empirical tests of asset pricing models

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:
Professor R.M. Stulz, Adviser
Professor G.M. Allenby
Professor G.A. Karolyi
Approved by
Adviser
Graduate Program in
Business Administration
ABSTRACT
The Capital Asset Pricing Model (CAPM) developed by Sharpe (1964) and Lint-
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…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 p erformance 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 predic-
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 a
robust 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
no evidence that these additional risk factors help to explain the cross-sectional vari-
ation in average returns. These results are consistent with the empirical predictions
of the CAPM.
The use of p ortfolios 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
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
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 em-
pirical 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, An-

drew 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
VITA
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
PUBLICATIONS
Research Publications
A. Abhyankar and P. Davies. "Market Timing and Economic Value: Evidence from
the Short Rate Revisited". Finance Letters 3, 1-9, 2005.
FIELDS OF STUDY
Major Field: Business Administration
Concentration: Finance
v
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
vi
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
vii
LIST OF TABLES
Table Page
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
viii
B.3 Empirical tests of asset pricing models: July 1963 - June 2005 . . . . 104
C.1 Transition Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
ix
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 . . . . . . . . . . . . . . 30
2.2 Posterior distributions for the intercept . . . . . . . . . . . . . . . . . 31
3.1 Posterior distribution plots for the risk premium, c
m
, in the simulation
study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 Posterior distribution plots for the risk premium, c
m
, at a return
horizon of 4 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3 The distribution of s at a 4 year return horizon . . . . . . . . . . . . 73
3.4 The distribution of …rm level s at a 4 year return horizon . . . . . . 74
3.5 The distribution of …rm level HML s at a 4 year return horizon . . . 75
3.6 Price indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.7 Posterior distribution plots for the Fama-French 3 factor model: July
1965 - June 1993 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
C.1 Di¤erences between pre-ranking and contemp oraneous s at a 4 year
return horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
x
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 path-
breaking 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 p ositive 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 under-
graduate 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
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 mo del 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 cross-
sectional 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
b
s. However,
2
b
s are estimated imprecisely, creating a measurement error problem when the
b
s 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 b oth 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 charac-
teristics. 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
imp ortant 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 1927 - June
2005.
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
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 1965 - June 2000. Portfolios are constructed using several
di¤erent approaches, but I focus on two of the most widely used sets of portfolios,
4
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 cross-
section 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 cau-
tioning 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. De-
pending on how the portfolios are formed, the underlying …rm level associations can
be masked, and the signs on risk premia reversed.
5
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 b ook-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
Three notable exceptions are Jagannathan and Wang (1996), Lettau and Ludvig-
son (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 im-
proved. 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 supp orting 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-to-
market 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
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 ag-
gregate 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
adopt a two-step estimation procedure. First, obtain estimates of ,
b
, then examine

whether there is a positive relation between average returns and
b
s. However,
b
s are
estimated imprecisely, creating a measurement error problem when the
b
s 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 1927 - 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

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 es-
timation 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
Finally, to assess the robustness of my …ndings across di¤erent time periods, I
split 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 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 ap-
proaches researchers have used to examine the CAPM for a large number of test
assets. A ‡exible statistical mo del 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 em-
pirical results and evaluate the performance of the CAPM. Section 2.5 concludes.
2.2 The CAPM
2.2.1 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
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 r
e
i
= r
i
 r
f
, we should …nd that c
0
= 0 and c
m
= E [r
m
 r
f
] > 0. Unfor-
tunately 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,
plug
n
b

i
o
into equation (2.2).
However, in a classical setting,
b
s for individual …rms are estimated imprecisely,
creating a measurement error problem when the
b
s are used to explain average re-
turns. This will result in a downward bias in the estimated co e¢ cient bc
m
. To improve
the precision of
b
s, 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
(1963 - 1990). 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 diver-
si…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; 
2
c
y

. r
i;t;y
denotes …rm i’s stock return in month t
during period y, and "
i;t;y
 N

0; 
2

i;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
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-b ook-to-
market 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 liter-
ature. 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).
2.2.2 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
Fama and French (1992) show that, after controlling for …rm size,  has no ex-
planatory 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 b e 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

SM B
i;y
+ c
hml;y

HML
i;y
+ "
i;y
(2.7)
r
i;t;y
= 
i;y
+ 

i;y
r
m;t;y
+ 
SM B
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