J.Lewellen Journal of Financial Economics 54 (1999)5-43 15 from 0.80%for the largest stocks to 1.24%for the smallest stocks,and the standard deviations of returns decrease monotonically with size,from 6.68%to 4.17%.Average returns for the book-to-market portfolios range from 0.76%for the second decile through 1.46%for the stocks with the highest B/M.Interest- ingly,the standard deviation of returns are U-shaped;they decrease monotoni- cally with B/M until the sixth decile,which has a standard deviation of 4.42%, and increase thereafter,to 6.86%for portfolio 10b. The statistics for B/M,like those for returns,reveal considerable cross- sectional differences in portfolio characteristics.Average B/M doubles from 0.40 for chemical firms to 0.82 for the transportation industry.A similar spread is shown for size portfolios,with B/M ranging from 0.51 for the largest stocks to 1.03 for the smallest stocks.The book-to-market portfolios,of course,have the greatest cross-sectional variation,with average B/M ranging from 0.15 for the low-B/M portfolio to 2.66 for the high-B/M portfolio.The standard deviations over time are also reasonably high,reflecting the volatility of stock returns.The time-series standard deviation of B/M is,on average,0.20 for the industries,0.24 for the size portfolios,and 0.29 for the book-to-market portfolios.Variation in B/M will be necessary for the time-series regressions to have power distinguish- ing between the competing hypotheses. Table 2 reports summary statistics for the Fama and French(1993)factors, which are described fully in the appendix.The market factor,RM,is the excess return on the CRSP value-weighted index,and the size and book-to-market factors,SMB and HML,are zero-investment portfolios designed to mimic underlying risk factors in returns.The average monthly return of Ry is 0.39%,of SMB is 0.30%,and of HML is 0.38%.The risk premium for each factor is measured by its mean return,so these averages imply positive compensation for bearing factor risk.As noted by Fama and French,the procedure used to construct SMB and HML appears to successfully control each factor for the influence of the other,as demonstrated by the low correlation between the factors,equal to -0.06.Also,SMB is positively correlated with RM(correlation of 0.36),while HML is negatively correlated with R(-0.35).Thus,the returns on the size and B/M factors are not independent of the market return,reflecting the fact that their construction did not control for differences in the betas of the underlying stocks. The CAPM and most empirical studies examine the relation between simple- regression market betas and expected returns.To enhance comparison with cross-sectional studies,I use size and B/M factors that are orthogonal to Rm. These factors,SMBO and HMLO,are constructed by adding the intercepts to the residuals when SMB and HML are regressed on a constant and the excess market return.From regression analysis (e.g.,Johnston,1984,p.238),the coefficients in the three-factor model will be unaffected by the change in variables,except that market betas will now be the simple-regression betas of the CAPM.Table 2 shows that the average return on the book-to-market factor
from 0.80% for the largest stocks to 1.24% for the smallest stocks, and the standard deviations of returns decrease monotonically with size, from 6.68% to 4.17%. Average returns for the book-to-market portfolios range from 0.76% for the second decile through 1.46% for the stocks with the highest B/M. Interestingly, the standard deviation of returns are U-shaped; they decrease monotonically with B/M until the sixth decile, which has a standard deviation of 4.42%, and increase thereafter, to 6.86% for portfolio 10b. The statistics for B/M, like those for returns, reveal considerable crosssectional di!erences in portfolio characteristics. Average B/M doubles from 0.40 for chemical "rms to 0.82 for the transportation industry. A similar spread is shown for size portfolios, with B/M ranging from 0.51 for the largest stocks to 1.03 for the smallest stocks. The book-to-market portfolios, of course, have the greatest cross-sectional variation, with average B/M ranging from 0.15 for the low-B/M portfolio to 2.66 for the high-B/M portfolio. The standard deviations over time are also reasonably high, re#ecting the volatility of stock returns. The time-series standard deviation of B/M is, on average, 0.20 for the industries, 0.24 for the size portfolios, and 0.29 for the book-to-market portfolios. Variation in B/M will be necessary for the time-series regressions to have power distinguishing between the competing hypotheses. Table 2 reports summary statistics for the Fama and French (1993) factors, which are described fully in the appendix. The market factor, RM , is the excess return on the CRSP value-weighted index, and the size and book-to-market factors, SMB and HML, are zero-investment portfolios designed to mimic underlying risk factors in returns. The average monthly return of RM is 0.39%, of SMB is 0.30%, and of HML is 0.38%. The risk premium for each factor is measured by its mean return, so these averages imply positive compensation for bearing factor risk. As noted by Fama and French, the procedure used to construct SMB and HML appears to successfully control each factor for the in#uence of the other, as demonstrated by the low correlation between the factors, equal to !0.06. Also, SMB is positively correlated with RM (correlation of 0.36), while HML is negatively correlated with RM (!0.35). Thus, the returns on the size and B/M factors are not independent of the market return, re#ecting the fact that their construction did not control for di!erences in the betas of the underlying stocks. The CAPM and most empirical studies examine the relation between simpleregression market betas and expected returns. To enhance comparison with cross-sectional studies, I use size and B/M factors that are orthogonal to RM . These factors, SMBO and HMLO, are constructed by adding the intercepts to the residuals when SMB and HML are regressed on a constant and the excess market return. From regression analysis (e.g., Johnston, 1984, p. 238), the coe$cients in the three-factor model will be una!ected by the change in variables, except that market betas will now be the simple-regression betas of the CAPM. Table 2 shows that the average return on the book-to-market factor J. Lewellen / Journal of Financial Economics 54 (1999) 5}43 15
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Table 1 Summary statistics for industry, size, and book-to-market portfolios, 5/64}12/94 Each month from May 1964 through December 1994, value-weighted portfolios are formed from all NYSE, Amex, and Nasdaq stocks on CRSP. Firms must also have Compustat data for the book-to-market portfolios. Book-to-market (B/M) is calculated as the ratio of book equity in the previous "scal year to market equity in the previous month for all stocks with Compustat data. The industry portfolios are based on two-digit SIC codes. The size portfolios are based on the market value of equity in the previous month, with breakpoints determined by NYSE deciles; portfolios 9 and 10 are further divided using the 85 and 95 percentiles of NYSE stocks. The book-to-market portfolios are based on B/M in the previous month, with breakpoints determined by NYSE deciles; portfolios 1 and 10 are further divided using the 5 and 95 percentiles of NYSE stocks Portfolio Return (%) Book-to-market Number of "rms Mean Std. dev. Mean Std. dev. Autocorr. Adj. R2 ! May 1964 Dec. 1994 Panel A: Industry portfolios Nat. resources 0.84 5.69 0.57 0.14 0.97 0.28 90 360 Construction 0.86 5.45 0.78 0.26 0.99 0.69 237 409 Food, tobacco 1.21 4.57 0.51 0.18 0.99 0.36 106 134 Consumer products 1.05 6.02 0.75 0.36 0.99 0.81 108 257 Logging, paper 0.98 5.35 0.54 0.15 0.98 0.54 74 190 Chemicals 0.96 4.78 0.40 0.13 0.99 0.28 102 392 Petroleum 1.08 5.23 0.74 0.20 0.98 0.38 30 32 Machinery, equipment 0.88 5.35 0.42 0.13 0.99 0.40 290 1222 Transportation 0.87 5.39 0.82 0.27 0.98 0.71 162 260 Utilities, telecom. 0.83 3.67 0.77 0.25 0.99 0.57 122 384 Trade 1.04 5.65 0.49 0.18 0.98 0.55 167 785 Financial 0.95 4.75 0.75 0.19 0.97 0.68 117 1747 Services and other 1.28 6.78 0.47 0.20 0.98 0.55 61 981 16 J. Lewellen / Journal of Financial Economics 54 (1999) 5 }43
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Panel B: Size portfolios Smallest 1.24 6.68 1.03 0.41 0.98 0.88 409 3338 2 1.16 6.17 0.90 0.32 0.97 0.79 175 932 3 1.15 6.08 0.85 0.30 0.97 0.77 154 634 4 1.16 5.91 0.84 0.31 0.97 0.80 149 463 5 1.21 5.70 0.76 0.26 0.96 0.80 136 426 6 1.22 5.41 0.72 0.23 0.97 0.88 130 333 7 1.07 5.19 0.68 0.19 0.97 0.88 132 310 8 1.09 5.13 0.67 0.19 0.98 0.86 130 274 9a 1.02 4.89 0.67 0.18 0.97 0.78 62 122 9b 0.97 4.71 0.66 0.20 0.97 0.78 64 111 10a 0.88 4.50 0.62 0.17 0.98 0.70 63 109 Largest 0.80 4.17 0.51 0.15 0.99 0.41 62 109 Panel C: Book-to-market portfolios Lowest 0.98 5.69 0.15 0.05 0.97 0.59 21 559 1b 0.83 5.15 0.24 0.07 0.97 0.64 19 328 2 0.76 5.04 0.34 0.10 0.97 0.82 40 493 3 0.79 4.78 0.46 0.14 0.98 0.93 39 470 4 0.83 4.63 0.57 0.18 0.98 0.94 39 469 5 0.82 4.47 0.67 0.21 0.98 0.94 42 492 6 0.98 4.42 0.78 0.24 0.98 0.94 40 467 7 1.17 4.54 0.89 0.28 0.98 0.94 42 461 8 1.25 4.72 1.04 0.32 0.98 0.95 43 460 9 1.43 5.20 1.28 0.40 0.98 0.96 43 574 10a 1.46 6.12 1.65 0.53 0.97 0.93 24 378 Highest 1.46 6.86 2.66 0.95 0.96 0.84 25 482 !Adjusted R2 from regressing the portfolio's B/M ratio on the value-weighted B/M ratio of all stocks that meet both CRSP and Compustat data requirements. J. Lewellen / Journal of Financial Economics 54 (1999) 5}43 17