Modeling and Estimating a Higher Systematic Co-Moment Asset Pricing Model in the Brazilian Stock Market

Transcription

1 Modeling and Estimating a Higher Systematic Co-Moment Asset Pricing Model in the Brazilian Stock Market André Carvalhal da Silva ABSTRACT. Many asset pricing models assume that only the secondorder systematic co-moment (CAPM beta) should be priced 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 paper examines the determinants of Brazilian stock returns from the CAPM beta, and Fama-French size and value factors, using higher-order systematic co-moments that encompass risks above the traditional CAPM beta. Our results indicate that the CAPM beta, and the Fama-French size and value factors appear to be important for explaining returns in Brazil, and they jointly provide statistically significant explanatory power across almost all the sample return periods. In general, adding a set of higher-order systematic co-moments has moderate explanatory power. RESUMEN. Muchos modelos para determinar el precio patrimonial presuponen que únicamente el co-momento sistémico de segundo orden (CAPM beta) debería tener su precio determinado por los inversores. Sin embargo, como los retornos patrimoniales no son normales, los inversores también están preocupados con los momentos más altos y co-momentos sistemáticos (co-sesgos, co-kurtosis, y así sucesivamente). Este documento examina las determinantes del retorno del capital accionario brasileño del CAPM beta, con factores de tamaño y valor Fama-French, André Carvalhal da Silva, DSc, is Professor at COPPEAD Graduate Business School, Federal University of Rio de Janeiro, Brazil ( andrec@coppead.ufrj.br). Latin American Business Review, Vol. 6(4) by The Haworth Press, Inc. All rights reserved. doi: /j140v06n04_05 85

2 86 LATIN AMERICAN BUSINESS REVIEW utilizando co-momentos sistemá ticos de orden más alto que agrupan los riesgos sobre el CAPM beta tradicional. Nuestros resultados indican que el CAPM beta y los factores de tamaño y valor Fama-French parecen ser importante para explicar los retornos en el Brasil y, juntos, suministran el poder explanatorio estadísticamente importante a lo largo de todos los períodos de retorno muestra, adicionando en general un conjunto de co-momentos sistemáticos de alto orden tiene un poder explanatorio moderado. RESUMO. Muitos modelos de precificação de ativos assumem que apenas o segundo co-momento (beta do CAPM) deve ser considerado pelos investidores. No entanto, como os retornos dos ativos não são normais, os investidores devem se preocupar também com momentos e co-momentos de ordem superior (co-assimetria, co-curtose etc). Esse artigo analisa os determinantes dos retornos das ações brasileiras a partir do beta do CAPM, dos fatores tamanho e valor de Fama-French e usando co-momentos de ordem superior que abrangem riscos além do tradicional beta do CAPM. Os resultados indicam que o beta do CAPM e os fatores de Fama-French são importantes para explicar os retornos no Brasil, e possuem poder explanatório conjunto estatisticamente significativo em quase todos os períodos. Em geral, os co-momentos de ordem superior possuem poder explanatório moderado. [Article copies available for a fee from The Haworth Document Delivery Service: HAWORTH. address: <docdelivery@haworthpress.com> Website: < by The Haworth Press, Inc. All rights reserved.] KEYWORDS. Asset pricing models, higher systematic co-moment, Brazil INTRODUCTION Many empirical studies have found that the Capital Asset Pricing Model (CAPM) beta has moderate or even insignificant explanatory power once the Fama and French factors are included. In contrast, Fama and French factors remain highly significant in explaining the crosssection of stock returns. Over the course of a series of papers, Fama and French (1992, 1993, 1996) argue that the size and value of a stock are additional risk factors that explain cross-sectional stock returns. The combination of this with beta is widely known as the Fama-French three-factor model.

3 André Carvalhal da Silva 87 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 factor, which stands for Small minus Big, is designed to measure the additional return investors have historically received by investing in stocks of companies with relatively small market capitalization. This additional return is often referred to as the size premium. The HML factor, which stands for High minus Low, has been constructed to measure the value premium provided to investors for investing in companies with high book-to-market values. There is also evidence in the literature [(Rubinstein (1973), Kraus and Litzenberger (1976), Friend and Westerfield (1980), Scott and Horvath (1980), Bonsal and Viswanathan (1993), Harvey and Siddique (2000), Kan and Wang (2001), Dittmar (2002), Hung, Shackleton and Xu (2003), Chung, Johnson, and Schill (2004)] that investors may be concerned with 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 priced, in addition to the first co-moment of stock returns, with the market return (beta). The CAPM suggests that only market risk should be priced by investors. One of the crucial assumptions 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 priced. Without normality, the CAPM is unlikely to hold, but when returns are normal, we only need the mean and variance to perfectly describe the distribution. However, an infinite number of moments are generally required to specify the tails completely. Investors are concerned about risk, and risk must be measured in terms of the entire probability distribution, which in turn can be measured with the moments of the distribution. Only in very special cases, such as quadratic utility or normality of returns, can we ignore the higher moments and systematic co-moments. Since asset returns are not normal, investors are also concerned about portfolio 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 (2000) have developed asset pricing models incorporating co-skewness terms, while Dittmar (2002) and Hung, Shackleton, and Xu (2003) have developed models incorporating co-kurtosis terms as well. Chung, Johnson, and Schill (2004) argue that higher-order co-moments matter to risk-averse investors, and they show that adding a set of

4 88 LATIN AMERICAN BUSINESS REVIEW systematic co-moments of order 3 through 10 reduces the explanatory power of the Fama-French factors to insignificance in almost every case. Higher-order systematic co-moments have been criticized for lacking intuition and for being unreliable. However, as Chung, Johnson, and Schill (2004) point 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 papers have tested higher-order pricing models for U.S. stock data [(Harvey and Siddique, (2000), Dittmar (2002), Barone-Adesi, Gagliardini and Urga (2002)], there has been little work outside the U.S [(Galagedera, Henry and Silvapulle (2002) in Australia, and Hung, Shackleton and Xu in the U.K. (2003)]. This paper examines the determinants of the cross-section of Brazilian stock returns from the CAPM beta, and Fama-French size and value factors, using higher-order asset pricing models that encompass systematic risks above the traditional CAPM beta. We also test the hypothesis that the Fama-French factors are simply a proxy for the pricing of higher-order co-moments. The growing economic importance 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 1990s, the Brazilian stock market has figured prominently among the star performers of the emerging markets. It seems worthwhile to take a close look at how predictable risk and returns have been on the Brazilian stock market in the last 15 years. The paper is organized as follows. The next section presents the data and methodology used in this paper. The third section reports the main empirical results. Conclusions are presented in the fourth section. DATA DESCRIPTION AND ESTIMATION We will now explore the return characteristics of Brazilian stocks. Brazil is the largest emerging market in Latin America, and one of the largest emerging markets in the world. Table 1 reports the market capitalization of selected emerging markets. Taiwan is the largest emerging market, with a total market capitalization of US$ 379 billion, followed by Korea (US$ 298 billion), India (US$ 252 billion), and Brazil (US$ 226 billion).

5 Market André Carvalhal da Silva 89 TABLE 1. Emerging Markets Profile as of December 2003 Market Capitalization in Emerging Countries at the End of 2003 Market Capitalization (US$ million) Latin America Argentina 34,994.7 Brazil 226,357.7 Chile 87,508.4 Mexico 122,533.0 Asia India 252,893.4 Indonesia 54,659.1 Korea 298,248.1 Malaysia 160,970.3 Singapore 148,502.6 Taiwan 379,060.4 Thailand 119,017.2 Developed Markets U.S (NYSE) 11,328,953.1 Japan 2,953,098.3 Source: Adapted from World Federation of Exchanges. Our sample consists of firms listed on the Sao Paulo stock exchange (Bovespa). The monthly closing prices were obtained from January 1990 through December Our proxy for the risk-free rate is the Interbank Certificate of Deposit (CDI) yield, and we collected information on stock prices, risk-free rates, accounting data, and market equity values from the Economatica database. To test the models, we could theoretically use the returns on the individual securities, but for estimating purposes we have to limit the number of parameters. We follow common practice in grouping the stocks into portfolios and testing the model on a small set of portfolios. We form equally weighted portfolios to analyze the return on a group of stocks instead of individual ones. Forming portfolios allows the factor of interest (size and book-to-market) to be concentrated by conducting a sort on that factor, reducing the error associated with misestimation of the stocks factor itself and the unexplained variance in the return prediction.

6 90 LATIN AMERICAN BUSINESS REVIEW Given the limited number of available firms in Brazil, five sizesorted portfolios and five book-to-market-sorted portfolios are constructed for the regressions (five is chosen as a trade-off between the number of stocks in each portfolio and the number of portfolio return observations in the regressions). The portfolios are formed at the end of December of every year from 1990 through In order to qualify for analysis in any period, a stock must have an observed market capitalization, book value, and monthly return data during that period. 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, 331 stocks satisfied the criterion and are analyzed, although not all stocks had a complete price or return history for the entire period. At the end of each calendar year, we rank all stocks by market capitalization, divide the sample into 5 portfolios of equal size, and compute a time series of equal-weighted and rebalanced portfolios returns. We repeat this procedure for the stratification using book-to-market ratio. The small size portfolio ( small ) consists of stocks in the smallest size quintile, and the large size portfolio ( large ) consists of stocks in the largest size quintile. The high book-to-market portfolio ( high ) consists of stocks in the highest quintile, and the low book-to-market portfolio ( low ) consists of stocks in the lowest quintile. Once constructed, the portfolios are held for one year, and the 12 monthly equally weighted portfolio returns are calculated. The full-period portfolio returns are calculated for the entire 168-month period, from January 1990 through December The SMB monthly factor is computed as the average return on the smallest stocks minus the average return on the largest stocks in that month. A positive SMB indicates that small cap stocks outperformed large cap stocks in that month, while a negative SMB indicates that large cap stocks outperformed small caps. Constructed in a fashion similar to that of SMB, HML is computed as the average returns on stocks with the highest book-to-market ratio minus the average return on stocks with the lowest book-to-market ratio each month. A positive HML indicates that value stocks outperformed growth stocks in that month, while a negative HML indicates that growth stocks outperformed value stocks. For each period t, we subtract the risk-free rate to obtain the excess portfolio return (R pt R ft ), and estimate a regression of excess portfolio

7 returns on the excess market returns (R mt -R ft ), squared excess market returns (R mt -R ft ) 2 and cubed excess market returns (R mt -R ft ) 3, in addition to the returns on the Fama-French small-minus-big (SMB) and highminus-low (HML) portfolios: R pt R ft p p (R mt R ft ) p (R mt R ft ) 2 p (R mt R ft ) 3 s p SMB h p HML p where the estimates b p, g p, d p, s p, h p represent the systematic covariance, co-skewness, and co-kurtosis, and the factor loadings for the SMB and HML, respectively. Kraus and Litzenberger (1976) state that it is trivial to extend the model to incorporate any number of higher co-moments. Chung, Johnson, and Schill (2004) argue that higher-order co-moments matter to risk-averse investors, and show that adding a set of systematic comoments of order 3 through 10 reduces the explanatory power of the Fama-French factors to insignificance in almost every case. In order to test the explanatory power of the SMB and HML factors, we also estimate the regression adding higher co-moments of order 3 through 10: 10 pt ft p pi mt ft i= 2 i R R = α + ( R R ) 1 + s SMB + h HML + p p p where the estimates pi, s p, h p represent the ith systematic co-moment and the factor loadings for the SMB and HML, respectively. For example, the 2nd systematic co-moment is the CAPM beta, and the 3rd systematic co-moment is the Kraus and Litzenberger (1976) systematic co-skewness. Empirical Results André Carvalhal da Silva 91 Table 2 provides summary statistics of the portfolio monthly returns in local currency from 1990 through Panel A gives the results for size-sorted portfolios, while Panel B gives the results for book-to-market-ratio-sorted portfolios. The small size portfolio ( small ) consists of stocks in the smallest size quintile, and the large size portfolio ( large ) consists of stocks in the largest size quintile. The high book-to-market portfolio ( high ) consists of stocks in the highest quintile, and the low book-to-market portfolio ( low ) consists of stocks in the lowest quintile.

8 92 LATIN AMERICAN BUSINESS REVIEW TABLE 2. Summary Statistics of Portfolio Returns in Brazil January 1990 Through December 2003 Small Large SMB Panel A: Size-Sorted Portfolios Mean Median Maximum Minimum 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 Notes: Descriptive statistics of monthly returns (%) in U.S. dollar on five equal-sized portfolios in Brazil from January 1990 through December Panel A gives the results for size-sorted portfolios, while Panel B gives the results for book-to-market-ratio-sorted portfolios. The small size portfolio ( small ) consists of stocks in the smallest size quintile, and the large size portfolio ( large ) consists of stocks in the largest size quintile. The high book-to-market portfolio ( 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) portfolios. Although simple average returns do not appear to decrease with portfolio size, the median returns seem to decrease with portfolio size. The small portfolio has a median monthly return of 0.78%, while the large portfolio has a median monthly return of 0.06%. The small minus big (SMB) portfolio has a median return of 0.38%. The portfolios skewness and kurtosis are consistently different from the corrected standard normal distribution, and the Jarque-Bera statistic strongly rejects normality.

9 André Carvalhal da Silva 93 In Panel B, although there is not a clear and monotonic relationship between average return and book-to-market, the high portfolio has a median monthly return of 0.25% (mean of 1.06%), while the low portfolio has a median monthly return of 1.56% (mean of 0.83%). The high minus low (HML) portfolio has a median return of 1.18% (mean of 0.23%). All series depart strongly from normality, as indicated by the skewness, kurtosis, and Jarque-Bera statistic. Next, we estimate a regression of excess portfolio returns on the excess market returns, the squared and cubed excess market returns, and on the returns of the Fama-French SMB and HML factors. Table 3 shows the results for size-sorted portfolios. The intercept terms are all insignificant, and beta is very significant in explaining the time-series return on portfolios from 1990 through Adding TABLE 3. Time-Series Regressions of Size-Sorted Portfolios January 1990 Through December 2003 R pt R ft a p b p (R mt R ft ) g p (R mt R ft ) 2 d p (R mt R ft ) 3 s p SMB h p HML p) Model Portfolio a p b p g p d p s p h p Adj. R 2 1 Small 0.02 (0.98) 0.50* (0.35) 0.47* (0.95) 0.50* (0.15) 0.72* 0.78 Large 0.02 (0.96) 0.86* Small 0.32 (0.76) 0.55* 0.11 (0.44) 0.10 (0.66) (0.27) 0.61* 0.04 (0.75) 0.48* (0.01) (0.65) 0.66* 0.17*** (0.08) 0.71* (0.24) 0.60* 0.08 (0.30) 0.51* 0.81 Large 0.04 (0.92) 0.70* 0.03 (0.53) 0.62* 0.91

10 94 LATIN AMERICAN BUSINESS REVIEW TABLE 3 (continued) Model Portfolio a p b p g p d p s p h p Adj. R 2 3 Small 0.00 (0.99) 0.80* 0.86* 0.05** (0.03) (0.30) 0.61* 0.37* 0.09** (0.03) (0.97) 0.61* 0.31* 0.06*** (0.09) (0.16) 0.74* 0.06 (0.15) 0.04 (0.20) 0.78 Large 0.00 (0.99) 0.80* 0.14* 0.05** (0.03) Small 0.00 (0.99) 0.69* 0.04 (0.45) 0.53* 0.90* 0.02 (0.43) (0.28) 0.65* 0.02 (0.87) 0.14 (0.38) 0.35* 0.08*** (0.06) (0.48) 0.69* 0.19** (0.02) 0.46* 0.29* 0.03 (0.33) (0.20) 0.61* 0.07 (0.35) 0.58* 0.10* (0.01) 0.00 (0.92) 0.81 Large 0.00 (0.99) 0.69* 0.04 (0.44) 0.53* 0.10* 0.02 (0.43) 0.92 Note: Time-series regression of size-sorted portfolio excess returns on the excess market returns, the squared and cubed excess market returns, and the returns of the Fama-French small-minus-big (SMB) and high-minus-low (HML) portfolios. The estimates b p, g p, d p, s p, h p represent the systematic covariance, co-skewness, and co-kurtosis, and the factor loadings for the SMB and HML, respectively. The coefficient estimates are reported with their p-values. *, **, and *** denote significance at the 1%, 5%, and 10% levels, respectively. co-moment terms (co-skewness and co-kurtosis) does increase the explanatory power 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 factors remain highly statistically significant, but the co-skewness and the HML factors are mostly insignificant.

11 André Carvalhal da Silva 95 Table 4 shows the results for the book-to-market-sorted portfolios. Beta is very significant in explaining time-series returns on portfolios, the co-skewness is not statistically significant, and the co-kurtosis is statistically significant in most models. The SMB and HML factors are statistically significant at the 1% level. Not surprisingly, the SMB factor does better in size-sorted portfolios and the HML does better in book-to-market-sorted portfolios. Next, following Chung, Johnson, and Schill (2004), we test to see if adding a set of systematic co-moments of order 3 through 10 reduces the explanatory power of the Fama-French factors. Table 5 shows the results for size-sorted portfolios. Adding a set of systematic co-moments of order 3 through 10 does not reduce the explanatory power of beta and the SMB factor. In every model they remain statistically significant, whereas the HML factor is not statistically significant, except for portfolio 2. Furthermore, the values of adjusted R 2 are largely unchanged when we add higher co-moments. Table 6 shows the results for the book-to-market-sorted portfolios. Adding a set of systematic co-moments of order 3 through 10 does not reduce the explanatory power of beta and the SMB factor. In every model they remain statistically significant, whereas the HML factor is statistically significant for most models. Not surprisingly, the HML does better in book-to-market-sorted portfolios than in size-sorted portfolios. Overall, the results indicate that beta is highly significant in explaining time-series returns in Brazil from 1990 through 2003, 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 explanatory power for portfolio returns, and, not surprisingly, the SMB factor does better in size-sorted portfolios, while the HML does better in book-to-market-sorted portfolios. Adding a set of systematic co-moments of order 3 through 10 does not seem to reduce the explanatory power of beta, and the SMB and HML factors. In order to check the robustness of our results, we also analyze three different sub-periods according to the Brazilian macro-economic situation 1 : high inflation ( ), stabilization with fixed exchange rate ( ), and stabilization with floating exchange rate ( ). The results are quite the same as those for the entire period. During the high inflation period, the beta, the SMB and HML factors, and the co-kurtosis remain significant. On the other hand, during the

12 96 LATIN AMERICAN BUSINESS REVIEW TABLE 4. Time-Series Regressions of Book-to-Market Portfolios January 1990 Through December 2003 R pt R ft a p b p (R mt R ft ) g p (R mt R ft ) 2 d p (R mt R ft ) 3 s p SMB h p HML p Model Portfolio a p b p g p d p s p h p Adj. R 2 1 High 0.34 (0.65) (0.65) (0.39) (0.06) Low 0.04 (0.97) 2 High 0.65 (0.43) (0.54) (0.44) (0.07) Low 0.38 (0.76) 3 High 0.27 (0.63) (0.50) (0.30) (0.06) Low 0.27 (0.63) 4 High 0.45 (0.45) (0.24) (0.24) (0.04) Low 0.45 (0.45) 0.53* 0.55* 0.52* 0.61* 0.64* 0.58* 0.64* 0.57* 0.56* 0.42* 0.69* 0.71* 0.68* 0.66* 0.69* 0.61* 0.69* 0.64* 0.59* 0.61* 0.10 (0.39) 0.08 (0.47) 0.00 (0.98) 0.00 (0.97) 0.21 (0.22) 0.09 (0.28) 0.10 (0.19) 0.04 (0.65) 0.01 (0.89) 0.09 (0.28) 0.13 (0.50) 0.38** (0.03) 0.17 (0.37) 0.19*** (0.10) 0.96* 0.39* (0.01) 0.03 (0.84) 0.12 (0.48) 0.33* (0.01) 0.39* (0.01) 0.38* 0.42* 0.45* 0.14* 0.38* 0.41* 0.44* 0.47* 0.17* 0.41* 0.18* 0.07* (0.01) 0.06 (0.15) 0.01 (0.66) 0.82* 0.20* 0.08* (0.01) 0.05 (0.25) 0.01 (0.75) 0.80* Note: Time-series regression of book-to-market portfolio excess returns on the excess market returns, the squared and cubed excess market returns, and the returns of the Fama-French small-minus-big (SMB) and high-minus-low (HML) portfolios. The estimates b p, g p, d p, s p, h p represent the systematic covariance, co-skewness, and co-kurtosis, and the factor loadings for the SMB and HML, respectively. The coefficient estimates are reported with their p-values. *, **, and *** denote significance at the 1%, 5%, and 10% levels, respectively

13 TABLE 5. Time-Series Regressions of Size-Sorted Portfolios January 1990 Through December 2003 Note: Time-series regression of size-sorted portfolio excess returns on the respective number of systematic co-moments, and on the returns of the Fama-French small-minus-big (SMB) and high-minus-low (HML) portfolios. The estimates pi, s p, h p represent the ith systematic co-moment, and the factor loadings for the SMB and HML, respectively. For example, the 2nd systematic co-moment is the CAPM beta and the 3rd systematic co-moment is the Kraus and Litzenberger (1976) systematic co-skewness. The coefficient estimates are reported with their p-values. *, **, and *** denote significance at the 1%, 5%, and 10% levels, respectively. 97

14 TABLE 6. Time-Series Regressions of Book-to-Market-Sorted Portfolios January 1990 Through December 2003 Note: Time-series regression of book-to-market-sorted portfolio excess returns on the respective number of systematic co-moments, and on the returns of the Fama-French small-minus-big (SMB) and high-minus-low (HML) portfolios. The estimates pi, s p, h p represent the ith systematic co-moment, and the factor loadings for the SMB and HML, respectively. For example, the 2nd systematic co-moment is the CAPM beta and the 3rd systematic co-moment is the Kraus and Litzenberger (1976) systematic co-skewness. The coefficient estimates are reported with their p-values. *, ** and *** denote significance at the 1%, 5% and 10% levels, respectively. 98

15 André Carvalhal da Silva 99 stabilization periods, beta, SMB and HML are significant, but neither the co-skewness nor the co-kurtosis is statistically significant. One possible explanation for the difference in the results for the high inflation and stabilization periods may be associated with the presence of extremely low and high stock returns in the Brazilian stock market during the high inflation period. Since Brazil was affected by high and variable inflation from the 1990 through 1993 period, portfolio returns appear to be more volatile during that time. Therefore, adding higher-order systematic co-moments provides better results, since the set of higher co-moments tends to measure more accurately the probability distribution and the likelihood of extreme outcomes. CONCLUSIONS There is evidence in the literature that the Capital Asset Pricing Model (CAPM) beta has moderate or even insignificant explanatory power 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 priced. When the returns are normal, we only need the mean and variance to describe the distribution completely. However, since asset returns are not normal, investors are also concerned about higher moments (skewness, kurtosis, and so on) and co-moments (co-skewness, co-kurtosis, and so on). This paper has examined the determinants of the cross-section of Brazilian stock returns from the CAPM beta, Fama-French size and value factors, using higher-order asset pricing models that encompass systematic risks above the traditional CAPM beta. We have also tested to see if adding a set of systematic co-moments of order 3 through 10 reduces the explanatory power of the Fama-French factors. Our results indicate that beta is highly significant in explaining timeseries returns in Brazil from 1990 through 2003, 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 explanatory power for portfolio returns, and, not surprisingly, the SMB factor does better in size-sorted portfolios, while the HML does better in book-to-market-sorted portfolios. Adding a set of systematic co-moments of order 3 through 10 does not seem to reduce the explanatory power of beta, and the SMB and HML factors.

16 100 LATIN AMERICAN BUSINESS REVIEW In order to check the robustness of our results, we have also analyzed three different sub-periods according to the Brazilian macro-economic situation. The results are quite the same as those for the entire period. During the high inflation period, the beta, the SMB and HML factors, and the co-kurtosis remain significant. On the other hand, during the stabilization periods, beta, SMB and HML are significant, but neither the co-skewness nor the co-kurtosis is statistically significant. One possible explanation for the difference in the results for the high inflation and stabilization periods may be associated with the presence of extremely low and high stock returns in the Brazilian stock market during the high inflation period. Since Brazil was affected by a high and variable inflation during the 1990 through 1993 period, portfolio returns appear to be more volatile during that time. Therefore, adding higher-order systematic co-moments provides better results, since the set of higher co-moments tends to measure more accurately the probability distribution and the likelihood of extreme outcomes. Overall, we can conclude that the CAPM beta, and the Fama-French size and value factors appear to be important for explaining returns in Brazil, and they jointly provide statistically significant explanatory power across almost all the sample return periods. In general, adding a set of higher-order systematic co-moments has moderate explanatory power. NOTE 1. The results are not reported here to conserve space, but are available upon request. REFERENCES Barone-Adesi, G., Gagliardini, P. and G. Urga (2002). Homogeneity Hypothesis in the Context of Asset Pricing Models: The Quadratic Market Model, City University Working Paper. Bonsal, R. and S. Viswanathan (1993). No Arbitrage and Arbitrage Pricing, Journal of Finance, Vol. 48, No. 4, pp Chung, Y., Johnson, H. and M. Schill (2004). Asset Pricing When Returns are Nonnormal: Fama-French Factors vs. Higher-Order Systematic Co-Moments, Journal of Business, forthcoming. Dittmar, R. (2002). Nonlinear Pricing Kernels, Kurtosis Preference, and Evidence from the Cross-Section of Equity Returns, Journal of Finance, Vol. 57, No. 1, pp

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