Modeling and Estimating a Higher Systematic Co-Moment Asset Pricing Model in the Brazilian Stock Market. Autoria: Andre Luiz Carvalhal da Silva

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1 Modeling and Estimating a Higher Systematic Co-Moment Asset Pricing Model in the Brazilian Stock Market Autoria: Andre Luiz Carvalhal da Silva Abstract Many asset ricing models assume that only the second-order systematic co-moment (CAPM beta) should be riced by investors. However, since asset returns are not normal, investors are also concerned about higher moments and systematic co-moments (co-skewness, co-kurtosis, and so on). This aer examines the determinants of Brazilian stock returns from the CAPM beta, Fama-French size and value factors, and using higher-order systematic co-moments that encomass risks above the traditional CAPM beta. Our results indicate that the CAPM beta and the Fama-French size and value factors aear to be imortant for exlaining returns in Brazil, and jointly rovide statistically significant exlanatory ower across almost all the samle return eriods. In general, adding a set of higher-order systematic co-moments has moderate exlanatory ower. 1 - Introduction Many emirical studies have found that the Caital Asset Pricing Model (CAPM) beta has moderate or even insignificant exlanatory ower once the Fama and French factors are included. In contrast, Fama and French factors remain highly significant in exlaining the cross-section of stock returns. Over a series of aer, Fama and French (199, 199, 1996) argue that the size and value of a stock are additional risk factors that exlain cross-sectional stock returns. Together with beta, they are widely known as the Fama-French three-factor model. The Fama and French three-factor model uses in addition to market risk two non-market risk factors: SMB (size risk) and HML (value risk). The SMB, which stands for Small minus Big, is designed to measure the additional return investors have historically received by investing in stocks for comanies with relatively small market caitalization. This additional return is oen referred to as the size remium. The HML, which stands for High minus Low, has been constructed to measure the value remium rovided to investors for investing in comanies with high book-to-market values. There is also evidence in the literature (Rubinstein (197), Kraus and Litzenberger (1976), Friend and Westerfield (1980), Scott and Horvath (1980), Bonsal and Viswanathan (199), Harvey and Siddique (000), Kan and Wang (001), Dittmar (00), Hung, Shackleton and Xu (00), Chung, Johnson and Schill (00)) that investors may care about higher moments (skewness, kurtosis, and so on), and higher co-moments (co-skewness, co-kurtosis, and so on). Kraus and Litzenberger (1976) were the first to suggest that higher co-moments may be riced, in addition to the first co-moment of stock returns with the market return (beta). The CAPM suggests that only market risk should be riced by investors. One of the crucial assuion of the CAPM is the normality of returns. If the CAPM holds, investors care only about two moments (mean and variance) and one co-moment (covariance). Therefore, only the second-order systematic co-moment (beta) should be riced. Without normality, the CAPM is unlikely to hold. When the returns are normal, we only need the mean and variance to describe the distribution erfectly. However, in general, an infinite number of moments are required to secify the tails comletely. Investors are concerned about risk, and risk must be measured in terms of the entire robability

2 distribution, which in turn can be measured with the moments of the distribution. Only in very secial cases, such as quadratic utility or normality of returns, we can ignore the higher moments and systematic co-moments. Since asset returns are not normal, investors are also concerned about ortfolio skewness and kurtosis, and each stock s contribution to systematic skewness (co-skewness) and kurtosis (co-kurtosis). Kraus and Litzenberger (1976), and Harvey and Siddique (000) develoed asset ricing models incororating co-skewness terms, while Dittmar (00) and Hung, Shackleton and Xu (00) develoed models incororating co-kurtosis terms as well. Chung, Johnson and Schill (00) argue that higher-order co-moments matter to riskaverse investors, and show that adding a set of systematic co-moments of order through 10 reduces the exlanatory ower of the Fama-French factors to insignificance in almost every case. Higher-order systematic co-moments have been criticized for lacking intuition and being unreliable. However, as Chung, Johnson and Schill (00) oint out, each co-moment may individually be unreliable, but the set of co-moments should not be, because it is a measure of the likelihood of extreme outcomes. Although a number of aers have tested higher order ricing models for U.S stock data (Harvey and Siddique, 000), Dittmar (00), Barone-Adesi, Gagliardini and Urga (00)), there has been little work outside the U.S (Galagedera, Henry and Silvaulle (00) in Australia, and Hung, Shackleton and Xu in the U.K. (00)). This aer examines the determinants of the cross-section of Brazilian stock returns from the CAPM beta, Fama-French size and value factors, and using higher order asset ricing models that encomass systematic risks above the traditional CAPM beta. We also test the hyothesis that the Fama-French factors simly roxy for the ricing of higher-order comoments. The growing economic imortance of the so-called emerging markets and the issue of financial asset return forecasting in the region have recently attracted the attention of investors and academics. Since the beginning of the 1990 s, the Brazilian stock market figures rominently among star erformers of the emerging markets. It seems worthwhile to take a close look at how redictable risk and returns have been on the Brazilian stock market in the last 15 years. The aer is organized as follows. The next section resents the data and methodology used in this aer. Section reorts the main emirical results. Conclusions are resented in Section.. Data Descrition and Estimation We exlore the return characteristics of Brazilian stock returns. Brazil is the largest emerging market in Latin America, and one the largest emerging markets in the world. Table 1 reorts the market caitalization of selected emerging markets. Taiwan is the largest emerging market, with a total market caitalization of US$ 79 billion, followed by Korea (US$ 98 billion), India (US$ 5 billion), and Brazil (US$ 6 billion). Our samle consists of firms listed in Sao Paulo Stock Exchange (Bovesa). The monthly closing rices were obtained from Jan 1990 through Dec 00. Our roxy for the risk-free rate is the Interbank Certificate of Deosit (CDI) yield. We collected information on stock rices, risk-free rate, accounting data and market value of equity from the Economatica database. To test the models, we could theoretically use the returns on the individual securities, but for estimating uroses we have to limit the number of arameters. We follow common

3 ractice in grouing the stocks into ortfolios and testing the model on a small set of ortfolios. We form equally-weighted ortfolios to analyze the return on a grou of stocks instead of individually. Forming ortfolios allows the factor of interest (size and book-to-market) to be concentrated by conducting a sort on that factor and reduces the error associated with misestimation of the stocks s factor itself and the unexlained variance in return rediction. Given the limited number of available firms in Brazil, five size-sorted ortfolios and five book-to-market-sorted ortfolios are constructed for the regressions (five is chosen as tradeoff between the number of stocks in each ortfolio and the number of ortfolio return observations in the regressions). The ortfolios are formed at the end of December every year from 1990 to 00. In order to qualify for analysis in any eriod, a stock must have an observed market caitalization, book value and monthly return data in that eriod. In calculating the book-to-market equity ratio, we use the book value for the latest fiscal year divided by the market value of equity. Stocks that have a negative book value to market ratio are excluded. In total, 1 stocks satisfied the criterion and are analyzed, although not all stocks had a comlete rice or return history during the entire eriod. At the end of each calendar year, we rank all stocks by market caitalization, divide the samle into 5 ortfolios of equal size, and comute a time series of equal-weighted and rebalanced ortfolios returns. We reeat this rocedure for the stratification using book-tomarket ratio. The small size ortfolio ( small ) consists of stocks in the smallest size quintile, and the large size ortfolio ( large ) consists of stocks in the largest size quintile. The high book-to-market ortfolio ( high ) consists of stocks in the highest quintile, and the low bookto-market ( low ) consists of stocks in the lowest quintile. Once constructed, the ortfolios are held for one year and the 1 monthly equally weighted ortfolio returns are calculated. The full-eriod ortfolio returns are calculated for the entire 168-month eriod, from Jan 1990 to Dec 00. The SMB monthly factor is comuted as the average return for the smallest stocks minus the average return of the largest stocks in that month. A ositive SMB indicates that small ca stocks outerformed large ca stocks in that month, while a negative SMB indicates the large ca stocks outerformed. Constructed in a fashion similar to that of SMB, HML is comuted as the average returns of stocks with the highest book-to-market ratio minus the average return of stocks with the lowest book-to-market ratio each month. A ositive HML indicates that value stocks outerformed growth stocks in that month, while a negative HML indicates that growth stocks outerformed. For each eriod t, we subtract the risk-free rate to obtain the excess ortfolio return (R t - R ), and estimate a regression of excess ortfolio returns on the excess market returns (R - R ), squared excess market returns (R - R ), cubed excess market returns (R - R ), and on the returns of the Fama-French small-minus-big (SMB) and high-minus-low (HML) ortfolios. R t = α + β ( R ) + γ ( R ) + δ ( R ) + s SMB + h HML + ε where the estimates β, γ, δ, s, h reresent the systematic covariance, co-skewness and co-kurtosis, and the factor loadings for the SMB and HML, resectively. Kraus and Litzenberger (1976) state that it is trivial to extend the model to incororate any number of higher co-moments. Chung, Johnson and Schill (00) argue that higher-order comoments matter to risk-averse investors, and show that adding a set of systematic comoments of order through 10 reduces the exlanatory ower of the Fama-French factors to insignificance in almost every case. In order to test the exlanatory ower of the SMB and

4 HML factors, we also estimated the regression adding higher co-moments of order through 10: R t = 10 i 1 α + φ i ( R ) + s SMB + h HML + i= ε where the estimates φ i, s, h reresent the i th systematic co-moment, and the factor loadings for the SMB and HML, resectively. For examle, the nd systematic co-moment is the CAPM beta and the rd systematic co-moment is the Kraus and Litzenberger (1976) systematic co-skewness.. Emirical Results Table rovides summary statistics of ortfolio monthly returns in local currency from 1990 to 00. Panel A gives the results for size-sorted ortfolios, while Panel B gives the results for book-to-market-ratio-sorted ortfolios. The small size ortfolio ( small ) consists of stocks in the smallest size quintile, and the large size ortfolio ( large ) consists of stocks in the largest size quintile. The high book-to-market ortfolio ( high ) consists of stocks in the highest quintile, and the low book-to-market ( low ) consists of stocks in the lowest quintile. Although simle average returns do not aear to decrease with the ortfolio size, the median returns seem to decrease with ortfolio size. The small ortfolio has a median monthly return of 0.78%, while the large ortfolio has a median monthly return of 0.06%. The small minus big (SMB) ortfolio has a median return of 0.8%. The ortfolios skewness and kurtosis are consistently different than the corrected standard normal distribution, and the Jarque-Bera statistic strongly rejects normality. In Panel B, although there is not a clear and monotonic relationshi between average return and book-to-market, the high ortfolio has a median monthly return of -0.5% (mean of 1.06%), while the low ortfolio has a median monthly return of -1.56% (mean of 0.8%). The high minus low (HML) ortfolio has a median return of 1.18% (mean of 0.%). All series deart strongly from normality, as indicated by the skewness, kurtosis, and Jarque-Bera statistic. Then, we estimate a regression of excess ortfolio returns on the excess market returns, squared and cubed excess market returns, and on the returns of the Fama-French SMB and HML factors. Table shows the results for size-sorted ortfolios. The intercet terms are all insignificant and beta is very significant in exlaining time-series return of ortfolios from 1990 to 00. Adding co-moment terms (co-skewness and co-kurtosis) does increase the exlanatory ower of the model. In most models, the co-skewness is not statistically significant, but the co-kurtosis is statistically significant at the 1% level. When the Fama-French factors are included without the higher order co-moment terms, the SMB factor is statistically significant at the 1% level, and the HML factor is statistically significant at the 5% and 10% levels. When the model includes beta, higher co-moments and the Fama-French factors, the beta, co-kurtosis and SMB factor remain highly statistically significant, but the co-skewness and the HML factor are mostly insignificant. Table shows the results for the book-to-market-sorted ortfolios. Beta is very significant in exlaining time-series return of ortfolios, the co-skewness is not statistically significant, and the co-kurtosis is statistically significant in most models. The SMB and HML factors are

5 statistically significant at the 1% level. Not surrisingly, the SMB factor does better in sizesorted ortfolios and the HML does better in book-to-market-sorted ortfolios. Then, following Chung, Johnson and Schill (00), we test if adding a set of systematic co-moments of order through 10 reduces the exlanatory ower of the Fama-French factors. Table 5 shows the results for size-sorted ortfolios. Adding a set of systematic co-moments of order through 10 does not reduce the exlanatory ower of beta and the SMB factor. In every model, they remain statistically significant, whereas the HML factor is not statistically significant, excet for ortfolio. Furthermore, the values of adjusted R are largely unchanged when we add higher co-moments. Table 6 shows the results for the book-to-market-sorted ortfolios. Adding a set of systematic co-moments of order through 10 does not reduce the exlanatory ower of beta and the SMB factor. In every model, they remain statistically significant, whereas the HML factor is statistically significant for most models. Not surrisingly, the HML does better in book-to-market-sorted ortfolios than in size-sorted ortfolios. Overall, the results indicate that beta is highly significant in exlaining time-series return in Brazil from 1990 to 00, even when we add higher-order co-moment terms and the Fama- French size and value factors. In most models, the co-skewness is insignificant, but the cokurtosis is statistically significant. At least one of the Fama-French factors has significant exlanatory ower for ortfolio returns, and, not surrisingly, the SMB factor does better in size-sorted ortfolios and the HML does better in book-to-market-sorted ortfolios. Adding a set of systematic co-moments of order through 10 seems not to reduce the exlanatory ower of beta and the SMB and HML factors. In order to check the robustness of our results, we also analyze three different sub-eriods according to the Brazilian macro-economic situation: high inflation ( ), stabilization with fixed exchange rate ( ), and stabilization with floating exchange rate ( ). The results, which are not reorted here to conserve sace, are quite the same as those for the entire eriod. During the high inflation eriod, the beta, the SMB and HML factors and the co-kurtosis remain significant. On the other hand, during the stabilization eriods, beta, SMB and HML are significant, but neither the co-skewness nor the co-kurtosis is statistically significant. One ossible exlanation for the difference in the results for the high inflation and stabilization eriods may be associated with the resence of extremely low and high stock returns in the Brazilian stock market during the high inflation eriod. Since Brazil was affected by a high and variable inflation from 1990 to 199 eriod, ortfolio returns aear to be more volatile during this eriod. Therefore, adding higher-order systematic co-moments rovides better results since the set of higher co-moments tend to measure more accurately the robability distribution and the likelihood of extreme outcomes.. Conclusions There is evidence in the literature that the Caital Asset Pricing Model (CAPM) beta has moderate or even insignificant exlanatory ower once the Fama and French size and value factors are included. If the CAPM holds, only the second-order systematic co-moment (beta) should be riced. When the returns are normal, we only need the mean and variance to describe the distribution comletely. However, since asset returns are not normal, investors are also concerned about higher moments (skewness, kurtosis, and so on) and co-moments (coskewness, co-kurtosis, and so on). This aer examines the determinants of the cross-section of Brazilian stock returns from the CAPM beta, Fama-French size and value factors, and using higher order asset ricing 5

6 models that encomass systematic risks above the traditional CAPM beta. We also test if adding a set of systematic co-moments of order through 10 reduces the exlanatory ower of the Fama-French factors. Our results indicate that beta is highly significant in exlaining time-series return in Brazil from 1990 to 00, even when we add higher-order co-moment terms and the Fama-French size and value factors. In most models, the co-skewness is insignificant, but the co-kurtosis is statistically significant. At least one of the Fama-French factors has significant exlanatory ower for ortfolio returns, and, not surrisingly, the SMB factor does better in size-sorted ortfolios and the HML does better in book-to-market-sorted ortfolios. Adding a set of systematic co-moments of order through 10 seems not to reduce the exlanatory ower of beta and the SMB and HML factors. In order to check the robustness of our results, we also analyze three different sub-eriods according to the Brazilian macro-economic situation. The results are quite the same as those for the entire eriod. During the high inflation eriod, the beta, the SMB and HML factors and the co-kurtosis remain significant. On the other hand, during the stabilization eriods, beta, SMB and HML are significant, but neither the co-skewness nor the co-kurtosis is statistically significant. One ossible exlanation for the difference in the results for the high inflation and stabilization eriods may be associated with the resence of extremely low and high stock returns in the Brazilian stock market during the high inflation eriod. Since Brazil was affected by a high and variable inflation from 1990 to 199 eriod, ortfolio returns aear to be more volatile during this eriod. Therefore, adding higher-order systematic co-moments rovides better results since the set of higher co-moments tend to measure more accurately the robability distribution and the likelihood of extreme outcomes. Overall, we can conclude that the CAPM beta and the Fama-French size and value factors aear to be imortant for exlaining returns in Brazil, and jointly rovide statistically significant exlanatory ower across almost all the samle return eriods. In general, adding a set of higher-order systematic co-moments has moderate exlanatory ower. 5. References Barone-Adesi, G., Gagliardini, P. and G. Urga (00), Homogeneity Hyothesis in the Context of Asset Pricing Models: The Quadratic Market Model, City University Working Paer. Bonsal, R. and S. Viswanathan (199), No Arbitrage and Arbitrage Pricing, Journal of Finance, v. 8, Chung, Y., Johnson, H. and M. Schill (00), Asset Pricing When Returns are Nonnormal: Fama-French Factors vs. Higher-Order Systematic Co-Moments, Journal of Business, forthcoming. Dittmar, R. (00), Nonlinear Pricing Kernels, Kurtosis Preference, and Evidence from the Cross-Section of Equity Returns, Journal of Finance, v. 57, n. 1, Fama, E. and K. French (199), The Cross-Section of Exected Stock Returns, Journal of Finance, v. 7, n., Fama, E. and K. French (199), Common Risk Factors in the Returns on Stocks and Bonds, Journal of Financial Economics, v., Fama, E. and K. French (1996), Multifactor Exlanations of Asset Pricing Anomalies, Journal of Finance, v. 51, n. 1, Friend, I. and R. Westerfield (1980), Co-Skewness and Caital Asset Pricing, Journal of Finance, v. 5,

7 Galagedera, D., Henry, D. and P. Silvaulle (00), Conditional Relation Between Higher Co-Moments and Stock Returns: Evidence from Australia, Monash/Latrobe University Working Paer. Harvey, C. and A. Siddique (000), Conditional Skewness in Asset Pricing Tests, Journal of Finance, v. 55, Hung, D, Shackleton, M. and X. Xu (00), CAPM, Higher Co-Moment and Factor Models of UK Stock Returns, Lancaster/Peking University Working Paer. Kan, R. and K. Wang (001), Non-Linear APT versus the Conditional CAPM: an Emirical Comarison, University of Toronto Working Paer. Kraus, A. and R. Litzenberger (1976), Skewness Preference and the Valuation of Risk Assets, Journal of Finance, v. 1, Rubinstein, M. (197), The Fundamental Theorem of Parameter-Preference Security Valuation, Journal of Financial and Quantitative Analysis, v. 8, Scott, R. and P. Horvath (1980), On the Direction of Preference for Moments of Higher Order than the Variance, Journal of Finance, v. 5, Table 1 Emerging Markets Profile as of December 00 Market Caitalization in Emerging Countries at the end of 00. Market Market Caitalization (US$ million) Latin America Argentina,99.7 Brazil 6,57.7 Chile 87,508. Mexico 1,5.0 Asia India 5,89. Indonesia 5,659.1 Korea 98,8.1 Malaysia 160,970. Singaore 18,50.6 Taiwan 79,060. Thailand 119,017. Develoed Markets U.S (NYSE) 11,8,95.1 Jaan,95,098. Source: World Federation of Exchanges Table Summary Statistics of Portfolio Returns in Brazil Jan 1990 to Dec 00 Descritive statistics of monthly returns (%) in local currency of five equal-sized ortfolios in Brazil from Jan 1990 to Dec 00. Panel A gives the results for size-sorted ortfolios, while Panel B gives the results for bookto-market-ratio-sorted ortfolios. The small size ortfolio ( small ) consists of stocks in the smallest size quintile, and the large size ortfolio ( large ) consists of stocks in the largest size quintile. The high book-tomarket ortfolio ( high ) consists of stocks in the highest quintile, and the low book-to-market ( low ) consists of stocks in the lowest quintile. The last columns contain the small minus big (SMB) and the high minus low (HML) ortfolios. Panel A: Size-Sorted Portfolios Small Large SMB Mean Median Maximum Minimum

8 Std. Dev Skewness Kurtosis Jarque-Bera (JB) JB Probability Panel B: Book-to-Market-Sorted Portfolios High Low HML Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera (JB) JB Probability Table Time-Series Regressions of Size-Sorted Portfolios Jan 1990 to Dec 00 Time-series regression of size-sorted ortfolio excess returns on the excess market returns, squared and cubed excess market returns, and on the returns of the Fama-French small-minus-big (SMB) and high-minus-low (HML) ortfolios. The estimates β, γ, δ, s, h reresent the systematic covariance, co-skewness and co-kurtosis, and the factor loadings for the SMB and HML, resectively. The coefficient estimates are reorted with their - values. *, ** and *** denote significance at the 1%, 5% and 10% levels, resectively. R = α + β ( R ) + γ ( R ) + δ ( R ) + s SMB + h HML + ε t Model Portfolio α β γ δ s h Adj. R Small * (0.98) () * (0.5) () * (0.95) () * (0.15) () 0.78 Large * (0.96) () 0.88 Small * (0.76) () (0.) (0.66) * (0.7) () (0.75) * 0.17*** -0.71* (0.65) () (0.08) () * * (0.) () (0.0) () 0.81 Large * * (0.9) () (0.5) () 0.91 Small 0.80* 0.86* -0.05** (0.99) () () (0.0) * 0.09** (0.0) () () (0.0) * 0.06*** (0.97) () () (0.09) * (0.16) () (0.15) (0.0)

9 Large Small Large (0.99) (0.99) 0.7 (0.8) -0.1 (0.8) (0.0) (0.99) 0.80* () 0.69* () 0.65* () 0.69* () () 0.69* () -0.0 (0.5) -0.0 (0.87) 0.19** (0.0) (0.5) -0.0 (0.) 0.5* () -0.1 (0.8) -0.6* () 0.58* () 0.5* () -0.1* () 0.90* () 0.5* () 0.9* () 0.10* -0.10* () -0.05** (0.0) -0.0 (0.) 0.0 (0.) - (0.9) -0.0 (0.) Table Time-Series Regressions of Book-to-Market Portfolios Jan 1990 to Dec 00 Time-series regression of book-to-market ortfolio excess returns on the excess market returns, squared and cubed excess market returns, and on the returns of the Fama-French small-minus-big (SMB) and high-minus-low (HML) ortfolios. The estimates β, γ, δ, s, h reresent the systematic covariance, co-skewness and co-kurtosis, and the factor loadings for the SMB and HML, resectively. The coefficient estimates are reorted with their - values. *, ** and *** denote significance at the 1%, 5% and 10% levels, resectively. R = α + β ( R ) + γ ( R ) + δ ( R ) + s SMB + h HML + ε t Model Portfolio α β γ δ s h Adj. R High * (0.65) () * (0.65) () * (0.9) () () 0.7 Low * (0.97) () 0.5 High * (0.) () (0.9) (0.50) * ** (0.5) () (0.7) (0.0) * (0.) () (0.98) (0.7) * *** (0.07) () (0.97) (0.10) 0.7 Low * * (0.76) () (0.) () 0.9 High * 0.8* 0.18* (0.6) () () () * 0.* 0.07* (0.50) () () * 0.5* (0.0) () () (0.15) * 0.1* - () () (0.66) 0.75 Low * 0.8* -0.8* (0.6) () () () 0.85 High * 0.1* 0.0* (0.5) () (0.8) () () * * 0.08* (0.) () (0.19) (0.8) ()

10 Low (0.) -1.0 (0.0) 0.5 (0.5) 0.6* () 0.59* () () 0.0 (0.65) (0.89) (0.8) 0.1 (0.8) 0.* 0.9* 0.7* () 0.17* () 0.1* () (0.5) (0.75) ()

11 Table 5 Time-Series Regressions of Size-Sorted Portfolios Jan 1990 to Dec 00 Time-series regression of size-sorted ortfolio excess returns on the resective number of systematic co-moments, and on the returns of the Fama-French small-minus-big i, s, h (SMB) and high-minus-low (HML) ortfolios. The estimates φ reresent the i th systematic co-moment, and the factor loadings for the SMB and HML, resectively. For examle, the nd systematic co-moment is the CAPM beta and the rd systematic co-moment is the Kraus and Litzenberger (1976) systematic co-skewness. The coefficient estimates are reorted with their -values. *, ** and *** denote significance at the 1%, 5% and 10% levels, resectively t = + φ i ( R ) + s SMB + h HML = R α + ε Portfolio Small Large Co-Moment β s h β s h β s h β s h β s h nd to rd 0.69* () nd to th 0.70* () nd to 5 th 0.71* () nd to 6 th 0.7* () nd to 7 th 0.8* () nd to 8 th 0.7* () nd to 9 th 0.70* () nd to 10 th 0.68* () 0.90* () 0.90* () 0.90* () 0.90* () 0.9* () 0.9* () 0.9* () 0.9* () -0.0 (0.) - (0.51) - (0.51) - (0.5) -0.0 (0.7) - (0.5) - (0.57) - (0.58) 0.65* () 0.6* () 0.55* () () 0.68* () () 0.6* () 0.71* () 0.5* () 0.5* () 0.* () 0.5* () 0.6* () 0.6* () 0.6* () 0.6* () * (0.08) * (0.07) 0.69* () 0.6* () 0.56* () () 0.6* () 0.60* () 0.60* () 0.6* () 0.9* () 0.8* () 0.7* () 0.8* () 0.8* () 0.8* () 0.8* () 0.8* () 0.0 (0.) 0.0 (0.5) 0.0 (0.51) 0.0 (0.6) 0.0 (0.6) 0.0 (0.6) 0.0 (0.6) 0.0 (0.6) () () 0.59* () 0.6* () 0.77* () 0.55* () 0.6* () 0.51* () 0.10* 0.10* 0.10* 0.10* 0.1* () 0.1* () 0.15* () 0.15* () (0.9) (0.88) (0.89) (0.95) (0.88) (0.99) (0.88) (0.88) 0.69* () 0.70* () 0.71* () 0.7* () 0.8* () 0.7* () 0.70* () 0.68* () -0.10* () -0.10* () -0.10* () -0.09* () -0.08* () -0.07* () - (0.0) - (0.0) -0.0 (0.) - (0.51) - (0.51) - (0.5) -0.0 (0.7) - (0.5) - (0.57) - (0.57) Adj. R i i

12 i, s, h Table 6 Time-Series Regressions of Book-to-Market-Sorted Portfolios Jan 1990 to Dec 00 Time-series regression of book-to-market-sorted ortfolio excess returns on the resective number of systematic co-moments, and on the returns of the Fama-French smallminus-big (SMB) and high-minus-low (HML) ortfolios. The estimates φ reresent the i th systematic co-moment, and the factor loadings for the SMB and HML, resectively. For examle, the nd systematic co-moment is the CAPM beta and the rd systematic co-moment is the Kraus and Litzenberger (1976) systematic co-skewness. The coefficient estimates are reorted with their -values. *, ** and *** denote significance at the 1%, 5% and 10% levels, resectively t = + φ i ( R ) + s SMB + h HML = R α + ε Portfolio High Low Co-Moment β s h β s h β s h β s h β s h nd to rd () nd to th 0.58* () nd to 5 th 0.51* () nd to 6 th 0.55* () nd to 7 th 0.58* () nd to 8 th 0.6* () nd to 9 th 0.1* () nd to 10 th 0.9* () 0.1* () 0.0* () 0.9* () 0.0* () 0.0* () 0.1* () 0.* () 0.* () 0.0* () 0.19* () 0.0* () 0.0* () 0.0* () 0.0* () 0.0* () 0.0* () 0.69* () 0.6* () () 0.6* () 0.60* () 0.60* () 0.68* () 0.77* () 0.* () 0.* () 0.* () 0.* () 0.* () 0.* () 0.1* () 0.1* () 0.08* (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) 0.6* () 0.6* () 0.58* () 0.65* () 0.81* () 0.5* () 0.* () 0.* () 0.7* () 0.7* () 0.6* () 0.7* () 0.9* () 0.5* () 0.5* () 0.5* () (0.5) (0.) (0.5) -0.0 (0.9) (0.5) -0.0 (0.9) -0.0 (0.5) -0.0 (0.5) 0.59* () 0.58* () 0.5* () 0.58* () 0.67* () 0.51* () 0.51* () 0.5* () 0.17* () 0.17* () 0.16* () 0.17* () 0.18* () 0.0* () 0.0* () 0.0* () (0.75) (0.79) (0.77) (0.71) (0.75) (0.66) (0.66) (0.66) () 0.58* () 0.51* () 0.55* () 0.58* () 0.6* () 0.1* () 0.9* () 0.1* () 0.0* () 0.9* () 0.0* () 0.0* () 0.1* () 0.* () 0.* () () -0.81* () () () () () () () Adj. R i i