McGraw-Hill/Irwin
CHAPTER 7
Capital Asset Pricing and
Arbitrage Pricing Theory
Copyright © 2008 The McGraw-Hill Companies, Inc., All Rights Reserved.

7.1 THE CAPITAL ASSET PRICING MODEL
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Capital Asset Pricing Model (CAPM)
Equilibrium model that underlies all modern financial theory
Derived using principles of diversification with simplified assumptions
Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development
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Assumptions
Individual investors cannot affect prices by their individual trades.
Single-period investment horizon.
Investors form portfolios from many publicly traded financial assets.
They can borrow at Rf and have unlimited borrowing and lending opportunities
No taxes and transaction costs.
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Assumptions (cont.)
Information is costless and available to all investors.
Investors are rational mean-variance optimizers.
They attempt to construct Markowitz’s efficient frontier.
There are homogeneous expectations. All investors analyse the securities in the same way and share the same economic view of the world. All investors use the same expected return, standard deviations, and correlations to generate the efficient frontier and the unique optimal risky portfolio.
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Resulting Equilibrium Conditions
1.
The proportion of the each security in all investors porfolios will be same as the proportion of each security in the market porfolio (consists of all securities in the market).
All investors will hold the same portfolio for risky assets – market portfolio
Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value
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Resulting Equilibrium Conditions (cont.)
2. The market portfolio will be on the efficient frontier. It will be the optimal portfolio, the tangency point of the CAL to the efficient frontier. As a result CML, the line from the rf through the market port, M is also the best attainable CAL.
All investors hold M as the optimal risky portfolio, differing only in the amount invested in it as compared to investment in risk-free asset.
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Resulting Equilibrium Conditions (cont.)
3. Risk premium on the market depends on the average risk aversion of all market participants
4. Risk premium on an individual security is a function of its covariance with the market
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Figure 7.1 The Efficient Frontier and the Capital Market Line
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The Risk Premium of the Market Portfolio
M = Market portfolio r f
E(r
M
) - r f
=
=
Risk free rate
Market risk premium
E(r
M
) - r f s
M
= Market price of risk =
= Slope of the CAPM
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Why all investors would hold the Market
Portfolio?
With the these assumptions, investors all must arrive at the same determination of the optimal risky portfolio. They all derive identical efficient frontiers and find the same portfolio for the CAL from T-bills to the frontier.
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If the weight of GM stock in each common risky port is 1%, then GM 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 port of all investors is simply a share of the market portfolio. Because each investor has the market portfolio for the optimal risky portfolio the CAL is called the CML
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Now suppose that the optimal portfolio of the our investors does not include the stock of say Delta Air
Lines when no investor is willing to had Delta stock, the demand is zero, and the stock price will fall. As Delta stock gets cheaper, it begins to look more attractive, while all other stocks look less attractive. Delta will reach a price at which it is desirable to include it in the optimal portfolio and all investors will buy. This price adjustment process guarantees that all stocks will be included in the optimal portfolio. The only issue is the price. If all investors had an identical risk portfolio, this portfolio must be the market portfolio.
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The Passive Strategy is Efficient
The passive strategy, using the CML as the optimal CAL, is a powerful alternative to an active strategy. In the simple world of
CAPM, all investors use precious resources in security analysis. A passive investor’s investing in the market portfolio benefits from the efficiency of that portfolio. An active investor who chooses any other portfolio will end on a CAL that is less efficient than the CML.
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Expected Returns
On Individual Securities
The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio
Individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio
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Expected Returns on Individual
Securities
• The contribution of one stock to the portfolio variance can be expressed as the sum of all the covariance terms.
• For example the contribution of GM stock to the variance of the market portfolio is: w
GM
 w
1
Cov ( r
1
, r
GM
)
 w
2
Cov ( r
2
, r
GM
)

.....
 w
GM
Cov ( r
GM
, r
GM
)

....
 w n
Cov ( r n
, r
GM

)
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Expected Returns on Individual Securities cont.
W e can best measure the stock’s contribution to the risk of the market portfolio by its covariance with the market portfolio
GM’s contr to mrk variance = w
GM
Cov(r
GM
,r m
)
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Expected Returns on Individual
Securities cont.
• The reward-to-risk ratio for investments in GM;
GM’s contribution to the risk premium/
GM’s contribution to the variance w
GM w
GM

E ( r
GM
)
Cov ( r
GM
,
 r m r f
)
 
E

( r
GM
Cov ( r
GM
)

, r m r f
)

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Expected Returns on Individual
Securities cont.
• The market port is the tangency (efficient meanvariance) port. The reward-to-risk ratio for investment in the market port.
• Market risk premium/market variance
(Market price of risk) = E ( r m s
)
2 m
 r f
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Expected Returns on Individual
Securities cont.
• A basic principal of equilibrium is that all investments should offer the same reward-torisk ratio. So that the reward-to-risk ratios of GM and the market port should be equal;

E ( r
GM
)
 r f
Cov ( r
GM
, r m
)


E ( r m s
)
2 m
 r f
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Expected Returns on Individual
Securities cont.
E ( r
GM
)
 r f

Cov ( r
GM s
2 m
, r m
)

E ( r m
)
 r f

E ( r
GM
)
 r f
 
GM

E ( r m
)
 r f

• This is expected return and beta relationship which is the expression of the CAPM.
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Figure 7.2 The Security Market Line and Positive Alpha Stock
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SML Relationships

= [COV(r i
,r m
)] / s m
2
E(r m
) – r f
= market risk premium
SML = r f
+

[E(r m
) - r f
]
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Sample Calculations for SML
E(r m
) - r f
= .08 r f
= .03
 x
= 1.25
E(r x
) = .03 + 1.25(.08) = .13 or 13%
 y
= .6
e(r y
) = .03 + .6(.08) = .078 or 7.8%
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Graph of Sample Calculations
E(r)
SML
.08
R x
=13%
R m
=11%
R y
=7.8%
3%
ß
.6
ß y
1.0
ß m
1.25
ß x
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Differences between CML and SML
The CML graphs the risk premiums of efficient portfolios
(ports composed of market and 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 graphs the risk premium of individual assets as a function of asset risk. The relevant measure of risk for individual assets is not the asset’s standard deviation instead the contribution of the asset to the portfolios standard deviation as measured by the asset’s beta. The
SML is valid both for portfolios and individual assets.
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E(r)
SML
15%
R m
=11% r f
=3%

1.0
1.25
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• Suppose a security with a  of 1.25 is offering expected return of 15%. (Actually expected return)
• According to SML, it should be 13%. (Fair
Expected Return)
• Under-priced: offering too high of a rate of return for its level of risk.
• α (Alpha) = 15%-13%=2% (positive for cheap securities).
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7.2 THE CAPM AND INDEX MODELS
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Estimating the Index Model
Using historical data on T-bills, S&P 500 and individual securities
Regress risk premiums for individual stocks against the risk premiums for the
S&P 500
Slope is the beta for the individual stock
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Table 7.1 Monthly Return Statistics for
T-bills, S&P 500 and General Motors
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Figure 7.3 Cumulative Returns for
T-bills, S&P 500 and GM Stock
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Figure 7.4 Characteristic Line for GM
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Table 7.2 Security Characteristic
Line for GM: Summary Output
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GM Regression: What We Can Learn
GM is a cyclical stock
Required Return: r f
 
( r
M
 r f
)
  
9.57%
Next compute betas of other firms in the industry
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Predicting Betas
The beta from the regression equation is an estimate based on past history
Betas exhibit a statistical property
– Regression toward the mean
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THE CAPM AND THE REAL WORLD
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CAPM and the Real World
The CAPM was first published by Sharpe in the Journal of Finance in 1964
Many tests of the theory have since followed including Roll’s critique in 1977 and the Fama and French study in 1992
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7.4 MULTIFACTOR MODELS AND THE
CAPM
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Multifactor Models
Limitations for CAPM
Market Portfolio is not directly observable
Research shows that other factors affect returns
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Fama French Three-Factor Model
Returns are related to factors other than market returns
Size
Book value relative to market value
Three factor model better describes returns
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Table 7.3 Summary Statistics for Rates of
Return Series
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Table 7.4 Regression Statistics for the
Single-index and FF Three-factor Model
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7.5 FACTOR MODELS AND THE
ARBITRAGE PRICING THEORY
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Arbitrage Pricing Theory
Arbitrage - arises if an investor can construct a zero beta investment portfolio with a return greater than the risk-free rate
If two portfolios are mispriced, the investor could buy the low-priced portfolio and sell the high-priced portfolio
In efficient markets, profitable arbitrage opportunities will quickly disappear
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Figure 7.5 Security
Line Characteristics
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APT and CAPM Compared
APT applies to well diversified portfolios and not necessarily to individual stocks
With APT it is possible for some individual stocks to be mispriced - not lie on the SML
APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio
APT can be extended to multifactor models
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