OBJECIEVES

Explain the theory behind the capital asset pricing model (CAPM).

Explain how to use the CAPM to establish benchmarks for measuring the performance of investment portfolios.

Explain how to infer from the CAPM the correct risk-adjusted discount rate to use in discounted cash now
valuation model.

Explain how the CAPM has been modified and supplemented by other theories to add greater realism.
CONTENTS
13.1
The Capital Asset Pricing Model in Brief
13.2
13.3
13.4
13.5
13.6
Determinants of the Risk Premium on the Market Portfolio
Beta and Risk Premiums on Individual Securities
Using the CAPM in Portfolio Selection
Valuation and Regulating Rates of Return
Modifications and Alternatives to the CAPM
The capital asset pricing model (CAPM) is a theory about equilibrium prices in the markets for risky assets. It builds on
the theory of portfolio selection developed in chapter 12 and derives the quantitative relations that must exist among
expected rates of return on risky assets on the assumption that asset prices adjust to equate supply and demand.
The CAPM is important for two reasons. First, it provides a theoretical justification for the widespread practice of
passive investing known as indexing. Indexing means holding a diversified portfolio in which securities are held in the
same relative proportions as in a broad market index such as the Standard & Pool’s 500 or the Morgan Stanley Index of
international stocks. Today many billions of dollars invested worldwide by pension funds, mutual funds, and other
institutions are managed passively by indexing, and indexing provides a simple feasible benchmark against which the
performance of active investment strategies are measured.
Second, the CAPM provides a way of estimating expected rates of return for use in a variety of financial applications.
For example, chapter 9 shows that risk adjusted expected rates of return are needed as inputs to discounted-cash-flow
valuation models for stocks. Chapter 16 shows how corporate managers use these models in making capital-budgeting
decisions. The CAPM is also used to establish “fair” rates of return on invested capital in regulated firms or in firms
that do business on a cost-plus basis.
13.1
THE CAPITAL ASSET PRICING MODEL IN BRIEF
The capital asset pricing model is an equilibrium theory that is based on the theory of portfolio selection
presented in chapter 12. The CAPM was developed in the early 1960s. It was derived by posing the question: What
would risk premiums on securities be in equilibrium if people had the same set of forecasts of expected returns and
risks and all chose their portfolios optimally according to the principle of efficient diversification?
The fundamental idea behind the CAPM is that in equilibrium the market rewards people for bearing risk.
Because people generally exhibit risk-averse behavior, the risk premium for the aggregate of all risky assets must be
positive to induce people to willingly hold ail of the risky assets that exist in the economy.
But the market does not reward people for holding inefficient portfolios——that is, for exposing themselves to
risks that could be eliminated by optimal diversification behavior. The risk premium on any individual security is,
therefore, not related to the security's "stand-alone" risk, but rather to its contribution to the risk of an efficiently
diversified portfolio.
Chapter 12showed that every efficient portfolio can be constructed by mixing just two particular assets: the
riskless asset and the optimal combination of risky assets (i.e., the tangency portfolio). To derive the CAPM, we need
two assumptions:
-Assumption l. · Investors agree in their forecasts of expected rates of 24~ standard deviations, and correlations
of the risky securities, and they, therefore, optimally hold risky assets in the same relative
proportions.
-Assumption2. · Investors generally behave optimally. In equilibrium, the prices of securities adjust so that
when investors are holding their optimal portfolios, the aggregate demand for each security
is equal to its supplyFrom these two assumptions, because every investor's relative holdings of risky assets is the same, the only way
the asset market can clear is if those optimal relative proportions arc the proportions in which they are valued in the
market place. A portfolio that holds all assets in proportion to their observed market values is called the market
portfolio. The composition of the market portfolio reflects the supplies of existing assets evaluated at their current
market prices.
Let us clarify what the market portfolio means. In the market portfolio, the fraction allocated to security i equals
the ratio of the market value of the ith security outstanding to the market value of all assets outstanding. Thus, for
simplicity suppose that there are only three assets: GM stock, Toyota stock, and the risk-free asset. The total market
values of each at current prices are $66 billion of GM, $22 billion of Toyota, and $12 billion of the risk-free asset. The
total market value of all assets is $100 billion. The composition of the market portfolio is, therefore, 66% GM stock,
22% Toyota stock, and 12% risk-free asset.
The CAPM says that in equilibrium any investor’s relative holdings of risky assets will be the same as in the
market portfolio. Depending on their risk aversion, investors hold different mixes of risk-free and risky assets, but the
relative holdings of risky assets are the same for all investors. Thus, in our simple example, all investors will hold GM
and Toyota stock in the proportions of 3 to 1 (i.e., 66/22). Another way to state this is to say that the composition of the
risky part of any investor's portfolio will be 75% GM stock and 25% Toyota stock.
Consider two investors, each with $100,000 to invest. Investor 1 has risk aversion equal to the average for
a11investors and, therefore, holds each asset in the same proportions as the market portfolio——$66,000 in GM,
$22,000 in Toyota stock, and $12,000 in the risk-free asset. Investor 2 is more risk averse than the average and,
therefore, chooses to invest $24,000 (twice as much as Investor 1) in the risk-free asset and $76,000 in risky assets.
Investor 2's investment in GM stock wi1l be .75 × $76,000 or $57,000, and the investment in Toyota stock will be 25
× $76,000 or $19,000. Thus, both investors will hold three times as much in GM stock as in Toyota stock.
The basic idea of the CAPM can also be explained with the help of Figure 13.1, which depicts the risk-reward
trade-off line facing each investor. Because the tangency portfolio or optimal combination of risky assets has the same
relative holdings of risky assets as the market portfolio, the market portfolio is located somewhere on the risk-return
trade-off line. In the CAPM, the trade-off line is called the capital market line (CML). In Figure 13.1, point M
represents the market portfolio, point F is the risk-free asset, and the CML is the straight line connecting these two
points.
The CAPM says that in equilibrium, the CML represents the best risk-reward combinations available to all
investors. Although everyone will strive to achieve points that are above the CML, the forces of competition will move
asset prices so that everyone achieves points that are on the line.
The CML’s formula is
E (r )  rf 
E (rM )  rf
M
The slope of the CML is, thus, the risk premium on the market portfolio divided by its standard deviation:
Slope of SML 
E (rM )  rf
M
The CAPM implies that most investors would do just as well to passively combine the risk-free asset with an
index fund holding risky assets in the proportions as in the market portfolio as they would by actively researching
securities and trying to “bear” the market. Especially diligent and competent investors do tend to earn rewards for
their efforts, but over time the competition among them to reduces those rewards to the minimum necessary to
induce them to perform their work. The rest of us can then benefit from their work by investing passively.
Another implication of the CAPM is that the risk premium on any individual security is proportional only to its
contribution to the risk of the market portfolio. The risk premium does not depend on the security's stand-alone risk.
Thus, according to the CAPM, in equilibrium, investors get rewarded with a higher expected return only for bearing
market risk. This is an irreducible or necessary risk that they must take to get their desired expected return.
The logic here is that because all efficient risk-reward combinations can be achieved simply by mixing the
market portfolio and the risk, free asset, the only risk an investor need bear to achieve an efficient portfolio is
market risk. Therefore, the market does not reward investors for bearing any non-market risk.
The market does not reward investors for choosing inefficient portfolio.
Sometimes this implication of the CAPM is emphasized by saying that only the market-related risk of a
security "matters”.
13.2 DETERMINANTS OF IHE RISK PREMIUM ON THE MARKET PORTFOIIO
According to the CAPM, the size of the risk premium of the market portfolio is determined by the aggregate risk
aversion of investors and the volatility of the market return. To be induced to accept the risk of the market portfolio,
investors must be offered an expected rate of return that exceeds the risk-free rate of interest. The greater the average
degree of risk aversion of the population, the higher the risk premium requiredIn the CAPM, the equilibrium risk premium on the market portfolio is equal to the variance of the market
portfolio times a weighted average of the degree of risk aversion of the holders of wealth (A):
E(rM )  rf  A M2
A should be thought of as an index of the degree of risk aversion in the economy.
Suppose that the standard deviation of the market portfolio is .20, and the average degree of risk aversion is 2.
Then the risk premium on the market portfolio is .08:
E(rM )  rf  2  .22  2  .04  .08
IT111s, according to the CAPM, the market risk premium can change over time either because the variance of
the market changes, because the degree of risk aversion changes, or both.
Note that the CAPM explains the difference between the riskless interest rate and the expected rate of return on
the market portfolio, but not their absolute levels. As discussed in chapter 4, the absolute level of the equilibrium
expected rate of return on the market portfolio is determined by factors such as the expected productivity of the capital
stock and household intertemporal preferences for consumption.
Given a particular level for the expected return on the market, the CAPM can be used to determine the riskless
rate of interest. In our numerical example, if the expected return on the market portfolio is 14 per year, then the CAPM
implies that the risk-free rate must be .06 per year.
Substituting these values into equation 13.1, the CML is given by the following formular:
E (r )  rf 
 .06  .40
E (rM )  rf
M
where the slope, the market reward to-risk ratio, is .40.
13.3
BETA AND RISK PREMIUMS ON INDIVIDUAL SECURITIES
By definition, equilibrium asset prices and expected returns arc such that knowledgeable investors willingly hold the
assets they have in their optimal portfolios. With the idea that investors must be compensated in terms of expected
return for bearing risk, we define the risk of a security by the size of its equilibrium expected return. Thus, the risk of
security A is larger than the risk of security B if in equilibrium the expected return on A exceeds the expected return
on B. By inspection of the CML in Figure 13.1, among optimal portfolios, the larger the standard deviation of its
return, the larger the equilibrium expected return E (r), and, therefore, the larger the risk. Hence, the risk of an
efficient portfolio is measured by σ. However, standard deviation of return does not measure generally the risk of
securities in the CAPM. Instead, the general measure of a security's risk is its beta (the Greek letter p). Technically,
beta describes the marginal contribution of that security's return to the standard deviation of the market portfolio's
return. The formula for the beta of security j is given by
 jM
2
denotes the covariance between the return on security j andM the return on the market portfolio.

where σ
According to the CAPM, in equilibrium, the risk premium on any asset is equal to its beta times the risk
premium on the market portfolio. The equation expressing this relation is
E (rj )  rf   [ E (rM )  rf ]
This is called the security market line (SML) relation, and it is depicted that in Figure 13.2, we plot the security's
beta on the horizontal axis and its expected return on the vertical. The slope of the SML is the risk premium on the
market portfolio. In our example, because the market risk premium is .08 or 8% per year, the SML relation is
E (rj )  rf  .08 j
Beta also provides a proportional measure of the sensitivity of a security’s realized return to the realized return on the
market portfolio. Thus, if the realized return on the market portfolio is Y% greater (less) than was expected, then the
realized return on security j will tend to b ×Y% greater (less) than was expected. Thus, securities with high betas
(greater than 1) are called “aggressive" because their returns tend to accentuate those of the overall market portfolio,
going up more in up markets and down more in down markets. Similarly, securities with low beta of 1, and securities
with a beta of 1 are said to have "average risk”.
If any security had an expected return and beta combination that was not on the SML, it would be a
contradiction of the CAPM. In particular, imagine a security with an expected return/beta combination represented by
point J in Figure 13.2. Because it lies below the SML, its expected return is "too low, to support equilibrium.
(Equivalent13.1, we can say that its market price is too high.)
The existence of such a situation contradicts the CAPM because it implies either that the market is not in
equilibrium or that investors do not agree on the distribution of returns or that investors arc not behaving as
mean-variance optimizers. Under the assumptions of the CAPM, investors could improve their portfolios by investing
less in security J and more in the other securities. Therefore, there is excess supply of security J and excess demand for
the other securities.
Any portfolio that lies on the CML (i.e. any portfolio formed by mixing the market portfolio and the riskless
asset) has a beta equal to the fraction of the portfolio invested in the market portfolio. For example, the beta of a
portfolio that is 0.75invested in the market portfolio and 0.25 invested in the risk-free asset is 0.75.
13.4
USING THE CAPM EN PORTEOIEO SELECTION
As we saw in section 133, the CAPM implies that the market portfolio of risky assets is an efficient portfolio. This
means that an investor will do as well by simply following a passive portfo1io selection strategy of combining a market
index fund and the risk-free asset as by following an active strategy of trying to beat the market.
Whether or not the CAPM applies to real-world asset prices, it nevertheless provides a rationale for a simple
passive portfolio strategy:
-Diversify your holdings of risky assets in the proportions of the market portfolio, and
-Mix this portfolio with the risk-free asset to achieve a desired risk-reward combination.
The same passive strategy can serve as a risk-adjusted benchmark for measuring the performance of active portfolio
selection strategies.
Let us illustrate. Suppose that you have $1 million to invest. You are deciding how to allocate it among two
risky asset classes: stocks and bonds and the risk-free asset. You know that in the economy as a whole, the net relative
supplies of these three asset classes are 60% in stocks, 40% in bonds, and 0% in the risk-free asset. This, therefore, is
the composition of the market portfolio.
If you have an average degree of risk aversion, then you will invest $600,000 in stocks, $400,000 in bonds, and
nothing in the risk-free asset. If you are more z, averse than average, you will invest some of your $1million in the
risk-free asset, and the rest in stocks and bonds. Whatever amount you invest in stocks and bonds will be allocated in
the proportions 60% in stocks and 40% in bonds.
In assessing the performance of portfolio managers on a risk-adjusted basis, the CAPM suggests a simple
benchmark based on the CML. It consists of comparing the rate of return earned on the managed portfolio to the rate of
return attainable by simply mixing the market portfolio and risk-free asset in proportion would have produced the same
volatility.
The method requires one to compute the volatility of the managed portfolio over the relevant period in the
past-for instance, the last 10 years, and then to figure out what the average rate of return would have been on a
strategy of mixing market portfolio and risk-free asset to produce a portfolio with that same volatility. Then compare
the managed portfolio's average rate of return to this simple benchmark portfolio's average rate of return.
In practice, the market portfolio actually used in measuring the performance of portfolio managers is a
well-diversified portfolio of stocks rather than the true market portfolio of all risky assets. It turns out that the simple
benchmark strategy has been a difficult one to beat. Studies of the performance of managed equity mutual funds
consistently find that the simple strategy outperforms around two-thirds of the funds. As a result, more households and
pension funds have been adopting the passive investment strategy used as the performance benchmark. This type of
strategy has come to be known as indexing, because the portfolio used as a proxy for the market portfolio often has the
same weights as well-known stock market indexes such as the Standard & Poor's 500.
Whether or not the CAPM is a valid theory, indexing is an attractive investment strategy for at least two reasons.
First, as an empirical matter, it has historically performed better than most actively managed portfolios. Second, it costs
less to implement than an active portfolio strategy, because one does not incur the costs of research to look for
mispriced securities, and the cost of transactions is typically much less.
As we have seen, the CML provides a convenient and challenging benchmark for measuring the performance of
an investor’s entire portfolio of assets. However, households and pension funds often use several different portfolio
managers, each of whom manages only a part of the whole portfolio. For measuring performance of such managers, the
CAPM suggests a different benchmark —— the SML.
As we saw in section 13.3, the CAPM holds that every security has a risk premium equal to its beta times the risk
premium on the market portfolio. The difference between the average rate of return on a security or a portfolio. The
difference between the average rate of return on a security or a portfolio of securities and its SML relation is called
alpha (the Greek letter?)
If a portfolio manager can consistently produce a positive alpha, then her performance is judged to be superior,
even if the managed portfolio does not outperform the CML as a stand-alone investment.
To understand this puzzle, consider how a fund with a positive alpha can be used by an investor in combination
with the market portfolio and the risk-free asset to create a total portfolio that outperforms the CML. Let us illustrate
the example.
Assume that the risk-free rate is 6% per year, the risk premium on the market portfolio is 8% per year, and the
standard deviation on the market portfolio is 20% per year. Suppose the Alpha Fund is a managed mutual fund with a
beta of 0.5, an alpha of 1% per year, and a standard deviation of 15%.
Figures 13.3 and 13.4 show the relation of Alpha Fund to the SML and the CML. In both figures, point Alpha
represents the Alpha Fund. In Figure 133,Alpha lies above the SML Alpha Fund's α is measured as the vertical
distance between Alpha and the SML.
In Figure 13.4, Alpha lies below the CML, and, therefore, is not efficient. Alpha Fund would never be held by
any investor as a total portfolio because investors could achieve lower risk and/or a higher expected return by mixing
the market portfolio and the risk-free asset. However, by combining Alpha Fund with the market portfolio in certain
optimal proportions, investors can achieve points that lie above the CML.
Point Q in Figure 13.4corresponds to the optimal combination of Alpha Fund and the market portfolio. By
mixing this portfolio with the risk-free asset, investors can achieve risk-return combinations anywhere along the line
connecting points F and Q, which lie above the CML. Thus, if you can find a portfolio manager with a positive α ,
you
beat the market (indexing investment strategy).
El---can
--i-13.5
VALUATION AND REGUEATENG RATES OF RETURN
In addition to their use in portfolio selection, risk premiums derived from the CAPM are employed in discounted cash
now (DCF) valuation models and in capital budgeting decisions of firms. They are also used to establish "fair" rates of
return on invested capital in regulated firms or in firms that do business on a cost-plus basis. In this section, we offer
brief examples of each of these applications.
13.5.1
Discounted Cash Flow Valuation Models
As we saw in chapter 7, some widely used methods of valuing a firm's stock view the price of a share as the present
value of all expected future dividends discounted at the market capitalization rate.
P0

Dt
D1
D2





2
t
(1  k ) (1  k )
t 1 (1  k )
where Dr is the expected dividend per share in period t and k is the risk-adjusted discount rate, which is the expected
rate of return that investors require to invest in the stock. In applying this formula, analysts often employ the CAPM to
compute k.
For example, Steadygrowth Corporation's dividends per share are expected to grow at a constant rate of 10% per
year .The expected stream of future dividends is
D1
$5
D2
$5.50
D3
$6.05
etc.
etc
As we showed in chapter 9, the present value of a perpetual stream of dividends growing at a constant rate, g, is
P0 
Dt
kg
With Steadygrowth's data, this implies that the price of the stock is
P0 
5
k  .10
One way to find k is to estimate Steadygrowth's beta and infer Steadygrowth's risk premium from the SML
relation:
k steady  rf   steady[ E (rM )  rf ]
Thus, suppose that the risk-free rate is.03, psteadv=15, and the risk premium on the market portfolio is.08. Then
k =. 15 per year. Substituting this value into the constant growth rate DDM, the estimated value of Steadygrowth stock
is
P0 
13.5.2
5
5

 100
k  .10 .12  .10
Cost of Capital
As we will see in chapter 16, corporate financial managers need to know their firm's cost of capital to make investment
(capital-budgeting) decisions. The firm's cost of capital is a weighted average of its cost of equity capital and debt.
Practitioners often use a CAPM-based method similar to the one we just demonstrated for Steady- growth Corporation
to estimate the cost of equity capital.
For example, suppose that you are the financial manager of ABC Corporation and you want to compute your
firm’s cost of equity capital. You compute the beta of ABC stock and find it to be 1.1. The current risk-free rate
is .06per year, and you assume that the market risk premium is .08 per year. men according to the SML the equilibrium
expected rate of return on ABC stock is
E (rABC )  rf   ABC [ E (rM )  rf ]  .06  1.1 .08  .148
Thus, .148 per year is ABC's cost of equity capital.
13.5.3
Regulation and Cost-Plus Pricing
Regulators use the CAPM to establish a "fair" rate of return on invested capital for public utilities and other firms
subject to price regulation. For example, a commission regulating an electric power company may have to establish a
price that the company is allowed to charge its customers for electricity. The commission will do so by computing the
cost of producing the electricity, including an allowance for the cost of capital.
Similarly, in situations in which a price is negotiated by two parties based on production cost, there is often a
need to decide on a fair allowance for the cost of capital. An example would be a noncompetitive (secret) contract to
develop z produce military equipment for a government.
In computing the cost of capital, a regulatory commission must compensate the providers of capital for the risk
they bear by investing in the electric utility. Because the investors are able to diversify their investment portfolios, the
only risk the regulators need to compensate them for is market risk, as measured by beta.
13.6 MODIFICATIONS AND ALTERNAIEVES TO THE CAPM
As early as the 1970s, researchers testing the empirical validity of the Security Market Line using the historical returns
of common stocks in the United States found that it did not seem to fit the data well enough to explain fully the
structure of expected returns on assets. Subsequent and currently ongoing research formulated and tested a variety of
enriched CAPM and alternative models using data from a variety of asset markets around the world. A consensus has
emerged that the original simple version of the CAPM needs to be modified.
Potential explanations for the apparent deviations from the CAPM fall into three categories. One such is that the
CAPM actually does hold, but the “market”portfolios used in the testing were incomplete and inadequate
representations of the true market portfolio. Another focuses on market imperfections not contemplated in the CAPM,
such as borrowing costs and constraints, short sale restrictions and costs, different tax treatments for various assets, and
the non-tradability of some important assets such as human capital. These elements are likely to change over time with
changes in technology, institutional structures, and regulations. A third approach has been to add greater realism to the
modeling assumptions, while maintaining the CAPM's basic methodology. This means retaining the fundamental
assumption of the CAPM that investors (or their agents) follow the principles of portfolio selection, and deriving the
equilibrium implications of such optimizing behavior in the presence of additional complicating factor. One such model
is the multifactor Intertemporal Capital Asset Pricing Model (ICAPM), in which equilibrium risk premiums on
securities in this dynamic model come from several dimensions of risks, reflected not only by their return sensitivities
or beta on the market portfolio but also by their sensitivity to other systematic risks such as changes in interest rates and
expected returns on assets and changes in consumption good prices. In this world, securities have a richer set of
hedging roles in addition to their place in the market portfolio.
Another line of research has been to develop alternative theories. The most prominent is the Arbitrage Pricing
Theory (APT). According to the APT a relation similar to the Security Market Line can exist even if investors are
not mean-variance optimizers. If there are enough different securities to "diversify away”, all but market risk, the
APT shows that an expected-return-to-beta relation will exist as a consequence of there not being any arbitrage
opportunities. Although the specific structure of asset risks in these models differs from the CAPM, the basic
insights of the CAPM—— that the risk premiums are related to broad systematic risk factors that matter to large
segments of the population—— still hold.
Summary
The CAPM has three main implications:
In equilibrium, everyone's relative holdings of risky assets are the same as in the market portfolio.
The size of the risk premium of the market portfolio is determined by the risk aversion of investors and the volatility
of the return.
The risk premium on any asset is equal to its beta times the risk premium on the market portfolio.
Whether or not the CAPM is strictly true, it provides a rational for a very simple passive portfolio strategy:
Diversify your holdings of risky assets according to the proportions of the market portfolio, and
Mix this portfolio with the risk-free asset to achieve a desired risk-reward combination
The CAPM is used in portfolio management primarily in two ways:
To establish a logical and convenient starting point in asset allocation and security selection, and
To establish a benchmark for evaluating portfolio-management ability on a risk-adjusted basis.
In corporate finance, the CAPM is used to determine the appropriate risk-adjusted discount rate in valuation models of
the firm and in capital-budgeting decisions. The CAPM is also used to establish a "fair" rate of return on invested
capital for regulated firms and in cost-plus pricing.
Today few financial scholars consider the CAPM in its simplest form to be an accurate model for fully
explaining or predicting risk premiums on risky assets. However, modified versions of the model are still a central
feature of the theory and practice of finance.
The APT gives a rationale for the expected return-beta relation that relies on the condition that there be no
arbitrage profit opportunities; the CAPM requires that investors be mean-variance portfolio optimizers. The APT and
CAPM are not incompatible; rather, they complement each other.