《金融投资学》教学资源（英文文献）The time-series relations

JOURNAL OF Financial ECONOMICS ELSEVIER Journal of Financial Economics 54 (1999)5-43 www.elsevier.com/locate/econbase The time-series relations among expected return,risk,and book-to-market* Jonathan Lewellen* William E.Simon Graduate School of Business Administration,University of Rochester,Rochester, NY14627,US4 Received 8 August 1997;received in revised form 20 October 1998 Abstract This paper examines the time-series relations among expected return,risk,and book- to-market (B/M)at the portfolio level.I find that B/M predicts economically and statistically significant time-variation in expected stock returns.Further,B/M is strongly associated with changes in risk,as measured by the Fama and French(1993)(Journal of Financial Economics,33,3-56)three-factor model.After controlling for risk,B/M provides no incremental information about expected returns.The evidence suggests that the three-factor model explains time-varying expected returns better than a character- istics-based model.C 1999 Elsevier Science S.A.All rights reserved. JEL classification:G12;G14 Keywords:Asset pricing;Book-to-market;Time-varying risk;Mispricing *Fax:+1-617-258-8408. E-mail address:lewellenjw@ssb.rochester.edu (J.Lewellen) *I am grateful to G.William Schwert,Jerold Warner,and especially Jay Shanken for guidance and encouragement.This paper has also benefited from the comments of Greg Bauer,Ken French (the referee).Christoph Hinkelmann,S.P.Kothari,John Long.Susan Shu,Peter Wysocki,and seminar participants at the London Business School,MIT,UCLA,UC-Berkeley,University of Illinois,University of Rochester,Yale School of Management,and the 1997 Southern Finance Association meetings. 0304-405X/99/S-see front matter C 1999 Elsevier Science S.A.All rights reserved. PI:S0304-405X(99)00030-6

Journal of Financial Economics 54 (1999) 5}43 The time-series relations among expected return, risk, and book-to-marketq Jonathan Lewellen* William E. Simon Graduate School of Business Administration, University of Rochester, Rochester, NY 14627, USA Received 8 August 1997; received in revised form 20 October 1998 Abstract This paper examines the time-series relations among expected return, risk, and book￾to-market (B/M) at the portfolio level. I "nd that B/M predicts economically and statistically signi"cant time-variation in expected stock returns. Further, B/M is strongly associated with changes in risk, as measured by the Fama and French (1993) (Journal of Financial Economics, 33, 3}56) three-factor model. After controlling for risk, B/M provides no incremental information about expected returns. The evidence suggests that the three-factor model explains time-varying expected returns better than a character￾istics-based model. ( 1999 Elsevier Science S.A. All rights reserved. JEL classixcation: G12; G14 Keywords: Asset pricing; Book-to-market; Time-varying risk; Mispricing *Fax: #1-617-258-8408. E-mail address: lewellenjw@ssb.rochester.edu (J. Lewellen) qI am grateful to G. William Schwert, Jerold Warner, and especially Jay Shanken for guidance and encouragement. This paper has also bene"ted from the comments of Greg Bauer, Ken French (the referee), Christoph Hinkelmann, S.P. Kothari, John Long, Susan Shu, Peter Wysocki, and seminar participants at the London Business School, MIT, UCLA, UC-Berkeley, University of Illinois, University of Rochester, Yale School of Management, and the 1997 Southern Finance Association meetings. 0304-405X/99/$ - see front matter ( 1999 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 5 X ( 9 9 ) 0 0 0 3 0 - 6

6 J.Lewellen Journal of Financial Economics 54 (1999)5-43 1.Introduction Empirical research consistently finds a positive cross-sectional relation be- tween average stock returns and the ratio of a firm's book equity to market equity(B/M).Stattman (1980)and Rosenberg et al.(1985)document the associ- ation between expected returns and B/M,which remains significant after con- trolling for beta,size,and other firm characteristics(Fama and French,1992). The explanatory power of B/M does not appear to be driven entirely by data snooping or survival biases;it is found in stock markets outside the United States(Chan et al,1991;Haugen and Baker,1996)and in samples drawn from sources other than Compustat(Davis,1994).As a whole,the evidence provides considerable support for the cross-sectional explanatory power of B/M. At least two explanations have been offered for the empirical evidence. According to asset-pricing theory,B/M must proxy for a risk factor in returns. The significance of B/M in competition with beta contradicts the capital asset pricing model (CAPM)of Sharpe(1964),Lintner(1965),and Black(1972),or more precisely,the mean-variance efficiency of the market proxy.However,the evidence might be consistent with the intertemporal models of Merton (1973) and Breeden (1979).In these models,the market return does not completely capture the relevant risk in the economy,and additional factors are required to explain expected returns.If a multifactor model accurately describes stock returns,and B/M is cross-sectionally correlated with the factor loadings,then the premium on B/M simply reflects compensation for risk. A positive relation between B/M and risk is expected for several reasons. Chan and Chen (1991)and Fama and French (1993)suggest that a distinct distress factor'explains common variation in stock returns.Poorly performing, or distressed,firms are likely to have high B/M.These firms are especially sensitive to economic conditions,and their returns might be driven by many of the same macroeconomic factors (such as variation over time in bankruptcy costs and access to credit markets).In addition,following the arguments of Ball (1978)and Berk(1995),B/M might proxy for risk because of the inverse relation between market value and discount rates.Holding book value constant in the numerator,a firm's B/M ratio increases as expected return,and consequently risk,increases. Alternatively,B/M might provide information about security mispricing.The mispricing view takes the perspective of a contrarian investor.A firm with poor stock price performance tends to be underpriced and have a low market value relative to book value.As a result,high B/M predicts high future returns as the underpricing is eliminated.Lakonishok et al.(1994)offer a rationale for the association between past performance and mispricing.They argue that investors naively extrapolate past growth when evaluating a firm's prospects. For example,investors tend to be overly pessimistic about a firm which has had low or negative earnings.On average,future earnings exceed the market's

1. Introduction Empirical research consistently "nds a positive cross-sectional relation be￾tween average stock returns and the ratio of a "rm's book equity to market equity (B/M). Stattman (1980) and Rosenberg et al. (1985) document the associ￾ation between expected returns and B/M, which remains signi"cant after con￾trolling for beta, size, and other "rm characteristics (Fama and French, 1992). The explanatory power of B/M does not appear to be driven entirely by data snooping or survival biases; it is found in stock markets outside the United States (Chan et al., 1991; Haugen and Baker, 1996) and in samples drawn from sources other than Compustat (Davis, 1994). As a whole, the evidence provides considerable support for the cross-sectional explanatory power of B/M. At least two explanations have been o!ered for the empirical evidence. According to asset-pricing theory, B/M must proxy for a risk factor in returns. The signi"cance of B/M in competition with beta contradicts the capital asset pricing model (CAPM) of Sharpe (1964), Lintner (1965), and Black (1972), or more precisely, the mean-variance e$ciency of the market proxy. However, the evidence might be consistent with the intertemporal models of Merton (1973) and Breeden (1979). In these models, the market return does not completely capture the relevant risk in the economy, and additional factors are required to explain expected returns. If a multifactor model accurately describes stock returns, and B/M is cross-sectionally correlated with the factor loadings, then the premium on B/M simply re#ects compensation for risk. A positive relation between B/M and risk is expected for several reasons. Chan and Chen (1991) and Fama and French (1993) suggest that a distinct &distress factor' explains common variation in stock returns. Poorly performing, or distressed, "rms are likely to have high B/M. These "rms are especially sensitive to economic conditions, and their returns might be driven by many of the same macroeconomic factors (such as variation over time in bankruptcy costs and access to credit markets). In addition, following the arguments of Ball (1978) and Berk (1995), B/M might proxy for risk because of the inverse relation between market value and discount rates. Holding book value constant in the numerator, a "rm's B/M ratio increases as expected return, and consequently risk, increases. Alternatively, B/M might provide information about security mispricing. The mispricing view takes the perspective of a contrarian investor. A "rm with poor stock price performance tends to be underpriced and have a low market value relative to book value. As a result, high B/M predicts high future returns as the underpricing is eliminated. Lakonishok et al. (1994) o!er a rationale for the association between past performance and mispricing. They argue that investors naively extrapolate past growth when evaluating a "rm's prospects. For example, investors tend to be overly pessimistic about a "rm which has had low or negative earnings. On average, future earnings exceed the market's 6 J. Lewellen / Journal of Financial Economics 54 (1999) 5}43

J.Lewellen Journal of Financial Economics 54 (1999)5-43 expectation,and the stock does abnormally well.Thus,the mispricing argument says that B/M captures biases in investor expectations. Fama and French(1993)provide evidence of a relation between B/M and risk. Using the time-series approach of Black et al.(1972),they examine a multifactor model consisting of market,size,and book-to-market factors,where the size and book-to-market factors are stock portfolios constructed to mimic underlying risk factors in returns.If the model explains cross-sectional variation in average returns,the intercepts will be zero when excess returns are regressed on the three factors.Fama and French find,as predicted by the risk-based view,that the model does a good job explaining average returns for portfolios sorted by size, B/M,earnings-price ratios,and other characteristics.Further,they document a strong association between a stock's B/M ratio and its loading on the book-to-market factor. More recently,Daniel and Titman(1997)argue in favor of a characteristics- based model,consistent with the mispricing view.They suggest that the three- factor model does not directly explain average returns.Instead,the model appears to explain average returns only because the factor loadings are corre- lated with firms'characteristics(size and B/M).To disentangle the explanatory power of the factor loadings from that of the characteristics,Daniel and Titman construct test portfolios by sorting stocks first on B/M ratios and then on factor loadings.This sorting procedure creates independent variation in the two variables.Consistent with the mispricing story,Daniel and Titman find a stronger relation between expected returns and B/M than between expected returns and factor loadings.Daniel and Titman conclude that firm charac- teristics,in particular B/M,and not covariances determine expected stock returns. In this paper,I provide further evidence on the risk-and characteristics-based stories.In contrast to Fama and French(1993)and Daniel and Titman (1997), I focus on the time-series relations among expected return,risk,and B/M. Specifically,I ask whether a portfolio's B/M ratio predicts time-variation in its expected return,and test whether changes in expected return can be explained by changes in risk.Recently,Kothari and Shanken(1997)and Pontiff and Schall (1998)find that B/M forecasts stock returns at the aggregate level,but the predictive ability of B/M for individual stocks or portfolios has not been explored. The time-series analysis is a natural alternative to cross-sectional regressions. An attractive feature of the time-series regressions is that they focus on changes in expected returns,not on average returns.The mispricing story suggests that a stock's expected return will vary over time with B/M,but it says little about average returns if mispricing is temporary.Cross-sectional regressions,however, can pick up a relation between average returns and B/M.The time-series regressions also highlight the interaction between B/M and risk,as measured by time-variation in market betas and the loadings on the Fama and French

expectation, and the stock does abnormally well. Thus, the mispricing argument says that B/M captures biases in investor expectations. Fama and French (1993) provide evidence of a relation between B/M and risk. Using the time-series approach of Black et al. (1972), they examine a multifactor model consisting of market, size, and book-to-market factors, where the size and book-to-market factors are stock portfolios constructed to mimic underlying risk factors in returns. If the model explains cross-sectional variation in average returns, the intercepts will be zero when excess returns are regressed on the three factors. Fama and French "nd, as predicted by the risk-based view, that the model does a good job explaining average returns for portfolios sorted by size, B/M, earnings-price ratios, and other characteristics. Further, they document a strong association between a stock's B/M ratio and its loading on the book-to-market factor. More recently, Daniel and Titman (1997) argue in favor of a characteristics￾based model, consistent with the mispricing view. They suggest that the three￾factor model does not directly explain average returns. Instead, the model appears to explain average returns only because the factor loadings are corre￾lated with "rms' characteristics (size and B/M). To disentangle the explanatory power of the factor loadings from that of the characteristics, Daniel and Titman construct test portfolios by sorting stocks "rst on B/M ratios and then on factor loadings. This sorting procedure creates independent variation in the two variables. Consistent with the mispricing story, Daniel and Titman "nd a stronger relation between expected returns and B/M than between expected returns and factor loadings. Daniel and Titman conclude that "rm charac￾teristics, in particular B/M, and not covariances determine expected stock returns. In this paper, I provide further evidence on the risk- and characteristics-based stories. In contrast to Fama and French (1993) and Daniel and Titman (1997), I focus on the time-series relations among expected return, risk, and B/M. Speci"cally, I ask whether a portfolio's B/M ratio predicts time-variation in its expected return, and test whether changes in expected return can be explained by changes in risk. Recently, Kothari and Shanken (1997) and Ponti! and Schall (1998) "nd that B/M forecasts stock returns at the aggregate level, but the predictive ability of B/M for individual stocks or portfolios has not been explored. The time-series analysis is a natural alternative to cross-sectional regressions. An attractive feature of the time-series regressions is that they focus on changes in expected returns, not on average returns. The mispricing story suggests that a stock's expected return will vary over time with B/M, but it says little about average returns if mispricing is temporary. Cross-sectional regressions, however, can pick up a relation between average returns and B/M. The time-series regressions also highlight the interaction between B/M and risk, as measured by time-variation in market betas and the loadings on the Fama and French J. Lewellen / Journal of Financial Economics 54 (1999) 5}43 7

8 J.Lewellen Journal of Financial Economics 54 (1999)5-43 (1993)size and book-to-market factors.Further,I can directly test whether the three-factor model explains time-varying expected returns better than the char- acteristics-based model.These results should help distinguish between the risk and mispricing stories. The empirical tests initially examine B/M's predictive ability without attempt- ing to control for changes in risk.I find that a portfolio's B/M ratio tracks economically and statistically significant variation in its expected return.An increase in B/M equal to twice its time-series standard deviation forecasts a 4.6%(annualized)increase in expected return for the typical industry port- folio,8.2%for the typical size portfolio,and 9.3%for the typical book-to- market portfolio.The average coefficient on B/M across all portfolios,0.99,is approximately double the cross-sectional slope,0.50,found by Fama and French(1992,p.439).B/M explains,however,only a small fraction of portfolio returns,generally less than 2%of total volatility. Return predictability indicates that either risk or mispricing changes over time.Of course,we cannot distinguish between these explanations without some model of risk.Following Daniel and Titman (1997),I examine B/M's explanatory power in competition with the Fama and French (1993)three- factor model.The multifactor regressions employ the conditional asset-pricing methodology of Shanken (1990),which allows both expected returns and factor loadings to vary over time with B/M.In these regressions,time-variation in the intercepts measures the predictive ability of B/M that cannot be explained by changes in risk.The mispricing view suggests that the intercepts will be positively related to B/M;the risk-based view implies that changes in the factor loadings will eliminate B/M's explanatory power,assum- ing the Fama and French factors are adequate proxies for priced risk in the economy. Empirically,the factors absorb much of the volatility of portfolio returns, which permits relatively powerful tests of the competing stories.I find that B/M explains significant time-variation in risk,but does not provide incremental information about expected return.In general,the loadings on the size and book-to-market factors vary positively with a portfolio's B/M ratio,and statistical tests strongly reject the hypothesis of constant risk.The results for market betas are more difficult to characterize:across different portfolios, B/M predicts both significant increases and significant decreases in beta. Overall,B/M contains substantial information about the riskiness of stock portfolios. In contrast,the intercepts of the three-factor model do not vary over time with B/M.For the industry portfolios,the average coefficient on B/M (that is, variation in the intercept)has the opposite sign predicted by the overreaction hypothesis and is not significantly different from zero.Across the 13 portfolios, eight coefficients are negative and none is significantly positive at conventional levels.The results are similar for size and book-to-market portfolios:the

(1993) size and book-to-market factors. Further, I can directly test whether the three-factor model explains time-varying expected returns better than the char￾acteristics-based model. These results should help distinguish between the risk and mispricing stories. The empirical tests initially examine B/M's predictive ability without attempt￾ing to control for changes in risk. I "nd that a portfolio's B/M ratio tracks economically and statistically signi"cant variation in its expected return. An increase in B/M equal to twice its time-series standard deviation forecasts a 4.6% (annualized) increase in expected return for the typical industry port￾folio, 8.2% for the typical size portfolio, and 9.3% for the typical book-to￾market portfolio. The average coe$cient on B/M across all portfolios, 0.99, is approximately double the cross-sectional slope, 0.50, found by Fama and French (1992, p. 439). B/M explains, however, only a small fraction of portfolio returns, generally less than 2% of total volatility. Return predictability indicates that either risk or mispricing changes over time. Of course, we cannot distinguish between these explanations without some model of risk. Following Daniel and Titman (1997), I examine B/M's explanatory power in competition with the Fama and French (1993) three￾factor model. The multifactor regressions employ the conditional asset-pricing methodology of Shanken (1990), which allows both expected returns and factor loadings to vary over time with B/M. In these regressions, time-variation in the intercepts measures the predictive ability of B/M that cannot be explained by changes in risk. The mispricing view suggests that the intercepts will be positively related to B/M; the risk-based view implies that changes in the factor loadings will eliminate B/M's explanatory power, assum￾ing the Fama and French factors are adequate proxies for priced risk in the economy. Empirically, the factors absorb much of the volatility of portfolio returns, which permits relatively powerful tests of the competing stories. I "nd that B/M explains signi"cant time-variation in risk, but does not provide incremental information about expected return. In general, the loadings on the size and book-to-market factors vary positively with a portfolio's B/M ratio, and statistical tests strongly reject the hypothesis of constant risk. The results for market betas are more di$cult to characterize: across di!erent portfolios, B/M predicts both signi"cant increases and signi"cant decreases in beta. Overall, B/M contains substantial information about the riskiness of stock portfolios. In contrast, the intercepts of the three-factor model do not vary over time with B/M. For the industry portfolios, the average coe$cient on B/M (that is, variation in the intercept) has the opposite sign predicted by the overreaction hypothesis and is not signi"cantly di!erent from zero. Across the 13 portfolios, eight coe$cients are negative and none is signi"cantly positive at conventional levels. The results are similar for size and book-to-market portfolios: the 8 J. Lewellen / Journal of Financial Economics 54 (1999) 5}43

J.Lewellen Journal of Financial Economics 54 (1999)5-43 9 average coefficients are indistinguishable from zero,and roughly half are negative.Importantly,the inferences from the multifactor regressions are not driven by low power.For all three sets of portfolios,statistical tests can reject economically large coefficients on B/M.In short,the three-factor model measures risk sufficiently well to explain time-variation in expected returns. As an aside,I find that the book-to-market factor,HML,explains common variation in returns that is unrelated to its industry composition.Daniel and Titman(1997)argue that HML does not proxy for a distinct risk factor,but explains return covariation only because similar types of firms become mis- priced at the same time.For example,a bank with high B/M will covary positively with HML simply because the factor is weighted towards underpriced financial firms.The time-series regressions provide evidence to the contrary.As an alternative to HML,I estimate the regressions with an 'industry-neutral' book-to-market factor.This factor is constructed by sorting stocks on their industry-adjusted B/M ratios,defined as the firm's B/M minus the industry average,so the factor should never be weighted towards particular industries. The results using the industry-neutral factor are similar to those with HML. Thus,HML's explanatory power does not appear to be driven by industry factors in returns. The remainder of the paper is organized as follows.Section 2 introduces the time-series regressions.Section 3 describes the data to be used in the empirical tests.Section 4 estimates the simple relation between expected returns and B/M, and Section 5 tests whether the predictive ability of B/M can be explained by changes in risk,as measured by the Fama and French(1993)three-factor model. Section 6 summarizes the evidence and concludes. 2.Distinguishing between characteristics and risk Book-to-market explains cross-sectional variation in average returns after controlling for beta.Fama and French(1993)provide evidence that B/M relates to common risk factors in returns.In contrast,Daniel and Titman(1997)argue that the Fama and French factors appear to be priced only because the loadings are correlated with firm characteristics,like B/M.This section introduces the time-series methodology used in the current paper and discusses,more gener- ally,asset-pricing tests of the risk and mispricing stories. I also replicate the empirical tests using size in place of B/M,with similar results.There is some evidence that size and expected returns are negatively related in time series.In conditional three-factor regressions,size captures significant time-variation in risk,but does not contain additional information about expected returns.Details are available on request.I thank Ken French for suggesting these tests

average coe$cients are indistinguishable from zero, and roughly half are negative. Importantly, the inferences from the multifactor regressions are not driven by low power. For all three sets of portfolios, statistical tests can reject economically large coe$cients on B/M. In short, the three-factor model measures risk su$ciently well to explain time-variation in expected returns.1 As an aside, I "nd that the book-to-market factor, HML, explains common variation in returns that is unrelated to its industry composition. Daniel and Titman (1997) argue that HML does not proxy for a distinct risk factor, but explains return covariation only because similar types of "rms become mis￾priced at the same time. For example, a bank with high B/M will covary positively with HML simply because the factor is weighted towards underpriced "nancial "rms. The time-series regressions provide evidence to the contrary. As an alternative to HML, I estimate the regressions with an &industry-neutral' book-to-market factor. This factor is constructed by sorting stocks on their industry-adjusted B/M ratios, de"ned as the "rm's B/M minus the industry average, so the factor should never be weighted towards particular industries. The results using the industry-neutral factor are similar to those with HML. Thus, HML's explanatory power does not appear to be driven by industry factors in returns. The remainder of the paper is organized as follows. Section 2 introduces the time-series regressions. Section 3 describes the data to be used in the empirical tests. Section 4 estimates the simple relation between expected returns and B/M, and Section 5 tests whether the predictive ability of B/M can be explained by changes in risk, as measured by the Fama and French (1993) three-factor model. Section 6 summarizes the evidence and concludes. 2. Distinguishing between characteristics and risk Book-to-market explains cross-sectional variation in average returns after controlling for beta. Fama and French (1993) provide evidence that B/M relates to common risk factors in returns. In contrast, Daniel and Titman (1997) argue that the Fama and French factors appear to be priced only because the loadings are correlated with "rm characteristics, like B/M. This section introduces the time-series methodology used in the current paper and discusses, more gener￾ally, asset-pricing tests of the risk and mispricing stories. 1 I also replicate the empirical tests using size in place of B/M, with similar results. There is some evidence that size and expected returns are negatively related in time series. In conditional three-factor regressions, size captures signi"cant time-variation in risk, but does not contain additional information about expected returns. Details are available on request. I thank Ken French for suggesting these tests. J. Lewellen / Journal of Financial Economics 54 (1999) 5}43 9

10 J.Lewellen Journal of Financial Economics 54 (1999)5-43 2.1.Time-series methodology The empirical tests initially examine the simple relation between expected returns and B/M.The explanations that have been offered for the cross-sectional evidence also suggest that expected returns will vary over time with B/M. According to the risk-based view,B/M should capture information about changes in risk,and consequently,expected return.The mispricing view says that B/M is related to biases in investor expectations,and will contain informa- tion about under-and overpricing.Thus,both explanations predict a positive slope coefficient in the regression R()=Yio+B/Mt-1)+e{), (1) where Ri is the portfolio's excess return and B/Mi is its lagged book-to- market ratio.Note that Eq.(1)specifies a separate time-series regression for each portfolio,with no constraint on the coefficients across different portfolios.The regressions focus only on the time-series relation between expected returns and B/M,and do not pick up any cross-sectional relation. Eq.(1)makes no attempt to understand the source of time-varying expected returns.According to traditional asset-pricing theory,a positive slope in Eq.(1) must be driven by an association between B/M and risk.It follows that the predictive power of B/M should be eliminated if the regressions control ad- equately for changes in risk.The characteristics-based story,on the other hand, suggests that B/M will capture information about expected returns that is unrelated to risk.To help distinguish between the two explanations,I examine the predictive power of B/M in competition with the Fama and French(1993) three-factor model. The multifactor regressions employ the conditional time-series methodology of Shanken(1990).Roughly speaking,these regressions combine the three-factor model with the simple regressions above.Fama and French estimate the unconditional model Ri(t)=ai+bi RM(t)+si SMB(t)+hi HML(t)+ext), (2) where RM is the excess market return,SMB (small minus big)is the size factor,and HML (high minus low)is the book-to-market factor.Uncondi- tional,here,refers to the implicit assumption that the coefficients of the model are constant over time.If this assumption is not satisfied,the estimates from Eq.(2)can be misleading.The unconditional intercepts and factor loadings could be close to zero,but might vary considerably over time. The conditional regressions allow both expected returns and factor loadings to vary with B/M.Suppose,for simplicity,that the coefficients of the three-factor

2.1. Time-series methodology The empirical tests initially examine the simple relation between expected returns and B/M. The explanations that have been o!ered for the cross-sectional evidence also suggest that expected returns will vary over time with B/M. According to the risk-based view, B/M should capture information about changes in risk, and consequently, expected return. The mispricing view says that B/M is related to biases in investor expectations, and will contain informa￾tion about under- and overpricing. Thus, both explanations predict a positive slope coe$cient in the regression Ri (t)"c i0 #c i1 B/Mi (t!1)#e i (t), (1) where Ri is the portfolio's excess return and B/Mi is its lagged book-to￾market ratio. Note that Eq. (1) speci"es a separate time-series regression for each portfolio, with no constraint on the coe$cients across di!erent portfolios. The regressions focus only on the time-series relation between expected returns and B/M, and do not pick up any cross-sectional relation. Eq. (1) makes no attempt to understand the source of time-varying expected returns. According to traditional asset-pricing theory, a positive slope in Eq. (1) must be driven by an association between B/M and risk. It follows that the predictive power of B/M should be eliminated if the regressions control ad￾equately for changes in risk. The characteristics-based story, on the other hand, suggests that B/M will capture information about expected returns that is unrelated to risk. To help distinguish between the two explanations, I examine the predictive power of B/M in competition with the Fama and French (1993) three-factor model. The multifactor regressions employ the conditional time-series methodology of Shanken (1990). Roughly speaking, these regressions combine the three-factor model with the simple regressions above. Fama and French estimate the unconditional model Ri (t)"a i #b i RM (t)#s i SMB(t)#h i HML(t)#e i (t), (2) where RM is the excess market return, SMB (small minus big) is the size factor, and HML (high minus low) is the book-to-market factor. Uncondi￾tional, here, refers to the implicit assumption that the coe$cients of the model are constant over time. If this assumption is not satis"ed, the estimates from Eq. (2) can be misleading. The unconditional intercepts and factor loadings could be close to zero, but might vary considerably over time. The conditional regressions allow both expected returns and factor loadings to vary with B/M. Suppose, for simplicity, that the coe$cients of the three-factor 10 J. Lewellen / Journal of Financial Economics 54 (1999) 5}43

J.Lewellen Journal of Financial Economics 54 (1999)5-43 11 model are linearly related to the firm's B/M ratio,or ait aio aiB/Mit-1),bit bio biB/Mit-1), (3 Sit Sio Si1 B/Mit -1),hit hio hi B/Mit-1) Substituting these equations into the unconditional regression yields a condi- tional version of the three-factor model: Ri=aio aiB/Mi+(bio biB/Mi)RM +(sio SiB/Mi)SMB (hio hiB/M)HML +ei, (4 where the time subscripts have been dropped to reduce clutter.Multiplying the factors through gives the regression equation for each portfolio.Thus,the conditional regressions contain not only an intercept and the three factors,but also four interactive terms with the portfolio's lagged B/M.2 Basically,Eq.(4)breaks the predictive power of B/M into risk and non-risk components.The coefficient an,the interactive term with the intercept, measures the predictive ability of B/M that is incremental to its association with risk in the three-factor model.A non-zero coefficient says that changes in the factor loadings,captured by the coefficients ba,si,and h,do not fully explain the time-series relation between B/M and expected return.Thus,rational asset- pricing theory predicts that ai will be zero for all stocks,assuming that the factors are adequate proxies for priced risk.The mispricing,or characteristics- based,view implies that B/M will forecast returns after controlling for risk and, consequently,an should be positive. 2.2.Discussion The conditional regressions directly test whether the three-factor model or the characteristic-based model better explains changes in expected returns.To interpret the regressions as a test of rational pricing,we must assume,of course, that the Fama and French factors capture priced risk in the economy.This assumption could be violated in two important ways(see Roll,1977).First,an equilibrium multifactor model might describe stock returns,but the Fama and French factors are not adequate proxies for the unknown risks.In this case,B/M can predict time-variation in expected returns missed by the three-factor model if it relates to the true factor loadings.Fortunately,this problem will not be 2Similar regressions appear in previous studies.Fama and French(1997)estimate regressions in which only the factor loadings on HML vary with B/M.He et al.(1996)estimate a model in the spirit of Eq.(4),but they constrain the intercepts and book-to-market coefficients to be the same across portfolios.Given previous cross-sectional evidence,the B/M coefficient will be non-zero in the absence of time-varying expected returns

model are linearly related to the "rm's B/M ratio, or a it"a i0 #a i1 B/Mi (t!1), b it"b i0 #b i1 B/Mi (t!1), (3) s it"s i0 #s i1 B/Mi (t!1), h it"h i0 #h i1 B/Mi (t!1). Substituting these equations into the unconditional regression yields a condi￾tional version of the three-factor model: Ri "a i0 #a i1 B/Mi #(b i0 #b i1 B/Mi )RM #(s i0 #s i1 B/Mi )SMB#(h i0 #h i1 B/Mi )HML#e i , (4) where the time subscripts have been dropped to reduce clutter. Multiplying the factors through gives the regression equation for each portfolio. Thus, the conditional regressions contain not only an intercept and the three factors, but also four interactive terms with the portfolio's lagged B/M.2 Basically, Eq. (4) breaks the predictive power of B/M into risk and non-risk components. The coe$cient a i1 , the interactive term with the intercept, measures the predictive ability of B/M that is incremental to its association with risk in the three-factor model. A non-zero coe$cient says that changes in the factor loadings, captured by the coe$cients b i1 , s i1 , and h i1 , do not fully explain the time-series relation between B/M and expected return. Thus, rational asset￾pricing theory predicts that a i1 will be zero for all stocks, assuming that the factors are adequate proxies for priced risk. The mispricing, or characteristics￾based, view implies that B/M will forecast returns after controlling for risk and, consequently, a i1 should be positive. 2.2. Discussion The conditional regressions directly test whether the three-factor model or the characteristic-based model better explains changes in expected returns. To interpret the regressions as a test of rational pricing, we must assume, of course, that the Fama and French factors capture priced risk in the economy. This assumption could be violated in two important ways (see Roll, 1977). First, an equilibrium multifactor model might describe stock returns, but the Fama and French factors are not adequate proxies for the unknown risks. In this case, B/M can predict time-variation in expected returns missed by the three-factor model if it relates to the true factor loadings. Fortunately, this problem will not be 2 Similar regressions appear in previous studies. Fama and French (1997) estimate regressions in which only the factor loadings on HML vary with B/M. He et al. (1996) estimate a model in the spirit of Eq. (4), but they constrain the intercepts and book-to-market coe$cients to be the same across portfolios. Given previous cross-sectional evidence, the B/M coe$cient will be non-zero in the absence of time-varying expected returns. J. Lewellen / Journal of Financial Economics 54 (1999) 5}43 11

12 J.Lewellen Journal of Financial Economics 54 (1999)5-43 a concern for the current paper because the three-factor model will,in fact, explain the predictability associated with B/M. Unfortunately,the assumption can also be violated in the opposite way: mispricing might explain deviations from the CAPM,but the size and book-to- market factors happen to absorb the predictive power of B/M.This possibility is a concern particularly because the factors are empirically motivated.Daniel and Titman (1997),for example,argue that the construction of HML,which is designed to mimic an underlying risk factor in returns related to B/M,could induce 'spurious'correlation between a portfolio's B/M ratio and its factor loading.HML is weighted,by design,towards firms with high B/M.If similar types of firms become mispriced at the same time,then we should expect that a firm will covary more strongly with HML when its B/M is high.As a result, apparent changes in risk might help explain B/M's predictive ability even under the mispricing story. In defense of the time-series regressions,it seems unlikely that changes in the factor loadings would completely absorb mispricing associated with B/M.More importantly,Daniel and Titman's argument cannot fully account for the rela- tion between B/M and risk.The argument suggests that the loadings on HML will tend to vary with B/M,but it does not say anything about the loadings on the market and size factors.We will see below,however,that B/M captures significant time variation in market betas and the loadings on SMB.Further, I provide evidence in Section 5.3 that the time-series relation between B/M and the factor loadings on HML is not driven by changes in the industry composi- tion of the factor.I estimate the conditional regressions with an 'industry neutral'factor,which prevents HML from becoming weighted towards particu- lar industries.When this factor is used in place of HML,we will continue to see a strong time-series relation between B/M and the factor loadings. Finally,it is useful to note that many industries have large unconditional factor loadings on HML,which suggests that HML does not simply capture mispric- ing in returns.Intuitively,Daniel and Titman's argument suggests that a given stock will sometimes vary positively and sometimes negatively with HML. Depending on the type of firms that are currently under-and overpriced,HML will be related to constantly changing micro-and macroeconomic factors.For example,HML will be sensitive to interest rate and inflation risk when it is weighted towards underpriced financial firms,but will be negatively related to these risks when financial firms are overpriced.Corresponding to the changes in HML,a stock will tend to covary positively with HML when similar firms are underpriced,but negatively when similar firms are overpriced.Over time, however,a firm's average factor loading on HML should be close to zero under the mispricing story,unless firms are persistently under-and overpriced (which seems unreasonable). This intuition can be formalized.Suppose that temporary overreaction ex- plains deviations from the CAPM,and that HML,because of its construction

a concern for the current paper because the three-factor model will, in fact, explain the predictability associated with B/M. Unfortunately, the assumption can also be violated in the opposite way: mispricing might explain deviations from the CAPM, but the size and book-to￾market factors happen to absorb the predictive power of B/M. This possibility is a concern particularly because the factors are empirically motivated. Daniel and Titman (1997), for example, argue that the construction of HML, which is designed to mimic an underlying risk factor in returns related to B/M, could induce &spurious' correlation between a portfolio's B/M ratio and its factor loading. HML is weighted, by design, towards "rms with high B/M. If similar types of "rms become mispriced at the same time, then we should expect that a "rm will covary more strongly with HML when its B/M is high. As a result, apparent changes in risk might help explain B/M's predictive ability even under the mispricing story. In defense of the time-series regressions, it seems unlikely that changes in the factor loadings would completely absorb mispricing associated with B/M. More importantly, Daniel and Titman's argument cannot fully account for the rela￾tion between B/M and risk. The argument suggests that the loadings on HML will tend to vary with B/M, but it does not say anything about the loadings on the market and size factors. We will see below, however, that B/M captures signi"cant time variation in market betas and the loadings on SMB. Further, I provide evidence in Section 5.3 that the time-series relation between B/M and the factor loadings on HML is not driven by changes in the industry composi￾tion of the factor. I estimate the conditional regressions with an &industry neutral' factor, which prevents HML from becoming weighted towards particu￾lar industries. When this factor is used in place of HML, we will continue to see a strong time-series relation between B/M and the factor loadings. Finally, it is useful to note that many industries have large unconditional factor loadings on HML, which suggests that HML does not simply capture mispric￾ing in returns. Intuitively, Daniel and Titman's argument suggests that a given stock will sometimes vary positively and sometimes negatively with HML. Depending on the type of "rms that are currently under- and overpriced, HML will be related to constantly changing micro- and macroeconomic factors. For example, HML will be sensitive to interest rate and in#ation risk when it is weighted towards underpriced "nancial "rms, but will be negatively related to these risks when "nancial "rms are overpriced. Corresponding to the changes in HML, a stock will tend to covary positively with HML when similar "rms are underpriced, but negatively when similar "rms are overpriced. Over time, however, a "rm's average factor loading on HML should be close to zero under the mispricing story, unless "rms are persistently under- and overpriced (which seems unreasonable). This intuition can be formalized. Suppose that temporary overreaction ex￾plains deviations from the CAPM, and that HML, because of its construction, 12 J. Lewellen / Journal of Financial Economics 54 (1999) 5}43

J.Lewellen Journal of Financial Economics 54 (1999)5-43 3 absorbs this mispricing (ignore the size factor for simplicity).To be more specific,assume that the proxy for the market portfolio,M,is not mean-variance efficient conditional on firms'B/M ratios.However,HML is constructed to explain the deviations from the CAPM,and Ry and HML together span the conditional tangency portfolio.The appendix proves that,in the time-series regression Ri(t)=ai +biRy(t)+hi HML(t)+ei(t), (5) the unconditional factor loading on HML,h,will equal zero if assets are correctly priced on average over time.3 This result reflects the idea that tempor- ary mispricing should not explain unconditional deviations from the CAPM.As noted above,however,many industries have large unconditional loadings on both SMB and HML,which therefore suggests that the factors do not simply capture mispricing in returns. In summary,the multifactor regressions test whether the three-factor model or the characteristic-based model explains time-variation in expected returns. The interpretation of the regressions,like the results for any asset-pricing test,is limited by our need to use a proxy for the unobservable equilibrium model of returns.Nevertheless,the regressions should help us understand whether the risk or mispricing story is a better description of asset prices. 3.Data and descriptive statistics The empirical analysis focuses on industry portfolios.These portfolios should exhibit cross-sectional variation in expected returns and risk,so the tests can examine a diverse group of portfolios.Industry portfolios are believed a priori to provide variation in expected returns and factor loadings,while sorting by other criteria is often motivated by previous empirical evidence.Hence,industry portfolios are less susceptible to the data-snooping issues discussed by Lo and MacKinlay (1990). As a robustness check,I also examine portfolios sorted by size and B/M.In cross-sectional studies,different sets of portfolios often produce vastly different estimates of risk premia.Of course,the time-series regressions in this paper might also be sensitive to the way portfolios are formed.Size portfolios have the advantage that they control for changes in market value,which has been shown to be associated with risk and expected returns,yet should be relatively stable 3 The result also requires that time-variation in b and h is uncorrelated with the factors'expected returns.This assumption seems reasonable since I am interested in the factor loadings changing over time with firm-specific variables,like B/M,not with macroeconomic variables.It is also consistent with the empirical evidence presented in Section 5

absorbs this mispricing (ignore the size factor for simplicity). To be more speci"c, assume that the proxy for the market portfolio, M, is not mean-variance e$cient conditional on "rms' B/M ratios. However, HML is constructed to explain the deviations from the CAPM, and RM and HML together span the conditional tangency portfolio. The appendix proves that, in the time-series regression Ri (t)"a i #b i RM (t)#h i HML(t)#e i (t), (5) the unconditional factor loading on HML, h i , will equal zero if assets are correctly priced on average over time.3 This result re#ects the idea that tempor￾ary mispricing should not explain unconditional deviations from the CAPM. As noted above, however, many industries have large unconditional loadings on both SMB and HML, which therefore suggests that the factors do not simply capture mispricing in returns. In summary, the multifactor regressions test whether the three-factor model or the characteristic-based model explains time-variation in expected returns. The interpretation of the regressions, like the results for any asset-pricing test, is limited by our need to use a proxy for the unobservable equilibrium model of returns. Nevertheless, the regressions should help us understand whether the risk or mispricing story is a better description of asset prices. 3. Data and descriptive statistics The empirical analysis focuses on industry portfolios. These portfolios should exhibit cross-sectional variation in expected returns and risk, so the tests can examine a diverse group of portfolios. Industry portfolios are believed a priori to provide variation in expected returns and factor loadings, while sorting by other criteria is often motivated by previous empirical evidence. Hence, industry portfolios are less susceptible to the data-snooping issues discussed by Lo and MacKinlay (1990). As a robustness check, I also examine portfolios sorted by size and B/M. In cross-sectional studies, di!erent sets of portfolios often produce vastly di!erent estimates of risk premia. Of course, the time-series regressions in this paper might also be sensitive to the way portfolios are formed. Size portfolios have the advantage that they control for changes in market value, which has been shown to be associated with risk and expected returns, yet should be relatively stable 3The result also requires that time-variation in b i and h i is uncorrelated with the factors' expected returns. This assumption seems reasonable since I am interested in the factor loadings changing over time with "rm-speci"c variables, like B/M, not with macroeconomic variables. It is also consistent with the empirical evidence presented in Section 5. J. Lewellen / Journal of Financial Economics 54 (1999) 5}43 13

14 J.Lewellen Journal of Financial Economics 54 (1999)5-43 over time.The book-to-market portfolios allow us to examine how the expected returns and risk of distressed,or high-B/M,firms change over time. The portfolios are formed monthly from May 1964 through December 1994, for a time series of 368 observations.The industry and size portfolios consist of all NYSE,Amex,and Nasdaq stocks on the Center for Research in Security Prices(CRSP)tapes,while the book-to-market portfolios consist of the subset of stocks with Compustat data.Stocks are sorted into 13 industry portfolios based on two-digit Standard Industrial Classification (SIC)codes as reported by CRSP.For the most part,the industries consist of consecutive two-digit codes, although some exceptions were made when deemed appropriate.4 The size portfolios are formed based on the market value of equity in the previous month,with breakpoints determined by NYSE deciles.To reduce the fraction of market value in any single portfolio,the largest two portfolios are further divided based on the 85th and 95th percentiles of NYSE stocks,for a total of 12 portfolios.Finally,the book-to-market portfolios are formed based on the ratio of book equity in the previous fiscal year to market equity in the previous month.Again,the breakpoints for these portfolios are determined by NYSE deciles.The lowest and highest deciles are further divided using the 5th and 95th percentiles of NYSE stocks,for a total of 12 portfolios. For all three sets of portfolios,value-weighted returns are calculated using all stocks with CRSP data,and value-weighted B/M ratios are calculated from the subset of stocks with Compustat data.To ensure that the explanatory power of B/M is predictive,I do not assume that book data become known until five months after the end of the fiscal year.Also,to reduce the effect of potential selection biases in the way Compustat adds firms to the database (see the discussion by Kothari et al.,1995),a firm must have three years of data before it is included in any calculation requiring book data.The time-series regressions use excess returns,calculated as returns minus the one-month T-bill rate,and the natural logarithm of B/M. Table 1 reports summary statistics for the portfolios.The average monthly returns for the industry portfolios range from 0.83%for utilities and telecommu- nications firms to 1.28%for the service industry (which includes entertainment, recreation,and services),for an annualized spread of 6.1%.Coincidentally,these industries also have the lowest(3.67%)and highest(6.78%)standard deviations, respectively.The size and book-to-market portfolios also exhibit wide variation in average returns and volatility.Average returns for the size portfolios vary Details available on request. s The stocks included in the calculation of B/M are a subset of those included in the calculation of returns,and we can interpret the estimate of B/M as a proxy for the entire portfolio.The inferences in this paper are unchanged when portfolio returns are based only on those stocks with Compustat data

over time. The book-to-market portfolios allow us to examine how the expected returns and risk of distressed, or high-B/M, "rms change over time. The portfolios are formed monthly from May 1964 through December 1994, for a time series of 368 observations. The industry and size portfolios consist of all NYSE, Amex, and Nasdaq stocks on the Center for Research in Security Prices (CRSP) tapes, while the book-to-market portfolios consist of the subset of stocks with Compustat data. Stocks are sorted into 13 industry portfolios based on two-digit Standard Industrial Classi"cation (SIC) codes as reported by CRSP. For the most part, the industries consist of consecutive two-digit codes, although some exceptions were made when deemed appropriate.4 The size portfolios are formed based on the market value of equity in the previous month, with breakpoints determined by NYSE deciles. To reduce the fraction of market value in any single portfolio, the largest two portfolios are further divided based on the 85th and 95th percentiles of NYSE stocks, for a total of 12 portfolios. Finally, the book-to-market portfolios are formed based on the ratio of book equity in the previous "scal year to market equity in the previous month. Again, the breakpoints for these portfolios are determined by NYSE deciles. The lowest and highest deciles are further divided using the 5th and 95th percentiles of NYSE stocks, for a total of 12 portfolios. For all three sets of portfolios, value-weighted returns are calculated using all stocks with CRSP data, and value-weighted B/M ratios are calculated from the subset of stocks with Compustat data.5 To ensure that the explanatory power of B/M is predictive, I do not assume that book data become known until "ve months after the end of the "scal year. Also, to reduce the e!ect of potential selection biases in the way Compustat adds "rms to the database (see the discussion by Kothari et al., 1995), a "rm must have three years of data before it is included in any calculation requiring book data. The time-series regressions use excess returns, calculated as returns minus the one-month T-bill rate, and the natural logarithm of B/M. Table 1 reports summary statistics for the portfolios. The average monthly returns for the industry portfolios range from 0.83% for utilities and telecommu￾nications "rms to 1.28% for the service industry (which includes entertainment, recreation, and services), for an annualized spread of 6.1%. Coincidentally, these industries also have the lowest (3.67%) and highest (6.78%) standard deviations, respectively. The size and book-to-market portfolios also exhibit wide variation in average returns and volatility. Average returns for the size portfolios vary 4 Details available on request. 5The stocks included in the calculation of B/M are a subset of those included in the calculation of returns, and we can interpret the estimate of B/M as a proxy for the entire portfolio. The inferences in this paper are unchanged when portfolio returns are based only on those stocks with Compustat data. 14 J. Lewellen / Journal of Financial Economics 54 (1999) 5}43