9
than its “fair” return given its risk. Second,
the model helps us to make an educated
guess as to the expected return on assets that
have not yet been traded in the marketplace.
For example, how do we price an initial public
offering of stock? How will a major new
investment project affect the return investors
require on a company’s stock? Although
the CAPM does not fully withstand empirical
tests, it is widely used because of the insight
it offers and because its accuracy is deemed
acceptable for important applications.
THE CAPITAL ASSET PRICING
MODEL
PARTIII
THE CAPITAL ASSET pricing model, almost
always referred to as the CAPM, is a centerpiece
of modern financial economics. The
model gives us a precise prediction of the
relationship that we should observe between
the risk of an asset and its expected return.
This relationship serves two vital functions.
First, it provides a benchmark rate of return
for evaluating possible investments. For
example, if we are analyzing securities, we
might be interested in whether the expected
return we forecast for a stock is more or less
The capital asset pricing model is a set of predictions concerning equilibrium expected
returns on risky assets. Harry Markowitz laid down the foundation of modern portfolio
management in 1952. The CAPM was developed 12 years later in articles by William
Sharpe,1
John Lintner,2
and Jan Mossin.3
The time for this gestation indicates that the leap
from Markowitz’s portfolio selection model to the CAPM is not trivial.
We will approach the CAPM by posing the question “what if,” where the “if” part refers
to a simplified world. Positing an admittedly unrealistic world allows a relatively easy leap
to the “then” part. Once we accomplish this, we can add complexity to the hypothesized
1
William Sharpe, “Capital Asset Prices: A Theory of Market Equilibrium,” Journal of Finance, September 1964.
2
John Lintner, “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and
Capital Budgets,” Review of Economics and Statistics, February 1965.
3
Jan Mossin, “Equilibrium in a Capital Asset Market,” Econometrica, October 1966.
9.1 THE CAPITAL ASSET PRICING MODEL
C H A P T E R N I N E
280 PART III Equilibrium in Capital Markets
environment one step at a time and see how the conclusions must be amended. This process
allows us to derive a reasonably realistic and comprehensible model.
We summarize the simplifying assumptions that lead to the basic version of the CAPM
in the following list. The thrust of these assumptions is that we try to ensure that individuals
are as alike as possible, with the notable exceptions of initial wealth and risk aversion.
We will see that conformity of investor behavior vastly simplifies our analysis.
1. There are many investors, each with an endowment (wealth) that is small compared
to the total endowment of all investors. Investors are price-takers, in that they act as
though security prices are unaffected by their own trades. This is the usual perfect
competition assumption of microeconomics.
2. All investors plan for one identical holding period. This behavior is myopic (shortsighted)
in that it ignores everything that might happen after the end of the singleperiod
horizon. Myopic behavior is, in general, suboptimal.
3. Investments are limited to a universe of publicly traded financial assets, such
as stocks and bonds, and to risk-free borrowing or lending arrangements. This
assumption rules out investment in nontraded assets such as education (human
capital), private enterprises, and governmentally funded assets such as town halls
and international airports. It is assumed also that investors may borrow or lend any
amount at a fixed, risk-free rate.
4. Investors pay no taxes on returns and no transaction costs (commissions and service
charges) on trades in securities. In reality, of course, we know that investors are
in different tax brackets and that this may govern the type of assets in which they
invest. For example, tax implications may differ depending on whether the income
is from interest, dividends, or capital gains. Furthermore, actual trading is costly,
and commissions and fees depend on the size of the trade and the good standing of
the individual investor.
5. All investors are rational mean-variance optimizers, meaning that they all use the
Markowitz portfolio selection model.
6. All investors analyze securities in the same way and share the same economic
view of the world. The result is identical estimates of the probability distribution
of future cash flows from investing in the available securities; that is, for any set
of security prices, they all derive the same input list to feed into the Markowitz
model. Given a set of security prices and the risk-free interest rate, all investors use
the same expected returns and covariance matrix of security returns to generate the
efficient frontier and the unique optimal risky portfolio. This assumption is often
referred to as homogeneous expectations or beliefs.
These assumptions represent the “if” of our “what if” analysis. Obviously, they ignore
many real-world complexities. With these assumptions, however, we can gain some powerful
insights into the nature of equilibrium in security markets.
We can summarize the equilibrium that will prevail in this hypothetical world of
securities and investors briefly. The rest of the chapter explains and elaborates on these
implications.
1. All investors will choose to hold a portfolio of risky assets in proportions that
duplicate representation of the assets in the market portfolio (M), which includes
all traded assets. For simplicity, we generally refer to all risky assets as stocks. The
proportion of each stock in the market portfolio equals the market value of the stock
CHAPTER 9 The Capital Asset Pricing Model 281
(price per share multiplied by the number of shares outstanding) divided by the
total market value of all stocks.
2. Not only will the market portfolio be on the efficient frontier, but it also will be the
tangency portfolio to the optimal capital allocation line (CAL) derived by each and
every investor. As a result, the capital market line (CML), the line from the riskfree
rate through the market portfolio, M, is also the best attainable capital allocation
line. All investors hold M as their optimal risky portfolio, differing only in the
amount invested in it versus in the risk-free asset.
3. The risk premium on the market portfolio will be proportional to its risk and the
degree of risk aversion of the representative investor. Mathematically,
E r r AM f M( ) Ϫ ϭ ␴2
where ␴M
2
is the variance of the market portfolio and A
Ϫ
is the average degree of
risk aversion across investors. Note that because M is the optimal portfolio, which is
efficiently diversified across all stocks, ␴M
2
is the systematic risk of this universe.
4. The risk premium on individual assets will be proportional to the risk premium on
the market portfolio, M, and the beta coefficient of the security relative to the market
portfolio. Beta measures the extent to which returns on the stock and the market
move together. Formally, beta is defined as
␤ ϭ
␴
i
i M
M
r rCov( , )
2
and the risk premium on individual securities is
E r r
r r
E r r E r ri f
i M
M
M f i M( )
( , )
[ ( ) ] [ ( )Ϫ ϭ
␴
Ϫ ϭ ␤ Ϫ
Cov
2 ff ]
Why Do All Investors Hold the Market Portfolio?
What is the market portfolio? When we sum over, or aggregate, the portfolios of all individual
investors, lending and borrowing will cancel out (because each lender has a corresponding
borrower), and the value of the aggregate risky portfolio will equal the entire
wealth of the economy. This is the market portfolio, M. The proportion of each stock in this
portfolio equals the market value of the stock (price per share times number of shares outstanding)
divided by the sum of the market values of all stocks.4
The CAPM implies that
as individuals attempt to optimize their personal portfolios, they each arrive at the same
portfolio, with weights on each asset equal to those of the market portfolio.
Given the assumptions of the previous section, it is easy to see that all investors will
desire to hold identical risky portfolios. If all investors use identical Markowitz analysis
(Assumption 5) applied to the same universe of securities (Assumption 3) for the same
time horizon (Assumption 2) and use the same input list (Assumption 6), they all must
arrive at the same composition of the optimal risky portfolio, the portfolio on the efficient
frontier identified by the tangency line from T-bills to that frontier, as in Figure 9.1. This
4
As noted previously, we use the term “stock” for convenience; the market portfolio properly includes all assets
in the economy.
282 PART III Equilibrium in Capital Markets
implies that if the weight of GE stock, for example,
in each common risky portfolio is 1%, then GE also
will comprise 1% of the market portfolio. The same
principle applies to the proportion of any stock in each
investor’s risky portfolio. As a result, the optimal risky
portfolio of all investors is simply a share of the market
portfolio in Figure 9.1.
Now suppose that the optimal portfolio of our
investors does not include the stock of some company,
such as Delta Airlines. When all investors avoid Delta
stock, the demand is zero, and Delta’s price takes a
free fall. As Delta stock gets progressively cheaper,
it becomes ever more attractive and other stocks look
relatively less attractive. Ultimately, Delta reaches a
price where it is attractive enough to include in the
optimal stock portfolio.
Such a price adjustment process guarantees that
all stocks will be included in the optimal portfolio. It
shows that all assets have to be included in the market
portfolio. The only issue is the price at which investors will be willing to include a stock in
their optimal risky portfolio.
This may seem a roundabout way to derive a simple result: If all investors hold an
identical risky portfolio, this portfolio has to be M, the market portfolio. Our intention,
however, is to demonstrate a connection between this result and its underpinnings, the
equilibrating process that is fundamental to security market operation.
The Passive Strategy Is Efficient
In Chapter 6 we defined the CML (capital market line) as the CAL (capital allocation line)
that is constructed from a money market account (or T-bills) and the market portfolio. Perhaps
now you can fully appreciate why the CML is an interesting CAL. In the simple world
of the CAPM, M is the optimal tangency portfolio on the efficient frontier, as shown in
Figure 9.1.
In this scenario, the market portfolio held by all investors is based on the common input
list, thereby incorporating all relevant information about the universe of securities. This
means that investors can skip the trouble of doing security analysis and obtain an efficient
portfolio simply by holding the market portfolio. (Of course, if everyone were to follow
this strategy, no one would perform security analysis and this result would no longer hold.
We discuss this issue in greater depth in Chapter 11 on market efficiency.)
Thus the passive strategy of investing in a market index portfolio is efficient. For this
reason, we sometimes call this result a mutual fund theorem. The mutual fund theorem
is another incarnation of the separation property discussed in Chapter 7. Assuming that
all investors choose to hold a market index mutual fund, we can separate portfolio selection
into two components—a technical problem, creation of mutual funds by professional
managers—and a personal problem that depends on an investor’s risk aversion, allocation
of the complete portfolio between the mutual fund and risk-free assets.
In reality, different investment managers do create risky portfolios that differ from the
market index. We attribute this in part to the use of different input lists in the formation
of the optimal risky portfolio. Nevertheless, the practical significance of the mutual fund
theorem is that a passive investor may view the market index as a reasonable first approximation
to an efficient risky portfolio.
F I G U R E 9.1 The efficient frontier and the
capital market line
E(r)
E(rM)
M
CML
rf
σ
σM
283
The nearby box contains a parable illustrating the argument for indexing. If the passive
strategy is efficient, then attempts to beat it simply generate trading and research costs with
no offsetting benefit, and ultimately inferior results.
CONCEPT
CHECK
1
If there are only a few investors who perform security analysis, and all others hold the market
portfolio, M, would the CML still be the efficient CAL for investors who do not engage in security
analysis? Why or why not?
The Risk Premium of the Market Portfolio
In Chapter 6 we discussed how individual investors go about deciding how much to invest
in the risky portfolio. Returning now to the decision of how much to invest in portfolio M
versus in the risk-free asset, what can we deduce about the equilibrium risk premium of
portfolio M?
THE PARABLE OF THE MONEY MANAGERS
WORDSFROMTHESTREET
Some years ago, in a land called Indicia, revolution led
to the overthrow of a socialist regime and the restoration
of a system of private property. Former government
enterprises were reformed as corporations, which
then issued stocks and bonds. These securities were
given to a central agency, which offered them for sale
to individuals, pension funds, and the like (all armed
with newly printed money).
Almost immediately a group of money managers
came forth to assist these investors. Recalling the
words of a venerated elder, uttered before the previous
revolution (“Invest in Corporate Indicia”), they invited
clients to give them money, with which they would buy
a cross-section of all the newly issued securities. Investors
considered this a reasonable idea, and soon everyone
held a piece of Corporate Indicia.
Before long the money managers became bored
because there was little for them to do. Soon they fell
into the habit of gathering at a beachfront casino where
they passed the time playing roulette, craps, and similar
games, for low stakes, with their own money.
After a while, the owner of the casino suggested a
new idea. He would furnish an impressive set of rooms
which would be designated the Money Managers’
Club. There the members could place bets with one
another about the fortunes of various corporations,
industries, the level of the Gross National Product, foreign
trade, etc. To make the betting more exciting, the
casino owner suggested that the managers use their
clients’ money for this purpose.
The offer was immediately accepted, and soon
the money managers were betting eagerly with one
another. At the end of each week, some found that they
had won money for their clients, while others found
that they had lost. But the losses always exceeded the
gains, for a certain amount was deducted from each
bet to cover the costs of the elegant surroundings in
which the gambling took place.
Before long a group of professors from Indicia U.
suggested that investors were not well served by the
activities being conducted at the Money Managers’
Club. “Why pay people to gamble with your money?
Why not just hold your own piece of Corporate Indicia?”
they said.
This argument seemed sensible to some of the
investors, and they raised the issue with their money
managers. A few capitulated, announcing that they
would henceforth stay away from the casino and use
their clients’ money only to buy proportionate shares
of all the stocks and bonds issued by corporations.
The converts, who became known as managers of
Indicia funds, were initially shunned by those who continued
to frequent the Money Managers’ Club, but in
time, grudging acceptance replaced outright hostility.
The wave of puritan reform some had predicted failed
to materialize, and gambling remained legal. Many
managers continued to make their daily pilgrimage
to the casino. But they exercised more restraint than
before, placed smaller bets, and generally behaved in
a manner consonant with their responsibilities. Even the
members of the Lawyers’ Club found it difficult to object
to the small amount of gambling that still went on.
And everyone but the casino owner lived happily
ever after.
Source: William F. Sharpe, “The Parable of the Money Managers,”
The Financial Analysts’ Journal 32 (July/August 1976), p. 4. Copyright
1976, CFA Institute. Reproduced from The Financial Analysts’ Journal
with permission from the CFA Institute. All rights reserved.
284 PART III Equilibrium in Capital Markets
We asserted earlier that the equilibrium risk premium on the market portfolio, E(rM) Ϫ rf,
will be proportional to the average degree of risk aversion of the investor population and
the risk of the market portfolio, ␴M
2
. Now we can explain this result.
Recall that each individual investor chooses a proportion y, allocated to the optimal
portfolio M, such that
y
E r r
A
M f
M
ϭ
Ϫ
␴
( )
2
(9.1)
In the simplified CAPM economy, risk-free investments involve borrowing and lending
among investors. Any borrowing position must be offset by the lending position of the
creditor. This means that net borrowing and lending across all investors must be zero, and
in consequence, substituting the representative investor’s risk aversion, A
Ϫ
, for A, the average
position in the risky portfolio is 100%, or yϪ ϭ 1. Setting y ϭ 1 in Equation 9.1 and
rearranging, we find that the risk premium on the market portfolio is related to its variance
by the average degree of risk aversion:
E r r AM f M( ) Ϫ ϭ ␴2 (9.2)
CONCEPT
CHECK
2
Data from the last eight decades (see Table 5.3) for the S&P 500 index yield the following
statistics: average excess return, 8.4%; standard deviation, 20.3%.
a. To the extent that these averages approximated investor expectations for the period, what
must have been the average coefficient of risk aversion?
b. If the coefficient of risk aversion were actually 3.5, what risk premium would have been
consistent with the market’s historical standard deviation?
Expected Returns on Individual Securities
The CAPM is built on the insight that the appropriate risk premium on an asset will
be determined by its contribution to the risk of investors’ overall portfolios. Portfolio
risk is what matters to investors and is what governs the risk premiums they
demand.
Remember that all investors use the same input list, that is, the same estimates of
expected returns, variances, and covariances. We saw in Chapter 7 that these covariances
can be arranged in a covariance matrix, so that the entry in the fifth row and third column,
for example, would be the covariance between the rates of return on the fifth and third
securities. Each diagonal entry of the matrix is the covariance of one security’s return with
itself, which is simply the variance of that security.
Suppose, for example, that we want to gauge the portfolio risk of GE stock. We measure
the contribution to the risk of the overall portfolio from holding GE stock by its
covariance with the market portfolio. To see why this is so, let us look again at the way
the variance of the market portfolio is calculated. To calculate the variance of the market
portfolio, we use the bordered covariance matrix with the market portfolio weights, as
discussed in Chapter 7. We highlight GE in this depiction of the n stocks in the market
portfolio.
CHAPTER 9 The Capital Asset Pricing Model 285
Portfolio
Weights w1 w2 . . . wGE . . . wn
w1 Cov(r1, r1) Cov(r1, r2) . . . Cov(r1, rGE) . . . Cov(r1, rn)
w2 Cov(r2, r1) Cov(r2, r2) . . . Cov(r2, rGE) . . . Cov(r2, rn)
…
…
…
…
…
w GE Cov(rGE, r1) Cov(rGE, r2) . . . Cov(rGE, rGE) . . . Cov(rGE, rn)
…
…
…
…
…
wn Cov(rn, r1) Cov(rn, r2) . . . Cov(rn, rGE) . . . Cov(rn, rn)
Recall that we calculate the variance of the portfolio by summing over all the elements
of the covariance matrix, first multiplying each element by the portfolio weights from the
row and the column. The contribution of one stock to portfolio variance therefore can be
expressed as the sum of all the covariance terms in the column corresponding to the stock,
where each covariance is first multiplied by both the stock’s weight from its row and the
weight from its column.5
For example, the contribution of GE’s stock to the variance of the market portfolio is
w w r r w r r wGE GE GE GECov Cov Cov[ ( , ) ( , ) . . . (1 1 2 2ϩ ϩ ϩ rr r
w r rn n
GE GE
GECov
, ) . . .
( , )]
ϩ
ϩ (9.3)
Equation 9.3 provides a clue about the respective roles of variance and covariance in
determining asset risk. When there are many stocks in the economy, there will be many
more covariance terms than variance terms. Consequently, the covariance of a particular
stock with all other stocks will dominate that stock’s contribution to total portfolio risk.
Notice that the sum inside the square brackets in Equation 9.3 is the covariance of GE with
the market portfolio. In other words, we can best measure the stock’s contribution to the
risk of the market portfolio by its covariance with that portfolio:
GE s contribution to variance CovGE GE’ ϭ w r rM( , ))
This should not surprise us. For example, if the covariance between GE and the rest
of the market is negative, then GE makes a “negative contribution” to portfolio risk: By
providing returns that move inversely with the rest of the market, GE stabilizes the return
on the overall portfolio. If the covariance is positive, GE makes a positive contribution to
overall portfolio risk because its returns reinforce swings in the rest of the portfolio.
To demonstrate this more rigorously, note that the rate of return on the market portfolio
may be written as
r w rM k k
k
n
ϭ
=
∑
1
5
An alternative approach would be to measure GE’s contribution to market variance as the sum of the elements
in the row and the column corresponding to GE. In this case, GE’s contribution would be twice the sum in Equation
9.3. The approach that we take in the text allocates contributions to portfolio risk among securities in a
convenient manner in that the sum of the contributions of each stock equals the total portfolio variance, whereas
the alternative measure of contribution would sum to twice the portfolio variance. This results from a type of
double-counting, because adding both the rows and the columns for each stock would result in each entry in the
matrix being added twice.
286 PART III Equilibrium in Capital Markets
Therefore, the covariance of the return on GE with the market portfolio is
Cov Cov CovGE GE( , ) , (r r r w r w rM k k
k
n
k kϭ ϭ
=
∑




1
,, )r
k
n
GE
=
∑
1
(9.4)
Notice that the last term of Equation 9.4 is precisely the same as the term in brackets
in Equation 9.3. Therefore, Equation 9.3, which is the contribution of GE to the variance
of the market portfolio, may be simplified to wGE Cov(rGE, rM). We also observe
that the contribution of our holding of GE to the risk premium of the market portfolio is
wGE [E(rGE) Ϫ rf].
Therefore, the reward-to-risk ratio for investments in GE can be expressed as
GE s contribution to risk premium
GE s contr
’
’ iibution to variance Cov
GE GE
GE
ϭ
Ϫw E r r
w r
f[ ( ) ]
( GGE
GE
GECov, )
( )
( , )r
E r r
r rM
f
M
ϭ
Ϫ
The market portfolio is the tangency (efficient mean-variance) portfolio. The reward-torisk
ratio for investment in the market portfolio is
Market risk premium
Market variance
ϭ
ϪE r rM f( )
␴␴M
2
(9.5)
The ratio in Equation 9.5 is often called the market price of risk6
because it quantifies the
extra return that investors demand to bear portfolio risk. Notice that for components of the
efficient portfolio, such as shares of GE, we measure risk as the contribution to portfolio
variance (which depends on its covariance with the market). In contrast, for the efficient
portfolio itself, its variance is the appropriate measure of risk.
A basic principle of equilibrium is that all investments should offer the same rewardto-risk
ratio. If the ratio were better for one investment than another, investors would rearrange
their portfolios, tilting toward the alternative with the better trade-off and shying
away from the other. Such activity would impart pressure on security prices until the ratios
were equalized. Therefore we conclude that the reward-to-risk ratios of GE and the market
portfolio should be equal:
E r r
r r
E r rf
M
M f
M
( )
( , )
( )GE
GECov
Ϫ
ϭ
Ϫ
␴2
(9.6)
To determine the fair risk premium of GE stock, we rearrange Equation 9.6 slightly to
obtain
E r r
r r
E r rf
M
M
M f( )
( , )
[ ( ) ]GE
GE
2
Cov
Ϫ ϭ
␴
Ϫ (9.7)
6
We open ourselves to ambiguity in using this term, because the market portfolio’s reward-to-volatility ratio
E r rM f
M
( ) Ϫ
␴
sometimes is referred to as the market price of risk. Note that because the appropriate risk measure of GE is its
covariance with the market portfolio (its contribution to the variance of the market portfolio), this risk is measured
in percent squared. Accordingly, the price of this risk, [E(rM) Ϫ rf]/␴2
, is defined as the percentage expected
return per percent square of variance.
CHAPTER 9 The Capital Asset Pricing Model 287
The ratio Cov /GE( , )r rM M␴2
measures the contribution of GE stock to the variance of the
market portfolio as a fraction of the total variance of the market portfolio. The ratio is
called beta and is denoted by ␤. Using this measure, we can restate Equation 9.7 as
E r r E r rf M f( ) [ ( ) ]GE GEϭ ϩ␤ Ϫ (9.8)
This expected return–beta relationship is the most familiar expression of the CAPM to
practitioners. We will have a lot more to say about the expected return–beta relationship
shortly.
We see now why the assumptions that made individuals act similarly are so useful. If
everyone holds an identical risky portfolio, then everyone will find that the beta of each
asset with the market portfolio equals the asset’s beta with his or her own risky portfolio.
Hence everyone will agree on the appropriate risk premium for each asset.
Does the fact that few real-life investors actually hold the market portfolio imply that the
CAPM is of no practical importance? Not necessarily. Recall from Chapter 7 that reasonably
well-diversified portfolios shed firm-specific risk and are left with mostly systematic
or market risk. Even if one does not hold the precise market portfolio, a well-diversified
portfolio will be so very highly correlated with the market that a stock’s beta relative to the
market will still be a useful risk measure.
In fact, several authors have shown that modified versions of the CAPM will hold true
even if we consider differences among individuals leading them to hold different portfolios.
For example, Brennan7
examined the impact of differences in investors’ personal tax
rates on market equilibrium, and Mayers8
looked at the impact of nontraded assets such as
human capital (earning power). Both found that although the market portfolio is no longer
each investor’s optimal risky portfolio, the expected return–beta relationship should still
hold in a somewhat modified form.
If the expected return–beta relationship holds for any individual asset, it must hold for
any combination of assets. Suppose that some portfolio P has weight wk for stock k, where
k takes on values 1, . . . , n. Writing out the CAPM Equation 9.8 for each stock, and multiplying
each equation by the weight of the stock in the portfolio, we obtain these equations,
one for each stock:
w E r w r w E r r
w E r w r
f M f
f
1 1 1 1 1
2 2 2
( ) [ ( ) ]
( )
ϭ ϩ ␤ Ϫ
ϩ ϭ ϩϩ ␤ Ϫ
ϩ ϭ
ϩ ϭ ϩ ␤
w E r r
w E r w r w E r
M f
n n n f n n M
2 2[ ( ) ]
( ) [ (
⋯ ⋯
)) ]
( ) [ ( ) ]
Ϫ
ϭ ϩ␤ Ϫ
r
E r r E r r
f
P f P M f
Summing each column shows that the CAPM holds for the overall portfolio because
E r w E rP
k
k k( ) ( )ϭ∑ is the expected return on the portfolio, and β βP
k
k kwϭ ∑ is the portfolio
beta. Incidentally, this result has to be true for the market portfolio itself,
E r r E r rM f M M f( ) [ ( ) ]ϭ ϩ␤ Ϫ
7
Michael J. Brennan, “Taxes, Market Valuation, and Corporate Finance Policy,” National Tax Journal, December
1973.
8
David Mayers, “Nonmarketable Assets and Capital Market Equilibrium under Uncertainty,” in Studies in the
Theory of Capital Markets, ed. M. C. Jensen (New York: Praeger, 1972). We will look at this model more closely
later in the chapter.
288 PART III Equilibrium in Capital Markets
Indeed, this is a tautology because ␤M ϭ 1, as we can verify by noting that
␤ ϭ
␴
ϭ
␴
␴
M
M M
M
M
M
r rCov( , )
2
2
2
This also establishes 1 as the weighted-average value of beta across all assets. If the market
beta is 1, and the market is a portfolio of all assets in the economy, the weighted-average
beta of all assets must be 1. Hence betas greater than 1 are considered aggressive in that
investment in high-beta stocks entails above-average sensitivity to market swings. Betas
below 1 can be described as defensive.
A word of caution: We are all accustomed to hearing that well-managed firms will provide
high rates of return. We agree this is true if one measures the firm’s return on investments
in plant and equipment. The CAPM, however, predicts returns on investments in the
securities of the firm.
Let us say that everyone knows a firm is well run. Its stock price will therefore be bid
up, and consequently returns to stockholders who buy at those high prices will not be
excessive. Security prices, in other words, already reflect public information about a firm’s
prospects; therefore only the risk of the company (as measured by beta in the context of
the CAPM) should affect expected returns. In an efficient market investors receive high
expected returns only if they are willing to bear risk.
Of course, investors do not directly observe or determine expected returns on securities.
Rather, they observe security prices and bid those prices up or down. Expected rates of
return are determined by the prices investors must pay compared to the cash flows those
investments might garner.
CONCEPT
CHECK
3
Suppose that the risk premium on the market portfolio is estimated at 8% with a standard
deviation of 22%. What is the risk premium on a portfolio invested 25% in GM and 75% in Ford,
if they have betas of 1.10 and 1.25, respectively?
The Security Market Line
We can view the expected return–beta relationship as a reward–risk equation. The beta of
a security is the appropriate measure of its risk because beta is proportional to the risk that
the security contributes to the optimal risky portfolio.
Risk-averse investors measure the risk of the optimal risky portfolio by its variance. In this
world we would expect the reward, or the risk premium on individual assets, to depend on the
contribution of the individual asset to the risk of the portfolio. The beta of a stock measures
its contribution to the variance of the market portfolio. Hence we expect, for any asset or
portfolio, the required risk premium to be a function of beta. The CAPM confirms this intuition,
stating further that the security’s risk premium is directly proportional to both the beta
and the risk premium of the market portfolio; that is, the risk premium equals ␤[E(rM) Ϫ rf].
The expected return–beta relationship can be portrayed graphically as the security
market line (SML) in Figure 9.2. Because the market’s beta is 1, the slope is the risk premium
of the market portfolio. At the point on the horizontal axis where ␤ ϭ 1, we can read
off the vertical axis the expected return on the market portfolio.
It is useful to compare the security market line to the capital market line. The CML
graphs the risk premiums of efficient portfolios (i.e., portfolios composed of the market and
the risk-free asset) as a function of portfolio standard deviation. This is appropriate because
standard deviation is a valid measure of risk for efficiently diversified portfolios that are
candidates for an investor’s overall portfolio. The SML, in contrast, graphs individual asset
CHAPTER 9 The Capital Asset Pricing Model 289
risk premiums as a function of asset risk. The relevant
measure of risk for individual assets held as parts of welldiversified
portfolios is not the asset’s standard deviation
or variance; it is, instead, the contribution of the
asset to the portfolio variance, which we measure by
the asset’s beta. The SML is valid for both efficient
portfolios and individual assets.
The security market line provides a benchmark for the
evaluation of investment performance. Given the risk of
an investment, as measured by its beta, the SML provides
the required rate of return necessary to compensate investors
for both risk as well as the time value of money.
Because the security market line is the graphic representation
of the expected return–beta relationship,
“fairly priced” assets plot exactly on the SML; that is,
their expected returns are commensurate with their risk.
Given the assumptions we made at the start of this section,
all securities must lie on the SML in market equilibrium.
Nevertheless, we see here how the CAPM may be
of use in the money-management industry. Suppose that
the SML relation is used as a benchmark to assess the
fair expected return on a risky asset. Then security analysis
is performed to calculate the return actually expected.
(Notice that we depart here from the
simple CAPM world in that some investors
now apply their own unique analysis
to derive an “input list” that may differ
from their competitors’.) If a stock is perceived
to be a good buy, or underpriced, it
will provide an expected return in excess
of the fair return stipulated by the SML.
Underpriced stocks therefore plot above
the SML: Given their betas, their expected
returns are greater than dictated by the
CAPM. Overpriced stocks plot below the
SML.
The difference between the fair and
actually expected rates of return on a
stock is called the stock’s alpha, denoted
by ␣. For example, if the market return
is expected to be 14%, a stock has a beta
of 1.2, and the T-bill rate is 6%, the SML
would predict an expected return on the
stock of 6 ϩ 1.2(14 Ϫ 6) ϭ 15.6%. If
one believed the stock would provide
an expected return of 17%, the implied
alpha would be 1.4% (see Figure 9.3).
One might say that security analysis
(which we treat in Part Five) is about uncovering
securities with nonzero alphas. This
analysis suggests that the starting point of
F I G U R E 9.2 The security market line
E(r)
E(rM)
E(rM) – rf = Slope of SML
SML
rf
βM =1.0
1
β
F I G U R E 9.3 The SML and a positive-alpha stock
E(r) (%)
SML
6
14
15.6
1.21.0
17
Stock
M
α
β
290
portfolio management can be a passive market-index portfolio. The portfolio manager will
then increase the weights of securities with positive alphas and decrease the weights of securities
with negative alphas. We showed one strategy for adjusting the portfolio weights in
such a manner in Chapter 8.
The CAPM is also useful in capital budgeting decisions. For a firm considering a new
project, the CAPM can provide the required rate of return that the project needs to yield,
based on its beta, to be acceptable to investors. Managers can use the CAPM to obtain this
cutoff internal rate of return (IRR), or “hurdle rate” for the project.
The nearby box describes how the CAPM can be used in capital budgeting. It also
discusses some empirical anomalies concerning the model, which we address in detail in
Chapters 11–13. The article asks whether the CAPM is useful for capital budgeting in light
of these shortcomings; it concludes that even given the anomalies cited, the model still can
be useful to managers who wish to increase the fundamental value of their firms.
TALES FROM THE FAR SIDE
Financial markets’ evaluation of risk determines the
way firms invest. What if the markets are wrong?
Investors are rarely praised for their good sense. But
for the past two decades a growing number of firms
have based their decisions on a model which assumes
that people are perfectly rational. If they are irrational,
are businesses making the wrong choices?
The model, known as the “capital-asset pricing
model,” or CAPM, has come to dominate modern
finance. Almost any manager who wants to defend a
project—be it a brand, a factory or a corporate merger
—must justify his decision partly based on the CAPM.
The reason is that the model tells a firm how to calculate
the return that its investors demand. If shareholders
are to benefit, the returns from any project must
clear this “hurdle rate.”
Although the CAPM is complicated, it can be
reduced to five simple ideas:
• Investors can eliminate some risks—such as the risk
that workers will strike, or that a firm’s boss will quit—
by diversifying across many regions and sectors.
• Some risks, such as that of a global recession,
cannot be eliminated through diversification. So
even a basket of all of the stocks in a stock market
will still be risky.
• People must be rewarded for investing in such a
risky basket by earning returns above those that
they can get on safer assets, such as Treasury bills.
• The rewards on a specific investment depend only
on the extent to which it affects the market basket’s
risk.
• Conveniently, that contribution to the market
basket’s risk can be captured by a single measure—
dubbed “beta”—which expresses the relationship
between the investment’s risk and the market’s.
Beta is what makes the CAPM so powerful. Although
an investment may face many risks, diversified
investors should care only about those that are related
to the market basket. Beta not only tells managers how
to measure those risks, but it also allows them to translate
them directly into a hurdle rate. If the future profits
from a project will not exceed that rate, it is not worth
shareholders’ money.
The diagram shows how the CAPM works. Safe
investments, such as Treasury bills, have a beta of zero.
Riskier investments should earn a premium over the
risk-free rate which increases with beta. Those whose
risks roughly match the market’s have a beta of one, by
definition, and should earn the market return.
So suppose that a firm is considering two projects,
A and B. Project A has a beta of ½: when the market
rises or falls by 10%, its returns tend to rise or
fall by 5%. So its risk premium is only half that of the
market. Project B’s risk premium is twice that of the
WORDSFROMTHESTREET
Return
Market
Return
Risk-Free
Return
A
B
1 2
β
rB
rA
Beta Power
1 2
291
market, so it must earn a higher return to justify the
expenditure.
NEVER KNOWINGLY UNDERPRICED
But there is one small problem with the CAPM: Financial
economists have found that beta is not much use
for explaining rates of return on firms’ shares. Worse,
there appears to be another measure which explains
these returns quite well.
That measure is the ratio of a firm’s book value (the
value of its assets at the time they entered the balance
sheet) to its market value. Several studies have found
that, on average, companies that have high book-tomarket
ratios tend to earn excess returns over long
periods, even after adjusting for the risks that are associated
with beta.
The discovery of this book-to-market effect has
sparked a fierce debate among financial economists.
All of them agree that some risks ought to carry greater
rewards. But they are now deeply divided over how risk
should be measured. Some argue that since investors are
rational, the book-to-market effect must be capturing an
extra risk factor. They conclude, therefore, that managers
should incorporate the book-to-market effect into their
hurdle rates. They have labeled this alternative hurdle
rate the “new estimator of expected return,” or NEER.
Other financial economists, however, dispute this
approach. Since there is no obvious extra risk associated
with a high book-to-market ratio, they say,
investors must be mistaken. Put simply, they are underpricing
high book-to-market stocks, causing them to
earn abnormally high returns. If managers of such firms
try to exceed those inflated hurdle rates, they will forgo
many profitable investments. With economists now at
odds, what is a conscientious manager to do?
Jeremy Stein, an economist at the Massachusetts
Institute of Technology’s business school, offers a
paradoxical answer.* If investors are rational, then
beta cannot be the only measure of risk, so managers
should stop using it. Conversely, if investors are irrational,
then beta is still the right measure in many cases.
Mr. Stein argues that if beta captures an asset’s fundamental
risk—that is, its contribution to the market
basket’s risk—then it will often make sense for managers
to pay attention to it, even if investors are somehow
failing to.
Often, but not always. At the heart of Mr. Stein’s
argument lies a crucial distinction—that between
(a) boosting a firm’s long-term value and (b) trying
to raise its share price. If investors are rational, these
are the same thing: any decision that raises long-term
value will instantly increase the share price as well. But
if investors are making predictable mistakes, a manager
must choose.
For instance, if he wants to increase today’s share
price—perhaps because he wants to sell his shares, or
to fend off a takeover attempt—he must usually stick
with the NEER approach, accommodating investors’
misperceptions. But if he is interested in long-term
value, he should usually continue to use beta. Showing
a flair for marketing, Mr. Stein labels this far-sighted
alternative to NEER the “fundamental asset risk”—or
FAR—approach.
Mr. Stein’s conclusions will no doubt irritate many
company bosses, who are fond of denouncing their
investors’ myopia. They have resented the way in which
CAPM—with its assumption of investor infallibility—has
come to play an important role in boardroom decisionmaking.
But it now appears that if they are right, and
their investors are wrong, then those same far-sighted
managers ought to be the CAPM’s biggest fans.
*Jeremy Stein, “Rational Capital Budgeting in an Irrational World,”
The Journal of Business, October 1996.
Source: “Tales from the FAR Side,” The Economist Group, Inc.
November 16, 1996, p. 8. © 1996 The Economist Newspaper Group,
Inc. Reprinted with permission. Further reproduction prohibited.
www. economist.com. All rights reserved.
EXAMPLE 9.1 Using the CAPM
Yet another use of the CAPM is in utility rate-making cases.9
In this case the issue is the
rate of return that a regulated utility should be allowed to earn on its investment in plant and
equipment. Suppose that the equityholders have invested $100 million in the firm and that
the beta of the equity is .6. If the T-bill rate is 6% and the market risk premium is 8%, then
the fair profits to the firm would be assessed as 6 ϩ .6 ϫ 8 ϭ 10.8% of the $100 million
investment, or $10.8 million. The firm would be allowed to set prices at a level expected to
generate these profits.
9
This application is fast disappearing, as many states are in the process of deregulating their public utilities and allowing
a far greater degree of free market pricing. Nevertheless, a considerable amount of rate setting still takes place.
292 PART III Equilibrium in Capital Markets
CONCEPT
CHECK
4 and 5
Stock XYZ has an expected return of 12% and risk of ␤ ϭ 1. Stock ABC has expected return of
13% and ␤ ϭ 1.5. The market’s expected return is 11%, and rf ϭ 5%.
a. According to the CAPM, which stock is a better buy?
b. What is the alpha of each stock? Plot the SML and each stock’s risk–return point on one
graph. Show the alphas graphically.
The risk-free rate is 8% and the expected return on the market portfolio is 16%. A firm considers
a project that is expected to have a beta of 1.3.
a. What is the required rate of return on the project?
b. If the expected IRR of the project is 19%, should it be accepted?
Actual Returns versus Expected Returns
The CAPM is an elegant model. The question is whether it has real-world value—whether
its implications are borne out by experience. Chapter 13 provides a range of empirical
evidence on this point, but for now we focus briefly on a more basic issue: Is the CAPM
testable even in principle? 9
For starters, one central prediction of the CAPM is that the market portfolio is a meanvariance
efficient portfolio. Consider that the CAPM treats all traded risky assets. To test
the efficiency of the CAPM market portfolio, we would need to construct a value-weighted
portfolio of a huge size and test its efficiency. So far, this task has not been feasible.
An even more difficult problem, however, is that the CAPM implies relationships among
expected returns, whereas all we can observe are actual or realized holding-period returns,
and these need not equal prior expectations. Even supposing we could construct a portfolio
to represent the CAPM market portfolio satisfactorily, how would we test its meanvariance
efficiency? We would have to show that the reward-to-volatility ratio of the market
portfolio is higher than that of any other portfolio. However, this reward-to-volatility
ratio is set in terms of expectations, and we have no way to observe these expectations
directly.
The problem of measuring expectations haunts us as well when we try to establish
the validity of the second central set of CAPM predictions, the expected return–beta
relationship. This relationship is also defined in terms of expected returns E(ri) and
E(rM):
E r r E r ri f i M f( ) [ ( ) ]ϭ ϩ␤ Ϫ (9.9)
The upshot is that, as elegant and insightful as the CAPM is, we must make additional
assumptions to make it implementable and testable.
The Index Model and Realized Returns
We have said that the CAPM is a statement about ex ante or expected returns, whereas in
practice all anyone can observe directly are ex post or realized returns. To make the leap
9.2 THE CAPM AND THE INDEX MODEL
CHAPTER 9 The Capital Asset Pricing Model 293
from expected to realized returns, we can employ the index model, which we will use in
excess return form as
R R ei i i M iϭ ␣ ϩ␤ ϩ (9.10)
We saw in Chapter 8 how to apply standard regression analysis to estimate Equation
9.10 using observable realized returns over some sample period. Let us now see
how this framework for statistically decomposing actual stock returns meshes with the
CAPM.
We start by deriving the covariance between the returns on stock i and the market
index. By definition, the firm-specific or nonsystematic component is independent of the
market wide or systematic component, that is, Cov(RM, ei) ϭ 0. From this relationship, it
follows that the covariance of the excess rate of return on security i with that of the market
index is
Cov( Cov
Cov Co
R R R R
R
i M i M i M
i M M
e
R
, ) ( , )
( , )
ϭ ␤ ϩ
ϭ ␤ ϩ vv( , )e RM
i M
i
ϭ ␤ ␴2
Note that we can drop ␣i from the covariance terms because ␣i is a constant and thus has
zero covariance with all variables.
Because Cov( , ) ,R Ri M i Mϭ ␤ ␴2
the sensitivity coefficient, ␤i, in Equation 9.10, which
is the slope of the regression line representing the index model, equals
␤ ϭ
␴
i
i M
M
R RCov( , )
2
The index model beta coefficient turns out to be the same beta as that of the CAPM
expected return–beta relationship, except that we replace the (theoretical) market portfolio
of the CAPM with the well-specified and observable market index.
The Index Model and the Expected Return–Beta Relationship
Recall that the CAPM expected return–beta relationship is, for any asset i and the (theoretical)
market portfolio,
E r r E r ri f i M f( ) [ ( ) ]Ϫ ϭ ␤ Ϫ
where ␤ ϭ ␴i i M MR RCov /( , ) .2
This is a statement about the mean or expected excess
returns of assets relative to the mean excess return of the (theoretical) market
portfolio.
If the index M in Equation 9.10 represents the true market portfolio, we can take the
expectation of each side of the equation to show that the index model specification is
E r r E r ri f i i M f( ) [ ( ) ]Ϫ ϭ ␣ ϩ␤ Ϫ
A comparison of the index model relationship to the CAPM expected return–beta relationship
(Equation 9.9) shows that the CAPM predicts that ␣i should be zero for all assets.
The alpha of a stock is its expected return in excess of (or below) the fair expected return as
predicted by the CAPM. If the stock is fairly priced, its alpha must be zero.
294 PART III Equilibrium in Capital Markets
We emphasize again that this is a statement about expected returns on a security. After
the fact, of course, some securities will do better or worse than expected and will have
returns higher or lower than predicted by the CAPM; that is, they will exhibit positive or
negative alphas over a sample period. But this superior or inferior performance could not
have been forecast in advance.
Therefore, if we estimate the index model for several firms, using Equation 9.10 as a
regression equation, we should find that the ex post or realized alphas (the regression intercepts)
for the firms in our sample center around zero. If the initial expectation for alpha
were zero, as many firms would be expected to have a positive as a negative alpha for some
sample period. The CAPM states that the expected value of alpha is zero for all securities,
whereas the index model representation of the CAPM holds that the realized value of alpha
should average out to zero for a sample of historical observed returns. Just as important,
the sample alphas should be unpredictable, that is, independent from one sample period to
the next.
Indirect evidence on the efficiency of the market portfolio can be found in a study by
Burton Malkiel,10
who estimates alpha values for a large sample of equity mutual funds.
The results, which appear in Figure 9.4, show that the distribution of alphas is roughly bell
shaped, with a mean that is slightly negative but statistically indistinguishable from zero.
On average, it does not appear that mutual funds outperform the market index (the S&P
500) on a risk-adjusted basis.1111
10
Burton G. Malkiel, “Returns from Investing in Equity Mutual Funds 1971–1991,” Journal of Finance 50 (June
1995), pp. 549–72.
11
Notice that the study included all mutual funds with at least 10 years of continuous data. This suggests the
average alpha from this sample would be upward biased because funds that failed after less than 10 years were
ignored and omitted from the left tail of the distribution. This survivorship bias makes the finding that the average
fund underperformed the index even more telling. We discuss survivorship bias further in Chapter 11.
F I G U R E 9.4 Estimates of individual mutual fund alphas, 1972–1991
This is a plot of the frequency distribution of estimated alphas for all-equity mutual funds with 10-year continuous
records.
Source: Burton G. Malkiel, “Returns from Investing in Equity Mutual Funds 1971–1991,” Journal of Finance 50 (June
1995), pp. 549–72. Reprinted by permission of the publisher, Blackwell Publishing, Inc.
Alpha (%)
Frequency
36
32
28
24
16
20
12
8
4
0
−3 −2 −1 0 1 2
CHAPTER 9 The Capital Asset Pricing Model 295
This result is quite meaningful. While we might expect realized alpha values of individual
securities to center around zero, professionally managed mutual funds might be
expected to demonstrate average positive alphas. Funds with superior performance (and
we do expect this set to be non-empty) should tilt the sample average to a positive value.
The small impact of superior funds on this distribution suggests the difficulty in beating
the passive strategy that the CAPM deems to be optimal.
There is yet another applicable variation on the intuition of the index model, the market
model. Formally, the market model states that the return “surprise” of any security is proportional
to the return surprise of the market, plus a firm-specific surprise:
r E r r E r ei i i M M iϪ ϭ ␤ Ϫ ϩ( ) [ ( )]
This equation divides returns into firm-specific and systematic components somewhat differently
from the index model. If the CAPM is valid, however, you can confirm that, substituting
for E(ri) from Equation 9.9, the market model equation becomes identical to the
index model. For this reason the terms “index model” and “market model” often are used
interchangeably.
CONCEPT
CHECK
6
Can you sort out the nuances of the following maze of models?
a. CAPM c. Single-index model
b. Single-factor model d. Market model
To discuss the role of the CAPM in real-life investments we have to answer two questions.
First, even if we all agreed that the CAPM were the best available theoretical model to
explain rates of return on risky assets, how would this affect practical investment policy?
Second, how can we determine whether the CAPM is in fact the best available model to
explain rates of return on risky assets?
Notice the wording of the first question. We don’t pose it as: “Suppose the CAPM perfectly
explains the rates of return on risky assets. . . .” All models, whether in economics
or science, are based on simplifications that enable us to come to grips with a complicated
reality, which means that perfection is an unreasonable and unusable standard. In our context,
we must clarify what “perfectly explains” would mean. From the previous section
we know that if the CAPM were valid, a single-index model in which the index includes
all traded securities (i.e., all risky securities in the investable universe as in Assumption 3)
also would be valid. In this case, “perfectly explains” would mean that all alpha values in
security risk premiums would be identically zero.
The notion that all alpha values can be identically zero is feasible in principle, but
such a configuration cannot be expected to emerge in real markets. This was demonstrated
by Grossman and Stiglitz, who showed that such an equilibrium may be one that
the real economy can approach, but not necessarily reach.12
Their basic idea is that the
12
Sanford J. Grossman and Joseph E. Stiglitz, “On the Impossibility of Informationally Efficient Markets,” American
Economic Review 70 (June 1981).
9.3 IS THE CAPM PRACTICAL?
296 PART III Equilibrium in Capital Markets
actions of security analysts are the forces that drive security prices to “proper” levels at
which alpha is zero. But if all alphas were identically zero, there would be no incentive
to engage in such security analysis. Instead, the market equilibrium will be characterized
by prices hovering “near” their proper values, at which alphas are almost zero, but
with enough slippage (and therefore reward for superior insight) to induce analysts to
continue their efforts.
A more reasonable standard, that the CAPM is the “best available model to explain
rates of return on risky assets,” means that in the absence of security analysis, one should
take security alphas as zero. A security is mispriced if and only if its alpha is nonzero—
underpriced if alpha is positive and overpriced if alpha is negative—and positive or
negative alphas are revealed only by superior security analysis. Absent the investment
of significant resources in such analysis, an investor would obtain the best investment
portfolio on the assumption that all alpha values are zero. This definition of the
superiority of the CAPM over any other model also determines its role in real-life
investments.
Under the assumption that the CAPM is the best available model, investors willing to
expend resources to construct a superior portfolio must (1) identify a practical index to
work with and (2) deploy macro analysis to obtain good forecasts for the index and security
analysis to identify mispriced securities. This procedure was described in Chapter 8
and is further elaborated on in Part Five (Security Analysis) and Part Seven (Applied Portfolio
Management).
We will examine several tests of the CAPM in Chapter 13. But it is important to explain
the results of these tests and their implications.
Is the CAPM Testable?
Let us consider for a moment what testability means. A model consists of (i) a set of
assumptions, (ii) logical/mathematical development of the model through manipulation
of those assumptions, and (iii) a set of predictions. Assuming the logical/mathematical
manipulations are free of errors, we can test a model in two ways, normative and positive.
Normative tests examine the assumptions of the model, while positive tests examine the
predictions.
If a model’s assumptions are valid, and the development is error-free, then the predictions
of the model must be true. In this case, testing the assumptions is synonymous
with testing the model. But few, if any, models can pass the normative test. In most cases,
as with the CAPM, the assumptions are admittedly invalid—we recognize that we have
simplified reality, and therefore to this extent are relying on “untrue” assumptions. The
motivation for invoking unrealistic assumptions is clear; we simply cannot solve a model
that is perfectly consistent with the full complexity of real-life markets. As we’ve noted,
the need to use simplifying assumptions is not peculiar to economics—it characterizes all
of science.
Assumptions are chosen first and foremost to render the model solvable. But we prefer
assumptions to which the model is “robust.” A model is robust with respect to an assumption
if its predictions are not highly sensitive to violation of the assumption. If we use
only assumptions to which the model is robust, the model’s predictions will be reasonably
accurate despite its shortcomings. The upshot of all this is that tests of models are
almost always positive—we judge a model on the success of its empirical predictions.
This standard brings statistics into any science and requires us to take a stand on what are
CHAPTER 9 The Capital Asset Pricing Model 297
acceptable levels of significance and power.13
Because the nonrealism of the assumptions
precludes a normative test, the positive test is really a test of the robustness of the model
to its assumptions.
The CAPM implications are embedded in two predictions: (1) the market portfolio is
efficient, and (2) the security market line (the expected return–beta relationship) accurately
describes the risk–return trade-off, that is, alpha values are zero. In fact, the second
implication can be derived from the first, and therefore both stand or fall together in a test
that the market portfolio is mean-variance efficient. The central problem in testing this
prediction is that the hypothesized market portfolio is unobservable. The “market portfolio”
includes all risky assets that can be held by investors. This is far more extensive
than an equity index. It would include bonds, real estate, foreign assets, privately held
businesses, and human capital. These assets are often traded thinly or (for example, in the
case of human capital) not traded at all. It is difficult to test the efficiency of an observable
portfolio, let alone an unobservable one. These problems alone make adequate testing of
the model infeasible.14
Moreover, even small departures from efficiency in the market portfolio
can lead to large departures from the expected return–beta relationship of the SML,
which would negate the practical usefulness of the model.
The CAPM Fails Empirical Tests
Because the market portfolio cannot be observed, tests of the CAPM revolve around the
expected return–beta relationship. The tests use proxies such as the S&P 500 index to stand
in for the true market portfolio. These tests therefore appeal to robustness of the assumption
that the market proxy is sufficiently close to the true, unobservable market portfolio.
The CAPM fails these tests, that is, the data reject the hypothesis that alpha values are
uniformly zero at acceptable levels of significance. For example, we find that, on average,
low-beta securities have positive alphas and high-beta securities have negative alphas.
It is possible that this is a result of a failure of our data, the validity of the market proxy,
or statistical method. If so, we would conclude the following: There is no better model out
there, but we measure beta and alpha values with unsatisfactory precision. This situation
13
To illustrate the meanings of significance and power, consider a test of the efficacy of a new drug. The agency
testing the drug may make two possible errors. The drug may be useless (or even harmful), but the agency may
conclude that it is useful. This is called a “Type I” error. The significance level of a test is the probability of a Type
I error. Typical practice is to fix the level of significance at some low level, for example, 5%. In the case of drug
testing, for example, the first goal is to avoid introducing ineffective or harmful treatments. The other possible
error is that the drug is actually useful, but the testing procedure concludes it is not. This mistake, called “Type
II” error, would lead us to discard a useful treatment. The power of the test is the probability of avoiding Type II
error (i.e., one minus the probability of making such an error), that is, the probability of accepting the drug if it
is indeed useful. We want tests that, at a given level of significance, have the most power, so we will admit effective
drugs with high probability. In social sciences in particular, available tests often have low power, in which
case they are susceptible to Type II error and will reject a correct model (a “useful drug”) with high frequency.
“The drug is useful” is analogous in the CAPM to alphas being zero. When the test data reject the hypothesis that
observed alphas are zero at the desired level of significance, the CAPM fails. However, if the test has low power,
the probability that we accept the model when true is not all that high.
14
The best-known discussion of the difficulty in testing the CAPM is now called “Roll’s critique.” See Richard
Roll, “A Critique of the Asset Pricing Theory’s Tests: Part I: On Past and Potential Testability of the Theory,”
Journal of Financial Economics 4 (1977). The issue is developed further in Richard Roll and Stephen A. Ross,
“On the Cross-Sectional Relation between Expected Return and Betas,” Journal of Finance 50 (1995); and
Schmuel Kandel and Robert F. Stambaugh, “Portfolio Inefficiency and the Cross-Section of Expected Returns,”
Journal of Finance 50 (1995).
298 PART III Equilibrium in Capital Markets
would call for improved technique. But if the rejection of the model is not an artifact of statistical
problems, then we must search for extensions to the CAPM, or substitute models.
We will consider several extensions of the model later in the chapter.
The Economy and the Validity of the CAPM
For better or worse, some industries are regulated, with rate commissions either setting or
approving prices. Imagine a commission pondering a rate case for a regulated utility. The
rate commission must decide whether the rates charged by the company are sufficient to
grant shareholders a fair rate of return on their investments. The normative framework of the
typical rate hearing is that shareholders, who have made an investment in the firm, are entitled
to earn a “fair” rate of return on their equity investment. The firm is therefore allowed to
charge prices that are expected to generate a profit consistent with that fair rate of return.
The question of fairness of the rate of return to the company shareholders cannot be
divorced from the level of risk of these returns. The CAPM provides the commission a
clear criterion: If the rates under current regulation are too low, then the rate of return to
equity investors would be less than commensurate with risk, and alpha would be negative.
As we pointed out in Example 9.1, the commissioner’s problem may now be organized
around arguments about estimates of risk and the security market line.
Similar applications arise in many legal settings. For example, contracts with payoffs
that are contingent on a fair rate of return can be based on the index rate of return and the
beta of appropriate assets. Many disputes involving damages require that a stream of losses
be discounted to a present value. The proper discount rate depends on risk, and disputes
about fair compensation to litigants can be (and often are) set on the basis of the SML,
using past data that differentiate systematic from firm-specific risk.
It may be surprising to find that the CAPM is an accepted norm in the U.S. and many
other developed countries, despite its empirical shortcomings. We can offer a twofold
explanation. First, the logic of the decomposition to systematic and firm-specific risk is
compelling. Absent a better model to assess nonmarket components of risk premiums,
we must use the best method available. As improved methods of generating equilibrium
security returns become empirically validated, they gradually will be incorporated into
institutional decision making. Such improvements may come either from extensions of the
CAPM and its companion, arbitrage pricing theory (discussed in the next chapter), or from
a yet- undiscovered new model.
Second, there is impressive, albeit less-formal, evidence that the central conclusion of
the CAPM—the efficiency of the market portfolio—may not be all that far from being
valid. Thousands of mutual funds within hundreds of investment companies compete for
investor money. These mutual funds employ professional analysts and portfolio managers
and expend considerable resources to construct superior portfolios. But the number of
funds that consistently outperform a simple strategy of investing in passive market index
portfolios is extremely small, suggesting that the single-index model with ex ante zero
alpha values may be a reasonable working approximation for most investors.
The Investments Industry and the Validity of the CAPM
More than other practitioners, investment firms must take a stand on the validity of the
CAPM. If they judge the CAPM invalid, they must turn to a substitute framework to guide
them in constructing optimal portfolios.
For example, the CAPM provides discount rates that help security analysts assess the
intrinsic value of a firm. If an analyst believes that some actual prices differ from intrinsic
values, then those securities have nonzero alphas, and there is an opportunity to construct
an active portfolio with a superior risk–return profile. But if the discount rate used to assess
CHAPTER 9 The Capital Asset Pricing Model 299
intrinsic value is incorrect because of a failure in the CAPM, the estimate of alpha will be
biased, and both the Markowitz model of Chapter 7 and the index model of Chapter 8 will
actually lead to inferior portfolios. When constructing their presumed optimal risky portfolios,
practitioners must be satisfied that the passive index they use for that purpose is satisfactory
and that the ratios of alpha to residual variance are appropriate measures of investment
attractiveness. This would not be the case if the CAPM is invalid. Yet it appears many practitioners
do use index models (albeit often with additional indexes) when assessing security
prices. The curriculum of the CFA Institute also suggests a widespread acceptance of the
CAPM, at least as a starting point for thinking about the risk–return relationship. An explanation
similar to the one we offered in the previous subsection is equally valid here.
The central conclusion from our discussion so far is that, explicitly or implicitly, practitioners
do use a CAPM. If they use a single-index model and derive optimal portfolios
from ratios of alpha forecasts to residual variance, they behave as if the CAPM is valid.15
If
they use a multi-index model, then they use one of the extensions of the CAPM (discussed
later in this chapter) or arbitrage pricing theory (discussed in the next chapter). Thus, theory
and evidence on the CAPM should be of interest to all sophisticated practitioners.
When assessing the empirical success of the CAPM, we must also consider our econometric
technique. If our tests are poorly designed, we may mistakenly reject the model. Similarly,
some empirical tests implicitly introduce additional assumptions that are not part of
the CAPM, for example, that various parameters of the model such as beta or residual variance
are constant over time. If these extraneous additional assumptions are too restrictive,
we also may mistakenly reject the model.15
To begin, notice that all the coefficients of a regression equation are estimated simultaneously,
and these estimates are not independent. In particular, the estimate of the intercept
(alpha) of a single- (independent) variable regression depends on the estimate of the slope
coefficient. Hence, if the beta estimate is inefficient and/or biased, so will be the estimate
of the intercept. Unfortunately, statistical bias is easily introduced.
An example of this hazard was pointed out in an early paper by Miller and Scholes,16
who
demonstrated how econometric problems could lead one to reject the CAPM even if it were
perfectly valid. They considered a checklist of difficulties encountered in testing the model
and showed how these problems potentially could bias conclusions. To prove the point,
they simulated rates of return that were constructed to satisfy the predictions of the CAPM
and used these rates to “test” the model with standard statistical techniques of the day.
The result of these tests was a rejection of the model that looks surprisingly similar to
what we find in tests of returns from actual data—this despite the fact that the “data” were
constructed to satisfy the CAPM. Miller and Scholes thus demonstrated that econometric
technique alone could be responsible for the rejection of the model in actual tests.
15
We need to be a bit careful here. On its face, the CAPM asserts that alpha values will equal zero in security market
equilibrium. But as we argued earlier, consistent with the vast amount of security analysis that actually takes
place, a better way to interpret the CAPM is that equilibrium really means that alphas should be taken to be zero
in the absence of security analysis. With private information or superior insight one presumably would be able to
identify stocks that are mispriced by the market and thus offer nonzero alphas.
16
Merton H. Miller and Myron Scholes, “Rates of Return in Relations to Risk: A Re-examination of Some Recent
Findings,” in Studies in the Theory of Capital Markets, Michael C. Jensen, ed. (New York: Praeger, 1972).
9.4 ECONOMETRICS AND THE EXPECTED
RETURN–BETA RELATIONSHIP
300 PART III Equilibrium in Capital Markets
There are several potential problems with the estimation of beta coefficients. First,
when residuals are correlated (as is common for firms in the same industry), standard beta
estimates are not efficient. A simple approach to this problem would be to use statistical
techniques designed for these complications. For example, we might replace OLS (ordinary
least squares) regressions with GLS (generalized least squares) regressions, which
account for correlation across residuals. Moreover, both coefficients, alpha and beta, as
well as residual variance, are likely time varying. There is nothing in the CAPM that
precludes such time variation, but standard regression techniques rule it out and thus may
lead to false rejection of the model. There are now well-known techniques to account for
time-varying parameters. In fact, Robert Engle won the Nobel Prize for his pioneering
work on econometric techniques to deal with time-varying volatility, and a good portion
of the applications of these new techniques have been in finance.17
Moreover, betas
may vary not purely randomly over time, but in response to changing economic conditions.
A “conditional” CAPM allows risk and return to change with a set of “conditioning
variables.”18
As importantly, Campbell and Vuolteenaho19
find that the beta of a security can be
decomposed into two components, one of which measures sensitivity to changes in corporate
profitability and another which measures sensitivity to changes in the market’s discount
rates. These are found to be quite different in many cases. Improved econometric
techniques such as those proposed in this short survey may help resolve part of the empirical
failure of the simple CAPM.
The CAPM uses a number of simplifying assumptions. We can gain greater predictive
accuracy at the expense of greater complexity by relaxing some of those assumptions. In
this section, we will consider a few of the more important attempts to extend the model.
This discussion is not meant to be exhaustive. Rather, it introduces a few extensions
of the basic model to provide insight into the various attempts to improve empirical
content.1718
The Zero-Beta Model19
Efficient frontier portfolios have a number of interesting characteristics, independently
derived by Merton and Roll.20
Three of these are
1. Any portfolio that is a combination of two frontier portfolios is itself on the
efficient frontier.
17
Engle’s work gave rise to the widespread use of so-called ARCH models. ARCH stands for autoregressive conditional
heteroskedasticity, which is a fancy way of saying that volatility changes over time, and that recent levels
of volatility can be used to form optimal estimates of future volatility.
18
There is now a large literature on conditional models of security market equilibrium. Much of it derives from
Ravi Jagannathan and Zhenyu Wang, “The Conditional CAPM and the Cross-Section of Expected Returns,”
Journal of Finance 51 (March 1996), vol pp. 3–53.
19
John Campbell and Tuomo Vuolteenaho, “Bad Beta, Good Beta,” American Economic Review 94 (December
2004), pp. 1249–75.
20
Robert C. Merton, “An Analytic Derivation of the Efficient Portfolio Frontier,” Journal of Financial and Quantitative
Analysis, 1972. Roll, see footnote 14.
9.5 EXTENSIONS OF THE CAPM
CHAPTER 9 The Capital Asset Pricing Model 301
2. The expected return of any asset can be expressed as an exact linear function of the
expected return on any two efficient-frontier portfolios P and Q according to the
following equation:
E r E r E r E r
r r r
i Q P Q
i P P
( ) ( ) [ ( ) ( )]
( , ) ( ,
Ϫ ϭ Ϫ
ϪCov Cov rr
r r
Q
P P Q
)
( , )␴ Ϫ2
Cov
(9.11)
3. Every portfolio on the efficient frontier, except for the global minimum-variance
portfolio, has a “companion” portfolio on the bottom (inefficient) half of the frontier
with which it is uncorrelated. Because it is uncorrelated, the companion portfolio
is referred to as the zero-beta portfolio of the efficient portfolio. If we choose
the market portfolio M and its zero-beta companion portfolio Z, then Equation 9.11
simplifies to the CAPM-like equation
E r E r E R E R
r r
Ei Z M Z
i M
M
i( ) ( ) [ ( ) ( )]
( , )
[Ϫ ϭ Ϫ
␴
ϭ ␤
Cov
2
(( ) ( )]r E rM ZϪ (9.12)
Equation 9.12 resembles the SML of the CAPM, except that the risk-free rate is
replaced with the expected return on the zero-beta companion of the market index
portfolio.
Fischer Black used these properties to show that Equation 9.12 is the CAPM equation
that results when investors face restrictions on borrowing and/or investment in the riskfree
asset.21
In this case, at least some investors will choose portfolios on the efficient
frontier that are not necessarily the market index portfolio. Because average returns on the
zero-beta portfolio are greater than observed T-bill rates, the zero-beta model can explain
why average estimates of alpha values are positive for low-beta securities and negative for
high-beta securities, contrary to the prediction of the CAPM. Despite this, the model is not
sufficient to rescue the CAPM from empirical rejection.
Labor Income and Nontraded Assets
An important departure from realism is the CAPM assumption that all risky assets are
traded. Two important asset classes that are not traded are human capital and privately held
businesses. The discounted value of future labor income exceeds the total market value of
traded assets. The market value of privately held corporations and businesses is of the same
order of magnitude. Human capital and private enterprises are different types of assets with
possibly different implications for equilibrium returns on traded securities.
Privately held business may be the lesser of the two sources of departures from the
CAPM. Nontraded firms can be incorporated or sold at will, save for liquidity considerations
that we discuss in the next section. Owners of private business also can borrow against
their value, further diminishing the material difference between ownership of private and
public business. Suppose that privately held business have similar risk characteristics as
those of traded assets. In this case, individuals can partially offset the diversification problems
posed by their nontraded entrepreneurial assets by reducing their portfolio demand
for securities of similar, traded assets. Thus, the CAPM expected return–beta equation may
not be greatly disrupted by the presence of entrepreneurial income.
To the extent that risk characteristics of private enterprises differ from those of traded
securities, a portfolio of traded assets that best hedges the risk of typical private business
21
Fischer Black, “Capital Market Equilibrium with Restricted Borrowing,” Journal of Business, July 1972.
302 PART III Equilibrium in Capital Markets
would enjoy excess demand from the population of private business owners. The price of
assets in this portfolio will be bid up relative to the CAPM considerations, and the expected
returns on these securities will be lower in relation to their systematic risk. Conversely,
securities highly correlated with such risk will have high equilibrium risk premiums and
may appear to exhibit positive alphas relative to the conventional SML. In fact, Heaton and
Lucas show that adding proprietary income to a standard asset-pricing model improves its
predictive performance.22
The size of labor income and its special nature is of greater concern for the validity of
the CAPM. The possible effect of labor income on equilibrium returns can be appreciated
from its important effect on personal portfolio choice. Despite the fact that an individual
can borrow against labor income (via a home mortgage) and reduce some of the
uncertainty about future labor income via life insurance, human capital is less “portable”
across time and may be more difficult to hedge using traded securities than nontraded business.
This may induce pressure on security prices and result in departures from the CAPM
expected return–beta equation. For one example, surely an individual seeking diversification
should avoid investing in his employer’s stock and limit investments in the same
industry. Thus, the demand for stocks of labor-intensive firms may be reduced, and these
stocks may require a higher expected return than predicted by the CAPM.
Mayers23
derives the equilibrium expected return–beta equation for an economy in
which individuals are endowed with labor income of varying size relative to their nonlabor
capital. The resultant SML equation is
E R E R
R R
P
P
R R
P
P
i M
i M
H
M
i H
M
H
( ) ( )
( , ) ( , )
ϭ
ϩ
␴ ϩ
Cov Cov
2
MM
M HR RCov( , )
(9.13)
where
PH ϭ value of aggregate human capital,
PM ϭ market value of traded assets (market portfolio),
RH ϭ excess rate of return on aggregate human capital.
The CAPM measure of systematic risk, beta, is replaced in the extended model by an
adjusted beta that also accounts for covariance with the portfolio of aggregate human capital.
Notice that the ratio of human capital to market value of all traded assets, P PH M/ ,
may well be greater than 1, and hence the effect of the covariance of a security with labor
income, Cov(Ri,RH), relative to the average, Cov(RM,RH), is likely to be economically significant.
When Cov(Ri,RH) is positive, the adjusted beta is greater when the CAPM beta is
smaller than 1, and vice versa. Because we expect Cov(Ri,RH) to be positive for the average
security, the risk premium in this model will be greater, on average, than predicted by
the CAPM for securities with beta less than 1, and smaller for securities with beta greater
than 1. The model thus predicts a security market line that is less steep than that of the
standard CAPM. This may help explain the average negative alpha of high-beta securities
and positive alpha of low-beta securities that lead to the statistical failure of the CAPM
equation. In Chapter 13 on empirical evidence we present additional results along these
lines.
22
John Heaton and Deborah Lucas, “Portfolio Choice and Asset Prices: The Importance of Entrepreneurial Risk,
Journal of Finance 55 (June 2000). This paper offers evidence of the effect of entrepreneurial risk on both portfolio
choice and the risk–return relationship.
23
See footnote 8.
CHAPTER 9 The Capital Asset Pricing Model 303
A Multiperiod Model and Hedge Portfolios
Robert C. Merton revolutionized financial economics by using continuous-time models to
extend many of our models of asset pricing.24
While his (Nobel Prize–winning) contributions
to option-pricing theory and financial engineering (along with those of Fischer Black
and Myron Scholes) may have had greater impact on the investment industry, his solo contribution
to portfolio theory was equally important for our understanding of the risk–return
relationship.
In his basic model, Merton relaxes the “single-period” myopic assumptions about investors.
He envisions individuals who optimize a lifetime consumption/investment plan, and
who continually adapt consumption/investment decisions to current wealth and planned
retirement age. When uncertainty about portfolio returns is the only source of risk and
investment opportunities remain unchanged through time, that is, there is no change in
the probability distribution of the return on the market portfolio or individual securities,
Merton’s so-called intertemporal capital asset pricing model (ICAPM) predicts the same
expected return–beta relationship as the single-period equation.25
But the situation changes when we include additional sources of risk. These extra risks
are of two general kinds. One concerns changes in the parameters describing investment
opportunities, such as future risk-free rates, expected returns, or the risk of the market portfolio.
For example, suppose that the real interest rate may change over time. If it falls in
some future period, one’s level of wealth will now support a lower stream of real consumption.
Future spending plans, for example, for retirement spending, may be put in jeopardy.
To the extent that returns on some securities are correlated with changes in the risk-free
rate, a portfolio can be formed to hedge such risk, and investors will bid up the price (and
bid down the expected return) of those hedge assets. Investors will sacrifice some expected
return if they can find assets whose returns will be higher when other parameters (in this
case, the risk-free rate) change adversely.
The other additional source of risk concerns the prices of the consumption goods that
can be purchased with any amount of wealth. Consider as an example inflation risk. In
addition to the expected level and volatility of their nominal wealth, investors must be
concerned about the cost of living—what those dollars can buy. Therefore, inflation risk
is an important extramarket source of risk, and investors may be willing to sacrifice some
expected return to purchase securities whose returns will be higher when the cost of living
changes adversely. If so, hedging demands for securities that help to protect against inflation
risk would affect portfolio choice and thus expected return. One can push this conclusion
even further, arguing that empirically significant hedging demands may arise for
important subsectors of consumer expenditures; for example, investors may bid up share
prices of energy companies that will hedge energy price uncertainty. These sorts of effects
may characterize any assets that hedge important extramarket sources of risk.
More generally, suppose we can identify K sources of extramarket risk and find K associated
hedge portfolios. Then, Merton’s ICAPM expected return–beta equation would generalize
the SML to a multi-index version:
E R E R E Ri iM M ik k
k
K
( ) ( ) ( )ϭ ␤ ϩ ␤
=
∑
1
(9.14)
where ␤iM is the familiar security beta on the market-index portfolio, and ␤ik is the beta on
the kth hedge portfolio.
24
Merton’s classic works are collected in Continuous-Time Finance (Oxford, U.K.: Basil Blackwell, 1992).
25
Eugene F. Fama also made this point in “Multiperiod Consumption-Investment Decisions,” American Economic
Review 60 (1970).
304 PART III Equilibrium in Capital Markets
Other multifactor models using additional factors that do not arise from extramarket
sources of risk have been developed and lead to SMLs of a form identical to that of the
ICAPM. These models also may be considered extensions of the CAPM in the broad sense.
We examine these models in the next chapter.
A Consumption-Based CAPM
The logic of the CAPM together with the hedging demands noted in the previous subsection
suggests that it might be useful to center the model directly on consumption. Such
models were first proposed by Mark Rubinstein, Robert Lucas, and Douglas Breeden.26
In a lifetime consumption plan, the investor must in each period balance the allocation of
current wealth between today’s consumption and the savings and investment that will support
future consumption. When optimized, the utility value from an additional dollar of consumption
today must be equal to the utility value of the expected future consumption that
can be financed by that additional dollar of wealth.27
Future wealth will grow from labor
income, as well as returns on that dollar when invested in the optimal complete portfolio.
Suppose risky assets are available and you wish to increase expected consumption
growth by allocating some of your savings to a risky portfolio. How would we measure the
risk of these assets? As a general rule, investors will value additional income more highly
during difficult economic times (when consumption opportunities are scarce) than in affluent
times (when consumption is already abundant). An asset will therefore be viewed as
riskier in terms of consumption if it has positive covariance with consumption growth—in
other words, if its payoff is higher when consumption is already high and lower when consumption
is relatively restricted. Therefore, equilibrium risk premiums will be greater for
assets that exhibit higher covariance with consumption growth. Developing this insight, we
can write the risk premium on an asset as a function of its “consumption risk” as follows:
E Ri iC C( ) ϭ ␤ RP (9.15)
where portfolio C may be interpreted as a consumption-tracking portfolio (also called a
consumption-mimicking portfolio), that is, the portfolio with the highest correlation with
consumption growth; ␤iC is the slope coefficient in the regression of asset i’s excess returns,
Ri, on those of the consumption-tracking portfolio; and, finally, RPC is the risk premium
associated with consumption uncertainty, which is measured by the expected excess return
on the consumption-tracking portfolio:
RPC C C fE R E r rϭ ϭ Ϫ( ) ( ) (9.16)
Notice how similar this conclusion is to the conventional CAPM. The consumptiontracking
portfolio in the CCAPM plays the role of the market portfolio in the conventional
CAPM. This is in accord with its focus on the risk of consumption opportunities
rather than the risk and return of the dollar value of the portfolio. The excess return on the
26
Mark Rubinstein, “The Valuation of Uncertain Income Streams and the Pricing of Options,” Bell Journal of
Economics and Management Science 7 (1976), pp. 407–25; Robert Lucas, “Asset Prices in an Exchange Economy,”
Econometrica 46 (1978), pp. 1429–45; Douglas Breeden, “An Intertemporal Asset Pricing Model with
Stochastic Consumption and Investment Opportunities,” Journal of Financial Economics 7 (1979), pp. 265–96.
27
Wealth at each point in time equals the market value of assets in the balance sheet plus the present value of
future labor income. These models of consumption and investment decisions are often made tractable by assuming
investors exhibit constant relative risk aversion, or CRRA. CRRA implies that an individual invests a constant
proportion of wealth in the optimal risky portfolio regardless of the level of wealth. You might recall that our prescription
for optimal capital allocation in Chapter 6 also called for an optimal investment proportion in the risky
portfolio regardless of the level of wealth. The utility function we employed there also exhibited CRRA.
CHAPTER 9 The Capital Asset Pricing Model 305
consumption-tracking portfolio plays the role of the excess return on the market portfolio,
M. Both approaches result in linear, single-factor models that differ mainly in the identity
of the factor they use.
In contrast to the CAPM, the beta of the market portfolio on the market factor of the
CCAPM is not necessarily 1. It is perfectly plausible and empirically evident that this beta
is substantially greater than 1. This means that in the linear relationship between the market
index risk premium and that of the consumption portfolio,
E R E RM M MC C M( ) ( )ϭ ␣ ϩ␤ ϩ ␧ (9.17)
where ␣M and ␧M allow for empirical deviation from the exact model in Equation 9.15, and
␤MC is not necessarily equal to 1.
Because the CCAPM is so similar to the CAPM, one might wonder about its usefulness.
Indeed, just as the CAPM is empirically flawed because not all assets are traded, so is the
CCAPM. The attractiveness of this model is in that it compactly incorporates consumption
hedging and possible changes in investment opportunities, that is, in the parameters of the
return distributions in a single-factor framework. There is a price to pay for this compactness,
however. Consumption growth figures are published infrequently (monthly at the
most) compared with financial assets, and are measured with significant error. Nevertheless,
recent empirical research28
indicates that this model is more successful in explaining
realized returns than the CAPM, which is a reason why students of investments should
be familiar with it. We return to this issue, as well as empirical evidence concerning the
CCAPM, in Chapter 13.
Standard models of asset pricing (such as the CAPM) assume frictionless markets, meaning
that securities can be traded costlessly. But these models actually have little to say
about trading activity. For example, in the equilibrium of the CAPM, all investors share all
available information and demand identical portfolios of risky assets. The awkward implication
of this result is that there is no reason for trade. If all investors hold identical portfolios
of risky assets, then when new (unexpected) information arrives, prices will change
commensurately, but each investor will continue to hold a piece of the market portfolio,
which requires no exchange of assets. How do we square this implication with the observation
that on a typical day, more than 3 billion shares change hands on the New York Stock
Exchange alone? One obvious answer is heterogeneous expectations, that is, beliefs not
shared by the entire market. Such private information will give rise to trading as investors
attempt to profit by rearranging portfolios in accordance with their now-heterogeneous
demands. In reality, trading (and trading costs) will be of great importance to investors.28
The liquidity of an asset is the ease and speed with which it can be sold at fair market
value. Part of liquidity is the cost of engaging in a transaction, particularly the bid–ask
spread. Another part is price impact—the adverse movement in price one would encounter
when attempting to execute a larger trade.Yet another component is immediacy—the ability
to sell the asset quickly without reverting to fire-sale prices. Conversely, illiquidity can be
measured in part by the discount from fair market value a seller must accept if the asset is to
be sold quickly. A perfectly liquid asset is one that would entail no illiquidity discount.
28
Ravi Jagannathan andYong Wang, “Lazy Investors, Discretionary Consumption, and the Cross-Section of Stock
Returns,” Journal of Finance 62 (August 2007), pp. 1633–61.
9.6 LIQUIDITY AND THE CAPM
306
Liquidity (or the lack of it) has long been recognized as an important characteristic that
affects asset values. For example, in legal cases, courts have routinely applied very steep
discounts to the values of businesses that cannot be publicly traded. But liquidity has not
always been appreciated as an important factor in security markets, presumably due to the
relatively small trading cost per transaction compared with the large costs of trading assets
such as real estate. The breakthrough came in the work of Amihud and Mendelson29
(see
the nearby box) and today, liquidity is increasingly viewed as an important determinant of
prices and expected returns. We supply only a brief synopsis of this important topic here
and provide empirical evidence in Chapter 13.
29
Yakov Amihud and Haim Mendelson, “Asset Pricing and the Bid–Ask Spread,” Journal of Financial Economics
17(1986). A summary of the ensuing large body of literature on liquidity can be found in Yakov Amihud,
Haim Mendelson, and Lasse Heje Pedersen, “Liquidity and Asset Prices,” Foundations and Trends in Finance 1,
no. 4 (2005).
STOCK INVESTORS PAY HIGH PRICE FOR LIQUIDITY
Given a choice between liquid and illiquid stocks, most
investors, to the extent they think of it at all, opt for
issues they know are easy to get in and out of.
But for long-term investors who don’t trade often—
which includes most individuals—that may be unnecessarily
expensive. Recent studies of the performance of
listed stocks show that, on average, less-liquid issues
generate substantially higher returns—as much as several
percentage points a year at the extremes.
ILLIQUIDITY PAYOFF
Among the academic studies that have attempted to
quantify this illiquidity payoff is a recent work by two
finance professors, Yakov Amihud of New York University
and Tel Aviv University, and Haim Mendelson of
the University of Rochester. Their study looks at New
York Stock Exchange issues over the 1961–1980 period
and defines liquidity in terms of bid–asked spreads as a
percentage of overall share price.
Market makers use spreads in quoting stocks to
define the difference between the price they’ll bid to
take stock off an investor’s hands and the price they’ll
offer to sell stock to any willing buyer. The bid price is
always somewhat lower because of the risk to the broker
of tying up precious capital to hold stock in inventory
until it can be resold.
If a stock is relatively illiquid, which means there’s
not a ready flow of orders from customers clamoring
to buy it, there’s more of a chance the broker will lose
money on the trade. To hedge this risk, market makers
demand an even bigger discount to service potential
sellers, and the spread will widen further.
The study by Profs. Amihud and Mendelson shows
that liquidity spreads—measured as a percentage discount
from the stock’s total price—ranged from less than
0.1%, for widely held International Business Machines
Corp., to as much as 4% to 5%. The widest-spread
group was dominated by smaller, low-priced stocks.
The study found that, overall, the least-liquid stocks
averaged an 8.5 percent-a-year higher return than the
most-liquid stocks over the 20-year period. On average,
a one percentage point increase in the spread was
associated with a 2.5% higher annual return for New
York Stock Exchange stocks. The relationship held after
results were adjusted for size and other risk factors.
An extension of the study of Big Board stocks done
at The Wall Street Journal’s request produced similar
findings. It shows that for the 1980–85 period, a one
percentage-point-wider spread was associated with
an extra average annual gain of 2.4%. Meanwhile, the
least-liquid stocks outperformed the most-liquid stocks
by almost six percentage points a year.
COST OF TRADING
Since the cost of the spread is incurred each time the
stock is traded, illiquid stocks can quickly become prohibitively
expensive for investors who trade frequently.
On the other hand, long-term investors needn’t worry
so much about spreads, since they can amortize them
over a longer period.
In terms of investment strategy, this suggests “that
the small investor should tailor the types of stocks he or
she buys to his expected holding period,” Prof. Mendelson
says. If the investor expects to sell within three
months, he says, it’s better to pay up for the liquidity
and get the lowest spread. If the investor plans to hold
the stock for a year or more, it makes sense to aim at
stocks with spreads of 3% or more to capture the extra
return.
Source: Barbara Donnelly, The Wall Street Journal, April 28, 1987,
p. 37. Reprinted by permission of The Wall Street Journal. © 1987
Dow Jones & Company, Inc. All Rights Reserved Worldwide.
WORDSFROMTHESTREET
CHAPTER 9 The Capital Asset Pricing Model 307
Early models of liquidity focused on the inventory management problem faced by security
dealers. Dealers in over-the-counter markets post prices at which they are willing to
buy a security (the bid price) or sell it (the ask price). The willingness of security dealers
to add to their inventory or sell shares from their inventory makes them crucial contributors
to overall market liquidity. The fee they earn for supplying this liquidity is the bid–ask
spread. Part of the bid–ask spread may be viewed as compensation for bearing the price
risk involved in holding an inventory of securities and allowing their inventory levels to
absorb the fluctuations in overall security demand. Assuming the fair price of the stock is
the average of the bid and ask prices, an investor pays half the spread upon purchase and
another half upon sale of the stock. A dealer on the other side of the transaction earns these
spreads. The spread is one important component of liquidity—it is the cost of transacting
in a security.
The advent of electronic trading has steadily diminished the role of dealers, but traders
still must contend with a bid–ask spread. For example, in electronic markets, the limitorder
book contains the “inside spread,” that is, the difference between the highest price
at which some investor will purchase any shares and the lowest price at which another
investor is willing to sell. The effective bid–ask spread will also depend on the size of the
desired transaction. Larger purchases will require a trader to move deeper into the limitorder
book and accept less-attractive prices. While inside spreads on electronic markets
often appear extremely low, effective spreads can be much larger, because the limit orders
are good for only small numbers of shares.
Even without the inventory problems faced by traditional securities dealers, the importance
of the spread persists. There is greater emphasis today on the component of the spread
that is due to asymmetric information. By asymmetric information, we mean the potential
for one trader to have private information about the value of the security that is not known
to the trading partner. To see why such an asymmetry can affect the market, think about the
problems facing someone buying a used car. The seller knows more about the car than the
buyer, so the buyer naturally wonders if the seller is trying to get rid of the car because it
is a “lemon.” At the least, buyers worried about overpaying will shave the prices they are
willing to pay for a car of uncertain quality. In extreme cases of asymmetric information,
trading may cease altogether.30
Similarly, traders who post offers to buy or sell at limit
prices need to be worried about being picked off by better-informed traders who hit their
limit prices only when they are out of line with the intrinsic value of the firm.
Broadly speaking, we may envision investors trading securities for two reasons. Some
trades are driven by “noninformational” motives, for example, selling assets to raise cash
for a big purchase, or even just for portfolio rebalancing. These sorts of trades, which are
not motivated by private information that bears on the value of the traded security, are
called noise trades. Security dealers will earn a profit from the bid–ask spread when transacting
with noise traders (also called liquidity traders because their trades may derive from
needs for liquidity, i.e., cash).
Other transactions are motivated by private information known only to the seller or
buyer. These transactions are generated when traders believe they have come across information
that a security is mispriced, and try to profit from that analysis. If an information
trader identifies an advantageous opportunity, it must be disadvantageous to the other party
in the transaction. If private information indicates a stock is overpriced, and the trader
decides to sell it, a dealer who has posted a bid price or another trader who has posted a
30
The problem of informational asymmetry in markets was introduced by the 2001 Nobel Laureate George
A. Akerlof and has since become known as the lemons problem. A good introduction to Akerlof’s contributions
can be found in George A. Akerlof, An Economic Theorist’s Book of Tales (Cambridge, U.K.: Cambridge
University Press, 1984).
308 PART III Equilibrium in Capital Markets
limit-buy order and ends up on the other side of the transaction will purchase the stock at
what will later be revealed to have been an inflated price. Conversely, when private information
results in a decision to buy, the price at which the security is traded will eventually
be recognized as less than fair value.
Information traders impose a cost on both dealers and other investors who post limit
orders. Although on average dealers make money from the bid–ask spread when transacting
with liquidity traders, they will absorb losses from information traders. Similarly, any
trader posting a limit order is at risk from information traders. The response is to increase
limit-ask prices and decrease limit-bid orders—in other words, the spread must widen. The
greater the relative importance of information traders, the greater the required spread to
compensate for the potential losses from trading with them. In the end, therefore, liquidity
traders absorb most of the cost of the information trades because the bid–ask spread that
they must pay on their “innocent” trades widens when informational asymmetry is more
severe.
The discount in a security price that results from illiquidity can be surprisingly large,
far larger than the bid–ask spread. Consider a security with a bid–ask spread of 1%. Suppose
it will change hands once a year for the next 3 years and then will be held forever
by the third buyer. For the last trade, the investor will pay for the security 99.5% or .995
of its fair price; the price is reduced by half the spread that will be incurred when the
stock is sold. The second buyer, knowing the security will be sold a year later for .995
of fair value, and having to absorb half the spread upon purchase, will be willing to pay
.995 Ϫ .005/1.05 ϭ .9902 (i.e., 99.02% of fair value), if the cost of trading is discounted
at a rate of 5%. Finally, the current buyer, knowing the loss next year, when the stock
will be sold for .9902 of fair value (a discount of .0098), will pay for the security only
.995 Ϫ .0098/1.05 ϭ .9857. Thus the discount has ballooned from .5% to 1.43%. In other
words, the present values of all three future trading costs (spreads) are discounted into the
current price.31
To extend this logic, if the security will be traded once a year forever, its
current illiquidity cost will equal immediate cost plus the present value of a perpetuity of
.5%. At an annual discount rate of 5%, this sum equals .005 ϩ .005/.05 ϭ .105, or 10.5%!
Obviously, liquidity is of potentially large value and should not be ignored in deriving the
equilibrium value of securities.
Consider three stocks with equal bid–ask spreads of 1%. The first trades once a year,
the second once every 2 years, and the third every 3 years. We have already calculated
the price discount due to illiquidity as the present value of illiquidity costs for the first as
10.5%. The discount for the second security is .5% plus the present value of a biannual
perpetuity of .5%, which at a discount rate of 5% amounts to .5 ϩ .5/(1.052
Ϫ 1) ϭ 5.38%.
Similarly, the cost for the security that trades only every 3 years is 3.67%. From this pattern
of discounts—10.5%, 5.38%, and 3.67%—it seems that for any given spread, the
price discount will increase almost in proportion to the frequency of trading. It also would
appear that the discount should be proportional to the bid–ask spread. However, trading
frequency may well vary inversely with the spread, and this will impede the response of
the price discount to the spread.
An investor who plans to hold a security for a given period will calculate the impact of
illiquidity costs on expected rate of return; liquidity costs will be amortized over the anticipated
holding period. Investors who trade less frequently therefore will be less affected
by high trading costs. The reduction in the rate of return due to trading costs is lower the
longer the security is held. Hence in equilibrium, investors with long holding periods will,
31
We will see another instance of such capitalization of trading costs in Chapter 13, where one explanation for
large discounts on closed-end funds is the substantial present value of a stream of apparently small per-period
expenses.
CHAPTER 9 The Capital Asset Pricing Model 309
on average, hold more of the illiquid securities, while short-horizon investors will more
strongly prefer liquid securities. This “clientele effect” mitigates the effect of the bid–ask
spread for illiquid securities. The end result is that the liquidity premium should increase
with the bid–ask spread at a decreasing rate. Figure 9.5 confirms this prediction.
So far, we have shown that the expected level of liquidity can affect prices, and therefore
expected rates of return. What about unanticipated changes in liquidity? Investors may
also demand compensation for liquidity risk. The bid–ask spread of a security is not constant
through time, nor is the ability to sell a security at a fair price with little notice. Both
depend on overall conditions in security markets. If asset liquidity fails at times when it is
most desired, then investors will require an additional price discount beyond that required
for the expected cost of illiquidity.32
In other words, there may be a systematic component
to liquidity risk that affects the equilibrium rate of return and hence the expected return–
beta relationship.
As a concrete example of such a model, Acharya and Pedersen33
consider the impacts
of both the level and the risk of liquidity on security pricing. They include three components
to liquidity risk—each captures the extent to which liquidity varies systematically
32
A good example of systematic effects in liquidity risk surrounds the demise of Long-Term Capital Management
in the summer of 1998. Despite extensive analysis that indicated its portfolio was highly diversified, many of its
assets went bad at the same time when Russia defaulted on its debt. The problem was that despite the fact that
short and long positions were expected to balance price changes based on normal market fluctuations, a massive
decline in the market liquidity and prices of some assets was not offset by increased prices of more liquid assets.
As a supplier of liquidity to others, LTCM was a large holder of less-liquid securities and a liquidity shock of
this magnitude was at that time an unimaginable event. While its portfolio may have been diversified in terms of
exposure to traditional business condition shocks, it was undiversified in terms of exposure to liquidity shocks.
33
V. V. Acharya and L. H. Pedersen, “Asset Pricing with Liquidity Risk,” Journal of Financial Economics 77
(2005).
F I G U R E 9.5 The relationship between illiquidity and average returns
Source: Derived from Yakov Amihud and Haim Mendelson, “Asset Pricing and the Bid–Ask Spread,” Journal of
Financial Economics 17 (1986), pp. 223–49.
1.2
0 0.5 1 1.5 2 2.5 3 3.5
Bid–Asked
Spread (%)
Average
Monthly Return
(% per month)
0
0.2
0.4
0.6
0.8
1
310 PART III Equilibrium in Capital Markets
with other market conditions. They identify three relevant “liquidity betas,” which measure
in turn: (i) the extent to which the stock’s illiquidity varies with market illiquidity;
(ii) the extent to which the stock’s return varies with market illiquidity; and (iii) the extent
to which the stock illiquidity varies with the market return. Therefore, expected return
depends on expected liquidity, as well as the conventional “CAPM beta” and three additional
liquidity-related betas:
E R kE Ci i L L L( ) ( ) ( )ϭ ϩ ␤ ϩ␤ Ϫ␤ Ϫ␤␭ 1 2 3 (9.18)
where
E(Ci) ϭ expected cost of illiquidity,
k ϭ adjustment for average holding period over all securities,
␭ ϭ market risk premium net of average market illiquidity cost, E(RM Ϫ CM),
␤ ϭ measure of systematic market risk,
␤L1, ␤L2, ␤L3 ϭ liquidity betas.
Compared to the conventional CAPM, the expected return–beta equation now has a predicted
firm-specific component that accounts for the effect of security liquidity. Such an
effect would appear to be an alpha in the conventional index model.
The market risk premium itself is measured net of the average cost of illiquidity, that is,
␭ ϭ E(RM Ϫ CM), where CM is the market-average cost of illiquidity.
The overall risk of each security now must account for the three elements of liquidity
risk, which are defined as follows:34
␤ ϭ
Ϫ
L
i M
M M
C C
R C
1
Cov
Var
( , )
( )
Measures the sensitivity of the security’s illiquidity to
market illiquidity. Investors want additional compensation
for holding a security that becomes illiquid when general
liquidity is low.34
␤ ϭ
Ϫ
L
i M
M M
R C
R C
2
Cov
Var
( , )
( )
Measures the sensitivity of the stock’s return to market
illiquidity. This coefficient appears with a negative sign
in Equation 9.18 because investors are willing to accept
a lower average return on stocks that will provide higher
returns when market illiquidity is greater.
␤ ϭ
Ϫ
L
i M
M M
C R
R C
3
Cov
Var
( , )
( )
Measures the sensitivity of security illiquidity to the
market rate of return. This sensitivity also appears with a
negative sign, because investors will be willing to accept
a lower average return on securities that can be sold
more easily (have low illiquidity costs) when the market
declines.
A good number of variations on this model can be found in the current (and rapidly
growing) literature on liquidity.35
What is common to all liquidity variants is that they
improve on the explanatory power of the CAPM equation and hence there is no doubt that,
sooner or later, practitioner optimization models and, more important, security analysis
will incorporate the empirical content of these models.
34
Several papers have shown that there is important covariance across asset illiquidity. See for example, T. Chordia,
R. Roll, and A. Subramanyam, “Commonality in Liquidity,” Journal of Financial Economics 56 (2000),
pp. 3–28 or J. Hasbrouck and D. H. Seppi “Common Factors in Prices, Order Flows and Liquidity,” Journal of
Financial Economics 59 (2001), pp. 383–411.
35
Another influential study of liquidity risk and asset pricing is L. Pastor and R. Stambaugh, “Liquidity Risk and
Expected Stock Returns,” Journal of Political Economy 111 (2003), pp. 642–85.
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CHAPTER 9 The Capital Asset Pricing Model 311
1. The CAPM assumes that investors are single-period planners who agree on a common input list
from security analysis and seek mean-variance optimal portfolios.
2. The CAPM assumes that security markets are ideal in the sense that:
a. They are large, and investors are price-takers.
b. There are no taxes or transaction costs.
c. All risky assets are publicly traded.
d. Investors can borrow and lend any amount at a fixed risk-free rate.
3. With these assumptions, all investors hold identical risky portfolios. The CAPM holds that in
equilibrium the market portfolio is the unique mean-variance efficient tangency portfolio. Thus
a passive strategy is efficient.
4. The CAPM market portfolio is a value-weighted portfolio. Each security is held in a proportion
equal to its market value divided by the total market value of all securities.
5. If the market portfolio is efficient and the average investor neither borrows nor lends, then the
risk premium on the market portfolio is proportional to its variance, ␴M
2
, and to the average
coefficient of risk aversion across investors, A:
E r r AM f M( ) Ϫ ϭ ␴2
6. The CAPM implies that the risk premium on any individual asset or portfolio is the product of
the risk premium on the market portfolio and the beta coefficient:
E r r E r ri f i M f( ) [ ( ) ]Ϫ ϭ ␤ Ϫ
where the beta coefficient is the covariance of the asset with the market portfolio as a fraction of
the variance of the market portfolio
␤ ϭ
␴
i
i M
M
r rCov( , )
2
7. When risk-free investments are restricted but all other CAPM assumptions hold, then the simple
version of the CAPM is replaced by its zero-beta version. Accordingly, the risk-free rate in
the expected return–beta relationship is replaced by the zero-beta portfolio’s expected rate of
return:
E r E r E r ri Z M i M Z M( ) [ ] [ ]ϭ ϩ␤ Ϫ( ) ( )
8. The simple version of the CAPM assumes that investors are myopic. When investors are
assumed to be concerned with lifetime consumption and bequest plans, but investors’ tastes and
security return distributions are stable over time, the market portfolio remains efficient and the
simple version of the expected return–beta relationship holds. But if those distributions change
unpredictably, or if investors seek to hedge nonmarket sources of risk to their consumption, the
simple CAPM will give way to a multifactor version in which the security’s exposure to these
nonmarket sources of risk command risk premiums.
9. The consumption-based capital asset pricing model (CCAPM) is a single-factor model in which
the market portfolio excess return is replaced by that of a consumption-tracking portfolio. By
appealing directly to consumption, the model naturally incorporates consumption-hedging considerations
and changing investment opportunities within a single-factor framework.
10. The Security Market Line of the CAPM must be modified to account for labor income and other
significant nontraded assets.
11. Liquidity costs and liquidity risk can be incorporated into the CAPM relationship. Investors
demand compensation for both expected costs of illiquidity as well as the risk surrounding
those costs.
SUMMARYSUMMARY
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Related Web sites for
this chapter are available
at www.mhhe.com/bkm
312 PART III Equilibrium in Capital Markets
1. What must be the beta of a portfolio with E(rP) ϭ 18%, if rf ϭ 6% and E(rM) ϭ 14%?
2. The market price of a security is $50. Its expected rate of return is 14%. The risk-free rate is 6%
and the market risk premium is 8.5%. What will be the market price of the security if its correlation
coefficient with the market portfolio doubles (and all other variables remain unchanged)?
Assume that the stock is expected to pay a constant dividend in perpetuity.
3. Are the following true or false? Explain.
a. Stocks with a beta of zero offer an expected rate of return of zero.
b. The CAPM implies that investors require a higher return to hold highly volatile securities.
c. You can construct a portfolio with beta of .75 by investing .75 of the investment budget in
T-bills and the remainder in the market portfolio.
4. You are a consultant to a large manufacturing corporation that is considering a project with the
following net after-tax cash flows (in millions of dollars):
Years from Now After-Tax Cash Flow
0 Ϫ40
1–10 15
The project’s beta is 1.8. Assuming that rf ϭ 8% and E(rM) ϭ 16%, what is the net present
value of the project? What is the highest possible beta estimate for the project before its NPV
becomes negative?
5. Consider the following table, which gives a security analyst’s expected return on two stocks for
two particular market returns:
Market Return Aggressive Stock Defensive Stock
5% Ϫ2% 6%
25 38 12
a. What are the betas of the two stocks?
b. What is the expected rate of return on each stock if the market return is equally likely to be
5% or 25%?
c. If the T-bill rate is 6% and the market return is equally likely to be 5% or 25%, draw the
SML for this economy.
d. Plot the two securities on the SML graph. What are the alphas of each?
e. What hurdle rate should be used by the management of the aggressive firm for a project with
the risk characteristics of the defensive firm’s stock?
For Problems 6 to 12: If the simple CAPM is valid, which of the following situations are
possible? Explain. Consider each situation independently.
6.
Portfolio
Expected
Return Beta
A 20 1.4
B 25 1.2
PROBLEM
SETS
PROBLEM
SETS
QuizQuiz
ProblemsProblems
homogeneous expectations
market portfolio
mutual fund theorem
market price of risk
beta
expected return–beta
relationship
security market line (SML)
alpha
market model
zero-beta portfolio
liquidity
illiquidity
KEY TERMSKEY TERMS
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CHAPTER 9 The Capital Asset Pricing Model 313
7.
Portfolio
Expected
Return
Standard
Deviation
A 30 35
B 40 25
8.
Portfolio
Expected
Return
Standard
Deviation
Risk-free 10 0
Market 18 24
A 16 12
9.
Portfolio
Expected
Return
Standard
Deviation
Risk-free 10 0
Market 18 24
A 20 22
10.
Portfolio
Expected
Return Beta
Risk-free 10 0
Market 18 1.0
A 16 1.5
11.
Portfolio
Expected
Return Beta
Risk-free 10 0
Market 18 1.0
A 16 0.9
12.
Portfolio
Expected
Return
Standard
Deviation
Risk-free 10 0
Market 18 24
A 16 22
For Problems 13 to 15 assume that the risk-free rate of interest is 6% and the expected
rate of return on the market is 16%.
13. A share of stock sells for $50 today. It will pay a dividend of $6 per share at the end of the year.
Its beta is 1.2. What do investors expect the stock to sell for at the end of the year?
14. I am buying a firm with an expected perpetual cash flow of $1,000 but am unsure of its risk. If I
think the beta of the firm is .5, when in fact the beta is really 1, how much more will I offer for
the firm than it is truly worth?
15. A stock has an expected rate of return of 4%. What is its beta?
16. Two investment advisers are comparing performance. One averaged a 19% rate of return and
the other a 16% rate of return. However, the beta of the first investor was 1.5, whereas that of the
second was 1.
a. Can you tell which investor was a better selector of individual stocks (aside from the issue of
general movements in the market)?
b. If the T-bill rate were 6% and the market return during the period were 14%, which investor
would be the superior stock selector?
c. What if the T-bill rate were 3% and the market return were 15%?
314 PART III Equilibrium in Capital Markets
17. Suppose the rate of return on short-term government securities (perceived to be risk-free) is
about 5%. Suppose also that the expected rate of return required by the market for a portfolio
with a beta of 1 is 12%. According to the capital asset pricing model:
a. What is the expected rate of return on the market portfolio?
b. What would be the expected rate of return on a stock with ␤ ϭ 0?
c. Suppose you consider buying a share of stock at $40. The stock is expected to pay $3 dividends
next year and you expect it to sell then for $41. The stock risk has been evaluated at
␤ ϭ Ϫ.5. Is the stock overpriced or underpriced?
18. Suppose that borrowing is restricted so that the zero-beta version of the CAPM holds. The
expected return on the market portfolio is 17%, and on the zero-beta portfolio it is 8%. What is
the expected return on a portfolio with a beta of .6?
19. a. A mutual fund with beta of .8 has an expected rate of return of 14%. If rf ϭ 5%, and you
expect the rate of return on the market portfolio to be 15%, should you invest in this fund?
What is the fund’s alpha?
b. What passive portfolio comprised of a market-index portfolio and a money market account
would have the same beta as the fund? Show that the difference between the expected rate of
return on this passive portfolio and that of the fund equals the alpha from part (a).
20. Outline how you would incorporate the following into the CCAPM:
a. Liquidity
b. Nontraded assets (Do you have to worry about labor income?)
Challenge
Problem
Challenge
Problem
1. a. John Wilson is a portfolio manager at Austin & Associates. For all of his clients, Wilson
manages portfolios that lie on the Markowitz efficient frontier. Wilson asks Mary Regan,
CFA, a managing director at Austin, to review the portfolios of two of his clients, the Eagle
Manufacturing Company and the Rainbow Life Insurance Co. The expected returns of the
two portfolios are substantially different. Regan determines that the Rainbow portfolio is
virtually identical to the market portfolio and concludes that the Rainbow portfolio must be
superior to the Eagle portfolio. Do you agree or disagree with Regan’s conclusion that the
Rainbow portfolio is superior to the Eagle portfolio? Justify your response with reference to
the capital market line.
b. Wilson remarks that the Rainbow portfolio has a higher expected return because it has
greater nonsystematic risk than Eagle’s portfolio. Define nonsystematic risk and explain
why you agree or disagree with Wilson’s remark.
2. Wilson is now evaluating the expected performance of two common stocks, Furhman Labs Inc.
and Garten Testing Inc. He has gathered the following information:
• The risk-free rate is 5%.
• The expected return on the market portfolio is 11.5%.
• The beta of Furhman stock is 1.5.
• The beta of Garten stock is .8.
Based on his own analysis, Wilson’s forecasts of the returns on the two stocks are 13.25% for
Furhman stock and 11.25% for Garten stock. Calculate the required rate of return for Furhman
Labs stock and for Garten Testing stock. Indicate whether each stock is undervalued, fairly valued,
or overvalued.
3. The security market line depicts:
a. A security’s expected return as a function of its systematic risk.
b. The market portfolio as the optimal portfolio of risky securities.
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CHAPTER 9 The Capital Asset Pricing Model 315
c. The relationship between a security’s return and the return on an index.
d. The complete portfolio as a combination of the market portfolio and the risk-free asset.
4. Within the context of the capital asset pricing model (CAPM), assume:
• Expected return on the market ϭ 15%.
• Risk-free rate ϭ 8%.
• Expected rate of return on XYZ security ϭ 17%.
• Beta of XYZ security ϭ 1.25.
Which one of the following is correct?
a. XYZ is overpriced.
b. XYZ is fairly priced.
c. XYZ’s alpha is Ϫ.25%.
d. XYZ’s alpha is .25%.
5. What is the expected return of a zero-beta security?
a. Market rate of return.
b. Zero rate of return.
c. Negative rate of return.
d. Risk-free rate of return.
6. Capital asset pricing theory asserts that portfolio returns are best explained by:
a. Economic factors.
b. Specific risk.
c. Systematic risk.
d. Diversification.
7. According to CAPM, the expected rate of return of a portfolio with a beta of 1.0 and an alpha of
0 is:
a. Between rM and rf.
b. The risk-free rate, rf.
c. ␤ (rM Ϫ rf).
d. The expected return on the market, rM.
The following table shows risk and return measures for two portfolios.
Portfolio
Average Annual
Rate of Return
Standard
Deviation Beta
R 11% 10% 0.5
S&P 500 14% 12% 1.0
8. When plotting portfolio R on the preceding table relative to the SML, portfolio R lies:
a. On the SML.
b. Below the SML.
c. Above the SML.
d. Insufficient data given.
9. When plotting portfolio R relative to the capital market line, portfolio R lies:
a. On the CML.
b. Below the CML.
c. Above the CML.
d. Insufficient data given.
316 PART III Equilibrium in Capital Markets
10. Briefly explain whether investors should expect a higher return from holding portfolio A versus
portfolio B under capital asset pricing theory (CAPM). Assume that both portfolios are fully
diversified.
Portfolio A Portfolio B
Systematic risk (beta) 1.0 1.0
Specific risk for each
individual security High Low
11. Joan McKay is a portfolio manager for a bank trust department. McKay meets with two clients,
Kevin Murray and Lisa York, to review their investment objectives. Each client expresses an
interest in changing his or her individual investment objectives. Both clients currently hold
well-diversified portfolios of risky assets.
a. Murray wants to increase the expected return of his portfolio. State what action McKay
should take to achieve Murray’s objective. Justify your response in the context of the CML.
b. York wants to reduce the risk exposure of her portfolio but does not want to engage in borrowing
or lending activities to do so. State what action McKay should take to achieveYork’s
objective. Justify your response in the context of the SML.
12. Karen Kay, a portfolio manager at Collins Asset Management, is using the capital asset pricing
model for making recommendations to her clients. Her research department has developed the
information shown in the following exhibit.
Forecast Returns, Standard Deviations, and Betas
Forecast Return Standard Deviation Beta
Stock X 14.0% 36% 0.8
Stock Y 17.0 25 1.5
Market index 14.0 15 1.0
Risk-free rate 5.0
a. Calculate expected return and alpha for each stock.
b. Identify and justify which stock would be more appropriate for an investor who wants to
i. add this stock to a well-diversified equity portfolio.
ii. hold this stock as a single-stock portfolio.
Go to www.mhhe.com/edumarketinsight and link to Company, then Population. Select
a company of interest to you and link to the Company Research page. Look for the Excel
Analytics section, and choose Valuation Data, then review the Profitability report. Find the
row that shows the historical betas for your firm. Is beta stable from year to year? Go back
to the Company Research page and look at the latest available S&P Stock Report for your
firm. What beta does the report indicate for your firm? Why might this be different from the
one in the Profitability Report? Based on current risk-free rates (available at finance.yahoo
.com), and the historical risk premiums discussed in Chapter 5, estimate the expected rate
of return on your company’s stock by using the CAPM.
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CHAPTER 9 The Capital Asset Pricing Model 317
SOLUTIONS TO CONCEPT CHECKS
1. We can characterize the entire population by two representative investors. One is the
“uninformed” investor, who does not engage in security analysis and holds the market portfolio,
whereas the other optimizes using the Markowitz algorithm with input from security analysis.
The uninformed investor does not know what input the informed investor uses to make portfolio
purchases. The uninformed investor knows, however, that if the other investor is informed, the
market portfolio proportions will be optimal. Therefore, to depart from these proportions would
constitute an uninformed bet, which will, on average, reduce the efficiency of diversification with
no compensating improvement in expected returns.
2. a. Substituting the historical mean and standard deviation in Equation 9.2 yields a coefficient of
risk aversion of
A
E r rM f
M
ϭ
Ϫ
␴
ϭ ϭ
( ) .
.
.2 2
084
203
2 04
b. This relationship also tells us that for the historical standard deviation and a coefficient of risk
aversion of 3.5 the risk premium would be
E r r AM f M( ) . . . . %Ϫ ϭ ␴ ϭ ϫ ϭ ϭ2 2
3 5 203 144 14 4
3. For these investment proportions, wFord, wGM, the portfolio ␤ is
␤ ϭ ␤ ϩ ␤
ϭ ϫ ϩ ϫ ϭ
P w wFord Ford GM GM
(. . ) (. . )75 1 25 25 1 10 11 2125.
As the market risk premium, E(rM) Ϫ rf, is 8%, the portfolio risk premium will be
E r r E r rP f P M f( ) [ ( ) ]
. . %
Ϫ ϭ ␤ Ϫ
ϭ ϫ ϭ1 2125 8 9 7
Beta and Security Returns
Fidelity provides data on the risk and return of its funds at www.fidelity.com.
Click on the Research link, then choose Mutual Funds from the submenu. In the Fund
Evaluator section, choose Advanced Search. Scroll down until you find the Risk/ Volatility
Measures section and indicate that you want to screen for funds with betas less
than or equal to .50. Click Search Funds to see the results. Click on the link that says
View All Matching Fidelity Funds. Select five funds from the resulting list and click
Compare. Rank the five funds according to their betas and then according to their
standard deviations. Do both lists rank the funds in the same order? How would you
explain any difference in the rankings? Note the 1-Year return for one of the funds
(use the load-adjusted return if it is available). Repeat the exercise to compare five
funds that have betas greater than or equal to 1.50.
E-Investments
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Visitusatwww.mhhe.com/bkm
318 PART III Equilibrium in Capital Markets
4. The alpha of a stock is its expected return in excess of that required by the CAPM.
␣ ϭ Ϫ ϩ␤ Ϫ
␣ ϭ Ϫ ϩ Ϫ
E r r E r rf M f
XYZ
( ) { [ ( ) ]}
[ . ( )12 5 1 0 11 5 ]] %
[ . ( )] %
ϭ
␣ ϭ Ϫ ϩ Ϫ ϭ Ϫ
1
13 5 1 5 11 5 1ABC
ABC plots below the SML, while XYZ plots above.
E(r), Percent
β
.5 1 1.5
14
αXYZ > 0
XYZ
Market
SML
0
αABC < 0
E(rM) = 11
rf = 5
ABC12
5. The project-specific required return is determined by the project beta coupled with the market
risk premium and the risk-free rate. The CAPM tells us that an acceptable expected rate of return
for the project is
r E r rf M fϩ␤ Ϫ ϭ ϩ Ϫ ϭ[ ( ) ] . ( ) . %8 1 3 16 8 18 4
which becomes the project’s hurdle rate. If the IRR of the project is 19%, then it is desirable. Any
project with an IRR equal to or less than 18.4% should be rejected.
6. The CAPM is a model that relates expected rates of return to risk. It results in the expected
return–beta relationship, where the expected risk premium on any asset is proportional to the
expected risk premium on the market portfolio with beta as the proportionality constant. As such
the model is impractical for two reasons: (i) expectations are unobservable, and (ii) the theoretical
market portfolio includes every risky asset and is in practice unobservable. The next three models
incorporate additional assumptions to overcome these problems.
The single-factor model assumes that one economic factor, denoted F, exerts the only common
influence on security returns. Beyond it, security returns are driven by independent, firm-specific
factors. Thus for any security, i,
r E r F ei i i iϭ ϩ␤ ϩ( )
The single-index model assumes that in the single-factor model, the factor F can be replaced by a
broad-based index of securities that can proxy for the CAPM’s theoretical market portfolio. The
index model can be stated as Ri ϭ ␣i ϩ ␤iRM ϩ ei.
At this point it should be said that many interchange the meaning of the index and market
models. The concept of the market model is that rate of return surprises on a stock are proportional
to corresponding surprises on the market index portfolio, again with proportionality constant ␤.