Performance evaluation and attribution of security portfolios

Performance
Evaluation and Attribution
of Security Portfolios


Introduction
to the Series

ii

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KENNETH J. ARROW and MICHAEL D. INTRILIGATOR


Performance
Evaluation and Attribution
of Security Portfolios
by
Bernd Fischer and Russell Wermers

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Preface

v
This book is intended to be the scientific state-of-the-art in performance evaluation—the
measurement of manager skills—and performance attribution—the measurement of all
of the sources of manager returns, including skill-based. We have attempted to include the
best and most promising scientific approaches to these topics, drawn from a voluminous
and quickly expanding literature.
Our objective in this book is to distill hundreds of both classic and the best cuttingedge academic and practitioner research papers into a unified framework. Our goal is
to present the most important concepts in the literature in order to provide a directed
study and/or authoritative reference that saves time for the practitioner or academic
researcher. Sufficient detail is provided, in most cases, such that the investment
practitioner can implement the approaches with data immediately, without consulting
the underlying literature. For the academic, we have provided enough detail to allow an
easy further study of the literature, as desired.
We have contributed in two dimensions in this volume—both of which, we believe,
are missing in currently available textbooks. Firstly, we provide a timely overview
of the most important performance evaluation techniques, which allow an accurate
assessment of the skills of a portfolio manager. Secondly, we provide an equally timely
overview of the most important and widely used performance attribution techniques,
which allow an accurate measure of all of the sources of investment returns, and which
are necessary for precise performance reporting by fund managers.
We believe that our text is timely. An estimated $71.3 trillion was invested in managed
portfolios worldwide, as of 2009 (source: www.thecityuk.com). Managing this money,
thus, is a business that draws perhaps $700 billion per year in management fees and
other expenses for asset managers, in addition to a perhaps similar magnitude in annual
trading costs accruing to brokers, market makers, and other liquidity providers (i.e., Wall
Street and other financial centers). Our book is the first comprehensive text covering the
latest science of measuring the main output of portfolio managers: their benchmarkrelative performance (alpha). Our hope is that investors use these techniques to
improve the allocation of their money, and that portfolio management firms use them
to better understand the quality of their funds’ output for investors.

We intend this book to be used in at least two ways:
First, as a useful reference source for investment practitioners—who may wish to read
only one or a few chapters. We have attempted to make chapters self-contained to meet
this demand. We have also included chapter-end questions that both test the reader’s


vi

Preface
understanding and provide examples of applications of each chapter’s concepts. The
audience for this use includes (at least) those studying for the CFA exams; performance
analysts; mutual fund and pension fund trustees; portfolio managers of mutual funds,
pension funds, hedge funds, and fund-of-funds; asset management ratings companies
(e.g., Lipper and Morningstar); quantitative portfolio strategists, regulators, financial
planners, and sophisticated individual investors.
Second, the book serves as an efficient way for mathematically advanced undergraduate,
masters, or Ph.D. students to undertake a thorough foundation in the science of
performance evaluation and attribution. After reading this book, students will be
prepared to handle new developments in these fields.
We have attempted to design each chapter of this book to contain enough detail to
bring the reader to a point of being able to apply the concepts therein, including the
chapter-end problems. In cases where further detail may be needed, we have cited the
most relevant source papers to allow further reading.
We have divided our book into two sections:
Part 1 of the book covers the area of performance evaluation.
Chapter 1 provides a short overview of the basics of empirical asset-pricing as applied
to performance assessment, including basic factor models, the CAPM, the Fama-French
three-factor model and the research on momentum, and the characteristic-based stock
benchmarking model of Daniel, Grinblatt, Titman, and Wermers.
Chapter 2 provides an overview of returns-based factor models, and the issues involved

in implementing them. Chapter 3 discusses the issue of luck vs. skill in generating
investment returns, and presents the fundamental performance evaluation measures,
including those based on the Chapter 2 factor models. In addition, extensions of these
factor models are introduced that contain factors that capture the ability of portfolio
managers to time the stock market or to time securities over the business cycle.
Chapter 4 presents the latest approaches to using portfolio holdings to more precisely
measure the skill of a portfolio manager. Chapter 5 provides a complete system for
evaluating the skills of a portfolio manager using her portfolio holdings and net returns.
Many managed portfolios generate non-normal returns. Chapter 6 shows how to apply
bootstrap techniques to generate more precise estimates of the statistical significance of
manager skills in the presence of non-normal returns and alphas.
Chapter 7 covers a very new topic: how to capture the time-varying abilities of a
portfolio manager (as briefly introduced in Chapter 3). Specifically, this chapter shows
how to predict which managers are most likely to generate superior alphas in the
current economic climate.
Chapter 8 also covers a very recent topic in performance evaluation: the assessment of
the proportion of a group of funds that are truly skilled using only their net returns.
This approach is very useful in assessing whether the highest alpha managers are truly
skilled, or are simply the luckiest in a large group of managers.


Preface

Finally, Chapter 9 is a “capstone chapter,” in that it provides an overview of the research
findings that use the principles outlined in the first 8 chapters. As such, it is a very useful
summary of what works (and what does not) when looking for a superior asset manager
(a “SAM”) and trying to avoid an inferior asset manager (an “IAM”).
Part 2 of the book primarily concerns performance attribution and related topics.
Since attribution analysis has become a crucial component within the internal control
system of investment managers and institutional clients, ample space is dedicated to a

thorough treatment of this field. The focus in this part lies on the practical applications
rather than on the discussions of the various approaches from an academic point of
view. This (practitioner’s) approach is accompanied by a multitude of examples derived
from practical experience in the investment industry. Great emphasis was also put
on the underlying mathematical detail, which is required for an implementation in
practice.
Chapter 10 provides an overview of the basic approaches for the measurement of
returns. In particular, the concepts of time-weighted return and internal rate of return, as
well as approximation methods for these measures are discussed in detail.
Attribution analysis, in practice, requires a deep understanding of the benchmarks
against which the portfolios are measured. Chapter 11 provides an introduction to the
benchmarks commonly used in practice, and their underlying concepts.
Chapter 12 covers fundamental models for the attribution analysis of equity portfolios
developed by Gary Brinson and others. Furthermore, basic approaches for the treatment
of currency effects and the linkage of performance contributions over multiple periods
are considered.
Chapter 13 contains an introduction to attribution analysis for fixed income portfolios
from a practitioner’s point of view. The focus lies on a methodology that is based on a
full valuation of the bonds and the option-adjusted spread. In addition, various other
approaches are described.
Based on the methodologies for equity and fixed income portfolios, Chapter 14 presents
different methodologies for the attribution analysis of balanced portfolios. This chapter
also illustrates the basic approaches for a risk-adjusted attribution analysis and covers
specific aspects in the analysis of hedge funds.
Chapter 15 describes the various approaches for the consideration of derivatives within
the common methodologies for attribution analysis.
The final chapter (Chapter 16) deals with Global Investment Performance Standards, a
globally applied set of ethical standards for the presentation of the performance results
of investment firms.
The authors are indebted to many dedicated academic researchers and tireless

practitioners for many of the insights in this book. Professor Wermers wishes to thank
the many investment practitioners that have provided data or insights into the topics

vii


viii

Preface
of this book, including through their professional investment management activities:
Robert Jones of Goldman Sachs Asset Management (now at System Two and Arwen),
Rudy Schadt of Invesco, Scott Schoelzel and Sandy Rufenacht of Janus (now retired,
and at Three Peaks Capital Management, respectively), Bill Miller and Ken Fuller
of Legg-Mason, Andrew Clark, Otto Kober, Matt Lemieux, Tom Roseen, and Robin
Thurston of Lipper, Don Phillips, John Rekenthaler, Annette Larson, and Paul Kaplan of
Morningstar, Sean Collins and Brian Reid of the Investment Company Institute.
Professor Wermers also wishes to thank all of the classes taught on performance
evaluation and attribution since 2001—at Chulalongkorn University (Bangkok);
the European Central Bank (Frankfurt); the Swiss Finance Institute/FAME Executive
Education Program (Geneva); Queensland University of Technology (Brisbane);
Stockholm University; the University of Technology, Sydney; and the University of
Vienna. Special thanks are due to students in that first class of the SFI/FAME program
during those dark days in September 2001, 10 days after the 9-11 attacks.
Professor Wermers is also indebted to his loving family, Johanna, Natalie, and
Samantha, for the endless hours spent away from them while preparing and teaching
this subject. He gratefully acknowledges Thomas Copeland and Richard Roll of UCLA
and Josef Lakonishok of University of Illinois (and LSV Asset Management) for early
inspiration, as well as Wayne Ferson, Robert Stambaugh, Lubos Pastor, and Mark
Carhart for their recent contributions to the field. In addition, he owes his career to the
brilliant mentoring of Mark Grinblatt and Sheridan Titman at UCLA, pioneers in the

subject of performance evaluation. This text would not have been possible from such
humble beginnings without their selfless support and guidance.
Dr. Fischer is indebted to his colleagues at IDS GmbH—Analysis and Reporting Services,
an international provider of operational investment controlling services. Over the
past years he has greatly benefited from numerous discussions surrounding practical
applications.
Thanks are also due to Dr. Fischer’s former team members at Cominvest Asset
Management GmbH. The design and the implementation of a globally applicable
attribution software from scratch, and the implementation of the Global Investment
Performance Standards were exciting experiences which left their mark on the current
treatise.
He also wishes to thank various colleagues (Markus Buchholz, Detlev Kleis, Ulrich
Raber, Carsten Wittrock, and others), with whom he co-authored papers in the past.
Several sections in this book are greatly indebted to the views expressed there.
Dr. Fischer is also indebted to the CFA institute and the Global Investment Performance
Committee for formative discussions surrounding the draft of the GIPS in 1998/1999
and during his official membership term from 2000 to 2004.
Both authors wish to thank J. Scott Bentley of Elsevier, whose vision it was to create such
a book, and whose patience it took to see it through.
To those whose contributions we have overlooked, our sincere apologies; such an
ambitious undertaking as condensing a huge literature necessitates that the authors


Preface
choose topics that are either most familiar to us or viewed by us as most widely useful.
Surely, we have missed some important papers, and we hope to have a chance to create a
second edition that expands on this one.
Finally, to the asset management practitioner: we dedicate this volume to you, and
hope that it is useful in furthering your goal of providing high-quality investment
management services!

ix


Chapter 1

An Introduction to Asset
Pricing Models
3

ABSTRACT
This chapter provides a brief overview of asset pricing models, with an emphasis
on those models that are widely used to describe the returns of traded financial
securities. Here, we focus on various models of stock returns and fixed-income
returns, and discuss the reasoning and assumptions that underlie the structure
of each of these models.

1.1  HISTORICAL ASSET PRICING MODELS
Individuals are born with a sense of the perils of risk, and they develop mental adjustments to penalize opportunities that involve more risk.1 For example,
farmers do not plant corn, which requires a great deal of rainfall (which may or
may not happen), unless the expected price of corn at harvest time is sufficiently
high. Currency traders will not take a long position in the Thai baht and short
the U.S. dollar unless they expect the baht to appreciate sufficiently. In essence,
the farmer and the currency trader are each applying a “personal discount
rate” to the expected return of planting corn or investing in baht. The farmer’s
­discount rate depends on his assessment of the risk of rainfall (which greatly
affects his total corn crop output) and the risk of a price change in the crop. The
currency trader’s discount rate depends on the relative economic health of Thailand and the U.S., and any potential government intervention against currency
1 Gibson and Walk (1960) performed a famous experiment that was designed to test for depth
perception possessed by infants as young as six months old. Infants were unwilling to crawl on a

transparent glass plate that was placed over a several-foot drop, proving that they possessed depth
perception at a very early age. Another inference which can be drawn from this experiment is that
infants already perceive physical risks and exhibit risk-averse behavior at a very early age (probably
before they are environmentally taught to avoid risk).
Performance Evaluation and Attribution of Security Portfolios. />© 2013 Elsevier Inc. All rights reserved.
For End-of-chapter Questions: © 2012. CFA Institute, Reproduced and republished with
permission from CFA Institute. All rights reserved.

Keywords
Asset Pricing
Models,
CAPM,
Factor Models,
Fama French
­three-factor model,
Carhart four-factor
model,
DGTW stock
­characteristics
model,
Estimating beta,
Expected return
and risk.


4

CHAPTER 1  An Introduction to Asset Pricing Models
speculation—both of which may carry large risks. Both e­ conomic agents' discount rates also depend on their personal aversion to risk, and, thus, may require
very different compensations to take similar risks.2, 3

Asset managers and investors also understand that some securities are less ­certain
in their payouts than others, and make adjustments to their investment plans
accordingly. Short-maturity bank certificates of deposit (CDs), while ­paying a
very low annual interest rate, are attractive because they return the principal
fairly quickly and guarantee (with insurance) a particular rate-of-return. Stocks,
with not even a promise that they will pay the next quarterly dividend, provide
much higher returns than CDs, on average. In general, greater levels of risk in
a security or security portfolio—especially those risks that cannot be inexpensively insured—require compensation by risk-averse investors in the form of
higher potential future returns.
The most basic approach to an “asset pricing model” that describes the compensation to investors for risk-taking simply ranks securities by the standard
deviation of their periodic (say, monthly) returns, then conjectures a particular
functional relation between this risk and the expected (average future) returns
of securities.4 But, should the relation be linear or non-linear between standard
deviation and expected return? Should there be any credit given to securities that
have counter-cyclical risk patterns (i.e., high returns during recessions)? How
can we account for offsetting risk patterns between a group of securities, even
within a bull market (e.g., technology vs. utility stocks)? Should risk that can be
diversified by holding many different investments be rewarded? These questions
are the focus of modern asset pricing theory.
The foundations of modern asset pricing models attempt to combine a few very
basic and simple axioms that appear to hold in society, including the following.
First, that investors prefer more wealth to less wealth. Second, that investors dislike risk in the payouts from securities because they prefer smooth patterns of
consumption of their wealth, and not “feast or famine” periods of time. And,
third, that investors should not be rewarded with extra return for taking on risk
that could be avoided through a smart and costless approach to mixing assets.
Our next sections briefly describe the most widely used asset pricing models of
today. In discussing these models, we focus on their application to describe the

2 The notion of creating a personal “price of risk”, or a required expected reward for taking on a unit
of risk, has its mathematical origins at least as long ago as 1738, when Daniel Bernoulli defined

the systematic process by which individuals make choices, and, in 1809, when Gauss discovered
the normal distribution. For an excellent discussion of the historical origins and development of
concepts of risk, see Bernstein (1996).
3 In cases where bankruptcy is possible, an economic agent may not take a risk that would otherwise
be attractive—if credit is not available to forestall the bankruptcy until the expected payoff from the
bet. This is the essence of Shleifer and Vishny’s (1997) “limits to arbitrage” argument (which might
be better referred to as “limits to risky arbitrage”).
4 An asset pricing model estimates the future required expected return that must be offered by a
­security or portfolio with certain observable characteristics, such as perceived future return volatility.


1.2  The Beginning of Modern Asset Pricing Models
evolution of returns for liquid securities—chiefly, stocks and bonds.5 However,
the ­usefulness of these models—with some modifications—goes far beyond
stocks and bonds to other securities, such as derivatives and less liquid assets
such as private equity and real estate.

1.2 THE BEGINNING OF MODERN ASSET PRICING
MODELS
A great deal of work has been done, over the past 60 years, to advance the
ability of statistical models to explain the returns on securities. Building on
Markowitz’s (1952) seminal work on efficient portfolio diversification, Sharpe
published his famous paper on the capital asset pricing model (CAPM) in 1964
(Sharpe, 1964).6 These two ideas shared the 1990 Nobel Prize in Economics.
The CAPM says that the expected (average) future excess return, Rt, is a linear
function of the systematic (or market-related) risk of a stock or portfolio, β:
E [Rt ] = β · E [RMRF t ] ,

(1.1)

where Rt = security or portfolio return minus riskfree rate, RMRFt = market
t ,RMRFt )
return minus riskfree rate, and β = cov(R
var(RMRFt ) is a measure of correlation of the
security or portfolio with the broad market portfolio.7
This relation is extremely simple and useful for relating the reward (expected
return) that is required of a stock with its level of market-based risk. For instance,
if market-based risk (β) is doubled, then expected return, in excess of the riskfree rate, must be doubled for the security or portfolio to be in equilibrium with
the market. If  T-bills pay 2%/year and a stock with a beta of one promises an average return of 7%, then a stock with a beta of two must promise an average of 12%.
Sharpe’s CAPM is simple and is an equilibrium theory, but it depends on several
unrealistic assumptions about the economy, including:
1. All investors have the exact same information about possible future expected
­earnings and their risks at each point in time.
2. Investors are risk-averse and behave perfectly rationally, meaning they do
not favor one type of security over another unless the calculated Net Present
Value of the first is higher.
3. The cost of trading securities is zero.
4. Investors are mean-variance optimizers (it is sufficient, but not necessary,
for this requirement that security returns are normally distributed).
5 For

a general review of asset pricing theories and empirical tests of the theories, see, for example,
Cochrane (2001) and Campbell et al. (1997).
6 Apparently, Bill Sharpe, a Ph.D. student in Economics at UCLA, visited Harry Markowitz at the
Rand Institute in Santa Monica, California during the early 1960s to discuss Markowitz’s paper and
Bill’s thoughts about an asset-pricing model. This led to Bill’s dissertation on the CAPM.
7 Note that the correlation coefficient between the excess return on a security or portfolio and the
excess return on the broad market is defined as ρ = √ cov(Rt ,RMRFt ) , which is close to the definivar(Rt )·var(RMRFt )
tion of β.

5


6

CHAPTER 1  An Introduction to Asset Pricing Models
5. All investors are myopic, and care only about one-period returns.
6. Investors are “price-takers”, meaning that their actions cannot influence
prices of securities.
7. There are no taxes on holding or trading securities.
8. Investors can trade any amount of an asset, no matter how small or large.
Several of these assumptions may not fit real-world markets, and many papers have
attempted, with some—but far from complete—success in extending the CAPM to
situations which eliminate one or more of these assumptions. Among these papers
are Merton’s (1973) intertemporal CAPM (ICAPM), which extends the CAPM to a
multiperiod model (to address #5). A good discussion of these extended CAPMs
can be found in several investments textbooks, such as Elton et al. (2011).
While there are many extensions of the CAPM that deal with dropping one
assumption at a time, it is not at all clear that dropping several assumptions
simultaneously still results in the CAPM being a good model that describes the
relation of returns to risk in real financial markets. Because of this, recent work
has focused on building practical models that “work” with data, even if they are
not based on a particular theoretical derivation. Although many attempts have
been made, with some success, at creating a new model of asset pricing, no theory has become as universally accepted as the CAPM once was. Hopefully, some
future financial economist will create such a new model that reflects real financial
markets well. In the meantime, we must rely on either empirical applications of
the CAPM, or on other models that have no particular equilibrium theory supporting them.

1.2.1  Estimating the CAPM Model
In reality, we do not know the true values of E [Rt ] , E [RMRFt ], and β, so we must

estimate them somehow from data. This is where a time-series version of the
CAPM (also called the Jensen model (Jensen, 1968)) can be used on return data
for a security or a portfolio of securities. The time-series version of the CAPM
can be written as
(1.2)

Rt = α + β · RMRFt + et ,

while its application to real-world data can be similarly written
rt = α + β · rmrft + ǫt ,

as:8
(1.3)

where we estimate the parameters α (the model intercept) and β (the model
slope) using historical values of Rt and RMRFt . (This model is more generally
called the “single-factor model”, as it does not require that the CAPM is exactly
correct to be implemented on real-world data.) A widely used method for
doing this is ordinary least squares (OLS), which fits the data with estimated
8 Note that, in probability and statistics, we use upper case to denote random variables and lower
case to denote realizations (outcomes) of these random variables. We will relax this in later chapters,
but will use this convention in this chapter to clarify the concepts.


1.2  The Beginning of Modern Asset Pricing Models
values of α and β, which are denoted as α and β , such that the sum of the
squared residuals from the “fitted OLS regression line” is minimized. Note
that Equation (1.1) implies that α = 0. We can either impose that restriction before estimating the model, or we can allow the model to estimate α ,
depending on our assumption about how strictly the CAPM model holds in
the real world. For instance, if we believe that the CAPM model is mostly correct, but that there are temporary deviations of stocks away from the model,

we would allow the intercept, α , to be estimated using real data. Even if the
CAPM holds exactly at the beginning of each period for, say, Apple, it is easy
to understand why there can be several unexpected positive surprises for
Apple over a several-month period (such as the unexpected introduction of
several innovative products). Such unexpected “shocks” can be captured by
the α estimate, which prevents them from affecting the precision of the β
estimate. In this discussion, we’ll stick with the model including an intercept
to accommodate such issues.
After we estimate the model, we write the resulting “fitted model” as
(1.4)

E [Rt ] = α + β · E [RMRFt ] ,

where we realize that α is just a temporary deviation, and we expect it to be zero
in the future. Using this expectation, we can use this model to forecast future
returns with:
(1.5)

E [Rt+1 ] = β · E [RMRFt+1 ] ,

where all we need to do is to estimate one value—the expected excess return of
the market portfolio of stocks, E [RMRFt+1 ]. One simple, but not very precise,
method of estimating this parameter is to use the average historical values over
the past T periods:9

E Rt+1 = β ·

1
T

t

rmrfj = β · rmrf .
j=t−T+1

Other methods of estimating E [RMRFt+1 ] include using the average return
forecast from professionals, such as security analysts, or deriving forecasts from
index futures or options markets.
We can also estimate the risk of holding a stock or portfolio—as well as decomposing this risk into market-based and idiosyncratic risk—with this one-factor
model by applying the rules of variances to Equation (1.2):
V [Rt ] = β 2 · V [RMRFt ] +
Systematic Risk

9 This

V [et ]

.

Idiosyncratic(stock or portfolio specific) risk

estimator is not precise because of the high variance of monthly values of rmrft.

(1.6)

7


8

CHAPTER 1  An Introduction to Asset Pricing Models
Here, we can again use the fitted regression, in conjunction with past values of
RMRF and the regression residuals, ǫt to estimate the future total risk:
2

V Rt+1 = β ·

1
T −1

t

2

rmrfj − rmrf
j=t−T+1

+

1
T −1

t
2

ǫˆj . (1.7)
j=t−T+1

Figure 1.1 and Tables 1.1 and 1.2 show an example of a fitted model using
­Chevron-Texaco (CVX) over the 2007–2008 period. Two approaches to fitting

the model of Equation (1.3) using OLS are presented in the graph and in the
tables: (1) the unrestricted model, and (2) the restricted model (where α is
forced to equal zero):
CVX Monthly Return,
Jan 2007-Dec 2008
0.15
0.10
0.05
-0.20

-0.15

-0.10

-0.05

-0.05

Unrestricted Model
Restricted Model

0.00

0.05 RMRF 0.10

-0.10
-0.15
-0.20

FIGURE 1.1

CAPM Regression Graph for Chevron-Texaco.

Table 1.1

Unrestricted Ordinary Least Squares CAPM Regression
Output for Chevron-Texaco
Regression Output (Unrestricted Model)

Intercept (α)
RMRF

Table 1.2

Coefficients

Standard Error

t Stat

P-value

0.012
0.58

0.012
0.22

1.0
2.58

0.32
0.01

Restricted Ordinary Least Squares CAPM Regression
Output for Chevron-Texaco
Regression Output (Restricted Model, a = 0 )

Intercept (α)
RMRF

Coefficients

Standard Error

t Stat

P-value

0
0.51

0.21

2.38

0.02


1.2  The Beginning of Modern Asset Pricing Models
Note that, if we restrict the intercept to equal zero, we get a lower estimate of

the slope coefficient on RMRF, β , since we force the fitted regression line to
pass through zero, as shown in the figure above.
In most cases, it is better to allow the intercept to be estimated, since it can
be non-zero by the randomness in stock returns, as illustrated by the Apple
example discussed previously.
Next, let’s model CVX over the following two years, 2009–2010, shown in
Table 1.3.
Table 1.3

Unrestricted Ordinary Least Squares CAPM Regression
Output for Chevron-Texaco, 2009–2010
Regression Output (Unrestricted Model)

Intercept (α)
RMRF

Coefficients

Standard Error

t Stat

P-value

−0.0024
0.81

0.01
0.16

−0.23
5.03

0.32
0.00005

Note that both α and β have changed from their values during 2007–2008.
Does this mean that these parameters actually change quickly for individual
stocks? In most cases, no—these changes are the result of “estimation error”,
which happens when we have a very “noisy” (volatile) y-variable, such as CVX
monthly returns,10 due again to randomness.
Besides using the above regression output in the context of Equation (1.5)
to estimate the expected (going-forward) return of CVX, we can also use the
regression output to estimate risk for CVX going forward, using Equation (1.6).
The results from the above two regression windows point out an important lesson to remember: individual stock betas are extremely difficult to estimate precisely, which makes the CAPM very difficult to use in modeling individual stocks.
There are several ways to attempt to correct these estimated betas while still using
the CAPM. One important example is a correction for stocks that respond slowly
to broad stock market forces, and might have a lag in their reaction due to their
illiquidity. Scholes and Williams (1977) describe an approach to correct for the
betas of these stocks by adding a lagged market factor to the CAPM regression,
rt = α + βˆ1 · rmrft−1 + βˆ2 · rmrft + ǫt .

10

(1.8)

One example of a case where these parameters could actually change quickly is when a company’s capital structure shifts dramatically, which might happen with an extreme stock return, a stock
repurchase, or a large issuance of equity or bonds. Theory predicts a change in the CAPM regression
slope, β, in all of these cases.

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CHAPTER 1  An Introduction to Asset Pricing Models
An improved estimate of the beta of a stock, the Scholes-Williams beta (β SW ),
is then computed by adding together the estimates of β1 and β2 (assuming rmrft
has trivial serial correlation):
β SW = β1 + β2

(1.9)

There are many other potential problems with estimated betas, and n
­ umerous
approaches to dealing with them. However, none of these methods, many of
which can be complicated to implement, fully correct for the problem of large
estimation errors for individual securities, such as stocks.11 As a result, ­one
should always be very careful about modeling an individual security. When possible, form portfolios of securities, then apply regression models.

1.3  EFFICIENT MARKETS
The notion of market prices efficiently reflecting all available (public) information
is likely as old as the notion of capitalism itself. Indeed, if prices swing wildly in
a way that is not consistent with the (unknown) expected intrinsic value of assets,
then a case can be made for government intervention. Examples of this are the two
rounds of “quantitative easing” (QE1 and QE2) that were implemented during
2009 and 2010, during and shortly after the financial crisis of 2008 and 2009.12
However, there are many shades of market efficiency, from completely informationally efficient markets to markets that are only “somewhat” informationally
efficient.13 In the world around us, we can easily see that many forms of information are fairly cheap to collect (such as announcements from the Federal
Reserve), while many other forms are expensive (such as buying a Bloomberg

terminal with all of its models). In their seminal paper, Grossman and Stiglitz
(1980) argued that, in a world of costly information, informed traders must earn
an excess return, or else they would have no incentive to gather and analyze
information to make prices more efficient (i.e., reflective of information). That is,
markets need to be “mostly but not completely efficient”, or else investors would
not make the effort to assess whether prices are “fair”. If that were to happen,
prices would no longer properly reflect all available and relevant information,
and markets would lose their ability to allocate capital efficiently. Thus, Grossman
and Stiglitz advocate that markets are likely “Grossman-Stiglitz efficient”, which
11 Bayesian

models can be very useful for controlling estimation error. A Bayesian prior can be based
on the CAPM, or another asset pricing model that is believed to be correct. However, they depend
on the researcher having some strong belief in the functional form of one of several possible asset
pricing models.

12 QE1 and QE2 involved the Federal Reserve purchasing long-term government bonds from the marketplace, which is, in essence, placing more money into circulation (i.e., the Fed “printed money”).
13 Informationally efficient markets are those that instantaneously reflect new information that
affects market prices, whether this information is freely available to the market or must be purchased
or processed using costly means. Such markets may not perfectly know the true value of a security,
which would require perfect information on the distribution of cashflows and the proper discount
rate, but they use current information properly to estimate these parameters in an unbiased way.


1.4  Studies That Attack the CAPM
means that costly information is not immediately and freely reflected in prices
available to all investors. Indeed, the idea of Grossman-Stiglitz efficient markets
is a very useful way for students to view real-world financial markets.
Behavioral finance academics, such as John Campbell and Robert Shiller, have
found evidence that markets do not behave “as if” investors are perfectly rational

in some Adam Smith “invisible hand” sense—in fact, they believe the evidence
makes the potential for efficient markets—Grossman-Stiglitz or other notions of
efficiency—very improbable in many areas of financial markets. This evidence
is somewhat controversial among academics, although investment practitioners
seem to have accepted the idea of behavioral finance more completely than
academics. While the field of behavioral finance has become immense, a full
discussion of the literature is beyond the scope of this book.14 However, in the
next section, we will discuss some research that documents return anomalies—
potentially driven by investor “misbehaviors”—that are directly related to the
models used to describe stock and bond returns today—so that the reader will
have a better understanding of the origin of these models.15

1.4 STUDIES THAT ATTACK THE CAPM
Many financial economists during the 1970s attempted, with some success, to
criticize the CAPM as a model that doesn’t reflect the real world of stock returns
and risk. The reader should note that no one doubted that the mathematics of
the CAPM were correct, given its many assumptions. Instead, the model was
attacked because it did not work well in the real world of stock, bond, and other
security and asset pricing, which means its assumptions were not realistic.
A few of the many famous papers are described here. Most CAPM criticisms have
focused on the stock market, mostly because stock price and return data have been
studied extensively by academic researchers and such data are of h
­ igh-quality (i.e.,
from the Center for Research in Security Prices–CRSP—at the University of Chicago).
First, Banz (1981) studied the returns of small capitalization stocks using
the CAPM model. Banz found that a size factor (one that reflects the return
­difference between stocks with low equity capitalization—price times shares
outstanding—and stocks with high equity capitalization) adds explanatory
­
power for the cross-section of future stock returns above the explanatory power

of market betas. He finds that average returns on small stocks are too high, even
controlling for their higher betas, and that average returns on large stocks are too
low, relative to the predictions of the CAPM.

14 Many contributions can be found in the articles and books of Kahneman and Tversky, Shiller,
Thaler, Campbell, Barber and Odean, Lo, and several others.
15 Studies that document anomalies in other markets are much more sparse, such as anomalies
in bond or futures markets. To some extent, this is due to the fact that academic researchers have
devoted the majority of their time to studying stock prices (due to the high-quality data and transparent markets for stocks, as well as the broad participation of individual investors in stock markets).

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CHAPTER 1  An Introduction to Asset Pricing Models
Bhandari (1988) found a positive relation between financial leverage (debt to
equity ratio) and the cross-section of future stock returns, even after controlling
for both size and beta. Basu (1983) finds that the earnings-to-price ratio (E/P)
predicts cross-sectional differences in future stock returns in models that include
size and beta as explanatory variables. High E/P stocks outperform low E/P stocks.
Keim (1983) finds that about 50% of the size factor return, during 1963–1979,
occurs in January. Further, over 50% of the January return occurs during the first
week of trading, in particular, the first trading day. And, Reinganum (1983) finds
similar results, and also finds that this “January effect” does not appear to be
completely explained by investor tax-loss selling in December and repurchasing
in January.

1.5 DOES PROVING THE CAPM WRONG = MARKET
INEFFICIENCY? OR, DO EFFICIENT MARKETS =

THE CAPM IS CORRECT?
Emphatically, no! This is often termed the “joint hypothesis problem”, since any
empirical test of the CAPM, such as the above-cited studies, is jointly testing the
validity of the model and whether violations to the model can be found. Often,
students of finance believe in the CAPM so thoroughly (probably through the
fault of their professors) that they equate the CAPM’s validity to the validity of
efficient markets. However, there is no such tie. Markets can be perfectly efficient,
and the CAPM model can simply be wrong—it’s just that it does not describe
the proper risk factors in the economy. For instance, if two risk factors drive the
economy, then the CAPM will not work.
If the CAPM is exactly correct, however, markets must be efficient—unless we
use an expanded notion of the CAPM that has two versions: one version that is
visible to everyone, and another that is visible only to the “informed investors”.
The CAPM modeled by Sharpe, however, has no such duality—there is one market portfolio and one beta for each security in the economy. In Sharpe’s CAPM
world, markets are perfectly efficient, and everyone has the same information.16

1.6  SMALL CAPITALIZATION AND VALUE STOCKS
In the early 1990s, Fama and French tried to settle the question of the usefulness of the CAPM in the face of all these apparent stock “anomalies”. In doing
so, Fama and French (FF; 1992) declared that “beta is dead”, meaning that the
CAPM was a somewhat useless model, at least for the stock market. Instead, FF
promoted the use of two new factors to model the difference in returns of different stocks: the market capitalization of the stock (also called “size”) and the
book-to-market ratio (BTM) of the stock—that is, the accounting book value of
equity divided by the market’s value of the equity (using the traded market price).
16 Dybvig

and Ross (1985), Mayers and Rice (1979), and Keim and Stambaugh (1986) were among
the first to expand the notion of the CAPM to one involving two types of investors, informed and
uninformed.

1.6  Small Capitalization and Value Stocks
FF used a clever approach to demonstate this argument. Most prior studies of the
CAPM first estimate individual stock or stock portfolio betas from the one-factor
regression of Equation (1.4), as we did for CVX above, then test whether these
betas forecast future stock returns. FF argued that small capitalization stocks tend
to have much higher betas than large capitalization stocks, so it might be that
small stocks simply have higher returns than large stocks, regardless of their betas.
First, FF estimated each stock’s beta with five years (60 months) of past returns,
using the one-factor regression model of Equation (1.3). Then, they ranked all
stocks by their market capitalization (size), from largest to smallest, then cut
these ranked stocks into 10 groups. The top decile group was the group of largest
stocks, while the bottom decile was the small stock group.
Next, FF ranked stocks—within each of these decile groups—by the betas of
the stocks that they had already computed. Then, FF took the highest 1/10th of
stocks, according to their betas, from each of the 10 size deciles (that 1/10th was
1/100 of all stocks)—then, recombined these 10 “high beta” subportfolios into
a high beta, mixed size portfolio. This was repeated for the 2nd highest 1/10th
of stocks in each portfolio to form the “2nd highest beta” subportfolio with
mixed size. And, so on, to the lowest beta 1/10th of stocks to form the “low beta”
subportfolio with mixed size. Finally, FF measured the equal-weighted returns
of each of these newly constructed 10 portfolios—each of which had stocks with
similar betas, but mixed size—during the following 7 years. The objective was
to separate the influence of size from beta by “mixing” the size of stocks with
similar betas. This procedure is depicted in Figure 1.2.

FIGURE 1.2
Fama-French’s “Beta is Dead” Slicing Test.

When FF regressed this 7-year future return, cross-sectionally, on the prior equalweighted betas of these 10 portfolios, they found no significant relation, where
the CAPM’s central prediction is a strong and positive relation between betas and

returns. Thus, according to FF, “beta was dead”.17 Then, FF presented evidence that
not only does size work well, but so does BTM ratio; together, they both worked
17 In fact, to provide a more statistically powerful test, they repeated this similar beta mixed size
portfolio construction at the end of each month during 1964 to 1989 to conclude that the evidence
of beta being important (or “priced”) was, at best, weak.

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CHAPTER 1  An Introduction to Asset Pricing Models
well, so they appear to be measuring different risks. Finally, FF looked at the returnon-equity (ROE) of small stocks and stocks with a high BTM ratio, and found that
the ROE of these stocks was quite low—indicating, perhaps, that they are under
financial distress and are at risk of bankruptcy. While not proving anything, FF
suggest that size and BTM may be a proxy for financial distress—small stocks with
high BTM, for instance, are highly stressed—and this may underlie the usefulness
of size and BTM. Simply put, investors demand higher returns for financially distressed stocks, as they are more likely to fail together during a recession.
The reception of the Fama French paper was one of controversy, which still exists
today. Most reseachers have admitted that Fama and French are right about what
works better in the real world of stocks, but they disagree about why. FF represent one camp with their rational investor, financial distress risk economic story.
Another camp believes that investors exhibit behavioral tendencies that color their
choice of stocks. Underlying this economic story is the fact that individuals tend to
overreact to longer-term trends in the economic fortunes of a corporation, and that
they believe that the fortunes of stocks that have become less profitable over the past
several years will continue to become worse—thus, they put sell pressure on small
stocks and value stocks (high BTM stocks). A third camp believes that small stocks
and value stocks have simply gone through a “lucky streak”, and that we should not
place too much importance on the experience of U.S. stocks in the past few decades.
In an attempt to further test the FF findings, Griffin (2002) studied size and

book-to-market as stock return predictors in the U.S., Japan, the U.K., and
Canada. He found evidence in all four countries that size and BTM forecast stock
returns, consistent with FF’s findings in U.S. stocks. However, he also found that
returns correlate poorly for size and BTM across these countries, which could
be evidence that they are risk-based or that they are due to irrational investor
behavior—and country stock markets are segmented, preventing investors from
arbitraging across differences in these factor returns across countries.

1.6.1  Momentum Stocks
Notably, Fama and French did not quite find all the important factors that drive
stock returns. Jegadeesh and Titman (JT; 1993) found that momentum, measured
as the one year past return of a stock is an important predictive variable for the
following year’s return. In fact, a simple sorting of stocks on their one-year past
return, followed by an equal-weighted long position in the top 10% “winners”
and a short position in the bottom 10% “losers” of last year provides an “arbitrage”
profit of almost 1% per month (i.e., about 10% during the following year).18
18 It

turns out that momentum, while known for decades by some practitioners and academics (e.g.,
Levy, 1967), was “discovered” by academics by accident. In conducting research for Grinblatt and
Titman’s (1993) study of mutual fund performance, a PhD student accidently measured the return of
mutual fund positions in stocks held today over the past year (rather than over the next year). The result
was that most U.S. domestic equity mutual fund managers were, to some extent, holding larger share
positions in last-years winners than in other stocks. Building on this finding, Grinblatt et al. (1995)
found that such “momentum-investing funds’ also outperformed market indexes in the future—indicating that the stocks that they were buying also outperformed—thus, stock momentum was discovered!


1.6  Small Capitalization and Value Stocks
Figure 1.3 illustrates the profitability over numerous portfolios formed over the
period 1965–1989. The monthly (not annualized) returns of the long-short portfolio over the 36 event months following the portfolio formation are shown first,

followed by the cumulated monthly returns over the same 36 months.19
Further evidence supporting momentum in U.S. stocks was found during 1941–
1964, although not quite as strong—shown in Figure 1.4.
However, JT found that the depression era did not support their “momentum theory”, and, instead, momentum stocks lost considerable money (see Figure 1.5).
JT explained that momentum likely did not work during the depression era
because of inconsistent monetary policy that artificially created reversals of stock
returns during that time. Specifically, when the stock market dropped, the Fed
eased monetary policy, and when it boomed, the Fed strongly tightened. Nevertheless, Daniel (2011) has shown, more recently, that momentum stocks outperformed during 1989–2007, but underperformed (badly) during the financial
crisis of 2008–2009.

Re lative Stre ngth Portfolios in Event Time

0.010

34

31

28

25

22

-0.005
-0.010

34

31

28

25

t

22

19

16

13

10

7

Re lative Stre ngth Portfolios in Event Time

4

0.100
0.090
0.080
0.070
0.060
0.050
0.040

0.030
0.020
0.010
0.000
-0.010

t

1

Cumulative Return

19

16

13

7

4

0.000

10

0.005
1

Monthly Return

0.015

FIGURE 1.3
Monthly and Cumulative Momentum Long/Short Portfolio Returns, 1965–1989.

19 This ranking and formation strategy is repeated using (overlapping) windows. Specifically, a new
portfolio is formed every month, giving (at any point in time) 36 simultaneous (overlapping) portfolio strategies.

15


CHAPTER 1  An Introduction to Asset Pricing Models
Relative Strength Portfolios in Event Time

0.010

34

31

28

25

22

-0.005
-0.010

t
Relative Strength Portfolios in Event Time

0.060
0.050
0.040
0.030
0.020

34

31

28

25

22

19

16

13

10

7

0.000

-0.010

4

0.010
1

Cumulative Return

19

16

13

7

4

0.000

10

0.005
1

Monthly Return

0.015

t

34

31

28

25

22

19

16

13

10

7

4

t

34

31

28

25

22

19

16

13

10

7

Relative Strength Portfolios in Event Time
4

0.000
-0.050
-0.100
-0.150
-0.200
-0.250
-0.300
-0.350
-0.400
-0.450

Relative Strength Portfolios in Event Time

1

0.010
0.005
0.000
-0.005
-0.010
-0.015
-0.020
-0.025
-0.030
-0.035
-0.040
-0.045
-0.050

1

Monthly Return

FIGURE 1.4
Monthly and Cumulative Momentum Long/Short Portfolio Returns, 1941–1964.

Cumulative Return

16

t

FIGURE 1.5
Monthly and Cumulative Momentum Long/Short Portfolio Returns, 1927–1940.


1.7  The Asset Pricing Models of Today
Further research by Rouwenhorst (1998) found that momentum exists in stocks
in Europe, but not in Asia. More recent research seems to find momentum even
in Japan (see Asness(2011)).
Today, although the evidence is, at times, inconsistent, momentum is strong enough
that most academic researchers appear to accept that it is a reality of markets. One
economic explanation of momentum is that investors underreact to short-term
news about companies, such as improving earnings or cashflows. Thus, a stock that
rises this year has a bright future next year—again, not always, but on average.20
Finally, Griffin et al. (GMJ; 2003) examined momentum in the U.S. and 39 other
countries, and found evidence that these factors work well in these markets, but
that momentum across different countries is only weakly correlated. Therefore,
country-level momentum factors work better in capturing momentum, rather
than a global momentum factor across all countries. This finding suggests that
whatever economics are at play in the risk of stocks, they work a little differently in
different countries, but with the same overall result: small stocks outperform large
stocks, value stocks outperform growth stocks, and momentum stocks outperform
contrarian stocks (all of this is for an average year, but the reverse can occur for
any single year or subset of years—such as the superior growth stock returns of the
technology boom during the 1990s). Finally, GMJ found that momentum profits
tend to reverse in the countries over the following one to four years.
Next, we will describe models that attempt to capture the multiple sources of
stock returns noted above. While academics and practitioners do not agree on
whether these sources of additional return represent systematic risks or simply
return “anomalies”, these models have been developed to better describe the

drivers of stock returns, regardless of the source of the factors” power.21

1.7  THE ASSET PRICING MODELS OF TODAY
The above studies have inspired researchers to add factors to the single-factor
model of  Equation (1.2) that is, itself, inspired by the CAPM theory. As opposed
to this “theory-inspired” single-factor model, almost all recent models are
“empirically inspired”, which means that they are chosen because they explain
the cross-section and/or time-series of security returns while still making economic sense. This means that we don’t simply try lots of factors until we find
some that work, as this can always be done (and often leads to a breakdown
of the model when we try to use it with other data). We carefully examine past
20 Momentum

might also be interpreted as a risk factor. See, for example, Chordia et al. (2002).
reader should note that there are many more recent papers documenting other anomalies in
stock returns. For instance, Sloan (1996) finds that stocks with high accruals—earnings minus cashflows—earn lower future returns than stocks with low accruals. Lee and Swaminathan (2000) find
that stocks with lower trading volume (less liquidity) have higher future returns than high trading
volume stocks. However, these anomalies are not yet accepted by academics to the point of revising
the models that we are about to present in the next section. Or, more accurately, there is not strong
agreement that these anomalies are strong enough and are independent of the existing factors to
warrant a more complicated model with additional factors.
21 The

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CHAPTER 1  An Introduction to Asset Pricing Models
research for both economic and econometric guidance on the factors that might
be used in a model. Fortunately, many researchers have already done this work

for us. Almost all models are “multifactor” models, meaning that more than one
x-variable (“risk factors”) is used to predict the y-variable (security or portfolio
excess returns).

1.7.1  Introduction to Multifactor Models
A multifactor model can be visualized as a simple extension of a single ­factor
model, such as the CAPM. However, by using multiple risk factors, we are implicitly rejecting the CAPM and its many assumptions about investors and markets.
The simplest multifactor model is a two-factor model. Let’s suppose that we
believe that, in addition to the broad stock market, the risk-premium to i­ nvesting
in small stocks drives security returns.
Then, the time-series model would be:
(1.10)

Rt = α + β · RMRFt + s · SMBt + ǫt ,

where s is the exposure of a security, or portfolio, to the “small-capitalization
risk-factor”. This regression for Chevron-Texaco, implemented using Excel during the 24-month period January 2009 to December 2010, results in the following output Table 1.4.
Table 1.4

Two-Factor Regression for CVX
Regression Output (Unrestricted Model)

Intercept
RMRF
SMB

Coefficients

Standard Error

t Stat

P-value

0.00031
0.92
−0.56

0.01
0.17
0.38

0.03
5.32
−1.47

0.98
0.00
0.16

The adjusted R2 from this regression is 0.54 (54%), while the adjusted R2 from
the single-factor regression of CVX excess returns on RMRF (from a prior section)
is 0.51.22 Therefore, in this case, the addition of a small-cap factor—to which
Chevron-Texaco is negatively correlated—does not matter much. However, since
we have estimated the two-factor model, and since its t-statistic is relatively close
to −1.645 (the two-tailed critical value for 10% significance), we’ll use it.
22

Note that these R2 values are very high for an individual stock—likely because CVX is very
large cap and had no big surprises during the period, thus, it roughly matched the stock market as

a whole. A regression of an individual stock return on the four-factor model usually gives an R2 of
only about 10–20%. As we will see in later chapters, a regression of a managed (long-only) portfolio, such as a mutual fund, using either the one-factor or four-factor models, usually gives an R2 in
excess of 90%. Thus, if you are applying your regression model correctly, you should generally
(but not always, as with CVX) see these levels of R2 values. This is a good diagnostic check of your
data work.