EMPIRICAL TESTS OF ASSET PRICING MODELS

Transcription

1 EMPIRICAL TESTS OF ASSET PRICING MODELS DISSERTATION Presented in Partial Ful llment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Philip R. Davies, B.Sc., M.Sc. * * * * * The Ohio State University 2007 Dissertation Committee: Approved by Professor R.M. Stulz, Adviser Professor G.M. Allenby Professor G.A. Karolyi Adviser Graduate Program in Business Administration

2 ABSTRACT The Capital Asset Pricing Model (CAPM) developed by Sharpe (1964) and Lintner (1965) is widely viewed as one of the most important contributions to our understanding of nance over the last 50 years. The CAPM predicts that non-diversi able risk () is the only risk that matters for the pricing of assets, and that an asset s expected return is a positive linear function of its non-diversi able risk. However, the empirical performance of the CAPM has been poor. This poor performance may re ect theoretical failings. Alternatively, it may be due to di culties in implementing valid tests of the model. This dissertation focuses on the second possibility. In the rst essay I develop a Bayesian approach to test the cross-sectional predictions of the CAPM at the rm level. Using a broad cross-section of NYSE, AMEX, and NASDAQ listed stocks over the period July June 2005, I nd evidence of a robust positive relation between and average returns. Fama and French (1993) propose two additional risk factors related to rm size and book-to-market equity. I nd no evidence that these additional risk factors help to explain the cross-sectional variation in average returns. These results are consistent with the empirical predictions of the CAPM. The use of portfolios as test assets in cross-sectional tests of asset pricing models is widespread, principally to help mitigate statistical problems. However, there is a considerable theoretical literature showing that the use of portfolios can make bad ii

3 models look good, and good models look bad. In the second essay I investigate whether inferences from portfolio level studies can be generalized to the rm level. Using the Bayesian approach developed in the rst essay, I nd that inferences at the portfolio level are closely linked to the way in which portfolios are formed, rather than the underlying rm level associations. These results raise questions about what we can really learn from empirical asset pricing studies that use portfolios as test assets. iii

4 ACKNOWLEDGMENTS I wish to thank my adviser, René Stulz, for his helpful comments, patience, and advice during my dissertation research. Andrew Karolyi introduced the eld of empirical asset pricing to me, and provided helpful comments and suggestions for my dissertation. I would also like to thank Greg Allenby for the time and e ort that he put into my education. His comments and encouragement have been invaluable. I hope that I will be able to inspire students in the same way that he has inspired me. Thanks also to Bernadette Minton for her help and advice throughout my time at Ohio State. My parents, Geo and Eleanor Davies, and my sister, Jo Davies, have supported me every step of the way, and it goes without saying that I would not have made it through the PhD program without the help and support of my friends and colleagues, Rei-Ning Chen, Chuan Liao, An Chee Low, Taylor Nadauld, Haoqing Pan, Robyn Scholl, and Jérôme Taillard. I also wish to thank Cli ord Ball, Long Chen, Eugene Fama, Satadru Hore, Andrew Snell, Ashish Tiwari, and seminar participants at Michigan State University, the Ohio State University, Southern Methodist University, SUNY Bu alo, the University of Colorado at Boulder, the University of Connecticut, the University of Edinburgh, the University of Iowa, the University of Warwick, and Vanderbilt University for helpful comments and suggestions. iv

5 VITA February 8, Born Bromley, United Kingdom B.Sc. Accounting and Finance University of Warwick M.Sc. Economics and Finance University of Warwick PUBLICATIONS Research Publications A. Abhyankar and P. Davies. "Market Timing and Economic Value: Evidence from the Short Rate Revisited". Finance Letters 3, 1-9, FIELDS OF STUDY Major Field: Business Administration Concentration: Finance v

6 TABLE OF CONTENTS Page Abstract Acknowledgments Vita List of Tables ii iv v viii List of Figures x Chapters: 1. Introduction Reviving the CAPM: A Bayesian approach for testing asset pricing models Introduction The CAPM Model Speci cation Testing the CAPM Model Estimation Evaluating competing model speci cations Data Results The CAPM at the rm level using portfolio s The CAPM at the rm level using rm-speci c s The Fama-French 3 Factor model at the rm level using rmspeci c s Robustness Conclusion vi

7 3. Testing Asset Pricing Models: Firms vs Portfolios Introduction Methodology Model Speci cation Model Estimation Simulation Study Data Results The CAPM Alternate Asset Pricing Models Model Fit Conclusion Conclusion Bibliography Appendices: A. Estimation Algorithm B. Additional Empirical Results for Chapter C. Additional material for Chapter C.1 Portfolio Formation Procedures C.2 Variation in rm level s over time vii

8 LIST OF TABLES Table Page 2.1 Summary Statistics Empirical tests of asset pricing models: July June Empirical tests of asset pricing models: July June Empirical tests of asset pricing models: July June Empirical tests of asset pricing models: Variance-Covariance Matrix The fully conditional CAPM: July June Empirical tests of the CAPM Empirical tests of the CAPM with Human Capital Empirical tests of the Consumption CAPM Empirical tests of the Fama-French 3 Factor Model Empirical tests of the Fama-French 3 Factor Model Firm characteristics Model Fit B.1 Empirical tests of asset pricing models: July June B.2 Empirical tests of asset pricing models: July June viii

9 B.3 Empirical tests of asset pricing models: July June C.1 Transition Matrix ix

10 LIST OF FIGURES Figure Page 2.1 Posterior distribution plots for the risk premium, c m, after controlling for rm size, at return horizons of 1-6 years Posterior distributions for the intercept Posterior distribution plots for the risk premium, c m, in the simulation study Posterior distribution plots for the risk premium, c m, at a return horizon of 4 years The distribution of s at a 4 year return horizon The distribution of rm level s at a 4 year return horizon The distribution of rm level HML s at a 4 year return horizon Price indices Posterior distribution plots for the Fama-French 3 factor model: July June C.1 Di erences between pre-ranking and contemporaneous s at a 4 year return horizon x

11 CHAPTER 1 INTRODUCTION Asset pricing refers to the process by which the prices of nancial assets are determined, and the resulting relationships between expected returns and the risks associated with those returns. Over four decades ago Sharpe (1964) and Lintner (1965) developed the Capital Asset Pricing Model (CAPM). Building on the pathbreaking work of Markowitz (1959), Sharpe (1964) and Lintner (1965) show that, in equilibrium, the aggregate wealth portfolio is mean-variance e cient. The e ciency of the aggregate wealth portfolio implies that 1) the only risk that matters for the pricing of nancial assets is non-diversi able risk, and 2) a nancial asset s expected return is a positive linear function of its non-diversi able risk. Today the CAPM is still widely used by academics and practitioners to estimate the cost of capital for rms, and to evaluate the performance of investment managers. Indeed, as Fama and French (2004) note, the CAPM is often the centerpiece of undergraduate and MBA investment courses. The reason behind the CAPM s widespread use is that it o ers powerful and intuitive predictions regarding how non-diversi able risk should be measured, and the relation between non-diversi able risk and expected returns. 1

12 However, the empirical performance of the CAPM has been poor. For example, in their in uential 1992 study, Fama and French nd that non-diversi able risk is unable to explain cross-sectional di erences in average returns. Further, building on their 1992 study, Fama and French (1993) propose two additional factors designed to capture the risks associated with rm size (SMB) and book-to-market equity (HML). They show that the empirical performance of their 3 factor model is superior to that of the CAPM. The poor empirical performance of the CAPM may re ect theoretical failings. Alternatively, it may be caused by di culties in implementing valid tests of the model. The focus of my dissertation is on the latter possibility. Researchers seeking to examine whether the CAPM is able to explain crosssectional di erences in average returns face two major di culties. First, the CAPM states that the risk of a stock should be measured relative to the aggregate wealth portfolio. However, the aggregate wealth portfolio is not observed by the researcher. Therefore, as Roll (1977) notes, tests of the CAPM can be interpreted as a joint test of two hypotheses: 1) the CAPM holds, and 2) returns on the aggregate wealth portfolio are a linear function of the returns on the proxy chosen by the researcher. The second major di culty facing researchers is that the non-diversi able risk of a rm, hereafter referred to as, is an unobserved, latent variable. Having chosen a proxy for aggregate wealth, the researcher must obtain estimates of s for rms to examine the prediction that average returns are positively related to s. Researchers typically adopt a two-step estimation procedure. First, obtain estimates of,, b then examine whether there is a positive relation between average returns and s. b However, 2

13 bs are estimated imprecisely, creating a measurement error problem when the s b are used to explain average returns. This will result in a downward bias in the estimated risk premium. Researchers have sought to develop techniques that minimize the measurement error problem while maximizing heterogeneity in s across both time and rms. The benchmark approach for estimating s at the rm level was developed by Fama and French (1992). Each year rms are assigned to 100 portfolios based on rm characteristics. Given the portfolio returns, Fama and French (1992) estimate portfolio s using a market model over the entire sample period. Estimates of s for diversi ed portfolios are more precise than estimates of s for individual rms. The portfolio s are then assigned to individual rms in each year. In chapter 2 I develop a Bayesian approach to examine the ability of the CAPM to explain the cross-sectional variation in average returns at the rm level. The principal advantage of the Bayesian approach is that it enables the researcher to assess just how important time and rm heterogeneity are in the estimation of s, while explicitly controlling for the inherent uncertainty associated with time varying rm-speci c s. I examine the empirical predictions of the CAPM using a broad cross-section of NYSE, AMEX, and NASDAQ listed stocks over the sample period July June When I use a portfolio approach similar to that of Fama and French (1992), I nd that is unable to explain the cross-sectional variation in average returns. However, when s are allowed to vary across both time and rms, I nd strong evidence supporting the main empirical prediction of the CAPM. There is a robust positive relation between average returns and. The estimated risk premium is 3

14 approximately 7% per year, which is economically plausible given that average excess returns on the stock market tend to range between 6% and 8% per year. Finally, the CAPM implies that the risk factors proposed by Fama and French (1993) should not be able to explain expected returns. Consistent with the predictions of the CAPM, I nd no robust evidence that risks associated with SMB and HML are able to help explain the di erences in average returns observed across rms. Although asset pricing models are supposed to work for individual rms as well as portfolios, over the past 40 years the majority of models have been estimated and tested only at the portfolio level. The principal reason for the use of portfolios as test assets in cross-sectional tests of asset pricing models is to reduce the impact of measurement error problems. However, cautions regarding the use of portfolios as test assets abound in the literature. Theoretical work shows that the use of portfolios can make bad asset pricing models look good (Roll 1977). On the other hand, Kan (2004) shows that the use portfolios can also make good asset pricing models look bad. Ultimately researchers are interested in how well asset pricing models explain returns at the rm level. In chapter 3 I examine whether inferences at the portfolio level can be generalized to the rm level. I use the Bayesian approach developed in chapter 2 to examine the performance of the CAPM, the CAPM with human capital, the consumption CAPM, and the Fama- French 3 Factor model at both the rm level and the portfolio level. The models are estimated using a broad cross-section of NYSE, AMEX, and NASDAQ listed stocks over the sample period July June Portfolios are constructed using several di erent approaches, but I focus on two of the most widely used sets of portfolios, 4

15 100 size-pre-ranking portfolios and 25 size-book-to-market equity portfolios. All portfolios are constructed from the same data used to conduct the rm level analysis. Consistent with past research, at the portfolio level, there is little evidence of a robust positive relation between stock market s and average returns. Similarly, human capital s, and consumption growth s are also unable to explain the crosssection of average portfolio returns. The rm level results paint a very di erent picture. I nd evidence that average returns increase linearly with both stock market s and consumption growth s. In addition, consistent with the ndings in chapter 2, there is little evidence that the additional factors proposed by Fama and French (1993), SMB and HML, are priced risk factors. The ndings across all four asset pricing models support the theoretical work cautioning researchers regarding the use of portfolios as test assets. I nd that inferences based on portfolio level tests are sensitive to the portfolio formation method. Depending on how the portfolios are formed, the underlying rm level associations can be masked, and the signs on risk premia reversed. 5

16 CHAPTER 2 REVIVING THE CAPM: A BAYESIAN APPROACH FOR TESTING ASSET PRICING MODELS 2.1 Introduction The capital asset pricing model of Sharpe (1964), Lintner (1965), and Black (1972) has shaped the way that academics and practitioners think about risk and return. The central prediction of the CAPM is that the aggregate wealth portfolio is mean-variance e cient. The e ciency of the aggregate wealth portfolio implies that 1) a security s expected return is a positive linear function of its sensitivity to non-diversi able risk, as measured by, and 2) s are su cient to describe the cross-section of expected returns. Early work by Black, Jensen, and Scholes (1972), Fama and MacBeth (1973), and Stambaugh (1982) nds that there is a positive relation between and average returns. However, in their in uential 1992 paper, Fama and French examine the ability of s, rm size, and book-to-market equity to explain average returns at the rm level. They conclude that, after controlling for rm size and book-to-market equity, is not able to help explain average stock returns. Since Fama and French (1992), very few papers examining the cross-sectional variation in average returns have found much, if any, support for the CAPM. 6

17 Three notable exceptions are Jagannathan and Wang (1996), Lettau and Ludvigson (2001), and Ferguson and Shockley (2003). Jagannathan and Wang (1996) nd that when a measure of human capital is included in the proxy for aggregate wealth, the performance of the CAPM (conditional and unconditional) is substantially improved. The conditional CAPM is able to explain over 50% of the cross-sectional variation in average returns. Lettau and Ludvigson (2001), using a similar approach to Jagannathan and Wang (1996), nd that the conditional consumption CAPM is able to explain the cross-sectional variation in average returns at least as well as the Fama-French 3 factor model. The Fama-French 3 factors are stock market returns, SMB, and HML. SMB and HML are factors designed to capture the risks associated with rm size and book-to-market equity. Finally, Ferguson and Shockley (2003) show that many empirical "anomalies" are actually consistent with the CAPM if researchers use an all equity proxy for the aggregate wealth portfolio. They propose two proxies to capture events in the debt markets, and nd evidence supporting the CAPM when the proxies for debt are incorporated in the empirical tests. However, Lewellen, Nagel, and Shanken (2006) highlight several methodological concerns with papers such as Jagannathan and Wang (1996), Lettau and Ludvigson (2001), and Ferguson and Shockley (2003). First, Lettau and Ludvigson (2001), and Ferguson and Shockley (2003) use the Fama-French 25 size-book-to-market portfolios. These portfolios are well known to have a strong factor structure. The Fama-French 3 factors can explain more than 75% of the cross-sectional variation in 25 size-book-tomarket portfolio returns. Thus, as long as a proposed factor is correlated with SMB or HML, a high R 2 will be obtained. When Lewellen, Nagel, and Shanken (2006) 7

18 extend Lettau and Ludvigson (2001) to portfolios other than the Fama-French 25 size-book-to-market portfolios, the results are weak, o ering little or no support for the conditional consumption CAPM. Second, the empirical tests of the conditional models proposed by Jagannathan and Wang (1996) and Lettau and Ludvigson (2001) ignore the theoretical restrictions on cross-sectional slope coe cients. Lewellen and Nagel (2006) argue that imposing such restrictions could greatly reduce the explanatory power of the proposed asset pricing models. Therefore it is not clear whether Jagannathan and Wang (1996), Lettau and Ludvigson (2001), and Ferguson and Shockley (2003) really do provide strong support for the CAPM, or the consumption CAPM. The CAPM s empirical problems may stem from two sources: 1) the theoretical model requires many simplifying assumptions, such as the existence of perfect capital markets, that are violated in reality, and 2) di culties in implementing valid empirical tests of the model. I focus on the latter possibility. There are two major di culties facing researchers seeking to test the CAPM. First, the CAPM states that the risk of a stock should be measured relative to the aggregate wealth portfolio. However, the aggregate wealth portfolio is not observed by the researcher. Therefore, as Roll (1977) notes, tests of the CAPM can be interpreted as a joint test of two hypotheses: 1) the CAPM holds, and 2) returns on the aggregate wealth portfolio are a linear function of the returns on the proxy chosen by the researcher. Second, s are unobserved, latent variables. Having settled on a proxy for aggregate wealth, the researcher must obtain estimates of s for rms to examine the prediction that average returns are positively related to s. Researchers typically 8

19 adopt a two-step estimation procedure. First, obtain estimates of,, b then examine whether there is a positive relation between average returns and s. b However, s b are estimated imprecisely, creating a measurement error problem when the s b are used to explain average returns. This will result in a downward bias in the estimated risk premium. Over the last 30 years researchers have sought to develop techniques that minimize the measurement error problem while maximizing heterogeneity in s across both time and rms. The benchmark approach for estimating s at the rm level was developed by Fama and French (1992). Each year rms are assigned to 100 portfolios based on rm characteristics. Given the portfolio returns, Fama and French (1992) estimate portfolio s using a market model over the entire sample period. The portfolio s are then assigned to individual rms in each year. As a rm transitions across portfolios, so its changes. In this chapter I propose a Bayesian approach to examine the ability of the CAPM to explain the cross-sectional variation in average returns at the rm level. The principal advantage of the Bayesian approach is that it enables the researcher to examine just how important time and rm heterogeneity are in the estimation of s, while explicitly controlling for the inherent uncertainty associated with time varying rm-speci c s. I use a broad cross-section of NYSE, AMEX, and NASDAQ listed stocks over the period July June 2005 to examine the empirical predictions of the CAPM. Consistent with previous studies, I assume that returns on the aggregate wealth portfolio are a linear function of returns on the CRSP value weighted stock market index. 9

20 Using a similar approach to Fama and French (1992) I nd that, after controlling for rm size, is unable to explain the cross-sectional variation in average returns. However, the Fama and French (1992) approach imposes two restrictions on the estimation of rm-speci c s. First, all rms assigned to the same portfolio must have the same, the portfolio. Second, portfolio s cannot vary across time periods. When s are allowed to vary across both time and rms, I nd strong evidence supporting the main empirical prediction of the CAPM. There is a positive relation between average returns and, which is robust to the inclusion of rm size. The mean of the posterior distribution for the risk premium is approximately 7% per year. This is consistent the standard textbook view that the risk premium is between 6% and 8% per year, and the actual data used in this study. Further, Fama and French (1993) propose two additional risk factors, SMB and HML. The CAPM implies that SMB and HML s should not be able to explain the cross-sectional variation in average returns left unexplained by stock market s. Consistent with the predictions of the CAPM, I nd no robust evidence that SMB and HML s are priced risk factors. Although is a priced risk factor, I nd that, contrary to the predictions of the CAPM, there is a robust negative association between rm size and average returns. Berk (1995) argues the the CAPM should not be rejected solely on the basis of the nding that rms size is negatively related to average returns, since such a relation will exist if an asset pricing model is misspeci ed in any way. Given that the majority of the evidence is actually consistent with the CAPM, I interpret the nding that rm size is negatively related to average returns as evidence that the CAPM is, in some way, empirically misspeci ed. For example, the stock market index may not be the best proxy for the aggregate wealth portfolio. 10

21 Finally, to assess the robustness of my ndings across di erent time periods, I split the sample period into two sub-periods, July June 1963, and July June In both sub-periods I nd evidence of a positive relation between s and average returns. This relation is robust to the inclusion of both rm size and the additional risk factors SMB and HML. Further, there is little evidence that SMB and HML s are priced risk factors in either sub-period. The chapter proceeds as follows. Section 2.2 brie y discusses the various approaches researchers have used to examine the CAPM for a large number of test assets. A exible statistical model is then developed to enable more precise tests of the CAPM, and the advantages and disadvantages of this new approach are discussed. Section 2.3 describes the data used to test the CAPM. In section 2.4 I report the empirical results and evaluate the performance of the CAPM. Section 2.5 concludes. 2.2 The CAPM Model Speci cation The Sharpe-Lintner version of the CAPM states that, in the cross-section, the following relation should hold, E [r i r f ] = (E [r m r f ]) i (2.1) where E [r i r f ] denotes the expected excess return for rm i over and above the risk free rate, r f. E [r m r f ] denotes the expected excess return on the aggregate wealth portfolio, and i = Cov(r i;r m) V ar(r m). 11

22 Given s, the Sharpe-Lintner CAPM implies that if we run a regression, r e i = c 0 + c m i + " i ; (2.2) where ri e = r i r f, we should nd that c 0 = 0 and c m = E [r m r f ] > 0. Unfortunately we do not observe s. They are latent variables. This problem prompted researchers to adopt a two-step estimation procedure. First, obtain estimates of, b, for rms using the market model, r i;t = i + i r m;t + " i;t ; (2.3) where r m denotes the return on a proxy for the aggregate wealth portfolio. Second, n o plug bi into equation (2.2). However, in a classical setting, s b for individual rms are estimated imprecisely, creating a measurement error problem when the s b are used to explain average returns. This will result in a downward bias in the estimated coe cient bc m. To improve the precision of s, b researchers such as Black, Jensen, and Scholes (1972), and Fama and MacBeth (1973) use well diversi ed portfolios rather than individual rms as test assets. Estimates of s for diversi ed portfolios are more precise than estimates of s for individual rms. Fama and French (1992) examine the CAPM at the rm level. They propose a new approach to estimate rm-speci c s. Each year rms are assigned to 100 portfolios based on size and pre-ranking b s. 1 Next, they calculate portfolio returns for each of the 100 portfolios. Given the portfolio returns Fama and French (1992) estimate s for each portfolio using the market model over their entire sample period 1 The pre-ranking s are estimated for each rm using 24 to 60 monthly returns in the 5 years prior to the portfolio formation month (July) each year. For more details refer to Fama and French (1992). 12

23 ( ). Using a long time series should help reduce the estimation error for the portfolio b s. The portfolio b s are then assigned to individual rms. As a rm transitions across portfolios, so its changes. This approach involves a bias-variance trade-o. Estimates of s for well diversi ed portfolios are more precise, but, to the extent that there is within portfolio heterogeneity in rm-speci c s, they are biased estimates of rm-speci c s. While the trade-o is acceptable to obtain accurate estimates of b, it is not appropriate for testing the CAPM. There will be a downward bias in the estimated coe cient bc m if the values of b assigned to each rm di er from the true values of rm-speci c s. I propose a direct approach to examine the relation between risk and average return which involves the estimation of a two equation system, r e i;y = c 0;y + c m;y i;y + " i;y (2.4) r i;t;y = i;y + i;y r m;t;y + " i;t;y ; (2.5) where r e i;y denotes the annualized average monthly excess return for rm i during time period y, and " i;y N 0; 2cy. r i;t;y denotes rm i s stock return in month t during period y, and " i;t;y N 0; 2i;y. The length of each time period, y, will be referred to as the return horizon in the remainder of the text. An implicit assumption of the model described by equations (2.4) and (2.5) is that average monthly excess returns in time period y are independent of any single monthly return during time period y. At short horizons, such as 6 months or 1 year, it is unlikely that the assumption of independence is appropriate. However, as the return horizon is extended to 2, 3, 4, 5, or 6 years, so the assumption becomes increasingly plausible. To investigate how sensitive the results are to this assumption, the model will be estimated using return horizons, ranging from 1 to 6 years, and 13

24 the results compared. An additional advantage of estimating the model at di erent return horizons is that I will also be able to examine whether inferences about the CAPM are sensitive to the return horizon chosen by the researcher (Levy (1984), Kothari, Shanken, and Sloan (1995)). Since Fama and French (1993), many studies, including Lettau and Ludvigson (2001), and Ferguson and Shockley (2003), use the Fama-French 25 size-book-tomarket portfolios as test assets rather than individual rms. I choose to examine the CAPM at the rm level rather than the portfolio level for three reasons. First, cautions regarding the use of portfolios as test assets abound in the literature. For example, Kan (2004) demonstrates that the use portfolios can not only make good asset pricing models look bad, but also make bad asset pricing models look good. Second, Lewellen, Nagel, and Shanken (2006) extend several studies, such as Lettau and Ludvigson (2001), to portfolios other than the 25 size-book-to-market portfolios. The results are much weaker when the models are tested on a wider set of portfolios, indicating that inferences may be sensitive to the choice of portfolios. Finally, estimating the CAPM at the rm level facilitates direct comparison with the ndings of Fama and French (1992) Testing the CAPM The Sharpe-Lintner version of the CAPM, speci ed in equation (2.1), generates three testable implications. First, the CAPM implies that there is a positive relation between expected return and risk, E [c m ] > 0. Second, the Sharpe-Lintner CAPM posits that E [c 0 ] = 0. Finally, the CAPM implies that is the only variable necessary to explain expected returns. 14

25 Fama and French (1992) show that, after controlling for rm size, has no explanatory power. Equation (2.4) can be modi ed to examine the impact of adding rm size ( ln (ME)) as a control variable, 2 r e i;y = c 0;y + c m;y i;y + c size;y ln(me i;y ) + " i;y : (2.6) If the CAPM holds, Fama and French (1992) argue that rm size should not be a priced risk factor, E [c size ] = 0. However, Berk (1995) shows that if an asset pricing model is misspeci ed in any way, rm size will be negatively associated with future returns. Berk (1995) argues that the observation that rm size explains part of the returns not explained by s, by itself, is not necessarily evidence that the CAPM is not able to price risk correctly. Rather, it could indicate that the empirical speci cation of the CAPM is not quite right. For example, the proxy for the aggregate wealth portfolio may be poor. Building on their 1992 paper, Fama and French (1993) propose two additional risk factors related to rm size (SMB), and book-to-market equity (HML). While Berk (1995) shows that the natural logarithm of rm size will be correlated with expected returns in the cross-section if the asset pricing model is misspeci ed, his paper does not imply that these associations can be captured by a stock s SMB and HML s. Equations (2.4) and (2.5) can be modi ed to incorporate these additional risk factors, r e i;y = c 0;y + c m;y i;y + c smb;y SMB i;y + c hml;y HML i;y + " i;y (2.7) r i;t;y = i;y + i;y r m;t;y + SMB i;y r smb;t;y + HML i;y r hml;t;y + " i;t;y ; (2.8) 2 Firm size for period y is the natural logarithm of a rm s market equity (in thousands of dollars) in the month immediately preceding the start of period y. 15

26 where r smb and r hml are the returns on portfolios constructed to mimic the risk factors related to size and book-to-market equity. The CAPM implies that the two additional risk factors should not be priced, E [c smb ] = 0 and E [c hml ] = Model Estimation Estimation of the model described in equations (2.4) and (2.5) is complicated by the fact that the latent variable i;y appears in both equations. Maximum likelihood estimation requires the researcher to integrate over the joint density to obtain the marginal density of the data. However, the joint density implied by equations (2.4) and (2.5) is complex and there is no closed form solution for the integral. Thus, it is a challenging problem to estimate the system of equations using maximum likelihood. An alternative estimation strategy would be to use the generalized method of moments (GMM). However, Ferson and Foerster (1994) demonstrate that GMM has rather poor nite sample properties, especially for problems with high dimensionality and many test assets. Speci cally they note that "in more complex models, the coef- cient estimates and standard errors can be biased by large amounts." Consequently, testing the CAPM at the rm level is not feasible using GMM. Instead, a Bayesian approach is adopted to estimate the model described in equations (2.4) and (2.5). Recent developments in statistical computing, in particular, Markov chain Monte Carlo (MCMC) methods, have made it possible to estimate models which would be di cult, if not impossible to estimate using non Bayesian methods. The Bayesian approach is likelihood based, and requires the speci cation of both a likelihood function and a prior. 16

27 There are several advantages to the Bayesian approach. First, researchers have struggled to develop techniques that minimize the measurement error problem while maximizing heterogeneity in s b across both time and rms. The Bayesian approach enables the researcher to examine just how important time and rm heterogeneity are in the estimation of s, while explicitly controlling for the inherent uncertainty associated with time varying rm-speci c s. Second, the Bayesian approach is able to overcome the problems with the classical two-step tests of asset pricing models identi ed by Kan and Zhang (1999). In an extreme setting where a risk factor is useless, de ned as being independent of all the asset returns, Kan and Zhang (1999) show that the second pass cross-sectional regressions tend to nd that the useless risk factor is priced more often than it should be. The true s of the assets with respect to the useless risk factor are zero, and the true risk premium for the risk factor is unde ned. However, in a classical setting, as point estimates of s approach zero, the absolute value of the risk premium goes towards in nity to "explain" the cross-sectional variation in average returns. This misspeci cation bias arises due to the estimation errors associated with point estimates of s. The Bayesian framework takes into account the parameter uncertainty associated with all the model parameters, thereby avoiding such a bias. The advantages of the Bayesian approach come at the cost of having to specify explicit priors. The priors for equation (2.4) are speci ed as, c y Normal (c; V c ) : (2.9) 17

28 c0;y where c y denotes c m;y c0 c =. In turn I specify proper and di use priors for c and V c, c m 0 Normal ; (2.10) V c Inverted Wishart (Nu c ; V 0;c ) (2.11) where Nu c = Nvar c + 3, V c;0 = (Nu c ) I. Nvar c denotes the number of independent variables in equation (2.4). 3 The CAPM provides a model for expected returns and risk. In reality we only observe realized returns and risk. Consequently, when examining the empirical predictions of the CAPM, inferences should be based on the posterior distribution of c. As Fama (1976) notes (pg 361), we would expect c 0;y and c 1;y to be quite variable through time, and even negative in some periods. While c y provides information about risk premia during time period y, c provides information about average risk premia across all time periods (y = 1; :::; Y ). In equation (2.9) I assume that c y is normally distributed with a mean c and a variance-covariance matrix V c. The normal distribution is a exible distribution, but it should be noted that in uence of the likelihood for each period may be attenuated for likelihoods centered a long way from the prior. Outliers will be shrunk towards the prior mean due to the thin tails of the normal distribution. I can further increase the exibility of the prior distribution by incorporating observable factors thought to be associated with time variation in risk premia by using a multivariate regression speci cation, c y Normal ( 0 z y ; V c ) : (2.12) 3 To estimate the model described by equations (2.7) - (2.8) this prior can extended such that c 0 = c 0 c m c smb c hml Normal(0; 100I) ; and Vc Inverted Wishart(Nu c ; V 0;c ) where Nvar c denotes the number of independent variables in equation (2.7). 18

29 where z represents a matrix of factors thought to be associated with time variation in the nominal risk premium. 4 Allowing the risk premium to vary across time results in a fully conditional version of the CAPM. A potential drawback to the approach outlined above is that the researcher may not have many observations of c y with which to make inferences about c (or ) and V c. For example, using a 6 year return horizon to compute average excess returns for equation (2.4) over the period will result in only 13 nonoverlapping observations of c y. Asymptotic approximations may not be accurate for such a small sample size. Fortunately, Bayesian inference is free from the use of asymptotic approximations and delivers exact nite sample inference. Finally, I must also specify priors for i;y and i;y in equation (2.5). I specify proper and di use priors which should not exert a strong in uence over the posterior distributions, i;y Normal ; V = Normal ; 0 10 (2.13) where 0 i;y = i;y i;y : 5 A detailed description of the estimation algorithm is provided in Appendix A. SMB i;y 4 See Chapter 5 in Rossi, Allenby, and McCulloch (2005) for a review of hierarchical Bayes models. 5 To estimate the model described by equations (2.7) - (2.8) priors must also be speci ed for, and HML i;y. I specify proper and di use priors as follows, i;y Normal ; V = ; C A where 0 i;y = i;y i;y smb i;y hml i;y Normal 4 19

30 2.2.4 Evaluating competing model speci cations To compare the relative performance of di erent empirical speci cations I compute the log marginal density using the importance sampling method of Newton and Raftery (1994). This is a Bayesian measure of model t which includes an implicit penalty for models with a large number of parameters. To get a better sense of how well di erent models perform in terms of in-sample explanatory power I also calculate the mean absolute error. On each iteration of the MCMC chain the mean absolute error is calculated and its value stored. The distribution for mean absolute error is then summarized to enable comparison across di erent empirical speci cations. 2.3 Data I use monthly stock returns on all corporations listed on the NYSE, AMEX, and NASDAQ over the period January June 2005 that are covered by the CRSP tapes. The 1 month T-Bill rate is used as a proxy for the risk free rate. Returns on the aggregate wealth portfolio are assumed to be a linear function of returns on the CRSP value weighted stock market index. Monthly return data for the risk factors proposed by Fama and French (1993) related to rm size (SMB), and book-to-market equity (HML) are taken from Kenneth French s website. 6 I examine the CAPM using non-overlapping return horizons ranging from 1-6 years. All time periods start in July and end in June. In order to compare the proposed approach with that of Fama and French (1992) it is necessary to create a data 6 I wish to thank Kenneth French for making the data available on his website: See Fama and French (1993) for more details on the contruction of the risk factors. 20

31 set which meets the requirements needed to implement both the Fama and French (1992) approach and the approach described in the previous section. Consequently, there are three data requirements for a rm to be included in the sample for a speci c time period. First, rms must have monthly returns for at least 24 of the 60 months preceding the start of the time period. This is necessary to obtain estimates of pre-ranking s. This requirement means that the actual sample period will be July June Second, to calculate rm size I require that rms must have stock price data in CRSP for the June immediately preceding the start of a time period. Firm size is de ned as the natural logarithm of a rm s market equity (in thousand dollars). Market equity for time period y is calculated using CRSP data on share prices and shares outstanding in the June immediately preceding the start of time period y. Finally, when examining the CAPM at an annual return horizon I require a rm to have 12 monthly returns during the 1 year time period in order to estimate the model described by equations (2.4) and (2.5). For return horizons of 2 or more years I require at least 24 monthly returns during the time period for a rm to be included in the sample during that period. Table 2.1 provides summary statistics. The average equally weighted excess return for rms in the sample is approximately 13% per year at all return horizons, while the returns on the CRSP value weighted market portfolio are 11% per year over the period The average returns on the risk factors proposed by Fama and French (1993), SMB and HML, are 3% and 5% per year. Finally, the average number of rms per time period ranges between 2,500 and 3,000 rms depending on the return horizon. 21

32 2.4 Results In table 2.2 I examine the performance of two asset pricing models, the CAPM and the Fama-French 3 factor model over the period July June 2005, using 1, 3, and 5 year return horizons. Results are reported with and without the inclusion of rm size as an explanatory variable The CAPM at the rm level using portfolio s The approach of Fama and French (1992) can be nested in the model described by equations (2.4) and (2.5) by imposing two restrictions on s. First, rms in the same size-pre-ranking b portfolio must have the same, the portfolio. Second, portfolio s cannot vary across time. When these restrictions are imposed (denoted by "FF" in column 1 of table 2.2), I obtain results consistent with Fama and French (1992). For example, at a 1 year return horizon, using the CAPM speci cation, I nd that is a priced risk factor. However, the relation between average returns and disappears when rm size is included in the speci cation. The results are similar for 3 and 5 year return horizons. Relaxing the constraint that portfolio s must remain constant across time periods (denoted by "TV FF" in column 1 of table 2.2) leads to an improvement in model t at all return horizons. However, relaxing this restriction does not change our inferences regarding the CAPM. After controlling for size there is no evidence, at conventional signi cance levels, that is able to explain the cross-sectional variation in average returns. 7 The results for 2, 4, and 6 year return horizons are reported in Appendix B. The results for even year return horizons are similar to those reported for odd year return horizons. 22

33 2.4.2 The CAPM at the rm level using rm-speci c s When s are allowed to vary across both time periods and rms (denoted by "Firm Level" in column 1 of table 2.2) there is a considerable improvement in the model t relative to the benchmark Fama and French (1992) approach. At a 1 year return horizon the log marginal density increases from -848,741 to -796,318 for the CAPM model, while the mean absolute pricing errors fall from to Similar improvements in model t are observed at all return horizons. Even if the Fama and French (1992) approach is supplemented by the addition of rm size as an explanatory variable, the simple CAPM model, in which s vary across time periods and rms, ts the data better at all return horizons. More importantly, when s are allowed to vary across time periods and rms, I nd strong evidence supporting the main empirical prediction of the CAPM. There is a positive relation between average returns and. This positive relation is robust to the inclusion of rm size as an explanatory variable. In gure 2.1 I plot the posterior distributions for the price of risk, c m, after controlling for rm size. Figure 2.1 shows that at return horizons of 3-6 years the posterior distributions for the risk premium are similar, with a mean of approximately 6% - 7%, and 95% con dence bounds ranging between 3% and 9%. This is consistent with the standard textbook view that the risk premium ranges between 6% and 8%. At 1 and 2 year return horizons the means of the posterior distributions are slightly higher at 11.5% and 8%, but the posterior distributions for the risk premium are more di use. The results in table 2.2 and gure 2.1 suggest that inferences regarding the CAPM are not especially sensitive to the choice of return horizon. 23

34 The reason the posterior distributions for the risk premium are more di use at shorter return horizons (1 and 2 years) is related to the model setup. At 1 and 2 year return horizons only 12 or 24 observations are used in the market model speci ed in equation (2.5). With so few observations the posterior distributions for rmspeci c s will be di use, re ecting the limited information provided by the data. The Bayesian approach automatically incorporates this parameter uncertainty into the posterior distributions for the risk premium. The Sharpe-Lintner CAPM posits that E [c 0 ] = 0. When s are allowed to vary across time periods and rms there is mixed evidence regarding this empirical prediction. Figure 2.2 plots the posterior distributions for c 0. At return horizons less than 5 years table 2.2 and gure 2.2 provide little evidence, at conventional signi cance levels, against the prediction of the Sharpe-Lintner CAPM, that E [c 0 ] = 0. However, at 5 and 6 year return horizons there is evidence that the posterior distributions for c 0 are focused above zero. While this may not be consistent with the Sharpe-Lintner CAPM, it is still consistent with Black s version of the CAPM in which there is restricted borrowing The Fama-French 3 Factor model at the rm level using rm-speci c s Table 2.2 shows that incorporating the additional factors, SMB and HML, leads to an improvement in the model t relative to the CAPM speci cation at all return horizons. However, the inclusion of the additional risk factors does not change our 24

35 inferences with regards to the price of stock market risk. Stock market risk is priced at all return horizons, with the mean of the posterior distributions for the risk premium ranging from 5% to 8% at returns horizons greater than 1 year. 8 There is no evidence, in table 2.2, that HML s help to explain the cross-sectional variation in average returns left unexplained by stock market s. However the evidence is mixed with regards to SMB s. When the rm characteristic, size, is not included in the speci cation, there is a positive relation between average returns and SMB s. In contrast, when rm size is included, there is no robust evidence across return horizons that SMB is a priced risk factor at conventional signi cance levels. While much of the evidence presented in table 2.2 is consistent with the empirical predictions of the CAPM, rm size is negatively related to average returns for all model speci cations. Berk (1995) argues that an asset pricing model should not be rejected solely on the basis of the nding that rm size is negatively related to average returns, since such a relation will exist if the asset pricing model is misspeci ed in any way. Given that the majority of the evidence is consistent with the CAPM, I interpret the relation between rm size and average returns not as evidence that the CAPM should be rejected outright, but rather as evidence that the CAPM is in some way empirically misspeci ed. For example, the assumption that returns on the aggregate wealth portfolio are a linear function of returns on the CRSP value weighted stock market index may not be correct. 8 At a 1 year return horizon the mean of the posterior distribution for the risk premium is slightly higher at 10.74%. However the posterior distribution is considerably more di use at the 1 year return horizon compared to longer return horizons. 25