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. http://dx.doi.org/10.1016/B978-0-08-092652-0.00001-7
© 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 economic agents”
discount rates also depend on their personal aversion to risk, and, thus, may
require very different compensations to take similar risks.2, 3
Markowitz
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. ten
Stocks,
in that they will pay the next quarterly dividend, provide
withone
not even a promise
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
.w.to
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
History of personal
risk
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
5
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:
excessreturn
E [Rt ] = β · E [RMRF t ] ,
(1.1)
factor model
single
Go'M
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
ere
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 nover 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).
nnha
d
5 For
r.sn
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.
correlation coefficient
ptyiji
6
CHAPTER 1 An Introduction to Asset Pricing Models
5Y M
Unusable
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. (2009).
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
i
Rt = α + β · RMRFt + et ,
(1.2)
while its application to real-world data can be similarly written as:8
rt = α + β · rmrft + #t ,
(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
!, such that the sum of the
α and β
values of α and β, which are denoted as !
squared residuals from the “fitted OLS regression line” is minimized. Note
that Equation (1.1) implies that α = 0. We can either impose that restricα,
tion before estimating the model, or we can allow the model to estimate !
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,
α , to be estimated using real data. Even if the
we would allow the intercept, !
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
!
α estimate, which prevents them from affecting the precision of the β
the !
estimate. In this discussion, we’ll stick with the model including an intercept
三
to accommodate such issues.
After we estimate the model, we write the resulting “fitted model” as
! · E [RMRFt ] ,
!
α+β
E [Rt ] = !
(1.4)
! · E [RMRFt+1 ] ,
!
E [Rt+1 ] = β
(1.5)
α is just a temporary deviation, and we expect it to be zero
where we realize that !
in the future. Using this expectation, we can use this model to forecast future
returns with:
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
t
1 "
! · rmrf .
rmrfj = β
T
j=t−T
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, !t to estimate the future total risk:
o
!
!2 ·
V [Rt+1 ] = β
1
T −1
t
"
$2
#
rmrfj − rmrf +
j=t−T
t
1 " 2
"ˆj .
T −1
(1.7)
j=t−T
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
ii
-0.10
-0.15
IT
zlnwould
-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)
Broke
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, α = 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
!, since we force the fitted regression line to
the slope coefficient on RMRF, β
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
! have changed from their values during 2007–2008.
α and β
Note that both !
Does this mean that these parameters actually change quickly for individual
stocks? In most cases, no—these changes are the result of “estimation error”,
rne
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 + β̂1 · rmrft−1 + β̂2 · rmrft + #t .
on
10
lag in
(1.8)
reaction
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.
9
10
CHAPTER 1 An Introduction to Asset Pricing Models
!SW ),
An improved estimate of the beta of a stock, the Scholes-Williams beta (β
is then computed by adding together the estimates of β1 and β2 (assuming rmrft
has trivial serial correlation):
!SW = β
!1 + β
!2
β
(1.9)
There are many other potential problems with estimated betas, and numerous
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
CAPM X pricing
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
mn.in
nhnEnn
n.nnnnnnn
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.
有
if itmed太多
1.4 Studies That Attack the
not educate
or
move themarket
11
CAPM
⼝
means that costly information is not immediately and freely reflected in prices 此
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.
l
still rational
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
worn
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
ī
6
Greatreturn
less risk
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.
critize
CAPMdecent
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 high-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).
iii
iii
A
i
smallcap risky
higher betalsyshis
SML PetaandExpected
Slope EIRMD.FI
retu
12
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.
lets
吴培
cnn.me
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.
ThejointhypEE c.AM
validity
EMT validity
factors
o
risk
egyz
CAPM is wrong 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”.
if CAPM iscorrect The CAPM modeled by Sharpe, however, has no such duality—there is one marIf the CAPM is exactly correct, however, markets must be efficient—unless we
Marketsmusttent ket portfolio and one beta for each security in the economy. In Sharpe’s CAPM
ifCAPMis tv
EMT is v
Fanaand Fend
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 ⼝
equity divided by the market’s value of the equity (using the traded market price).
cut
器叶
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.
size分成10但
1 咀 rankbyp
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.
No Significant
diff
O
⼼⼼
higherreturns
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
皉
lowerreturns
FIGURE 1.2
Fama-French’s “Beta is Dead” Slicing Test.
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.
13
啊
14
CHAPTER 1 An Introduction to Asset Pricing Models
Result
I
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
p
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, 1964), 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
15
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.
long
ĪTÉTÉÉ
Re lative Stre ngth Portfolios in Event Time
34
31
28
25
22
19
16
13
10
7
4
0.000
months
-0.005
t
Re lative Stre ngth Portfolios in Event Time
34
31
28
25
t
22
19
16
13
10
7
4
o
1
0.100
0.090
0.080
0.070
0.060
0.050
0.040
0.030
0.020
0.010
0.000
-0.010
FIGURE 1.3
Monthly and Cumulative Momentum Long/Short Portfolio Returns, 1965–1989.
19 This
Niort
bottom lot
Hedgefund
0.005
-0.010
Cumulative Return
Top10
0.010
1
Monthly Return
0.015
⼼的 ⼼ 知
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.
if they
have similar
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
same pattern
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
17
18
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 Icould skip
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 investing
in small stocks drives security returns.
Then, the time-series model would be:
(1.10)
Rt = α + β · RMRFt + s · SMBt + #t ,
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
0.56
−1.35
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.
1.7 The Asset Pricing Models of Today
Once the model is fitted, the next-period estimated expected return is:
! · E [RMRFt+1 ] +!
!
s · E [SMBt+1 ] ,
E [Rt+1 ] = β
or, using the fitted regression from above,
!
E [Rt+1 ] = 0. 92 · E [RMRFt+1 ] − 0. 56 · E [SMBt+1 ] .
(1.11)
(1.12)
Note that this is an equation of a plane in three-dimensional space, where
!
E [Rt+1 ] is the vertical axis. The residuals, !t, are the vertical distance from this
plane of the actual month-by-month outcomes, rt, from the model-predicted
values of Equation (1.12).
The next-period estimated total risk, which contains a term for the covariance
between RMRF and SMB, is
!
V [Rt+1 ]
" #$ %
Total Risk (Variance)=
! ·!
!2 · V [RMRFt+1 ] +!
s · C [RMRFt+1 , SMBt+1 ]
s 2 · V [SMBt+1 ] + 2β
=β
#$
%
"
Systematic Risk
+
V ["t+1 ]
" #$ %
Idiosyncratic Risk
,
(1.13)
and the next-period estimated systematic (risk-factor related) risk is:
!2 · V [RMRFt+1 ] +!
! ·!
!
s 2 · V [SMBt+1 ] + 2β
s · C [RMRFt+1 , SMBt+1 ] .
VS [Rt+1 ] = β
(1.14)
Again, following the simple approach of using historical sample data to estimate
the above expected returns and variances, the equations for expected return,
total, and systematic-only risk become
! · rmrf +!
!
s · smb,
E [Rt+1 ] = β
and
2
2
!2 · !
! ·!
!
σRMRF
+!
s2 ·!
σSMB
+ 2β
s ·!
σRMRF, SMB + !
σ#2 ,
V [Rt+1 ] = β
2
2
!2 · !
! ·!
!
σRMRF
+!
s2 ·!
σSMB
+ 2β
s ·!
σRMRF, SMB ,
VS [Rt+1 ] = β
!
where rmrf = T1 Tt=1 rmrft,
1 !T
smbt ,
smb =
t=1
T
$2
1 !T #
2
"
σRMRF
=
rmrft − rmrf ,
t=1
T −1
$2
#
!
T
1
2
"
σSMB =
smbt − smb ,
t=1
T −1
$#
$
#
1 "T
and !
.23
σRMRF,SMB = T−1
rmrf
smb
−
smb
rmrf
−
t
t
t=1
(1.15)
(1.16)
(1.17)
23 These sampling statistics are easy to compute in Excel, using the sample mean, variance, and covari-
ance functions applied over the time-series of historical data.
19
20
CHAPTER 1 An Introduction to Asset Pricing Models
1.7.2 Models of Stock Returns
why Peta ti
Regression-Based Models Fama and French (1993) designed a widely used
multifactor model which adds both the small-capitalization factor (SMB) and a
“value stock factor” (HML) to the single-factor model of Equation (1.2).24
LNalue
M
BH
rfs
Rt = α + β · RMRF
t + s · SMBt + h · HMLt + et.
(1.18)
However, the most widely used returns-based model for analyzing equities is the
four-factor model of Carhart (1997),
growt C
Rt = α + β · RMRFt + s · SMBt + h · HMLt + u · UMDt + #t ,
(1.19)
who added a momentum factor (UMDt ) to the three-factor model of Fama and
French.25
Let’s estimate the “Carhart model” for CVX, during 2009–2010 in Table 1.5.
Table 1.5
potsit.ve
relation momentu
Four-Factor Regression for CVX
Regression Output (Unrestricted Model)
Intercept
RMRF
SMB
HML
UMD
Coefficients
Standard Error
t Stat
P-value
0.0046
1.1
−0.61
0.00028 0
0.31
0.0097
0.20
0.34
0.31
0.11
0.64
5.5
−1.8
0.00090
2.8
0.98
.000026
0.086
0.99 X
0.012
0
How did the addition of HML and UMD affect the estimated coefficients on
! and !
! from 0.92 to 1.1, and decreased
s )? They increased β
RMRF and SMB (β
!
s from −0.56 to −0.61. Why did these changes occur with the addition of HML
and UMD? The answer is that these two new regressors must be correlated, to
o
o
o
negative exposure
24
One might wonder why Fama and French added back the RMRF factor, when their 1992 paper
found that beta did not affect stock returns. The reason is that their tests were cross-sectional, meaning that one can assume that the betas of all stocks are unity without much error. In the crosssection, RMRF then washes out of differences in stock returns. However, in the time-series, RMRF
matters for each individual stock or portfolio return. Why don’t we force beta to equal one in the
time-series regression? For practical reasons, among them, managed funds often carry cashholdings,
while others leverage their portfolios, which even Fama and French would admit moves the portfolio beta away from one.
25 A more detailed description:
Rt is the month-t excess return on the stock (net return minus T-bill
return), RMRFt is the month-t excess return on a value-weighted aggregate market proxy portfolio,
and SMBt, HMLt, and UMDt are the month-t returns on value-weighted, zero-investment factormimicking portfolios for size, book-to-market equity, and one-year momentum in stock returns,
respectively. This model is based on empirical research by Fama and French (1992, 1993, 1996) and
Jegadeesh and Titman (1993) that finds these factors closely capture the cross-sectional and timeseries variation in stock returns.
1.7 The Asset Pricing Models of Today
21
some extent, with RMRF and SMB, thus “stealing” some (pretty small) explanatory power from them, and changing their relation with the predicted variable, Rt.
Also, the four-factor model shows that RMRF and UMD are the most statistically
significant explanatory variables, with SMB close behind. HML has no significance, since its p-value is equal to 99% (meaning that the chances of observing
a coefficient of |0.00028| or larger by pure randomness, when its actual value is
zero, is 99%). So, we conclude that CVX, during 2009–2010, has a beta close to 1
(typical for a stock), is a very large capitalization stock (since its “loading” on SMB
is very negative and statistically significant), and it has significant momentum
(meaning the prior-year return is high over the period 2009–2010—consistent
with increasing oil prices!). Note that, in general, coefficients in this model that
are close to (or slightly exceed) one have a large exposure to that risk factor.
However, even coefficients at the level of 0.2 or 0.3 indicate a substantial exposure to
a certain risk factor.
coeff close1
么
些
exposure
A Stock Characteristic-Based Model Another approach to modeling stocks that
is based on the findings noted above (i.e., that market capitalization, value, and
momentum drive stock returns) uses the characteristics (observable features) of
stocks to assemble them into groups or portfolios of stocks with similar characteristics. Daniel and Titman (1997) found empirical evidence that suggests
that characteristics provide better ex-ante forecasts than regression models of
the cross-sectional patterns of future stock returns. This evidence indicates that
stock factors like equity book-to-market ratio at least partially relate to future
stock returns due to investors having behavioral biases against certain types of
stocks (e.g., those stocks with recent bad news, which pushes the BTM ratio up
“too much”).
Following Daniel and Titman, in the characteristic benchmarking approach, the
average return of the similar characteristic portfolio is used as a more precise
proxy for the expected return of the stock during the same time period. Any
deviation of a single stock from this expected return is the stock’s “residual”, or
Leniency
unexpected return. Daniel et al. (1997) developed such an approach
for U.S.
three
equities, and many other researchers have replicated their approach in other
stock markets.
First, all stocks (listed on NYSE, AMEX, or Nasdaq) having at least two years of
book value of equity information available in the Compustat database, and stock
return and market capitalization of equity data in the CRSP database, are ranked, at
the end of each June, by their market capitalization. Quintile portfolios are formed
(using NYSE size quintile breakpoints), and each quintile portfolio is further subdivided into book-to-market quintiles, based on their most recently available fiscal
year-end book-to-market data as of the end of June of the ranking year.26 Here,
we “industry-normalize” the book-to-market ratio, since we would like to classify
26 This
usually involves allowing a 30 to 60-day delay in disclosure of fiscal results by corporations.
嚚7125
22
CHAPTER 1 An Introduction to Asset Pricing Models
stocks by how much they deviate from their “industry norms”.27,28 Finally, each
of the resulting 25 fractile portfolios are further subdivided into quintiles based on
the 12-month past return of stocks through the end of May of the ranking year.
This three-way ranking procedure results in 125 fractile portfolios, each having a
distinct combination of size, book-to-market, and momentum characteristics.29
The three-way ranking procedure is repeated at the end of June of each year, and
the 125 portfolios are reconstituted at that date.
Figure 1.6 illustrates this process.
A modification of this procedure is to reconstitute these portfolios at the end
of each calendar quarter, rather than only once per year on June 30, using
updated size, BTM, and momentum data. While the annual sort is closer to an
implementable strategy that is an alternative to holding a particular stock, the
quarterly sort allows us to more accurately control for the changing characteristics
of the stock. For example, the momentum, defined as the prior 12-month return
of a stock, can change quickly.
Value-weighted returns are computed for each of the 125 fractile portfolios, and
the benchmark for each stock during a given quarter is the buy-and-hold return
of the fractile portfolio of which that stock is a member during that quarter.
Therefore, the benchmark-adjusted return for a given stock is computed as the
buy-and-hold stock return minus the buy-and-hold value-weighted benchmark
return during the same quarter. i
1.7.3 Models of Bond Returns
Fama and French (1993) found a set of five risk factors that worked well in
modeling both stock and bond returns. This includes three stock market factors
and two bond market factors:
1. stock market return (RMRF),
2. size factor (small cap return minus large cap return) (SMB),
3. value factor (high book-to-market stock return minus low BTM stock return)
(HML),
27 Specifically,
j
j
we compute the book-to-market characteristic as !ln(BTMi,t )−ln(BTMt ) ", where
j
j
σj ln(BTMi,t )−ln(BTMt )
j
BTMi,t is the book-to-market ratio of stock i, which belongs to industry j on June 30th of year t, and
j
ln(BTMt ) is the log book-to-market ratio
! of industry j at year "t (the aggregate book-value divided by
j
j
the aggregate market value). Also, σj ln(BTMi,t ) − ln(BTMt ) is the cross-sectional standard devia-
tion of the adjusted book-to-market ratio across industry. This approach was suggested by Cohen
and Polk (1998), as well as by discussions with Christopher Polk.
28 We could industry-normalize the size and momentum of a stock as well, and some researchers
have followed this approach. However, the most common approach is to industry-normalize only
the book-to-market.
29 Thus, a stock belonging to size portfolio one, book-to-market portfolio one, and prior return
portfolio one is a small, low book-to-market stock having a low prior-year return.
1.7 The Asset Pricing Models of Today
1
4. bond market maturity premium (10-year Treasury yield minus 30-day T-bill
yield) (TERM), and
5. default risk premium (Moody’s Baa-rated bond yield minus 10-year Treasury yield) (DEFAULT).
Panel A
! Rank all NYSE stocks by Mkt. Cap. -
Divide into 5 Quintiles
! Rank Quintiles = Book Value/Market Value (BTM)
Subdivide into 5 more quintiles
! Rank the 25 fractiles by past year stock return
Subdivide into 5 more quintile
A rank of:
Size=5,
Large Cap
BTM=5,
High BTM
PR1YR=5
High Past Return
Panel B
Panel C
FIGURE 1.6
Daniel, Grinblatt, Titman, and Wermers stock benchmarking procedure.
23
24
CHAPTER 1 An Introduction to Asset Pricing Models
It is very important to note that Fama and French (1993) modeled the time-series
of returns on stocks and bonds, where Fama and French (1992) modeled the crosssectional (across-stock) differences in returns on stocks—which is why the stock
market return is included in the above group, but not in the 1992 paper’s factors.
In essence, the 1992 paper says that we can assume that beta=1 for all stocks (without a huge amount of error), and, therefore, beta only affects stock returns over
time. There is no difference in different stock returns at the same period of time,
since they all have assumed betas of one, according to Fama and French (1992).
0
Fama and French (1993) also find that stock and bond returns are linked together
through the correlation of the stock market return with the return on the two bond
factors. Interestingly, a large body of other research since then, including Kandel
and Stambaugh (1996) has found that broad macroeconomic factors, including
the two bond factors noted above, help to forecast the stock market return.
Gruber, Elton, Agrawal, and Mann (2001) find that the three stock risk factors
above (1–3) are also useful in modeling corporate bonds—in addition to
exposure to potential default and taxation of bond income. Finally, Cornell
and Green (1991) find that stock market returns are even more important than
government bond market yields in modeling high-yield (junk) bonds.
1
The above research on bond markets suggest that a five-factor model should be
used to model bonds:
Rt = α + β · RMRFt + s · SMBt + h · HMLt + m · TERMt + d · DEFAULTt + #t .
(1.20)
Note that there is no momentum factor for bond markets, although some recent
papers have also challenged this.
1.8 CHAPTER-END PROBLEMS
1. Download the monthly returns for Exxon-Mobil (XOM) during 2009 and
2010 from CRSP, Yahoo Finance, or another source. Also, download the
30-day Treasury Bill return and the monthly factor returns for RMRF, SMB,
HML, and UMD from Ken French’s website, http://mba.tuck.dartmouth.
edu/pages/faculty/ken.french/.
A. Using Excel or a statistics package, run a single-factor linear regression
(ordinary least squares) for XOM (the y-variable is the excess return of
XOM, which is the XOM return minus T-Bill return, while the x-variable
is the monthly return on RMRF). How does your regression output compare with that of CVX shown in this chapter—what are the differences in
the two stocks according to this output?
B. Repeat, using a two-factor model that includes RMRF and SMB. How
does your regression output compare with that of CVX shown in this
chapter—what are the differences in the two stocks according to this
output?
References1.8 Chapter-End Problems
C. Repeat, using the Carhart four-factor model. How does your regression
output compare with that of CVX shown in this chapter—what are the
differences in the two stocks according to this output?
2. Download monthly returns for Apple (AAPL) during 2009 and 2010, and
run a single-factor regression on the S&P 500 as the “market factor”. What
are the resulting alpha and beta?
3. Using the AAPL data from problem #2, run a four-factor model. What are
the coefficients on each factor, and what do they tell you about Apple’s
stock?
4. Starting with the model of Equation (1.2), derive the risk model shown by
Equation (1.6).
5. Starting with the model of Equation (1.10), derive the risk model shown by
Equation (1.13).
6. Describe the empirical tests that find violations of the CAPM in stock
returns.
7. Describe the empirical approach that Fama and French (1992) used to find
that “beta is dead”.
8. Discuss each of the assumptions of the CAPM. For each assumption, provide
some brief evidence from financial markets that indicates that the assumption may not be correct.
9. Suppose that an institution holds Portfolio K. The institution wants to use
Portfolio L to hedge its exposure to inflation. Specifically, it wants to combine K and L to reduce its inflation exposure to zero. Portfolios K and L are
well diversified, so the manager can ignore the risk of individual assets and
assume that the only source of uncertainty in the portfolio is the surprises
in the two factors. The returns to the two portfolios are:
RK = 0.12 + 0.5FINFL + 1.0FGDP
RL = 0.11 + 1.5FINFL + 2.5FGDP
Calculate the weights that a manager should have on K and L to achieve this
goal.
10. Portfolio A has an expected return of 10.25 percent and a factor sensitivity of
0.5. Portfolio B has an expected return of 16.2 percent and a factor sensitivity of 1.2. The risk-free rate is 6 percent, and there is one factor. Determine
the factor’s price of risk (see Tables 1.4 and 1.5).
REFERENCES
Bernstein, Peter, 1996. Against the Gods: The Remarkable Story of Risk. John Wiley & Sons, Inc.
Campbell, John, Lo, Andrew, MacKinlay, Craig, 1997. The Econometrics of Financial Markets and
Craig MacKinlay. Princeton University Press.
Cochrane, John, 2001. Asset Pricing. Princeton University Press.
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