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 information about under- and overpricing. Thus, both explanations predict a positive slope coe$cient in the regression Ri (t)"c i0 #c i1 B/Mi (t!1)#e i (t), (1) where Ri is the portfolio's excess return and B/Mi is its lagged book-tomarket ratio. Note that Eq. (1) speci"es a separate time-series regression for each portfolio, with no constraint on the coe$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 adequately 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. Unconditional, 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 conditional 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$cients b i1 , s i1 , and h i1 , do not fully explain the time-series relation between B/M and expected return. Thus, rational assetpricing theory predicts that a i1 will be zero for all stocks, assuming that the factors are adequate proxies for priced risk. The mispricing, or characteristicsbased, view implies that B/M will forecast returns after controlling for risk and, consequently, a i1 should be positive. 2.2. Discussion The conditional regressions directly test whether the three-factor model or the characteristic-based model better explains changes in expected returns. To interpret the regressions as a test of rational pricing, we must assume, of course, that the Fama and French factors capture priced risk in the economy. This assumption could be violated in two important ways (see Roll, 1977). First, an equilibrium multifactor model might describe stock returns, but the Fama and French factors are not adequate proxies for the unknown risks. In this case, B/M can predict time-variation in expected returns missed by the three-factor model if it relates to the true factor loadings. Fortunately, this problem will not be 2 Similar regressions appear in previous studies. Fama and French (1997) estimate regressions in which only the factor loadings on HML vary with B/M. He et al. (1996) estimate a model in the spirit of Eq. (4), but they constrain the intercepts and book-to-market coe$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-tomarket 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 relation 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 composition of the factor. I estimate the conditional regressions with an &industry neutral' factor, which prevents HML from becoming weighted towards particular 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 mispricing 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 explains 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 temporary 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 telecommunications "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