Paulo Maio 1. First draft: October This version: January

Transcription

1 Do stock return factors outperform other risk factors? Evidence from a large cross-section of anomalies Paulo Maio 1 First draft: October 2014 This version: January Hanken School of Economics, Department of Finance and Statistics. paulofmaio@gmail.com 2 I thank Frederico Belo for helpful discussions. I am grateful to Kenneth French, Amit Goyal, Robert Shiller, Robert Stambaugh, and Lu Zhang for providing stock market data. A previous version was titled Do equity factors explain (several) stock market anomalies?. Any remaining errors are mine.

2 Abstract I compare the explanatory power of equity multifactor models (in which the factors are excess returns) with ICAPM models (containing factors that are not returns). I use a large cross-section of equity portfolio associated with 22 market anomalies and introduce a new cross-sectional R 2, which imposes the restriction that risk prices should be equal to the factor sample means. Several multifactor models do not outperform significantly the CAPM and ICAPM in explaining several dimensions of cross-sectional returns. This is especially true when the anomalies are not trivially related with the equity factors. Overall, the models from Carhart (1997) and Hou, Xue, and Zhang (2014a) show the best performance, while the other equity models clearly underperform. This suggests that the performance and usefulness of several prominent factor models is overstated. Keywords: asset pricing models; equity risk factors; stock market anomalies; cross-section of stock returns; linear multifactor models; factor risk premia; ICAPM; macro factors JEL classification: G10; G12

3 1 Introduction For some time the empirical asset pricing literature has documented patterns in the crosssection of average excess stock returns that are significant both in statistical and economic terms. These patterns are denominated by market or CAPM anomalies in the sense that the CAPM from Sharpe (1964) and Lintner (1965) is not able to explain these spreads in average returns. Historically, the most prominent anomalies have been the value premium (e.g., Rosenberg, Reid, and Lanstein (1985) and Fama and French (1992)) and momentum (Jegadeesh and Titman (1993)). The value premium refers to the evidence showing that value stocks (stocks with high book-to-market ratios, (BM)) outperform growth stocks (low BM). Momentum designates the pattern in which stocks with high prior short-term returns outperform stocks with low prior returns. These patterns in risk premia still remain robust nowadays (see, for example, Fama and French (2014) and Hou, Xue, and Zhang (2014a)). Another anomaly is the long-term reversal in returns (De Bondt and Thaler (1985)), which refers to stocks with low returns in the long past having higher subsequent returns, while past winners have lower returns. On the other hand, the investment anomaly refers to a pattern in which stocks of firms that invest more in physical capital tend to have lower average returns than the stocks of firms that invest less (Titman, Wei, and Xie (2004), Cooper, Gulen, and Schill (2008), Fama and French (2008)). Furthermore, there is evidence that stocks of firms that experimented positive surprises in their earnings tend to have higher average returns than stocks associated with firms with low or negative earnings surprises, which represents the earnings momentum anomaly (Chan, Jegadeesh, and Lakonishok (1996)). In response to the failure of the CAPM in explaining these (and other patterns) in crosssectional average stock returns, the last decades observed the emergence of a series of new multifactor asset pricing models that seek to outperform the CAPM in pricing the crosssection of equity risk premia. Among these, the most successful models have been empirical factor model in which the factors represent excess stock returns. For a long time, the most prominent factor models in the empirical asset pricing literature have been the three-factor 1

4 model from Fama and French (1993) (FF3) and the four-factor model from Carhart (1997) (C4). Yet, these models have been shown more successful in pricing portfolios that are mechanically related with their factors. Specifically, the FF3 model is able to price portfolios sorted on size and BM, which are trivially related with the SMB and HML factors. For example, the model does not outperform the CAPM in explaining the momentum anomaly (e.g., Fama and French (1996)). On the other hand, the C4 model is successful in explaining momentum portfolios only due to the presence of the momentum factor (UMD), which is mechanically related with those portfolios. In response to these findings, some recent models aim at explaining other market anomalies beyond the traditional value and momentum spreads in returns (e.g., Fama and French (2014) (FF5) and Hou, Xue, and Zhang (2014a) (HXZ4)). Essentially, these new models add investment (IA and CM A) and profitability (ROE and RMW ) factors to the traditional size and value factors. This paper revaluates the explanatory power of the empirical multifactor models presented in the literature for a large number of market anomalies. The paper departures from the existent literature in three important ways. First, I assess the explanatory power of the models for several anomalies simultaneously, rather than just explaining each anomaly separately. Given the abundance of anomalies in the literature (e.g., Hou, Xue, and Zhang (2014a)) it is natural to assess the joint explanatory power. Moreover, it is expected that the factor risk price estimates will differ significantly among cross-sectional tests with different portfolios, thus, one should investigate whether a common set of estimated risk prices is able to explain several portfolio sorts simultaneously. Second, the paper introduces a new goodness-of-fit measure to evaluate multifactor models in which the factors are excess stock returns. The new measure is designated by constrained cross-sectional R 2 (RC 2 ). This measure uses pricing errors that are obtained from a constrained cross-sectional regression in which the risk price estimates are equal to the corresponding factor means, rather than being freely estimated from an OLS crosssectional regression. Hence, this metric imposes the restriction that the risk prices are not 2

5 free parameters when the factors are excess stock returns, something that is neglected in the empirical asset pricing literature. Third, I compare the performance of the equity factor models against alternative multifactor models in which the factors are not excess stock returns (Intertemporal CAPM (ICAPM) models and the macro model from Chen, Roll, and Ross (1986)). I test five alternative multifactor models on a large cross-section of stock returns containing decile portfolios associated with 22 different anomalies. The results can be summarized as follows. First, one concludes that the R 2 from an OLS cross-sectional regression overstates the true explanatory power of the models since this measure relies on estimated risk prices that can be significantly different from the correct estimates (factor sample means). This is especially evident when the models are forced to price portfolios that are not trivially related with their factors. Second, only two factor models C4 and HXZ4 have a satisfactory performance in explaining the large cross-section of stock returns. The other multifactor models proposed in the literature either do not outperform significantly the baseline CAPM in explaining jointly the 22 market anomalies (as is the case with FF3) or have a rather weak explanatory power (as is the case with FF5). Third, in what concerns to explaining each of the 22 anomalies individually there are several anomalies in which all factor models cannot explain the cross-sectional risk premia, and thus improve the baseline CAPM. Again, in the single-anomaly tests, both C4 and HXZ4 outperform the FF5 model (as well as the alternative models) by a good margin. In particular, both models do a relatively good job in pricing several portfolio sorts that are not mechanically related with their factors. When one conducts a comparison with models in which the factors are not excess returns, the results show that some ICAPM models can outperform the equity factor models in explaining several individual market anomalies. Moreover, my results show that when the models are forced to price a large cross-section of anomalies the best performing factor models (C4 and HXZ4) outperform the macro models. Yet, the ICAPM models compare favorably with several equity factor models, including the FF3 and FF5 models, when it 3

6 comes to price a large cross-section of stock returns. These results are remarkable given that the alternative factors are by construction much less correlated with equity portfolio returns than the equity factors. Overall, my results suggest that the performance of several important empirical equity factor models (in which the factors are excess stock returns) in terms of explaining several CAPM anomalies is overstated. This is especially true when one forces the models to explain a large number of anomalies jointly, and when several of these anomalies (portfolios) are not trivially related with the factors in the models. Moreover, the traditional cross-sectional coefficient of determination, frequently used in the literature to evaluate asset pricing models, overstates the true explanatory power of these models for the cross-section of equity risk premia since it relies on implausible estimates of the factor risk prices. The paper proceeds as follows. Section 2 describes the new cross-sectional R 2 metric, while Section 3 shows the evaluation results for the equity multifactor models. Section 4 conducts a comparison with alternative models in which the factors are not excess returns. Finally, Section 5 concludes. 2 Methodology 2.1 Constrained regression approach The standard approach used in the asset pricing literature to test multifactor models is the two-step time-series/cross-sectional regression procedure (see Black, Jensen, and Scholes (1972), Jagannathan and Wang (1998), Cochrane (2005), Brennan, Wang, and Xia (2004), and Campbell and Vuolteenaho (2004), among many others). This method allows one to compare models containing factors that are excess stock returns (e.g., the baseline CAPM from Sharpe (1964) and Lintner (1965)) with models in which (some of) the factors are not excess returns (e.g., the ICAPM from Merton (1973) or macro models like the Consumption- CAPM from Breeden (1979)). In comparison, the time-series regressions approach is only 4

7 suitable to test multifactor models in which all the factors are excess returns (see Cochrane (2005), Chapter 12). To illustrate the two-pass regression approach, let s consider the case of the Fama and French (1993) three-factor model (FF3). In the first step, the factor betas are estimated from the time-series multivariate regressions for each test asset, R i,t+1 R f,t+1 = δ i + β i,m RM t+1 + β i,smb SMB t+1 + β i,hml HML t+1 + ε i,t+1, (1) where R i denotes the return on asset i; R f stands for the risk-free rate; RM, SMB, and HML denote the excess market return, size premium, and value premium, respectively, while β i,m, β i,smb, and β i,hml stand for the corresponding multivariate factor loadings. In the second step, the expected return-beta representation of the three-factor model, E(R i,t+1 R f,t+1 ) = λ M β i,m + λ SMB β i,smb + λ HML β i,hml, (2) is estimated by an OLS cross-sectional regression, R i R f = λ M β i,m + λ SMB β i,smb + λ HML β i,hml + α i,ols, (3) where R i R f represents the average time-series excess return for asset i. The regression above allows one to obtain estimates for factor risk prices ( λ) and pricing errors for the testing assets (ˆα i,ols ). 1 The t-statistics for the factor risk prices are usually based on Shanken (1992) standard errors, which incorporate a correction for the estimation error in the factor betas (from the time-series regressions), thus accounting for the error- 1 I do not include an intercept in the cross-sectional regression, which means that an asset that has zero betas against all factors should earn a zero risk premium (relative to the risk-free rate). Excluding the intercept also allows one to prevent the multicollinearity problem (between the intercept and some of the factor betas) arising from small cross-sectional variation in the betas. This often leads to economically implausible factor risk price estimates, like a negative market price of risk (see Jagannathan and Wang (2007) and Kan, Robotti, and Shanken (2013)). Moreover, my focus is on explaining the cross-section of equity risk premia rather than in fitting the risk-free rate. 5

8 in-variables bias into the cross-sectional regression (see Cochrane (2005), Chapter 12, for details). A simple and robust measure of the global fit of a given model for the cross-section of average excess returns, frequently used in the empirical asset pricing literature, is the cross-sectional OLS coefficient of determination, R 2 OLS = 1 Var N(ˆα i,ols ) Var N (R i R f ), (4) where Var N ( ) stands for the cross-sectional variance, with N denoting the number of test assets. R 2 OLS represents a proxy for the proportion of the cross-sectional variance of average excess returns on the test assets explained by the factor loadings associated with a given model. Since I do not include an intercept in the cross-sectional regression, this R 2 measure can assume negative values. A negative estimate means that the regression including the factor loadings (associated with a given model) as regressors does worse than a simple regression containing just a constant. In other words, the factor betas underperform the cross-sectional average risk premium in explaining cross-sectional variation in risk premia, that is, the model performs worse than a model that predicts constant risk premia in the cross-section of average returns. 2 Nevertheless, despite is prominence, the traditional cross-sectional regression approach outlined above has one important limitation when the factors are excess returns, as is the case with some of the models studied in this paper. Indeed, when the factors are excess returns the model should also price its factors, thus implying that the risk price estimates 2 Similar R 2 measures are used in Lettau and Ludvigson (2001), Campbell and Vuolteenaho (2004), Yogo (2006), Maio (2013a), Lioui and Maio (2014), among others. Kan, Robotti, and Shanken (2013) use an alternative measure in which the denominator is the sum of squared excess returns (rather than squared deviations to the cross-sectional mean) in order to bound the R 2 between 0 and 1 when the intercept is not included in the cross-sectional regression. However, this alternative measure has the disadvantage that one may obtain a large value even if the model does not explain any cross-sectional dispersion in risk-premia, just by fitting the cross-sectional average risk premium. For example, they report R 2 estimates associated with both the CAPM and CCAPM above 80% when estimated on the 25 size/book-to-market portfolios (see their Table 5), which stems from these models explaining the cross-sectional mean risk premium since it is well known that they cannot price these portfolios (e.g., Fama and French (1993), Lettau and Ludvigson (2001), Lioui and Maio (2014), among others). 6

9 correspond to the sample means of the factors rather than being freely estimated in a crosssectional regression. To see this, take the example of SMB. The pricing equation for SMB is as follows: E(SMB t+1 ) = λ M β SMB,M + λ SMB β SMB,SMB + λ HML β SMB,HML. (5) By definition, SMB has a multivariate beta of one against itself (β SMB,SMB = 1) and zero betas against the other factors (β SMB,M = β SMB,HML = 0). Substituting back in the pricing equation above, one obtains, E(SMB t+1 ) = λ SMB, (6) and thus the risk price estimate for SMB should be equal to the mean of the factor (for the sample in analysis), SMB. A similar reasoning implies that λ M = E(RM t+1 ) and λ HML = E(HML t+1 ). Therefore, one can define a constrained cross-sectional regression in which the risk price estimates are numerically equal to the factor means, R i R f = RMβ i,m + SMBβ i,smb + HMLβ i,hml + α i,c, (7) where RM, SMB, and HML denote the sample means of RM, SMB, and HML, respectively, and α i,c denote the pricing errors from this constrained regression. These are the correct pricing errors in the sense that they are based on the correct risk price estimates. To assess the fit of a specific model for cross-sectional risk premia, I define the constrained cross-sectional R 2, R 2 C = 1 Var N(ˆα i,c ) Var N (R i R f ), (8) which is similar to ROLS 2, but is based on the pricing errors from the constrained regression. If the risk prices estimated from the cross-sectional regression differ significantly from 7

10 the factor sample means, then almost sure the R 2 C estimate will be significantly smaller than the OLS counterpart. On the other hand, the two goodness-of-fit measures will be relatively similar if the risk price estimates are similar to the factor means, in other words, the constraint is not binding. It is important to note that, in contrast to R 2 OLS, R2 C does not necessarily increase by adding factors to the pricing equation (e.g., FF3 in comparison to the CAPM). The reason is that the risk price estimates are fixed by the factor means rather than allowed to minimize the sum of squared pricing errors (as in the cross-sectional regression). Therefore, adding a new factor can lead the model further away into explaining dispersion in cross-sectional risk premia (e.g., spreads in average returns among extreme portfolios like growth and value portfolios), if the dispersion in betas associated with the new factor (multiplied by the respective risk price) goes in the wrong direction. As noted in Cochrane (2005), one way to obtain the correct risk price estimates when the factors are (excess) returns is by conducting a GLS cross-sectional regression that includes the factors as test assets. The GLS cross-sectional regression can be represented in matrix form as Σ 1 2 r = ( Σ 1 2 β ) λ + α, (9) where r(n 1) is a vector of average excess returns; β(n K) is a matrix of K factor loadings for the N test assets; λ(k 1) is a vector of risk prices; and Σ E(ɛ t ɛ t) denotes the variance-covariance matrix associated with the residuals from the time-series regressions (see Cochrane (2005), Shanken and Zhou (2007), Lewellen, Nagel, and Shanken (2010), among others). In this approach, the test assets with a lower variance of the residuals (from the time-series regressions) will receive more weight in the cross-sectional regression. This method enables us to obtain standard errors for the correct risk price estimates. However, it is not valid to evaluate the explanatory power of a given model for a set of portfolios (e.g., book-to-market portfolios). The reason is that the GLS regression will attempt to price the factors in the model, which will have small pricing errors, thereby neglecting the original portfolios that one seeks to explain. 8

11 Specifically, the cross-sectional GLS coefficient of determination is given by R 2 GLS = 1 ˆα Σ 1 ˆα r Σ 1 r, (10) where r denotes the N 1 vector of (cross-sectionally) demeaned average excess returns. This metric measures the fraction of the cross-sectional variation in risk premia among the transformed portfolios explained by the factors associated with a given model. Since the factors are including in the test assets, the values for this metric will tend to be fairly large in most cases. Thus, a given factor model may have a very large value for RGLS 2, even if it fails completely in pricing the original test assets (portfolios) of economic interest (see Cochrane (2005) and Ludvigson (2012) for a related discussion). 2.2 Factor models In the empirical analysis conducted in the next section, I employ six factor models commonly used in the cross-sectional asset pricing literature. The first two models are the baseline CAPM, E(R i,t+1 R f,t+1 ) = λ M β i,m, (11) and the FF3 model presented above. The third model is the four-factor model from Carhart (1997) (C4, henceforth). This model adds a momentum factor (U M D, up-minus-down short-term past returns) to the FF3 model: 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. (12) The fourth model is the four-factor model from Pástor and Stambaugh (2003) (PS4), 9

12 which replaces U M D by a stock liquidity factor (LIQ, high-minus-low liquidity): 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 LIQ β i,liq. (13) The fifth model is the four-factor model from Hou, Xue, and Zhang (2014a) (HXZ4). This model includes an investment factor (IA, low-minus-high investment-to-assets ratio) and a profitability factor (ROE, high-minus-low return on equity) in addition to the market and size (ME) factors: E(R i,t+1 R f,t+1 ) = λ M β i,m + λ ME β i,me + λ IA β i,ia + λ ROE β i,roe. (14) Finally, I use the five-factor model from Fama and French (2014) (FF5), which adds an investment (CMA) and a profitability (RMW ) factor (both constructed in a different way than in Hou, Xue, and Zhang (2014a)) to the FF3 model: 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 CMA β i,cma + λ RMW β i,rmw. (15) 3 Factor models In this section, I test the multifactor models presented in the previous section by using several equity portfolios that are associated with different market anomalies. Since the objective is to access whether the models can explain jointly several anomalies, I force the models to price several portfolio classes simultaneously. 3.1 Data The data on the equity factors, RM, SMB, HML, and UMD are obtained from Kenneth French s data library, while LIQ is retrieved from Robert Stambaugh s webpage. The data 10

13 associated with M E, IA, ROE, CM A, and RM W are obtained from Lu Zhang. The sample is 1972:01 to 2013:12. The descriptive statistics for the equity factors are displayed in Table 1. The equity factor with the largest mean is UMD (0.71% per month), followed by ROE and RM, both with means above 0.50% per month. The factor with the lowest mean is SM B (0.24% per month), confirming evidence that the size premium has declined in the last decades. 3 The most volatile factors are the equity premium and the momentum factor, with standard deviations around or above 4.5% per month. The investment factors (IA and CM A) show the lowest volatility, with standard deviations slightly above 1.8% per month. All the factors exhibit very low serial correlation, as shown by the first-order autoregressive coefficients (below 20% in all cases), as usually is the case with stock returns. The Fama-French profitability factor (RM W ) shows the highest autocorrelation (0.18). The pairwise correlations presented in Panel B show that the two versions of the size (SMB and ME) and investment (IA and CMA) factors are strongly correlated as indicated by the correlation coefficients above 0.90 in both cases. The two profitability factors (ROE and RM W ) also show a positive, although not as strong, correlation (0.67). Both investment factors are weakly negatively correlated with RM (around -0.38) and largely positively correlated with HM L (around 0.70). Both profitability factors show weak negative correlations with both SMB and ME as shown by the correlation coefficients below On the other hand, ROE is positively correlated with UMD (0.50), but the same does not occur with RM W. Hence, the two profitability factors might have quite different explanatory power for several dimensions of the cross-section of stock returns (e.g., momentum portfolios). Moreover, the value factor shows a weak negative correlation with the market factor (-0.33). The portfolio return data used in the cross-sectional asset pricing tests are associated with some of the most relevant so-called CAPM or market anomalies. These represent patterns in the cross-section of average stock returns that are not explained by the standard CAPM. 3 The size factor used by Hou, Xue, and Zhang (2014a) has a slightly higher mean (0.31%). 11

14 I employ a total of 22 anomalies or portfolio sorts, which represent a subset of the anomalies considered in Hou, Xue, and Zhang (2014a) (see their Table 4). Table 2 contains the list and description of the anomalies included in my analysis. Following Hou, Xue, and Zhang (2014a), these anomalies can be broadly classified in strategies related with value (BM and DUR), momentum (MOM, SUE, ABR, and IM), Investment (IA, NSI, CEI, PIA, IG, IVC, IVG, NOA, OA, POA, and PTA), profitability (ROE, GPA, and NEI), and intangibles (OCA and OL). All the portfolios are value-weighted and all the sorts include decile portfolios, except IM and NEI with nine portfolios each. Compared to the portfolio groups employed in Hou, Xue, and Zhang (2014a), I do not use portfolios sorted on earnings-to-price and cashflow-to-price ratios since these deciles are strongly correlated with the book-to-market (BM) deciles. Similarly, I do not consider the return on assets deciles because they are strongly correlated with the return on equity deciles (ROE). Moreover, I use only one measure of price momentum (MOM), earnings surprise (SUE), and cumulative abnormal stock returns around earnings announcements (ABR). This is in contrast to Hou, Xue, and Zhang (2014a) who perform the empirical analysis for two different portfolio sorts (involving different holding periods or different horizons in prior returns) associated with each of these anomalies. I also exclude all portfolio sorts used in Table 4 of Hou, Xue, and Zhang (2014a) that start after 1972:01. 4 All the portfolio return data are obtained from Lu Zhang. The one-month Treasury bill rate used to construct portfolio excess returns is collected from French s webpage. Table 3 presents the descriptive statistics for high-minus-low spreads in returns between the last and first decile among each portfolio class. The anomaly with the largest spread in average returns is price momentum (MOM) with a gap above 1% per month, which is highly significant from an economic point of view. The spreads in returns associated with BM, ROE, net stock issues (NSI), and ABR are also strongly significant in economic terms 4 Several other market anomalies are not included in my analysis since Hou, Xue, and Zhang (2014a) show that these are not statistically significant (see their Table 3). These anomalies include, for example, the long-term return reversal (De Bondt and Thaler (1985)), idiosyncratic volatility (Ang et al. (2006)), size (Banz (1981)), short-term return reversal (Jegadeesh (1990)), abnormal corporate investment (Titman, Wei, and Xie (2004)), total accruals (Richardson et al. (2005)), and illiquidity (Amihud (2002)), among others. 12

15 with (absolute) means around 0.70% per month. The anomalies that seem less pervasive are operating accruals (OA), gross profits-to-assets (GPA), inventory growth (IVG), earnings increases (NEI), investment growth (IG), operating leverage (OL), and net operating assets (NOA), all with average gaps in returns below 0.40% in magnitude. Price momentum is the anomaly with most volatile spreads in returns (standard deviation above 7% per month), followed by ROE and industry momentum (IM), both with volatilities above 5%. On the other hand, IG and NEI are associated with the least volatile spreads in returns (below 3%). The pairwise correlations among the high-minus-low spreads are depicted in Table 4. We can see that there is not an excessive degree of overlapping between the different portfolio sorts. The larger positive correlations occur between MOM and IM (0.78) and between NSI and composite issuance (CEI, 0.60). On the other hand, BM and equity duration are strongly negatively correlated (-0.71), thus indicating that the (reverse) duration strategy resembles to a large extent the value strategy. We can also observe a large positive correlation between the spreads associated with ROE and NEI (0.59) and those associated with IA and changes in property, plant, and equipment scaled by assets (PIA), with a correlation of Not surprisingly, IVG and inventory changes (IVC) also present some degree of overlapping (0.54). 3.2 Factor risk premia Before analysing the explanatory power of the different multifactor models, I focus on the factor risk price estimates. Table 5 presents the constrained risk price estimates for the equity factors. These estimates are obtained from a GLS cross-sectional regression in which the factors are included in the set of test assets. As noted above, these estimates are numerically equal to the sample means of the factors already reported in Table 1. The t-ratios are based on Shanken (1992) standard errors. We can see that most estimates are statistically significant at the 1% or 5% level. The sole exception is the estimate for λ SMB, which is only significant at the 10% level, consistent with the evidence that the size premium is less pervasive than the other factors. 13

16 The estimates for the unrestricted risk prices, obtained from the traditional two-pass regression approach, are displayed in Table 6. I start by forcing the models to price simultaneously several market anomalies, thus, the cross-sectional tests contain several classes of decile portfolios. I consider four groups of test assets. The first cross-sectional test (Panel A) contains BM, MOM, IA, and ROE. Thus, this test involves portfolios that are (nearly) mechanically related with the equity factors HML, UMD, IA, ROE, RMW, and CMA. The second test contains all remaining portfolio sorts (a total of 18 groups). In Panel C, I consider the most important anomalies, defined as the portfolio sorts exhibiting average high-minus-low return spreads above 0.50% in magnitude. These include, BM, MOM, ROE, NSI, organizational capital-to-assets (OCA), ABR, CEI, DUR, and IM. In principle, all else equal, these portfolio groups will be more challengeable to price than the portfolio sorts showing lower average spreads in returns. The reason is that one needs a larger spread in loadings associated with (some of) the factors in a given model to match the larger spread in average excess returns. Finally, in Panel D, I conduct a test including all 22 portfolio sorts for a total of 218 equity portfolios. There are some remarkable discrepancies between the risk price estimates and the factor sample means presented above. Specifically, the estimates for λ SMB within the FF3, C4, and PS4 models are negative in all four cross-sectional tests. On the other hand, the estimates for the size factor within the FF5 model are positive, but the magnitudes are quite different from the sample mean of SMB (0.24%). Specifically, the size risk price estimate in the first sample of anomalies is about twice as big as the correct estimate, whereas in the second and fourth samples the corresponding estimates are around zero. The estimates for λ LIQ also have the wrong sign in all cases except in the second cross-sectional test, in which case the estimated risk price has half the magnitude of the factor sample mean. All the other risk price estimates have the correct sign, yet, the magnitudes are in several cases quite different than the corresponding factor means. Specifically, the risk price estimates associated with HML within FF5 are below the correct estimate of 0.40% by more 14

17 than 20 basis points in all four cross-sectional tests. A similar pattern holds for λ ME and λ ROE in the second cross-sectional test. On the other hand, the estimates for the RMW risk price are too high in the first sample (0.53% versus 0.31%), while an inverse pattern holds in the second test (0.10%). Several of these risk price estimates are not statistically significant. Nonetheless, the divergence of these estimates to the factor sample means suggests that the OLS cross-sectional R 2, widely used in the empirical asset pricing literature, may overstate the fit of multifactor models in which the factors represent excess stock returns. 3.3 R 2 estimates I turn now the attention to the main focus of the paper the explanatory power of the different multifactor models for the joint cross-section of CAPM anomalies. The estimates for both the unrestricted (OLS) and restricted cross-sectional R 2 are displayed in Table 7. As a consequence of the divergence between the risk price estimates from the crosssectional regression and the correct estimates, documented above, one should expect that the estimates of R 2 C will lie below the corresponding estimates for R2 OLS in several cases. Indeed, the estimates of R 2 C associated with the FF3 and PS4 models are in most cases below the corresponding R 2 OLS values by more than 20 percentage points, the exception being the second cross-sectional test in which the gap is below that level. A similar pattern holds for the FF5 model in the second test as indicated by the R 2 C of 3% compared to 28% for ROLS 2. Thus, the restriction that the factor risk price estimates should be equal to the corresponding factor sample means is in several cases binding in a significant way. The estimates of R 2 C are closer to the corresponding R2 OLS values in the cases of the C4 and HXZ4 models, especially in the second and fourth samples (in the case of C4) and first and third samples (in the case of HXZ4) in which the decline in explanatory power is below 10 percentage points. Yet, the other model in which the values for the two goodnessof-fit measures are more similar is the CAPM. This stems from the fact that this model already exhibits a low explanatory power in the traditional approach, that is, the role of 15

18 the market factor is mainly of fitting the cross-sectional average rather than explaining any cross-sectional dispersion in risk premia. Despite the evidence showing that R 2 OLS overstates the explanatory power of the multifactor models, it turns out that both C4 and HXZ4 do a good job in pricing the joint cross-section of market anomalies, with both models showing a similar fit in most crosssectional tests. In the first test, including the BM, MOM, IA, and ROE deciles, the HZX4 model outperforms marginally C4, while a converse relation holds in the third sample of anomalies. Still, in both samples the explanatory ratios are around or above 50% for both models. In the test including all 22 portfolio sorts, the R 2 C estimates are 40% and 33% for C4 and HXZ4, respectively. However, in the test containing other portfolios than BM, MOM, IA, and ROE (Panel B), the C4 model does relatively better with an explanatory ratio of 32% versus 18% for HXZ4. These R 2 C estimates are about half the corresponding values in the test including the four portfolio sorts, which suggests that the two models perform significantly better in pricing anomalies that are mechanically related with the respective factors. Surprisingly, the FF5 model lags behind both models by a significant margin: the RC 2 estimates for the five-factor model vary between 3% (second sample) and 24% (first test). In the estimation with all 22 anomalies, the fit of the five-factor model is only 9%, quite below the corresponding fit of both C4 and HXZ4. These results suggest that the types of risks measured by both RMW and CMA differ significantly from ROE and IA, respectively. In other words, there is no relevant overlapping between the two versions of the investment and profitability factors, with the Fama-French factors underperforming when it comes to price a large cross-section of stock returns. Regarding the other multifactor models, both FF3 and PS4 cannot price the large crosssections of stock returns as the R 2 C estimates are negative in all cases, similarly to the case of the CAPM. This means that these models perform worse than a trivial model that predicts constant expected excess returns in the cross-section of equity portfolios. Hence, it 16

19 is interesting to note that both the FF3 and PS4 models do not significantly outperform the CAPM in pricing the large cross-section of stock returns. This suggests that (some of) the additional factors in those models SM B, HM L, LIQ produce a gap in fitted risk premia that is not enough to match the original spreads in average excess returns that we need to explain. The results of this subsection can be summarized as follows. First, I conclude that the R 2 from an OLS cross-sectional regression, traditionally used in the literature, overstates the true explanatory power of the multifactor models since this measure relies on estimated risk prices that can be significantly different from the correct estimates (factor sample means). This is especially evident when the models are forced to price portfolios that are not trivially related with their factors. Second, only two factor models C4 and HXZ have a satisfactory performance in explaining the large cross-section of stock returns. The other multifactor models proposed in the literature either do not outperform significantly the baseline CAPM in explaining the 22 market anomalies (as is the case with FF3 and PS4) or have a rather weak explanatory power (as is the case with FF5). 3.4 Explaining single anomalies I proceed by examining the ability of the equity multifactor models to explain each portfolio sort separately. This represents a much less stringent test than the multiple-anomaly tests conducted above. The goal is to assess which anomalies are explained by each model. For clarity of exposition, I concentrate the analysis in the best performing multifactor factor models as shown above (C4, HXZ4, and to a lower degree FF5). Table 8, which is identical to Table 7, reports the R 2 C estimates associated with the cross-sectional tests for each of the individual 22 portfolio sorts. Table 9 reports several model evaluation and comparison criteria associated with the R 2 C estimates. I define a cutoff estimate of 50% for R 2 C to conclude that a given model has a good explanatory power for a given set of deciles. Notice that, ceteris paribus, with ten portfolios in each cross-sectional 17

20 test it should be easier for multifactor model to attain a large fit than in the joint-anomaly tests. The C4 and HXZ4 models continue to outperform significantly the other models when it comes to explain the single anomalies. HXZ4 and C4 produce explanatory ratios above 50% in eight and seven (out of the 22 anomalies), respectively. In 13 of the 22 portfolio sorts, the C4 model dominates all other models (in terms of RC 2 ), although in some cases by a low margin, whereas the HXZ4 model dominates in five anomalies. Not surprisingly, the C4 model is able to price the portfolios broadly classified as momentum strategies (MOM, IM) or value strategies (BM and DUR). This explanatory ability is driven by the UMD and HM L factors, respectively. More interestingly, the C4 model can explain several anomalies classified as investment strategies (IA, IVC, and IVG), which are not a priori trivially related with the factors in the model. Regarding the HXZ4 model, as expected, the model drives the ROE deciles. More interesting is the fact that this model can price several sorts not trivially related with the respective factors. These include several momentum strategies (MOM, SUE, and IM), value anomalies (BM and DUR), and also some investment-related anomalies (NSI and IVC). On the other hand, it is quite surprising that this model cannot price the IA deciles, which are mechanically related with its investment factor, as indicated by the explanatory ratio of -28%. This result should be linked with a non-monotonic behavior in the average returns of these deciles, and is consistent with the rejection of the four-factor model (when tested on these portfolios) by the GRS test (Gibbons, Ross, and Shanken (1989)) according to Hou, Xue, and Zhang (2014a) (p-values of 1%, see their Table 4). Hence, the fit of the HXZ4 model for the large cross-section of stock returns, documented above, is not overstated by the presence of the IA deciles. On the negative side, there are several anomalies that these two models cannot explain, as indicated by the negative R 2 C estimates. These include the GPA, NOA, and percent operating accruals (POA) anomalies in the case of C4. On the other hand, the HXZ4 model performs worse than a model with just an intercept when it comes to price deciles sorted on 18

21 GPA, ABR, PIA, IVG, OA, and POA, in addition to the already mentioned IA deciles. In this sense there is more volatility in the performance of the HXZ4 model since it performs negatively in more anomalies than the C4 model (seven versus three). Hence, there are several anomalies related to profitability, investment, and momentum that are left explained by the C4, and especially HXZ4 models. Nevertheless, when one restricts the analysis to the anomalies in which the average spread in returns is above 0.50% per month (Panel B of Table 9), the performance of the two four-factor models is quite similar: the HXZ4 model produces an explanatory ratio above 50% in six (out of the nine) anomalies, compared to four portfolio sorts for C4. On the other hand, only for one anomaly (ABR) do one observes a negative R 2 C estimate associated with the HXZ4 model. Therefore, the performance of the two models is comparable when it comes to explain the more important anomalies. The performance of the FF5 model is significantly weaker than the two four-factor models. Only for four anomalies do one obtains R 2 estimates above 50%. Not surprisingly, the fivefactor model can explain the value strategies (BM and DUR) due to the presence of the HML factor. More interesting is the fact that the model has pricing ability for the NSI and IG deciles, with R 2 C estimates around 60%. On the other hand, for ten (out of the 22) anomalies the model produces negative explanatory ratios. In other words, the model performs worse than a model that predicts constant expected returns within each of these ten groups of deciles. In particular, the model fails completely in explaining momentum-related anomalies (MOM, ABR, and IM) as well as several investment strategies (IA, PIA, IVG, OA, and POA). Thus, one key factor behind the outperformance of the C4 and HXZ4 models (against FF5) relies on the capacity of explaining momentum-related anomalies and also price several investment-related strategies, which is consistent with the evidence provided in Hou, Xue, and Zhang (2014b). The five-factor model also cannot explain two of the profitability anomalies (GPA and NEI). Regarding the FF3 and PS4 models, only for the BM and DUR deciles one has explanatory ratios above 50%, which is due to the presence of the HML factor. However, this adds 19

22 little or nothing in terms of the economic or fundamentals sources of cross-sectional risk premia. It just tells us that there is an approximate monotonic pattern in average returns within the deciles associated with those two portfolio sorts. In other words, stocks that are closer to growth stocks have average returns that are more similar to those of growth stocks than to value stocks. In fact, these two models produce negative R 2 C estimates in 12 out of the 22 anomalies. Hence, my results indicate that when these two models (and also the FF5 model to some degree) are forced to price portfolios that are not trivially related with the respective factors, they outperform the CAPM by a smaller magnitude or simply don t outperform at all. 4 Alternative multifactor models In this section, I compare the performance of the equity multifactor models discussed in the previous sections with alternative multifactor models in which the factors are not excess stock returns. The alternative models are mainly retrieved from the ICAPM literature, in which the factors (apart from the market factor) represent the innovations in state variables that forecast the changes in future investment opportunities. Most candidate state variables in empirical applications of the ICAPM represent variables commonly used as predictors of the aggregate equity premium. The innovation in each state variable (z) is obtained from an AR(1) process: z t+1 ε t+1 = z t+1 ψ φz t. (16) The first model is the three-factor model from Hahn and Lee (2006) (HL3), E(R i,t+1 R f,t+1 ) = λ M β i,m + λ T ERM β i, T ERM + λ DEF β i, DEF, (17) where λ T ERM and λ DEF denote the risk prices associated with the innovations in the term spread ( T ERM t+1 ) and default spread ( DEF t+1 ), respectively. 20

23 The second model is the five-factor ICAPM from Petkova (2006) (P5), which adds to HL3 the innovations in the log dividend yield ( DY t+1 ) and T-bill rate ( RF t+1 ): E(R i,t+1 R f,t+1 ) = λ M β i,m + λ T ERM β i, T ERM + λ DEF β i, DEF + λ DY β i, DY + λ RF β i, RF. (18) The third model is an unrestricted version of the ICAPM from Campbell and Vuolteenaho (2004) (CV4), 5 E(R i,t+1 R f,t+1 ) = λ M β i,m + λ T ERM β i, T ERM + λ P E β i, P E + λ V S β i, V S, (19) in which λ P E and λ V S represent the risk prices associated with the innovations in the aggregate price-earnings ratio ( P E t+1 ) and the value spread ( V S t+1 ), respectively. The ICAPM from Campbell et al. (2014) (CGPT6) adds to CV4 the innovations in DEF and in the stock market variance ( SV AR t+1 ): E(R i,t+1 R f,t+1 ) = λ M β i,m + λ T ERM β i, T ERM + λ P E β i, P E + λ V S β i, V S + λ DEF β i, DEF + λ SV AR β i, SV AR. (20) The fifth model corresponds to a restricted version of the factor model from Chen, Roll, and Ross (1986) (CRR4), in which the factors are the CPI inflation rate (INF ) and the growth in industrial production (IP G), in addition to T ERM and DEF : E(R i,t+1 R f,t+1 ) = λ T ERM β i,t ERM + λ DEF β i,def + λ IP G β i,ip G + λ INF β i,inf. (21) In this specification, all the factors represent the levels (rather than innovations) of the macro 5 The factors in Campbell and Vuolteenaho (2004) are cash-flow and discount rate news, which represent linear functions of the original state variables used in a first-order VAR. Hence, the two specifications are equivalent (see Campbell (1996) and Maio (2013b) for details). 21

24 variables. T ERM is computed as the yield spread between the ten-year and the one-year Treasury bonds, and DEF denotes the yield spread between BAA and AAA corporate bonds from Moody s. The bond yield data are available from the St. Louis Fed Web page. RF corresponds to the one-month T-bill rate, available from Kenneth French s website. DY is computed as the log ratio of annual dividends to the level of the S&P 500 index, while P E is the log price-earnings ratio associated with the same index, where the earnings measure is based on a 10-year moving average of annual earnings. The data on both P E and the components of DY are retrieved from Robert Shiller s website. V S represents the difference in the log book-to-market ratios of small-value and small-growth portfolios, where the bookto-market data are from French s data library. 6 The time-series of the realized stock market variance is from Amit Goyal s webpage. INF and IP G represent the log differences in the consumer price index and industrial production index, respectively, which are obtained from St. Louis Fed. The R 2 OLS estimates associated with the alternative factor models are displayed in Table 10. HL3, P5, and CRR4 do not outperform significantly the CAPM in pricing the joint cross-section of anomalies as the R 2 estimates are negative in most cases. The sole exception is P5 in the sample with four anomalies with a positive, but modest, explanatory ratio (14%). On the other hand, the performance of both CV4 and CGPT6 is significantly better in comparison to the other three models, with positive R 2 OLS estimates in all four crosssectional tests. 7 Specifically, the fit of both models is similar to both C4 and HXZ4 in the test with four sorts (slightly above 50%). In the last two cross-sectional tests the two ICAPM models underperform both C4 and HXZ4 as indicated by the explanatory ratios between 19% and 34%. However, it is remarkable that these two models outperform the FF5 model in all four samples of anomalies. Specifically, in the last two cross-sectional tests the 6 See Campbell and Vuolteenaho (2004) for details on the construction of V S. 7 The performance of a restricted version of CGPT6 that excludes the volatility factor is very similar to that of the six-factor model. 22

25 fit of the two ICAPM models is about twice to that of FF5. The R 2 OLS estimates for the alternative factor models in the case of the single-anomaly tests are reported in Table 11. We can see that the R 2 estimates associated with the four ICAPM models are positive in most cases. The few exceptions are HL3 (in the tests with OCA, CEI, PIA, IG, and PTA), P5 (CEI), and CV4 (OCA and NOA). In fact, in many of the single-anomaly tests one observes explanatory ratios above 50%, suggesting that the ICAPM models have a significant pricing ability for several individual anomalies. The performance of the CRR4 model is significantly worse than the ICAPM models, as indicated by the negative explanatory ratios for nine out of the 22 portfolio sorts. Still, this four-factor model has large explanatory power (R 2 OLS above 50%) for several anomalies (BM, MOM, SUE, OL, NOA, and IM). Hence, this macro model is successful in pricing momentum-related anomalies, which is consistent with the evidence in Liu and Zhang (2008) that industrial production helps to explain the cross-sectional risk premia associated with price-momentum portfolios. Why these models perform better in pricing individual portfolio sorts than a large crosssection of anomalies? The reason lies on the fact that the factor risk price estimates vary widely across different portfolio sorts, making it difficult to obtain a large fit when the models are forced to price simultaneously several anomalies. Still, the ICAPM models compare favorably with the equity factor models in pricing individual anomalies. Table 12 reports the single anomalies in which the four ICAPM models beat the best performing factor models (C4, HXZ4, and FF5) by at least 10 percentage points (in terms of R 2 ). We can see that three of the ICAPM models (P5, CV4, and CGPT6) compare favorably with the three factor models in explaining most individual market anomalies. In sum, the results from this section show that the common wisdom suggesting that the factor models outperform models in which the factors are not excess returns is probably overstated. Several ICAPM models can outperform the factor models in explaining several individual market anomalies. Moreover, my results show that when the models are forced to price a large cross-section of anomalies, the best performing factor models (C4 and HXZ4) 23