
50 Table 1: Descriptive statistics for (C)ICAPM factors This table reports descriptive statistics for the risk factors from the (C)ICAPM and alternative factor models. RM t+1, RM t+1 z t and z t+1 denote the market, scaled and intertemporal risk factors from the (C)ICAPM. The conditioning variables (z) are the Fed funds rate (F F R) and the relative Tbill rate (RREL). SMB t+1, HML t+1, and UMD t+1 denote the size, value, and momentum factors, respectively. The sample is 1963: :12. φ designates the first order autocorrelation coefficient. The correlations between the state variables are presented in Panel B. Panel A Mean (%) Stdev. (%) Min. (%) Max. (%) φ RM t SMB t HML t UMD t RM t+1 F F R t F F R t RM t+1 RREL t RREL t Panel B SMB t+1 HML t+1 UMD t+1 RM t+1 F F R t F F R t+1 RM t+1 RREL t RREL t+1 RM t SMB t HML t UMD t RM t+1 F F R t F F R t RM t+1 RREL t RREL t
51 Table 2: Factor risk premia for (C)ICAPM This table reports the estimation and evaluation results for the threefactor Conditional Intertemporal CAPM ((C)ICAPM). The estimation procedure is the timeseries/crosssectional regressions approach. The test portfolios are the 25 size/booktomarket portfolios (SBM25, Panel A) and 25 size/momentum portfolios (SM25, Panel B). λ M denotes the beta risk price estimate for the market factor; λ M,z denotes the risk price associated with the scaled factor; and λ z represents the risk price associated with the intertemporal risk factor. The conditioning variables (z) are the Fed funds rate (F F R) and the relative Tbill rate (RREL). Below the risk price estimates (in %) are displayed tstatistics based on Shanken standard errors (in parenthesis). The column labeled MAE(%) presents the mean absolute pricing error (in %), and the column labeled R 2 OLS denotes the crosssectional OLS R2. The column labeled χ 2 presents the χ 2 statistic (first line), and associated asymptotic pvalues (in parenthesis) for the test on the joint significance of the pricing errors. The sample is 1963: :12. Italic, underlined and bold numbers denote statistical significance at the 10%, 5% and 1% levels, respectively. λ M λ M,z λ z χ 2 MAE(%) Panel A (SBM25) ROLS 2 F F R (2.22) ( 0.20) ( 2.83) (0.06) RREL (2.57) (1.05) ( 2.90) (0.09) Panel B (SM25) F F R (3.22) (4.50) ( 1.54) (0.00) RREL (2.32) (3.23) ( 2.08) (0.04) 49
52 Table 3: Factor risk premia for alternative factor models This table reports the estimation and evaluation results for alternative models the CAPM (Row 1), the FamaFrench threefactor model (Row 2) and the Carhart fourfactor model (Row 3). The estimation procedure is the timeseries/crosssectional regressions approach. The test portfolios are the 25 size/booktomarket portfolios (SBM25, Panel A) and 25 size/momentum portfolios (SM25, Panel B). λ M, λ SMB, λ HML, λ UMD denote the beta risk price estimates for the market, size, value and momentum factors, respectively. Below the risk price estimates (in %) are displayed tstatistics based on Shanken standard errors (in parenthesis). The column labeled MAE(%) presents the mean absolute pricing error (in %), and the column labeled ROLS 2 denotes the crosssectional OLS R2. The column labeled χ 2 presents the χ 2 statistic (first line), and associated asymptotic pvalues (in parenthesis) for the test on the joint significance of the pricing errors. The sample is 1963: :12. Italic, underlined and bold numbers denote statistical significance at the 10%, 5% and 1% levels, respectively. Row λ M λ SMB λ HML λ UMD χ 2 MAE(%) Panel A (SBM25) ROLS (2.94) (0.00) (2.07) (1.62) (3.81) (0.00) (2.43) (1.59) (3.67) (3.72) (0.05) Panel B (SM25) (2.56) (0.00) (2.53) (3.05) ( 2.82) (0.00) (2.64) (1.40) (2.13) (4.37) (0.00) 50
53 Table 4: Average risk premia across booktomarket and momentum quintiles This table reports the average risk premium (average beta times (beta) risk price) for each factor, across quintiles for booktomarket (BM) and prior shortterm returns (momentum, M). The model is the threefactor (C)ICAPM when the state variables (z) are the Fed funds rate (F F R) and the relative Tbill rate (RREL). E(R) denotes the average excess return for each BM and M quintile, and α represents the average pricing error per quintile. RM t+1, RM t+1 z t and z t+1 denote the market, scaled and intertemporal risk factors from the (C)ICAPM. All the values are presented in percentage points. BM1 and M1 denote the lowest BM and M quintile, respectively, and Dif. denotes the difference across extreme quintiles. The sample is 1963: :12. E(R) RM t+1 RM t+1 z t z t+1 ᾱ Panel A (F F R) BM BM Dif M M Dif Panel B (RREL) BM BM Dif M M Dif Table 5: Average risk premia across BM and momentum quintiles: Alternative models This table reports the average risk premium (average beta times (beta) risk price) for each factor, across quintiles for booktomarket (BM) and prior shortterm returns (momentum, M). The models are the FamaFrench model (FF3, Panel A), and the Carhart model (C4, Panel B). E(R) denotes the average excess return for each BM and M quintile, and α represents the average pricing error per quintile. RM, SMB, HML, and UMD denote the market, size, value, and momentum factors, respectively. All the values are presented in percentage points. BM 1 and M 1 denote the lowest BM and M quintile, respectively, and Dif. denotes the difference across extreme quintiles. The sample is 1963: :12. Panel A: FF3 E(R) RM t+1 SMB t+1 HML t+1 ᾱ BM BM Dif M M Dif Panel B: C4 E(R) RM t+1 SMB t+1 HML t+1 UMD t+1 ᾱ BM BM Dif M M Dif
54 Table 6: Timeseries regressions for HML and UMD This table reports the results from timeseries regressions of HML (Panel A) and UMD (Panel B) on the (C)ICAPM factors, RM t+1, RM t+1 z t and z t+1. The conditioning variables (z) are the Fed funds rate (F F R) and the relative Tbill rate (RREL). Below the coefficient estimates are displayed heteroskedasticityrobust tstatistics (in parenthesis). The column labeled R 2 denotes the adjusted coefficient of determination. The sample is 1963: :12. Italic, underlined and bold numbers denote statistical significance at the 10%, 5% and 1% levels, respectively. const. RM t+1 RM t+1 z t z t+1 R 2 Panel A (HML t+1 ) F F R (3.88) ( 5.54) ( 4.56) ( 3.80) RREL (4.07) ( 6.06) ( 1.48) ( 3.77) Panel B (UMD t+1 ) F F R (5.04) ( 2.48) (3.45) (1.47) RREL (5.09) ( 1.81 ) (4.01) (2.15) Table 7: Factor risk premia for (C)ICAPM: Bond returns This table reports the estimation and evaluation results for the threefactor Conditional Intertemporal CAPM ((C)ICAPM). The estimation procedure is the timeseries/crosssectional regressions approach. The test portfolios are 7 Treasury bond returns plus 25 equity portfolios. The equity portfolios are the 25 size/booktomarket portfolios (SBM25, Panel A) and 25 size/momentum portfolios (SM25, Panel B). λ M denotes the beta risk price estimate for the market factor; λ M,z denotes the risk price associated with the scaled factor; and λ z represents the risk price associated with the intertemporal risk factor. The conditioning variables (z) are the Fed funds rate (F F R) and the relative Tbill rate (RREL). Below the risk price estimates (in %) are displayed tstatistics based on Shanken standard errors (in parenthesis). The column labeled MAE(%) presents the mean absolute pricing error (in %), and the column labeled R 2 OLS denotes the crosssectional OLS R2. The column labeled χ 2 presents the χ 2 statistic (first line), and associated asymptotic pvalues (in parenthesis) for the test on the joint significance of the pricing errors. The sample is 1963: :12. Italic, underlined and bold numbers denote statistical significance at the 10%, 5% and 1% levels, respectively. λ M λ M,z λ z χ 2 MAE(%) Panel A (SBM25) ROLS 2 F F R (2.29) ( 1.34) ( 3.19) (0.01) RREL (3.05) ( 1.15) ( 2.66) (0.00) Panel B (SM25) F F R (3.33) (4.31) (0.51) (0.00) RREL (2.57) (3.86) ( 1.33) (0.00) 52
55 Table 8: Factor risk premia for (C)ICAPM (30 portfolios) This table reports the estimation and evaluation results for the threefactor Conditional Intertemporal CAPM ((C)ICAPM). The estimation procedure is the timeseries/crosssectional regressions approach. The test portfolios are 10 portfolios sorted on size, 10 portfolios sorted on booktomarket and 10 momentum portfolios. λ M denotes the beta risk price estimate for the market factor; λ M,z denotes the risk price associated with the scaled factor; and λ z represents the risk price associated with the intertemporal risk factor. The conditioning variables (z) are the Fed funds rate (F F R) and the relative Tbill rate (RREL). Below the risk price estimates (in %) are displayed tstatistics based on Shanken standard errors (in parenthesis). The column labeled MAE(%) presents the mean absolute pricing error (in %), and the column labeled R 2 OLS denotes the crosssectional OLS R2. The column labeled χ 2 presents the χ 2 statistic (first line), and associated asymptotic pvalues (in parenthesis) for the test on the joint significance of the pricing errors. The sample is 1963: :12. Italic, underlined and bold numbers denote statistical significance at the 10%, 5% and 1% levels, respectively. λ M λ M,z λ z χ 2 MAE(%) ROLS 2 F F R (2.59) (3.14) ( 2.90) (0.13) RREL (2.45) (3.46) ( 2.31) (0.19) Table 9: Factor risk premia for (C)ICAPM: Estimation by GMM This table reports the estimation and evaluation results for the threefactor Conditional Intertemporal CAPM ((C)ICAPM). The estimation procedure is firststage GMM with equally weighted errors. The test portfolios are the 25 size/booktomarket portfolios (SBM25, Panel A) and 25 size/momentum portfolios (SM25, Panel B). γ M denotes the (covariance) risk price estimate for the market factor; γ M,z denotes the risk price associated with the scaled factor; and γ z represents the risk price associated with the intertemporal risk factor. The conditioning variables (z) are the Fed funds rate (F F R) and the relative Tbill rate (RREL). The first line associated with each row presents the covariance risk price estimates, and the second line reports the asymptotic GMM robust tstatistics (in parenthesis). The column labeled MAE(%) presents the mean absolute pricing error (in %), and the column labeled ROLS 2 denotes the crosssectional OLS R 2. The column labeled χ 2 presents the χ 2 statistic (first line), and associated asymptotic pvalues (in parenthesis) for the test on the joint significance of the pricing errors. The sample is 1963: :12. Italic, underlined and bold numbers denote statistical significance at the 10%, 5% and 1% levels, respectively. γ M γ M,z γ z χ 2 MAE(%) Panel A (SBM25) ROLS 2 F F R ( 0.12) ( 1.23) ( 2.17) (0.01) RREL (0.03) (1.13) ( 3.19) (0.06) Panel B (SM25) F F R ( 0.13) (2.87) ( 0.92) (0.00) RREL (0.57) (2.51) ( 1.85 ) (0.02) 53
56 Table 10: Factor risk premia for nested models This table reports the estimation and evaluation results for nested models of the (C)ICAPM. The two nested models are the twofactor conditional CAPM (Row 1) and the twofactor ICAPM (Row 2). The estimation procedure is the timeseries/crosssectional regressions approach. The test portfolios are the 25 size/booktomarket portfolios (SBM25, Panels A and C) and 25 size/momentum portfolios (SM25, Panels B and D). λ M denotes the beta risk price estimate for the market factor; λ M,z denotes the risk price associated with the scaled factor; and λ z represents the risk price associated with the intertemporal risk factor. The conditioning variables (z) are the Fed funds rate (F F R) and the relative Tbill rate (RREL). Below the risk price estimates (in %) are displayed tstatistics based on Shanken standard errors (in parenthesis). The column labeled MAE(%) presents the mean absolute pricing error (in %), and the column labeled R 2 OLS denotes the crosssectional OLS R2. The column labeled χ 2 presents the χ 2 statistic (first line), and associated asymptotic pvalues (in parenthesis) for the test on the joint significance of the pricing errors. The sample is 1963: :12. Italic, underlined and bold numbers denote statistical significance at the 10%, 5% and 1% levels, respectively. Row λ M λ M,z λ z χ 2 MAE(%) Panel A (SBM25, F F R) ROLS (2.48) ( 3.43) (0.00) (2.17) ( 2.96) (0.25) Panel B (SM25, F F R) (3.31) (4.20) (0.00) (2.66) (0.54) (0.00) Panel C (SBM25, RREL) (3.21) ( 1.68 ) (0.00) (2.68) ( 2.81) (0.00) Panel D (SM25, RREL) (2.60) (3.79) (0.00) (2.54) (2.19) (0.00) 54
57 Table 11: Factor risk premia for (C)ICAPM (size/longterm reversal portfolios) This table reports the estimation and evaluation results for the threefactor Conditional Intertemporal CAPM ((C)ICAPM). The estimation procedure is the timeseries/crosssectional regressions approach. The test portfolios are the 25 size/longterm reversal portfolios (SLTR25). λ M denotes the beta risk price estimate for the market factor; λ M,z denotes the risk price associated with the scaled factor; and λ z represents the risk price associated with the intertemporal risk factor. The conditioning variables (z) are the Fed funds rate (F F R) and the relative Tbill rate (RREL). Below the risk price estimates (in %) are displayed tstatistics based on Shanken standard errors (in parenthesis). The column labeled MAE(%) presents the mean absolute pricing error (in %), and the column labeled ROLS 2 denotes the crosssectional OLS R 2. The column labeled χ 2 presents the χ 2 statistic (first line), and associated asymptotic pvalues (in parenthesis) for the test on the joint significance of the pricing errors. The sample is 1963: :12. Italic, underlined and bold numbers denote statistical significance at the 10%, 5% and 1% levels, respectively. λ M λ M,z λ z χ 2 MAE(%) ROLS 2 F F R (2.54) ( 0.61) ( 2.81) (0.02) RREL (2.47) (1.99) ( 2.63) (0.63) 55
58 Panel A: F F R Panel B: RREL Figure 1: Shortterm interest rates This figure plots the timeseries for the monthly changes in the Fed funds rate ( F F R) and the relative Tbill rate ( RREL). The sample is 1963: :12. The vertical lines indicate the NBER recession periods. 56
59 Panel A: pricing errors Panel B: tstatistics Figure 2: Individual pricing errors (F F R): SBM25 This figure plots the pricing errors (in %, Panel A), and respective tstatistics (Panel B) of the 25 size/booktomarket portfolios (SBM25) from the (C)ICAPM (version based on F F R); FamaFrench model (FF3); and the Carhart model (C4). The pricing errors are obtained from an OLS crosssectional regression of average excess returns on factor betas. ij designates a portfolio associated with the ith size quintile and jth booktomarket quintile. 57
60 Panel A: pricing errors Panel B: tstatistics Figure 3: Individual pricing errors (F F R): SM25 This figure plots the pricing errors (in %, Panel A), and respective tstatistics (Panel B) of the 25 size/momentum portfolios (SM25) from the (C)ICAPM (version based on F F R); FamaFrench model (FF3); and the Carhart model (C4). The pricing errors are obtained from an OLS crosssectional regression of average excess returns on factor betas. ij designates a portfolio associated with the ith size quintile and jth prior return quintile. 58
61 Panel A: RM t+1 F F R t Panel B: RM t+1 RREL t Panel C: F F R t+1 Panel D: RREL t+1 Figure 4: Regression betas for SBM25 This figure plots the multiple regression beta estimates associated with the SBM25 portfolios from (C)ICAPM. The factors are the scaled factor (RM t+1 F F R t, RM t+1 RREL t ) and the innovations in the state variable ( F F R t+1, RREL t+1 ). ij designates a portfolio associated with the ith size quintile and jth booktomarket quintile. The sample is 1963: :12. 59
62 Panel A: RM t+1 F F R t Panel B: RM t+1 RREL t Panel C: F F R t+1 Panel D: RREL t+1 Figure 5: Regression betas for SM25 This figure plots the multiple regression beta estimates associated with the SM25 portfolios from (C)ICAPM. The factors are the scaled factor (RM t+1 F F R t, RM t+1 RREL t ) and the innovations in the state variable ( F F R t+1, RREL t+1 ). ij designates a portfolio associated with the ith size quintile and jth prior return quintile. The sample is 1963: :12. 60
63 Panel A: All periods, F F R Panel B: All periods, RREL Panel C: Periods with high F F R Panel D: Periods with high RREL Panel E: Periods with low F F R Panel F: Periods with low RREL Figure 6: Average conditional market betas for SM25 This figure plots the average conditional market beta estimates associated with the SM25 portfolios from (C)ICAPM, β i,m + β i,m,z E(z t ). In Panels A and B all the periods are used, whereas in Panels C,D (E,F) only the61 periods in which F F R, RREL are 1.5 standard deviations above (below) the mean are used. ij designates a portfolio associated with the ith size quintile and jth prior return quintile. The sample is 1963: :12.
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