empirical tests of asset pricing models

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 v

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

The Capital Asset Pricing Model (CAPM) developed by Sharpe (1964) and ner (1965) is widely viewed as one of the most important contributions to our under-standing of …nance over the last 50 years The CAPM predicts that non-diversi…ablerisk ( ) is the only risk that matters for the pricing of assets, and that an asset’sexpected 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 mayre‡ect theoretical failings Alternatively, it may be due to di¢ culties in implementingvalid tests of the model This dissertation focuses on the second possibility

Lint-In the …rst essay I develop a Bayesian approach to test the cross-sectional tions of the CAPM at the …rm level Using a broad cross-section of NYSE, AMEX,and NASDAQ listed stocks over the period July 1927 - June 2005, I …nd evidence of arobust positive relation between and average returns Fama and French (1993) pro-pose two additional risk factors related to …rm size and book-to-market equity I …nd

predic-no evidence that these additional risk factors help to explain the cross-sectional ation in average returns These results are consistent with the empirical predictions

vari-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 aconsiderable theoretical literature showing that the use of portfolios can make bad

models look good, and good models look bad In the second essay I investigatewhether 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 theportfolio level are closely linked to the way in which portfolios are formed, rather thanthe underlying …rm level associations These results raise questions about what wecan really learn from empirical asset pricing studies that use portfolios as test assets.

I wish to thank my adviser, René Stulz, for his helpful comments, patience, andadvice during my dissertation research Andrew Karolyi introduced the …eld of em-pirical asset pricing to me, and provided helpful comments and suggestions for mydissertation I would also like to thank Greg Allenby for the time and e¤ort that heput into my education His comments and encouragement have been invaluable Ihope 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 atOhio 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 itthrough 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, RobynScholl, and Jérôme Taillard

I also wish to thank Cli¤ord Ball, Long Chen, Eugene Fama, Satadru Hore, drew Snell, Ashish Tiwari, and seminar participants at Michigan State University, theOhio State University, Southern Methodist University, SUNY Bu¤alo, the University

An-of Colorado at Boulder, the University An-of Connecticut, the University An-of Edinburgh,the University of Iowa, the University of Warwick, and Vanderbilt University forhelpful comments and suggestions

February 8, 1979 Born — Bromley, United Kingdom

2001 B.Sc Accounting and Finance —

Uni-versity of Warwick

2002 M.Sc Economics and Finance —

Uni-versity of Warwick

PUBLICATIONSResearch Publications

A Abhyankar and P Davies "Market Timing and Economic Value: Evidence fromthe Short Rate Revisited" Finance Letters 3, 1-9, 2005

FIELDS OF STUDY

Major Field: Business Administration

Concentration: Finance

TABLE OF CONTENTS

Page

Abstract ii

Acknowledgments iv

Vita v

List of Tables viii

List of Figures x

Chapters: 1 Introduction 1

2 Reviving the CAPM: A Bayesian approach for testing asset pricing models 6 2.1 Introduction 6

2.2 The CAPM 11

2.2.1 Model Speci…cation 11

2.2.2 Testing the CAPM 14

2.2.3 Model Estimation 16

2.2.4 Evaluating competing model speci…cations 20

2.3 Data 20

2.4 Results 22

2.4.1 The CAPM at the …rm level using portfolio s 22

2.4.2 The CAPM at the …rm level using …rm-speci…c s 23

2.4.3 The Fama-French 3 Factor model at the …rm level using …rm-speci…c s 24

2.4.4 Robustness 26

2.5 Conclusion 29

3 Testing Asset Pricing Models: Firms vs Portfolios 42

3.1 Introduction 42

3.2 Methodology 47

3.2.1 Model Speci…cation 47

3.2.2 Model Estimation 47

3.2.3 Simulation Study 49

3.3 Data 52

3.4 Results 54

3.4.1 The CAPM 55

3.4.2 Alternate Asset Pricing Models 61

3.4.3 Model Fit 68

3.5 Conclusion 70

4 Conclusion 90

Bibliography 92

Appendices: A Estimation Algorithm 95

B Additional Empirical Results for Chapter 2 99

C Additional material for Chapter 3 105

C.1 Portfolio Formation Procedures 105

C.2 Variation in …rm level s over time 106

LIST OF TABLES

2.1 Summary Statistics 33

2.2 Empirical tests of asset pricing models: July 1927 - June 2005 35

2.3 Empirical tests of asset pricing models: July 1927 - June 1963 37

2.4 Empirical tests of asset pricing models: July 1963 - June 2005 39

2.5 Empirical tests of asset pricing models: Variance-Covariance Matrix 40 2.6 The fully conditional CAPM: July 1927 - June 2005 41

3.1 Empirical tests of the CAPM 79

3.2 Empirical tests of the CAPM with Human Capital 81

3.3 Empirical tests of the Consumption CAPM 83

3.4 Empirical tests of the Fama-French 3 Factor Model 85

3.5 Empirical tests of the Fama-French 3 Factor Model 87

3.6 Firm characteristics 88

3.7 Model Fit 89

B.1 Empirical tests of asset pricing models: July 1927 - June 2005 100

B.2 Empirical tests of asset pricing models: July 1927 - June 1963 102

B.3 Empirical tests of asset pricing models: July 1963 - June 2005 104C.1 Transition Matrix 109

LIST OF FIGURES

2.1 Posterior distribution plots for the risk premium, cm, after controllingfor …rm size, at return horizons of 1 - 6 years 302.2 Posterior distributions for the intercept 31

3.1 Posterior distribution plots for the risk premium, cm, in the simulationstudy 71

3.2 Posterior distribution plots for the risk premium, cm, at a returnhorizon of 4 years 723.3 The distribution of s at a 4 year return horizon 733.4 The distribution of …rm level s at a 4 year return horizon 743.5 The distribution of …rm level HML s at a 4 year return horizon 753.6 Price indices 76

3.7 Posterior distribution plots for the Fama-French 3 factor model: July

1965 - June 1993 77C.1 Di¤erences between pre-ranking and contemporaneous s at a 4 yearreturn horizon 108

CHAPTER 1

INTRODUCTION

Asset pricing refers to the process by which the prices of …nancial assets aredetermined, and the resulting relationships between expected returns and the risksassociated with those returns Over four decades ago Sharpe (1964) and Lintner(1965) developed the Capital Asset Pricing Model (CAPM) Building on the path-breaking work of Markowitz (1959), Sharpe (1964) and Lintner (1965) show that, inequilibrium, 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 thepricing of …nancial assets is non-diversi…able risk, and 2) a …nancial asset’s expectedreturn is a positive linear function of its non-diversi…able risk

Today the CAPM is still widely used by academics and practitioners to estimatethe 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 under-graduate and MBA investment courses The reason behind the CAPM’s widespreaduse is that it o¤ers powerful and intuitive predictions regarding how non-diversi…ablerisk should be measured, and the relation between non-diversi…able risk and expectedreturns

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 isunable to explain cross-sectional di¤erences in average returns Further, building ontheir 1992 study, Fama and French (1993) propose two additional factors designed tocapture 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 theoreticalfailings Alternatively, it may be caused by di¢ culties in implementing valid tests ofthe model The focus of my dissertation is on the latter possibility

Researchers seeking to examine whether the CAPM is able to explain sectional di¤erences in average returns face two major di¢ culties First, the CAPMstates that the risk of a stock should be measured relative to the aggregate wealthportfolio 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 jointtest of two hypotheses: 1) the CAPM holds, and 2) returns on the aggregate wealthportfolio 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

cross-a …rm, herecross-after referred to cross-as , is an unobserved, latent variable Having chosen aproxy for aggregate wealth, the researcher must obtain estimates of s for …rms toexamine the prediction that average returns are positively related to s Researcherstypically adopt a two-step estimation procedure First, obtain estimates of , b, thenexamine whether there is a positive relation between average returns and bs However,

bs are estimated imprecisely, creating a measurement error problem when the bs areused to explain average returns This will result in a downward bias in the estimatedrisk premium.

Researchers have sought to develop techniques that minimize the measurementerror problem while maximizing heterogeneity in s across both time and …rms Thebenchmark approach for estimating s at the …rm level was developed by Fama andFrench (1992) Each year …rms are assigned to 100 portfolios based on …rm charac-teristics Given the portfolio returns, Fama and French (1992) estimate portfolio susing a market model over the entire sample period Estimates of s for diversi…edportfolios are more precise than estimates of s for individual …rms The portfolio sare then assigned to individual …rms in each year

In chapter 2 I develop a Bayesian approach to examine the ability of the CAPM toexplain the cross-sectional variation in average returns at the …rm level The principaladvantage of the Bayesian approach is that it enables the researcher to assess just howimportant time and …rm heterogeneity are in the estimation of s, while explicitlycontrolling 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 ofNYSE, AMEX, and NASDAQ listed stocks over the sample period July 1927 - June2005

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 strongevidence supporting the main empirical prediction of the CAPM There is a robustpositive relation between average returns and The estimated risk premium is

approximately 7% per year, which is economically plausible given that average excessreturns on the stock market tend to range between 6% and 8% per year Finally, theCAPM 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 helpexplain 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 andtested only at the portfolio level The principal reason for the use of portfolios astest assets in cross-sectional tests of asset pricing models is to reduce the impact ofmeasurement error problems However, cautions regarding the use of portfolios astest assets abound in the literature Theoretical work shows that the use of portfolioscan 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 lookbad Ultimately researchers are interested in how well asset pricing models explainreturns at the …rm level In chapter 3 I examine whether inferences at the portfoliolevel can be generalized to the …rm level

I use the Bayesian approach developed in chapter 2 to examine the performance ofthe 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 areestimated using a broad cross-section of NYSE, AMEX, and NASDAQ listed stocksover the sample period July 1965 - June 2000 Portfolios are constructed using severaldi¤erent approaches, but I focus on two of the most widely used sets of portfolios,

100 size-pre-ranking portfolios and 25 size-book-to-market equity portfolios Allportfolios 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 cross-section of average portfolio returns The …rm level results paint a very di¤erentpicture 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 tioning researchers regarding the use of portfolios as test assets I …nd that inferencesbased on portfolio level tests are sensitive to the portfolio formation method De-pending on how the portfolios are formed, the underlying …rm level associations can

cau-be masked, and the signs on risk premia reversed

CHAPTER 2

REVIVING THE CAPM: A BAYESIAN APPROACH FOR

TESTING ASSET PRICING MODELS

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 Thecentral prediction of the CAPM is that the aggregate wealth portfolio is mean-variancee¢ cient The e¢ ciency of the aggregate wealth portfolio implies that 1) a security’sexpected 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 expectedreturns

Early work by Black, Jensen, and Scholes (1972), Fama and MacBeth (1973),and Stambaugh (1982) …nds that there is a positive relation between and averagereturns However, in their in‡uential 1992 paper, Fama and French examine theability 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-marketequity, is not able to help explain average stock returns Since Fama and French(1992), very few papers examining the cross-sectional variation in average returnshave found much, if any, support for the CAPM

Three notable exceptions are Jagannathan and Wang (1996), Lettau and son (2001), and Ferguson and Shockley (2003) Jagannathan and Wang (1996) …ndthat when a measure of human capital is included in the proxy for aggregate wealth,the performance of the CAPM (conditional and unconditional) is substantially im-proved The conditional CAPM is able to explain over 50% of the cross-sectionalvariation in average returns Lettau and Ludvigson (2001), using a similar approach

Ludvig-to Jagannathan and Wang (1996), …nd that the conditional consumption CAPM isable to explain the cross-sectional variation in average returns at least as well as theFama-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 associatedwith …rm size and book-to-market equity

Finally, Ferguson and Shockley (2003) show that many empirical "anomalies" areactually consistent with the CAPM if researchers use an all equity proxy for theaggregate wealth portfolio They propose two proxies to capture events in the debtmarkets, and …nd evidence supporting the CAPM when the proxies for debt areincorporated in the empirical tests

However, Lewellen, Nagel, and Shanken (2006) highlight several methodologicalconcerns with papers such as Jagannathan and Wang (1996), Lettau and Ludvigson(2001), and Ferguson and Shockley (2003) First, Lettau and Ludvigson (2001), andFerguson 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 3factors can explain more than 75% of the cross-sectional variation in 25 size-book-to-market portfolio returns Thus, as long as a proposed factor is correlated with SMB

or HML, a high R2 will be obtained When Lewellen, Nagel, and Shanken (2006)

extend Lettau and Ludvigson (2001) to portfolios other than the Fama-French 25size-book-to-market portfolios, the results are weak, o¤ering little or no support forthe conditional consumption CAPM.

Second, the empirical tests of the conditional models proposed by Jagannathanand Wang (1996) and Lettau and Ludvigson (2001) ignore the theoretical restrictions

on cross-sectional slope coe¢ cients Lewellen and Nagel (2006) argue that imposingsuch restrictions could greatly reduce the explanatory power of the proposed assetpricing models Therefore it is not clear whether Jagannathan and Wang (1996),Lettau and Ludvigson (2001), and Ferguson and Shockley (2003) really do providestrong support for the CAPM, or the consumption CAPM

The CAPM’s empirical problems may stem from two sources: 1) the theoreticalmodel requires many simplifying assumptions, such as the existence of perfect capitalmarkets, that are violated in reality, and 2) di¢ culties in implementing valid empiricaltests 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 theaggregate wealth portfolio However, the aggregate wealth portfolio is not observed bythe 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 aggregatewealth portfolio are a linear function of the returns on the proxy chosen by theresearcher

Second, s are unobserved, latent variables Having settled on a proxy for gregate wealth, the researcher must obtain estimates of s for …rms to examine theprediction that average returns are positively related to s Researchers typically

ag-adopt a two-step estimation procedure First, obtain estimates of , b, then examinewhether there is a positive relation between average returns and bs However, bs areestimated imprecisely, creating a measurement error problem when the bs are used

to explain average returns This will result in a downward bias in the estimated riskpremium

Over the last 30 years researchers have sought to develop techniques that minimizethe measurement error problem while maximizing heterogeneity in s across both timeand …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) estimateportfolio s using a market model over the entire sample period The portfolio s arethen 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 Theprincipal advantage of the Bayesian approach is that it enables the researcher toexamine 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 theperiod July 1927 - June 2005 to examine the empirical predictions of the CAPM.Consistent with previous studies, I assume that returns on the aggregate wealthportfolio are a linear function of returns on the CRSP value weighted stock marketindex

Using a similar approach to Fama and French (1992) I …nd that, after controllingfor …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 es-timation of …rm-speci…c s First, all …rms assigned to the same portfolio must havethe 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 evidencesupporting the main empirical prediction of the CAPM There is a positive relationbetween 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% and8% 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 SMBand HML s should not be able to explain the cross-sectional variation in averagereturns left unexplained by stock market s Consistent with the predictions of theCAPM, 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 theCAPM, 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 relationwill 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 …rmsize is negatively related to average returns as evidence that the CAPM is, in someway, empirically misspeci…ed For example, the stock market index may not be thebest proxy for the aggregate wealth portfolio

Finally, to assess the robustness of my …ndings across di¤erent time periods, Isplit the sample period into two sub-periods, July 1927 - June 1963, and July 1963

- June 2005 In both sub-periods I …nd evidence of a positive relation between sand average returns This relation is robust to the inclusion of both …rm size and theadditional risk factors SMB and HML Further, there is little evidence that SMB andHML s are priced risk factors in either sub-period

The chapter proceeds as follows Section 2.2 brie‡y discusses the various proaches researchers have used to examine the CAPM for a large number of testassets A ‡exible statistical model is then developed to enable more precise tests ofthe 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 em-pirical results and evaluate the performance of the CAPM Section 2.5 concludes

V ar(r m )

Given s, the Sharpe-Lintner CAPM implies that if we run a regression,

where re

i = ri rf, we should …nd that c0 = 0 and cm = E [rm rf] > 0 tunately we do not observe s They are latent variables This problem promptedresearchers to adopt a two-step estimation procedure First, obtain estimates of ,

Unfor-b, for …rms using the market model,

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 100portfolios based on size and pre-ranking bs.1 Next, they calculate portfolio returnsfor 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).

(1963 - 1990) Using a long time series should help reduce the estimation error forthe portfolio bs The portfolio bs are then assigned to individual …rms As a …rmtransitions across portfolios, so its changes.

This approach involves a bias-variance trade-o¤ Estimates of s for well si…ed portfolios are more precise, but, to the extent that there is within portfolioheterogeneity in …rm-speci…c s, they are biased estimates of …rm-speci…c s Whilethe trade-o¤ is acceptable to obtain accurate estimates of b, it is not appropriate fortesting the CAPM There will be a downward bias in the estimated coe¢ cient bcm ifthe values of b assigned to each …rm di¤er from the true values of …rm-speci…c s

diver-I propose a direct approach to examine the relation between risk and averagereturn which involves the estimation of a two equation system,

rei;y = c0;y+ cm;y i;y + "i;y (2.4)

the results compared An additional advantage of estimating the model at di¤erentreturn horizons is that I will also be able to examine whether inferences about theCAPM 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-to-market portfolios as test assets rather than individual …rms I choose to examine theCAPM 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 ature For example, Kan (2004) demonstrates that the use portfolios can not onlymake good asset pricing models look bad, but also make bad asset pricing modelslook good Second, Lewellen, Nagel, and Shanken (2006) extend several studies, such

liter-as Lettau and Ludvigson (2001), to portfolios other than the 25 size-book-to-marketportfolios 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)

The Sharpe-Lintner version of the CAPM, speci…ed in equation (2.1), generatesthree testable implications First, the CAPM implies that there is a positive relationbetween expected return and risk, E [cm] > 0 Second, the Sharpe-Lintner CAPMposits that E [c0] = 0 Finally, the CAPM implies that is the only variable necessary

to explain expected returns

Fama and French (1992) show that, after controlling for …rm size, has no planatory power Equation (2.4) can be modi…ed to examine the impact of adding

ex-…rm size ( ln (M E)) as a control variable,2

rei;y = c0;y + cm;y i;y+ csize;yln(M Ei;y) + "i;y: (2.6)

If the CAPM holds, Fama and French (1992) argue that …rm size should not be apriced risk factor, E [csize] = 0 However, Berk (1995) shows that if an asset pricingmodel is misspeci…ed in any way, …rm size will be negatively associated with futurereturns Berk (1995) argues that the observation that …rm size explains part of thereturns not explained by s, by itself, is not necessarily evidence that the CAPM is notable 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 wealthportfolio may be poor

Building on their 1992 paper, Fama and French (1993) propose two additional riskfactors 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 expectedreturns in the cross-section if the asset pricing model is misspeci…ed, his paper doesnot 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,

rei;y = c0;y+ cm;y i;y+ csmb;y SM Bi;y + chml;y HM Li;y + "i;y (2.7)

ri;t;y = i;y+ i;yrm;t;y+ SM Bi;y rsmb;t;y + HM Li;y rhml;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.

where rsmband rhml are the returns on portfolios constructed to mimic the risk factorsrelated to size and book-to-market equity The CAPM implies that the two additionalrisk factors should not be priced, E [csmb] = 0 and E [chml] = 0.

Estimation of the model described in equations (2.4) and (2.5) is complicated bythe fact that the latent variable i;y appears in both equations Maximum likelihoodestimation requires the researcher to integrate over the joint density to obtain themarginal 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 achallenging problem to estimate the system of equations using maximum likelihood

An alternative estimation strategy would be to use the generalized method ofmoments (GMM) However, Ferson and Foerster (1994) demonstrate that GMM hasrather poor …nite sample properties, especially for problems with high dimensionalityand 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 tions (2.4) and (2.5) Recent developments in statistical computing, in particular,Markov chain Monte Carlo (MCMC) methods, have made it possible to estimatemodels which would be di¢ cult, if not impossible to estimate using non Bayesianmethods The Bayesian approach is likelihood based, and requires the speci…cation

equa-of both a likelihood function and a prior

There are several advantages to the Bayesian approach First, researchers havestruggled to develop techniques that minimize the measurement error problem whilemaximizing heterogeneity in bs across both time and …rms The Bayesian approachenables the researcher to examine just how important time and …rm heterogeneityare in the estimation of s, while explicitly controlling for the inherent uncertaintyassociated with time varying …rm-speci…c s.

Second, the Bayesian approach is able to overcome the problems with the sical two-step tests of asset pricing models identi…ed by Kan and Zhang (1999) In

clas-an extreme setting where a risk factor is useless, de…ned as being independent of allthe asset returns, Kan and Zhang (1999) show that the second pass cross-sectionalregressions 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 thetrue 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 goestowards in…nity to "explain" the cross-sectional variation in average returns Thismisspeci…cation bias arises due to the estimation errors associated with point esti-mates of s The Bayesian framework takes into account the parameter uncertaintyassociated with all the model parameters, thereby avoiding such a bias

The advantages of the Bayesian approach come at the cost of having to specifyexplicit priors The priors for equation (2.4) are speci…ed as,

where cy denotes c0;y

cm;y In turn I specify proper and di¤use priors for c and Vc,

The CAPM provides a model for expected returns and risk In reality we onlyobserve realized returns and risk Consequently, when examining the empirical pre-dictions of the CAPM, inferences should be based on the posterior distribution of c

As Fama (1976) notes (pg 361), we would expect c0;y and c1;y to be quite variablethrough time, and even negative in some periods While cy provides informationabout risk premia during time period y, c provides information about average riskpremia across all time periods (y = 1; :::; Y )

In equation (2.9) I assume that cy is normally distributed with a mean c and avariance-covariance matrix Vc The normal distribution is a ‡exible distribution, but

it should be noted that in‡uence of the likelihood for each period may be attenuatedfor likelihoods centered a long way from the prior Outliers will be shrunk towardsthe prior mean due to the thin tails of the normal distribution

I can further increase the ‡exibility of the prior distribution by incorporatingobservable factors thought to be associated with time variation in risk premia byusing a multivariate regression speci…cation,

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 V c Inverted Wishart(N u c ; V 0;c ) where Nvar c denotes the number of independent variables in equation (2.7).

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 maynot have many observations of cy with which to make inferences about c (or )and Vc For example, using a 6 year return horizon to compute average excessreturns for equation (2.4) over the period 1927 - 2005 will result in only 13 non-overlapping observations of cy Asymptotic approximations may not be accurate forsuch a small sample size Fortunately, Bayesian inference is free from the use ofasymptotic approximations and delivers exact …nite sample inference

Finally, I must also specify priors for i;y and i;y in equation (2.5) I specifyproper and di¤use priors which should not exert a strong in‡uence over the posteriordistributions,

pro-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

5 ;

2 6 4

1 C

A where 0i;y = i;y i;y smbi;y hmli;y .

2.2.4 Evaluating competing model speci…cations

To compare the relative performance of di¤erent empirical speci…cations I pute the log marginal density using the importance sampling method of Newton andRaftery (1994) This is a Bayesian measure of model …t which includes an implicitpenalty for models with a large number of parameters

com-To get a better sense of how well di¤erent models perform in terms of in-sampleexplanatory power I also calculate the mean absolute error On each iteration ofthe MCMC chain the mean absolute error is calculated and its value stored Thedistribution for mean absolute error is then summarized to enable comparison acrossdi¤erent empirical speci…cations

I use monthly stock returns on all corporations listed on the NYSE, AMEX, andNASDAQ over the period January 1925 - June 2005 that are covered by the CRSPtapes The 1 month T-Bill rate is used as a proxy for the risk free rate Returns onthe aggregate wealth portfolio are assumed to be a linear function of returns on theCRSP value weighted stock market index Monthly return data for the risk factorsproposed by Fama and French (1993) related to …rm size (SMB), and book-to-marketequity (HML) are taken from Kenneth French’s website.6

I examine the CAPM using non-overlapping return horizons ranging from 1 - 6years All time periods start in July and end in June In order to compare theproposed 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: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html See Fama and French (1993) for more details on the contruction of the risk factors.

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…ctime period.

First, …rms must have monthly returns for at least 24 of the 60 months precedingthe 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 1927 - June 2005.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 andshares 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 themodel 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 returnfor …rms in the sample is approximately 13% per year at all return horizons, whilethe returns on the CRSP value weighted market portfolio are 11% per year over theperiod 1927 - 2005 The average returns on the risk factors proposed by Fama andFrench (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 returnhorizon

2.4 Results

In table 2.2 I examine the performance of two asset pricing models, the CAPMand the Fama-French 3 factor model over the period July 1927 - 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.7

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 thesame 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 …ndthat is a priced risk factor However, the relation between average returns anddisappears 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 atall return horizons However, relaxing this restriction does not change our inferencesregarding the CAPM After controlling for size there is no evidence, at conventionalsigni…cance levels, that is able to explain the cross-sectional variation in averagereturns

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.

2.4.2 The CAPM at the …rm level using …rm-speci…c sWhen 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 themodel …t relative to the benchmark Fama and French (1992) approach At a 1 yearreturn horizon the log marginal density increases from -848,741 to -796,318 for theCAPM model, while the mean absolute pricing errors fall from 31.25 to 25.91 Similarimprovements in model …t are observed at all return horizons Even if the Fama andFrench (1992) approach is supplemented by the addition of …rm size as an explanatoryvariable, 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 tothe inclusion of …rm size as an explanatory variable In …gure 2.1 I plot the posteriordistributions for the price of risk, cm, after controlling for …rm size

Figure 2.1 shows that at return horizons of 3 - 6 years the posterior distributionsfor 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 standardtextbook view that the risk premium ranges between 6% and 8% At 1 and 2 yearreturn 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 Theresults in table 2.2 and …gure 2.1 suggest that inferences regarding the CAPM arenot especially sensitive to the choice of return horizon

The reason the posterior distributions for the risk premium are more di¤use atshorter return horizons (1 and 2 years) is related to the model setup At 1 and 2 yearreturn 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 speci…c s will be di¤use, re‡ecting the limited information provided by the data.The Bayesian approach automatically incorporates this parameter uncertainty intothe posterior distributions for the risk premium

…rm-The Sharpe-Lintner CAPM posits that E [c0] = 0 When s are allowed to varyacross time periods and …rms there is mixed evidence regarding this empirical predic-tion Figure 2.2 plots the posterior distributions for c0 At return horizons less than

5 years table 2.2 and …gure 2.2 provide little evidence, at conventional signi…cancelevels, against the prediction of the Sharpe-Lintner CAPM, that E [c0] = 0 However,

at 5 and 6 year return horizons there is evidence that the posterior distributions for

c0 are focused above zero While this may not be consistent with the Sharpe-LintnerCAPM, it is still consistent with Black’s version of the CAPM in which there isrestricted borrowing

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 returnhorizons However, the inclusion of the additional risk factors does not change our

inferences with regards to the price of stock market risk Stock market risk is priced atall return horizons, with the mean of the posterior distributions for the risk premiumranging 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-sectionalvariation in average returns left unexplained by stock market s However the evi-dence is mixed with regards to SMB s When the …rm characteristic, size, is notincluded in the speci…cation, there is a positive relation between average returns andSMB s In contrast, when …rm size is included, there is no robust evidence acrossreturn 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 empiricalpredictions of the CAPM, …rm size is negatively related to average returns for allmodel speci…cations Berk (1995) argues that an asset pricing model should not berejected solely on the basis of the …nding that …rm size is negatively related to averagereturns, since such a relation will exist if the asset pricing model is misspeci…ed in anyway Given that the majority of the evidence is consistent with the CAPM, I interpretthe relation between …rm size and average returns not as evidence that the CAPMshould be rejected outright, but rather as evidence that the CAPM is in some wayempirically misspeci…ed For example, the assumption that returns on the aggregatewealth portfolio are a linear function of returns on the CRSP value weighted stockmarket 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.

2.4.4 Robustness

Fama and French (1992) …nd no evidence that is a priced risk factor over theperiod 1963 - 1990 However, when they examine the CAPM over a longer period,

1941 - 1990, they …nd that is a priced risk factor when …rm size is not included as

a control variable The results in table 2.2 are based on the period 1927 - 2005 The

…nding that is a priced risk factor could being driven by data prior to 1963, while

in more recent periods may not be priced

In addition, the period between the 1920s and the 1950s contains several largesystematic events, including the great depression and the second world war, whichcould have an undue in‡uence on the results Thus it is important to examine whetherthe …ndings are robust across sub-periods I split the sample period into two sub-periods, July 1927 - June 1963, and July 1963 - June 2005

The summary statistics for the sub-periods, in table 2.1, reveal that average excessreturns are considerably more volatile in the …rst sub-period, particularly at shorterreturn horizons Average value weighted market returns, and average returns on theSMB and HML portfolios are similar across the two sub-periods, but the returns onthe market and the HML portfolios are more volatile at shorter return horizons inthe …rst sub-period The other main di¤erence across the sub-periods is the number

of …rms in the sample Prior to 1963 there are 400 - 1,000 …rms in each time period,while after 1963 there are 1,000 - 7,700 …rms in each time period

The results for the …rst sub-period, July 1927 - June 1963, are reported in table2.3 When s are allowed to vary across both time and …rms there is a positive relationbetween average returns and This relation is robust to the inclusion of …rm size as a

control variable Incorporating the SMB and HML factors leads to an improvement inmodel …t at all return horizons, but there is no evidence, at conventional signi…cancelevels, that SMB and HML s are priced risk factors.

If the …rst sub-period is driving the results in table 2.2 for the whole sampleperiod, will not be a priced risk factor in the second sub-period, July 1963 - June

2005 Table 2.4 reports the results for the second sub-period The results are similar

to those in tables 2.2 and 2.3 When s are allowed to vary across time periods and

…rms, I …nd a strong positive relation between average returns and , which is robust

to the inclusion of …rm size The inclusion of SMB and HML as additional risk factorsagain leads to an improvement in model …t, but there is little evidence across all thereturn horizons that SMB and HML s are priced risk factors, either with or withoutthe inclusion of …rm size as a control variable

The posterior distributions for the various risk factors are similar across the twosub-periods This is surprising given that the …rst sub-period includes the greatdepression and the second world war In table 2.5 I compare the posterior distributionsfor the variance-covariance matrix, Vc, across both sub-periods using a 1 year returnhorizon Given the summary statistics in table 2.1 I would expect to observe muchgreater volatility in the market risk premium and the HML risk premium in the …rstsub-period Table 2.5 con…rms this The realized risk premia on stock market s andHML s in the …rst sub-period exhibit substantially more volatility relative to thesecond sub-period

Jagannathan and Wang (1996) and Lettau and Ludvigson (2001) …nd that tional versions of the CAPM and the consumption CAPM perform more e¤ectivelythan unconditional versions To estimate a fully conditional model equation (2.9)

condi-must be replaced with equation (2.12) The conditional version of the CAPM requiresthe nominal risk premium to be a function of observable macroeconomic conditioningvariables I use in‡ation, and the default premium as conditioning variables Thisdata is obtained from the Federal Reserve Economic Database at the Federal ReserveBank of St Louis.

The results for the conditional CAPM over the full sample period are reported

in table 2.6 Allowing for a conditional risk premium improves the model …t onlyvery slightly relative to the results in table 2.2 Further, none of the conditioningvariables are strongly associated with time variation in the risk premium The reasonfor the lack of improvement in the model …t relates to the in‡uence of the prior onthe posterior distributions for the risk premium The model is estimated at the …rmlevel for 1,000’s of …rms, so, in any given time period there is considerable informationabout the parameter vector cy contained in the likelihood Given the strength of thisinformation, the in‡uence of the prior on the posterior is negligible.9 As such, thevalue of using a fully conditional model appears, in this setting, to be limited.Finally, in results not reported, I conduct a sensitivity analysis to investigatewhether the posterior distributions are sensitive to the prior speci…cations I …ndthat the posterior distributions are not sensitive to large changes in the speci…cation

of the priors for either equation (2.4) or (2.5) For example, the posterior distributionsare virtually identical regardless of whether the prior variance for c is 100I; 50I; or150I

9 Other macro variables were experimented with, including the term premium, GDP growth, and changes in industrial production However, the results were not in‡uenced by the choice of macro variables.

2.5 Conclusion

In this chapter I propose a Bayesian approach to examine the empirical predictions

of the CAPM at the …rm level Using a broad cross-section of NYSE, AMEX, andNASDAQ listed stocks over the sample period July 1927 - June 2005 I …nd support forthe empirical predictions of the CAPM There is a strong positive relation between sand average returns, which is una¤ected by the inclusion of …rm size as an explanatoryvariable Further, there is no robust evidence that SMB and HML s are able toexplain the cross-sectional variation in average returns left unexplained by traditionalstock market s

While the majority of the evidence is consistent with the CAPM, there is a tive relation between …rm size and average returns According to the CAPM, aloneshould be su¢ cient to explain the cross-sectional variation in average returns How-ever, Berk (1995) shows that if an asset pricing model is misspeci…ed in any way, sizewill be negatively related to average returns Consequently, I do not reject the CAPMoutright based on the …nding that …rm size can explain part of the average returnsnot explained by s As Berk (1995) notes, the CAPM could price risk correctly,but the empirical speci…cation may not be quite right Overall, the evidence in thischapter suggests that the CAPM is doing a much better job than people currentlygive it credit for

nega-Figure 2.1: Posterior distribution plots for the risk premium, cm, after controllingfor …rm size, at return horizons of 1 - 6 years

The MCMC algorithm is run for 10,000 iterations The …rst 5,000 iterationsare discarded as a "burn in" period The posterior distributions are based on theremaining 5,000 iterations The 0 - 2.5th percentiles are shaded in black, as are the97.5th - 100th percentiles