Asset pricing models: a comparison

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1 Applied Financial Economics, 007, 17, Asset pricing models: a comparison Edward R. Lawrence a, *, John Geppert b and Arun J. Prakash a a Department of Finance, College of Business Administration, Florida International University, Miami, FL 33199, USA b University of Nebraska, Lincoln, USA We empirically test and compare the performance of the traditional capital asset pricing model (CAPM), the three-moment CAPM and the Fama French (FF) three-factor model using the FF 5 portfolios data. Based on the time-series and the cross-sectional tests, the FF three-factor model outperforms the other models. In the cross-sectional tests, the threemoment CAPM has a higher R than CAPM but in the time-series regression, the performances of CAPM and the three-moment CAPM are comparable. I. Introduction The publication of the seminal article on capital asset pricing by Sharpe (1964), revolutionized the theory as well as the applicability of modern techniques in finance. Be it the investment or the corporate side of finance, the development of the capital asset pricing model (CAPM) gave theoretically sound tool for long-term resource allocation under conditions of risk. The use of in portfolio construction, use of CAPM in computing the cost of capital for capital budgeting etc. is known to all the students of finance. Even though, the original CAPM, also referred to as two-moment CAPM developed under theoretically sound base, empirically it does not fit well with real data. Its empirical validity has been questioned by several studies. The empirical findings by Friend and Blume (1970), Black et al. (197), Miller and Scholes (197), Fama and MacBeth (1973) and Blume and Friend (1973), etc. show that the two-moment CAPM overestimate (underestimate) the real returns for low (high) stocks. Several studies show that factors other than successfully explain the portion of security returns not captured by. Basu (1977) finds that low price/earnings portfolios show higher rate of returns than what could be explained using CAPM. Banz (1981) and Reinganum (1981) find that smaller firms show high abnormal returns. Litzenberger and Ramaswamy (1979) find higher rate of returns on equities with high dividend yields. The ability of other variables in explaining the portion of returns unexplained by CAPM indicates that CAPM is misspecified and needs some additional factors to explain security returns. Arditti (1967) argues that the investors do not base their decisions solely on the first two-moments on the rates of return probability distributions, but show a preference for positive skewness. Based on this argument, Kraus and Litzenberger (1976) extend the two-moment CAPM to incorporate the effect of this positive preference for skewness. Empirically, they find that the threemoment CAPM is a better predictor of returns than its two-moment counterpart. More recently, Fama and French (FF) (199) work confirms the inadequacy of two-moment CAPM. Their empirical work indicates that book-to-market ratio and market capitalization (firm size) explain the cross-section variation of average returns more appropriately than. Fama and French (1993) propose a model wherein the excess return version of CAPM is modified by adding two more items namely SMB (small minus big; the difference between the return on a portfolio *Corresponding author. elawrenc@fiu.edu Applied Financial Economics ISSN print/issn online ß 007 Taylor & Francis DOI: /

2 934 E. R. Lawrence et al. of small stocks and the return on a portfolio of large stocks) and HML (high minus low; the difference between the return on a portfolio of high book-tomarket stocks and the return on a portfolio of low book-to-market stocks). Using the data from July 1963 to December 1993 (366 months) FF (1996) form portfolios on past returns to test their model and find that their model is fully able to explain the crosssectional equilibrium pricing mechanism as the coefficient of determination (R ) averages 0.93 and the regression intercepts are close to zero. Fama and French (1996) compared the performance of CAPM and the FF three-factor model over a period of 366 months using the time-series tests and find that their three-factor model outperforms CAPM. Pastor and Stambaugh (000) compare the Sharpe Lintner CAPM, the FF three-factor model and the characteristic-based model of Daniel and Titman (1997). They compare asset pricing models from the perspective of investors who center their prior beliefs on the models and then update those beliefs with data for the 1963 to 1997 period. They find that the largest losses to investors occur with CAPM belief whereas the losses with the other two models are comparable. Hodrick and Zhang (000) compare six asset pricing models namely the CAPM, the Consumption CAPM, the Jagannathan and Wang (1996) conditional CAPM, the Campbell (1996) dynamic asset pricing model, the Cochrane (1996) production-based model and the Fama and French (1993) three-factor and five-factor models. They use the methodology of Hansen and Jagannathan (1997) to compare the models and test the models using the returns on the FF 5 portfolios from Only Campbell s (1996) model passes the test of HJ-distance equals zero, but it fails to pass the stability tests. Thus, none of the models they test correctly prices returns. Mishra et al. (005) test the Kraus and Litzenberger threemoment CAPM and the FF three-factor model over a period of 1936 to 00 and find that for sizesorted portfolios, both the FF three-factor model and the three-moment CAPM significantly explain the expected returns. None of the studies done so far compare the two-moment CAPM, the three-moment CAPM and the FF three-factor models together and rank them according to their return predicting accuracies. In this article, we empirically test and compare the performance of the traditional two-moment CAPM, the three-moment CAPM and the FF three-factor model using the FF 5 portfolios data. Based on the time-series tests and the Fama MacBeth cross-sectional tests, the FF threefactor model outperforms the CAPM and the threemoment CAPM models. In the cross-sectional test, the three-moment CAPM has a higher R than CAPM but in the time-series regression, the performance of CAPM and the three-moment CAPM is comparable. The article is organized as follows. We provide details on data in Section II. In Section III, we describe and empirically test the three asset pricing models using the FF 5 portfolio data. In Section IV, we compare the three models. The article ends with a brief conclusion. II. Data Description We collect the FF 5 portfolio return data and the data of explanatory variables R m (market rate of return), SMB (the difference between the return on a portfolio of small stocks and the return on a portfolio of large stocks) and HML (the difference between the return on a portfolio of high book-to-market stocks and the return on a portfolio of low book-to-market stocks) from the web site of Dr Kenneth French 1 and data for the risk-free rate, R f, from the CRSP. We use the FF 5 portfolio data for 474 months starting from July 1963 to December 00 to test the three asset pricing models. III. Model Description and Testing The CAPM The CAPM states that in equilibrium, the rate of return on any risky asset is a linear function of their covariance with the market portfolio. According to CAPM, the expected return for any asset i at any time t is given by the following equation: E t r i,tþ1 ¼ i,t E t r M,tþ1 ð1þ where ( i ) is defined as, i ¼ Covðr i, r M Þ M In the above equation, r is the returns in excess of the risk-free return and M represents the market portfolio. The econometric restriction of this model 1 The construction procedure for the portfolios is detailed in FF (1996).

3 Asset pricing models: a comparison 935 imposes that in the time-series regression of the excess returns on the market excess return, the intercept should be 0, the betas should be significant and the market risk premium estimate should be the same across all the assets. In the cross-sectional regression of the excess returns on the betas, the slope (the market risk premium) should be significantly different from zero. Table 1 provides the results of the time-series regression given by Equation 1 on FF 5 portfolios. Beta is significant for all 5 portfolios but 1 out of 5 portfolios show a significant constant term also (which is inconsistent with CAPM). The average R value for the 5 portfolios is 0.7. Another technique of testing CAPM empirically is given by Fama and MacBeth (1973). The methodology is based on two regressions. For each stock, a is estimated from a rolling time-series regression of historical stock returns (R it ) on the market returns (R mt ): R it R ft ¼ i ðr mt R ft Þþ" t where R ft is the risk-free rate. In the first step, we estimate the betas and then we perform cross-sectional regressions on to recover market risk premia: R it R ft þ i þ " i ðþ ð3þ Table 1, Panel C reports the results for the above test. The results show that the market risk premium, is insignificant, the constant term is significant whereas the R is 0.6, suggesting weak results for CAPM in cross-sectional tests. Table 1. Results of the traditional CAPM (Equation 1) regression for monthly percent excess returns on 5 FF portfolios Panel A Book-to-market equity (BE/ME) quintiles Size Low 3 4 High Low 3 4 High a t(a) Small Big t() Small Big R s(e) Small Big Panel B # of significant Model Range of R Mean R b CAPM Panel C Model Mean R t() t() CAPM Panel A has the descriptive statistics of the estimated parameters for the 5 portfolios whereas, the average statistics for the 5 portfolios is reported in Panel B. Panel C has the results of CAPM using Fama MacBeth process (Equation 3).

4 936 E. R. Lawrence et al. The three-moment CAPM Kraus and Litzenberger (1976) developed the threemoment CAPM which was later refined by Harvey and Siddique (000). Assuming the existence of a conditionally risk-free asset, Harvey and Siddique (000) derive the following results: E t r i,tþ1 ¼ 1,t Cov t r i,tþ1, r M,tþ1 h i þ, t Cov t r i,tþ1, r M,tþ1 ð4þ where 1,t ¼ Var t r M,tþ1 Et ½r M,tþ1 Š Skew t ½r M,tþ1 ŠE t r M,tþ1 Var t ½r M,tþ1 ŠVar t r M,tþ1 ðskewt ½r M,tþ1 ŠÞ,t ¼ Var t½r M,tþ1 ŠE t r M,tþ1 Skewt ½r M,tþ1 ŠE t ½r M,tþ1 Š Var t ½r M,t 1 ŠVar t r M,tþ1 ðskewt ½r M,tþ1 ŠÞ In the above equations, Cov stands for the covariance, Var stands for the variance and Skew is the skewness. The restriction the above model imposes on a cross section of assets is that 1,t and,t are same across all the assets and are statistically different from 0. This is the conditional version of the three-moment CAPM first proposed by Kraus and Litzenberger (1976). According to Harvey and Siddique (000), Equation 4 can be rewritten as: E t ri,tþ1 ¼ At E t rm,tþ1 þ Bt E t r M,tþ1 ð5þ where A t and B t are the functions of the market variance, skewness, covariance and coskewness. A t and B t illustrate the relation between the Harvey and Siddique (000) model and the Kraus and Litzenberger (1976) three-moment CAPM. Equation 5 is an empirically testable restriction imposed on the cross section of expected asset returns by the asset pricing model incorporating skewness and is an alternative to Equation 1. Panel A of Table shows the time-series regression of Equation 5 for all FF 5 portfolios. The summarized result in Panel B shows a significant constant for 14 portfolios, a significant (parameter for the market risk premium) for all 5 portfolios and an insignificant gamma (parameter for skewness) for 11 out of 5 portfolios. The average R for the 5 portfolios (0.7) remains approximately the same as with the traditional CAPM (reported in Table 1). These results indicate that three-moment CAPM does not do any better than CAPM when tested on FF portfolio returns. Panel C in Table reports the results of threemoment CAPM using the Fama MacBeth procedure. The R for the regression is Both the market risk premium and the premium for skewness are insignificant whereas the constant term is significant. The time-series and cross-sectional tests indicate weak results for the three-moment CAPM when tested using the FF 5 portfolios. The Fama French three-factor model Fama and French (1993) propose a three-factor model which says that the expected return on a portfolio in excess of the risk-free rate is explained by the sensitivity of its return to three-factors: (1) The excess return on a broad market portfolio (R m R f ). () The difference between the return on a portfolio of small stocks and the return on a portfolio of large stocks (SMB). (3) The difference between the return on a portfolio of high book-to-market stocks and the return on a portfolio of low book-tomarket stocks (HML). Algebraically, the expected excess return on a portfolio i is given by: EðR i Þ R f ¼ b i EðR m Þ R f þ si EðSMBÞ þ h i EðHMLÞ ð6þ where E(R m ) R f, E(SMB) and E(HML) are expected premiums. The ex-post version of FF model is given by the following equation: R i R f ¼ i þ b i R M R f þ si SMB þ h i HML þ " i where i is the constant term in the regression equation and b i, s i, h i are the parameters in the timeseries regression. Fama and French (1993) show that the model (6) is a good description of returns on portfolios formed based on size and BE/ME. According to FF (1993, 1994, 1995), SMB and HML mimic combinations of two underlying risk factors or state variables of special hedging concern to investors. Table 3 reports estimates of the three-factor time-series regression for Equation 7. Panel A has the descriptive statistics of the estimated parameters for the 5 portfolios whereas the average statistics for the 5 portfolios is in Panel B. The average of the 5 regression R is The small average absolute intercept of 0.015% per month illustrates that the model captures most of the variation in the average returns on the portfolios. The summary statistic in Panel B shows that the parameter for ð7þ

5 Asset pricing models: a comparison 937 Table. Results of the three-moment CAPM regression (Equation 5) for monthly percent excess returns on 5 FF portfolios Panel A Book-to-market equity (BE/ME) quintiles Size Low 3 4 High Low 3 4 High a t(a) Small Big t() Small Big t() Small Big R s(e) Small Big Panel B # of significant Model Range of R Mean R b 3M-CAPM Panel C Model Mean R t() t() t() 3M-CAPM Panel A has the descriptive statistics of the estimated parameters for the 5 portfolios whereas the average statistics for the 5 portfolios is reported in Panel B. Panel C has the results of three-moment CAPM using Fama MacBeth process. the market risk premium is significant for all 5 portfolios, the parameter for SMB is significant for 4 portfolios and the parameter for HML is significant for all 5 portfolios whereas the constant term is significant for only 6 out of 5 portfolios. Panel C reports the results of cross-sectional regressions, using the Fama MacBeth procedure. The R for the regression is 0.54, the market risk premium and the premium for HML is significant and the constant is significant, while the premium for SMB is insignificant. These results are consistent with the findings of Harvey and Siddique (000). IV. Comparative Evaluation of the Asset Pricing Models In this section, we compare the three models of asset pricing namely, the traditional CAPM, the Harvey and Siddique report the result for FF three-factor model in Panel A of their Table 4. In the full sample, they find that the market risk premium and the premium for HML are significant whereas the premium for SMB is insignificant.

6 938 E. R. Lawrence et al. Table 3. Results of the FF three-factor model regression (Equation 7) for monthly percent excess returns on 5 FF portfolios Panel A Book-to-market equity (BE/ME) quintiles Size Low 3 4 High Low 3 4 High Means SD Small Big a t(a) Small Big b t(b) Small Big s t(s) Small Big h t(h) Small Big R s(e) Small Big Panel B # of significant Model Range of R Mean R b s h FF three-factor Panel C Model Mean R t() t() s t(s) h t(h) FF three-factor Panel A has the descriptive statistics of the estimated parameters for the 5 portfolios whereas the average statistics for the 5 portfolios is reported in Panel B. Panel C has the results of FF three-factor model using Fama MacBeth process.

7 Asset pricing models: a comparison 939 Table 4. Time-series regression results for the traditional CAPM, the three-moment CAPM and the FF three-factor model # of significant Model Range of R Mean R b s h CAPM M-CAPM FF three-factor The R for CAPM and three-moment CAPM are nearly same whereas the R for FF three-factor model surpasses the R of the other two models. The parameter for market risk premium is significant in 1 portfolios for CAPM, 15 portfolios for three-moment CAPM and on all 5 portfolios in FF three-factor model. The parameter on skewness is significant on 14 out of 5 portfolios, the parameter for SMB is significant for all 5 portfolios and the parameter for HML is significant for 4 out of 5 portfolios. The constant term is significant on 1 out of 5 portfolios in CAPM, 15 out of 5 in three-moment CAPM whereas it is significant for only 6 out of 5 portfolios in FF three-factor model. Table 5. Cross-sectional two pass Fama MacBeth test results for the traditional CAPM, the three-moment CAPM and the FF three-factor model Model Mean R t() t() t() s t(s) h t(h) CAPM M-CAPM FF three-factor The R for FF three-factor model is the highest followed by three-moment CAPM and then CAPM. The market risk premium is insignificant in both CAPM and three-moment CAPM whereas it is highly significant in the FF three-factor model. The premium for skewness and SMB are insignificant whereas the premium for HML is highly significant. three-moment CAPM and the FF three-factor model. Similar to Fama and French (1993) we test the asset pricing models using the time-series regression approach of Black et al. (197), because it captures the variation in the different factor loadings. Moreover, the time-series regression slopes have the clear interpretation as risk-factor sensitivities. In a time-series regression, a correctly specified model produces intercepts that are not significantly different from 0. Thus, the estimated intercepts provide a simple formal test for comparing the performance of different asset pricing models. Table 4 shows the time-series regression of the three models. The R for CAPM and three-moment CAPM are nearly the same (0.7) whereas the R for FF three-factor model (0.89) surpasses the R of the other two models. The parameter for the market risk premium is significant in all 5 portfolios for CAPM, three-moment CAPM and the FF three-factor model. The parameter on skewness is significant on 14 out of 5 portfolios, the parameter for SMB is significant for 5 portfolios and the parameter for HML is significant for 4 out of 5 portfolios. The constant term is significant on 1 out of 5 portfolios in CAPM, 15 out of 5 in three-moment CAPM but is significant for only 6 out of 5 portfolios in the FF three-factor model. Based on these results, the FF three-factor model ranks the highest. There is no significant difference in the ranking of the other two models relative to each other, i.e. the three-moment CAPM and the traditional CAPM perform similarly. Table 5 presents the cross-sectional two pass Fama MacBeth test for the CAPM, the threemoment CAPM and the FF three-factor model. The R for FF three-factor model is the highest (0.54) followed by three-moment CAPM (0.40) and then CAPM (0.6). The market risk premium is insignificant in both CAPM and three-moment CAPM whereas it is highly significant in the FF three-factor model. The premium for skewness is insignificant in the three-moment CAPM and SMB is insignificant whereas the premium for HML is highly significant in the FF three-factor model. The above discussion indicates the dominance of the FF three-factor model

8 940 E. R. Lawrence et al. over the other two asset pricing models. Based on R, the three-moment CAPM cross-sectionally does a better job than CAPM; the market risk premium is more significant in the three-moment CAPM than in CAPM. 3 V. Conclusion In this study, we compare the performance of the traditional CAPM, the three-moment CAPM and the FF three-factor model. In both the time-series tests and the Fama MacBeth cross-sectional tests, the FF three-factor model outperforms the CAPM and the three-moment CAPM models. Based on the crosssectional test, the three-moment CAPM has a higher R than CAPM but in the time-series regression, the performance of CAPM and the three-moment CAPM is comparable. References Arditti, F. D. (1967) Risk and required return on equity, Journal of Finance,, Banz, R. W. (1981) The relationship between return and market value of common stocks, Journal of Financial Economics, 9, Basu, S. (1977) Investment performance of common stocks in relation to their price earnings ratios: a test of the Efficient Market hypothesis, Journal of Finance, 3, Black, F., Jensen, M. C. and Scholes, M. (197) The capital asset pricing model: some empirical tests, in Theory of Capital Markets (Ed.) M. C. Jensen, Praeger, New York, pp Blume, M. and Friend, I. (1973) A new look at the capital asset pricing model, Journal of Finance, 8, Campbell, J. Y. (1996) Understanding risk and returns, Journal of Political Economy, 104, Cochrane, J. H. (1996) A cross sectional test of an investment-based asset pricing model, Journal of Political Economy, 104, Daniel, K. and Titman, S. (1997) Evidence on the characteristics of cross sectional variation in stock returns, Journal of Finance, 5, Fama, E. F. and French, K. R. (199) The cross-section of expected stock returns, Journal of Finance, 47, Fama, E. F. and French, K. R. (1993) Common risk factors in the returns on stocks and bonds, Journal of Financial Economics, 33, Fama, E. F. and French, K. R. (1994) Industry cost of equity, Working Paper, Graduate School of Business, University of Chicago, Chicago IL, revised July Fama, E. F. and French, K. R. (1995) Size and book-tomarket factors in earnings and returns, Journal of Finance, 50, Fama, E. F. and French, K. R. (1996) Multifactor explanations of asset pricing anomalies, Journal of Finance, 51, Fama, E. F. and Macbeth, J. D. (1973) Risk, return and equilibrium: empirical tests, Journal of Political Economy, 81, Friend, I. I. and Blume, M. (1970) Measurement of portfolio performance under uncertainty, American Economic Review, 60, Hansen, L. P. and Jagannathan, R. (1997) Assessing specification errors in stochastic discount factor models, Journal of Finance, 5, Hodrick, R. J. and Zhang, X. (000) Evaluating the specification errors of asset pricing models, NBER working paper No Harvey, C. and Siddique, A. (000) Conditional skewness in asset pricing tests, Journal of Finance, 55, Jagannathan, R. and Wang, Z. (1996) The conditional CAPM and the cross-section of expected returns, Journal of Finance, 51, Kraus, A. and Litzenberger, R. (1976) Skewness preference and the valuation of risk assets, Journal of Finance, 38, Miller, M. H. and Scholes, M. (197) Rate of return in relation to risk: a reexamination of some recent findings, in Studies in the Theory of Capital Markets (Ed.) M. C. Jensen, Praeger Publishers, New York, pp Mishra, S., DeFusco, R. A. and Prakash, A. J. (005) Skewness preference, value and size effects, Working Paper. Pastor, L. and Stambaugh, R. (000) Comparing asset pricing models: an investment perspective, Journal of Financial Economics, 56, Reinganum, M. R. (1981) The arbitrage pricing theory: some empirical results, Journal of Finance, 36, Sharpe, W. (1964) Capital asset prices: a theory of market equilibrium under conditions of risk, Journal of Finance, 19, The data on the monthly returns of the 5 FF portfolios and the explanatory variables R m, SMB and HML data are taken from the web site of Dr Kenneth French. ken.french/data_library.html 3 The market risk premium is insignificant for both CAPM and three-moment CAPM at 10% significance.

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