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Chapter 7
Capital Asset Pricing
and Arbitrage
Pricing Theory
McGraw-Hill/Irwin
Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
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7.1 THE CAPITAL ASSET PRICING MODEL
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Capital Asset Pricing Model (CAPM)
• The CAPM is a centerpiece of modern financial
economics, which was proposed by William Sharpe,
who was awarded the 1990 Nobel Prize for
economics
CAPM tells us
1) what is the price of risk?
(Market price of risk)
2) what is the risk of asset i?
(quantity of risk asset i)
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Capital Asset Pricing Model (CAPM)
• It is an “equilibrium” model derived using
principles of diversification and some simplified
assumptions for the behavior of investors and the
market condition
– The market equilibrium refers to a condition in which
for all securities, market prices are established to
balance the demand of buyers and the supply of
sellers. These prices are called equilibrium prices
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Capital Asset Pricing Model (CAPM)
• The CAPM is a model that relates the expected
required rate of return for any security to its risk as
measured by beta
• Specifically:
Total risk = systematic risk + unsystematic risk
CAPM says:
(1)Unsystematic risk can be diversified away. It can be
avoided by diversifying at NO cost, the market will not
reward the holder of unsystematic risk at all.
(2)Systematic risk cannot be diversified away without
cost. investors need to be compensated by a certain risk
premium for bearing systematic risk
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Capital Asset Pricing Model (CAPM
Diversification and Beta
Beta measures systematic risk
– Investors differ in the extent to which they will take
risk, so they choose securities with different betas
• E.g., an aggressive investor could choose a portfolio with a
beta of 2.0
• E.g., a conservative investor could choose a portfolio with a
beta of 0.5
A measure of the sensitivity of a stock’s return to
the returns on the market portfolio
βi = Cov(Ri, Rm)/Var(Rm)
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Capital Asset Pricing Model (CAPM
Cov(R i , R M )
E[R i ]  R F 
[E[R M ]  R F ]
Var(R M )
E[R i ]  R F   i [E[R M ]  R F ]
Number of units of
systematic risk ()
Market Risk Premium
or the price per unit risk
So E(Ri)=Rf + βi(E(Rm) – Rf)
Rf + Units × Price.
• If we know the expected rate of return of a security, the theoretical
price of this security can be derived by discounting the cash flows
generated from this security at this expected rate of return
• So, this expected return-beta relationship is viewed as a kind of asset
pricing model
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Assumptions for CAPM
• Single-period investment horizon
• Investors can invest in the universal set of publicly
traded financial assets
• Investors can borrow or lend at the risk-free rate
unlimitedly
• No taxes and transaction costs
• Information is costless and available to all investors
• Assumptions associated with investors
– Investors are price takers (there is no sufficiently
wealthy investor such that his will or behavior can
influence the whole market and thus security prices)
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Assumptions for CAPM
– All investors have the homogeneous expectations about
the expected values, variances, and correlations of
security returns
– All investors attempt to construct efficient frontier
portfolios, i.e., they are rational mean-variance
optimizers
(Investors are all very similar except their initial wealth
and their degree of risk aversion)
Several assumptions are unrealistic
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Resulting Equilibrium Conditions
• Identical efficient frontier
– All investors are mean-variance optimizers and face the
same universal set of securities, so they all derive the
identical efficient frontier and the same tangent portfolio
(O) and the corresponding CAL given the current risk-free
rate
• The market portfolio is the tangent portfolio O
– All investors will put part of their wealth on the same
risky portfolio O and the rest on the risk-free asset
– The market portfolio is defined as the aggregation of the
risky portfolios held by all investors
– Hence, the composition of the market portfolio must be
identical to that of the tangent portfolio O, and thus E(rM)
= E(rO) and σM = σO
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Resulting Equilibrium Conditions (cont.)
• The capital market line (CML) As a result, all investors will
hold the same portfolio of risky assets–market portfolio,
which contains all publicly traded risky assets in the economy
– The market portfolio is of course on the efficient frontier,
and the line from the risk-free rate through the market
portfolio is called the capital market line (CML)
• We call this result the mutual fund theorem :Only
one mutual fund of risky assets–the market
portfolio–is sufficient to satisfy the investment
demands of all investors
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The Efficient Frontier and the Capital Market Line
※ Note that the CML is
on the E(r)-σ plane
M = Market portfolio
rf = Risk free rate
E(rM) - rf = Market risk premium
[E(rM) – rf] / σM = Slope of the CML
= Sharpe ratio for the market portfolio or for all
combined portfolios on the CML
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Evaluating the CAPM
theoretically the CAPM is untestable because Huge
measurability problems because the market
portfolio is unobservable.
•However, practically the CAPM is testable and could still be a
useful predictor of expected returns. Empirical testing shows
that the CAPM works reasonably well
Betas are not as useful at predicting returns as other
measurable factors may be.
• More advanced versions of the CAPM that do a better
job at estimating the market portfolio are useful at
predicting stock returns.
• Still widely used and well understood.
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CAPM and the Real World
the principles we learn from the CAPM are still
entirely valid
– Investors should diversify (invest in the market
portfolio)
– Differences in risk tolerances can be handled by
changing the asset allocation decisions in the
complete portfolio
– Systematic risk is the only risk that matters (thus we
have the relationship between the expected return and
the beta of each security)
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Expected Returns On Individual
Securities
The risk premium, defined as the expected return in
excess of rf , reflect the compensation for
securities holders
In the equilibrium, the ratio of risk premium to beta
should be the same for any two securities or
portfolios (including the market portfolio)
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Expected Returns On Individual Securities
– Therefore, for all securities,
E (rM )  rf
M

E (rM )  rf
1

E (ri )  rf
i
※The competition among investors for pursuing the securities with
higher risk premiums and smaller betas will result in the above
equality
– Rearranging gives us the CAPM’s expected return-beta
relationship
E (ri )  rf  i [ E (rM )  rf ] or
E (ri )  rf  i [ E (rM )  rf ]
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Expected Returns On Portfolios
• Since the expected return-beta relationship
according to the CAPM is linear and holds not only
for ALL INDIVIDUAL ASSETS but also for ANY
PORTFOLIO, the beta of a portfolio is simply the
weighted average of the betas of the assets in the
portfolio
βP = Wi βi
If you put half your money in a stock with a beta of 1.5 and
30% of your money in a stock with a beta of 0.9 and the
rest in T-bills, what is the portfolio beta?
• βP = 0.50(1.5) + 0.30(0.9) + 0.20(0) = 1.02
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Security Market Line (SML) Relationships
E(ri) = rf + βi [E(rM) – rf]
βi = cov(Ri,RM) / var(RM)
E(rM) – rf = market risk premium
※ SML: graphical representation of the expected return-beta relationship of
the CAPM (on the E(r)-beta plane)
For example: E(rM) – rf = 8% and rf = 3%
βx = 1.25  E(rx) = 3% + 1.25 × (8%) = 13%
βy = 0.6  E(ry) = 3% + 0.6 × (8%) = 7.8%
※ For the stock with a higher beta, since it is with higher systematic risk, it
needs to offer a higher expected return to attract investors
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Graph of Security Market Line
E(r)
SML
slope is 0.08,
which is the
market risk
premium
E(rx)=13%
E(rM)=11%
E(ry)=7.8%
3%
0.6 1 1.25
βy βM βX
β
※ The CAPM implies that all securities or portfolios should lie on this SML
※ Note that the SML is on the E(r)-β plane, and CML is on E(r)-σ plane
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Applications of the CAPM
• In reality, not all securities lie on the SML in the
economy
• Underpriced (overpriced) stocks plot above (below)
the SML: Given their betas, their expected rates of
return are higher (lower) than the predication by
the CAPM and thus the securities are underpriced
(overpriced)
• The difference between actually rate of return on a
security and the expected and is the abnormal rate
of return on this security, which is often denoted
as alpha (α)
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Disequilibrium Example
E(r)
SML
15%
Rm=11%
rf=3%

1.0
1.25
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Disequilibrium Example
• Suppose a security with a  of 1.25 is offering
expected return of 15%.
• According to SML, it should be 13%.
• Under-priced: offering too high of a rate of
return for its level of risk.
• This kind of security is a more attractive
investment target
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More on alpha and beta
E(rM) =14%
βS
=1.5
rf
= 5%
Required return = rf + β S [E(rM) – rf]
=5 + 1.5 [14 – 5] = 18.5%
If you believe the stock will actually provide a return of 17%,
what is the implied alpha(abnormal return=actual- expected)?
 = 17% - 18.5% = -1.5%
A stock with a negative alpha plots below the
SML & gives the buyer a negative abnormal
return
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Problem 1
CAPM: E(ri) = rf + β(E(rM)-rf)
a. CAPM: E(ri) = 5% + β(14% -5%)
 E(rX)
= 5% + 0.8(14% – 5%) = 12.2%
 X
= 14% – 12.2% = 1.8%
 E(rY)
= 5% + 1.5(14% – 5%) = 18.5%
 Y
= 17% – 18.5% = –1.5%
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Problem 1
X = 1.8%
Y = -1.5%
b. Which stock?
i. Well diversified:
Relevant Risk Measure?
β: CAPM Model
Best Choice?
Stock X with the
positive alpha
b. Which stock?
ii. Held alone:
Relevant Risk Measure?

Best Choice?
Calculate Sharpe
ratios
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Problem 1
b. (continued) Sharpe Ratios
Sharpe Ratio 
E(r)  rf
σ
ii. Held Alone:
Sharpe Ratio X = (0.14 – 0.05)/0.36 = 0.25
Better Sharpe Ratio Y = (0.17 – 0.05)/0.25 = 0.48
Sharpe Ratio Index = (0.14 – 0.05)/0.15 = 0.60
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Problem 2
E(rP) = rf + [E(rM) – rf]
20% = 5% + (15% – 5%)
 = 15/10 = 1.5
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Problems 5 & 6
5.
6.
5. Not possible. Portfolio A has a higher beta than Portfolio B, but the
expected return for Portfolio A is lower.
Possible.
6. Portfolio A's lower expected rate of return can be paired with a higher
standard deviation, as long as Portfolio A's beta is lower than that of
Portfolio B.
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Problem 7
7.
Sharpe Ratio 
E(r)  rf
σ
7. Calculate Sharpe ratios for both portfolios:
Sharpe M 
.18  .10
 0.33
.24
Sharpe A 
.16  .10
 0.5
.12
Not possible. The reward-to-variability ratio for Portfolio A is better
than that of the market, which is not possible according to the CAPM,
since the CAPM predicts that the market portfolio is the portfolio with
the highest return per unit of risk.
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Problem 9
9.
9. Given the data, the SML is:
E(r) = 10% + (18% – 10%)
A portfolio with beta of 1.5 should have an expected return of:
E(r) = 10% + 1.5(18% – 10%) = 22%
Not Possible: The expected return for Portfolio A is 16% so that
Portfolio A plots below the SML (i.e., has an  = –6%), and hence is an
overpriced portfolio. This is inconsistent with the CAPM.
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Problem 11
11.
11. Sharpe A = (16% - 10%) / 22% = .27
Sharpe M = (18% - 10%) / 24% = .33
Possible: Portfolio A's ratio of risk premium to standard
deviation is less attractive than the market's. This situation
is consistent with the CAPM. The market portfolio should
provide the highest reward-to-variability ratio.
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Problem 13
a.
b.
r1 = 19%; r2 = 16%; 1 = 1.5; 2 = 1.0
We can’t tell which adviser did the better job selecting stocks because
we can’t calculate either the alpha or the return per unit of risk.
CAPM: ri = 6% + β(14%-6%)
r1 = 19%; r2 = 16%; 1 = 1.5; 2 = 1.0, rf = 6%; rM = 14%
1 = 19% – [6% + 1.5(14% – 6%)] = 19% – 18% = 1%
2 = 16% – [6% + 1.0(14% – 6%)] = 16% – 14% = 2%
The second adviser did the better job selecting stocks (bigger + alpha)
Part c?
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Problem 13
c.
CAPM: ri = 3% + β(15%-3%)
r1 = 19%; r2 = 16%; 1 = 1.5; 2 = 1.0, rf = 3%; rM = 15%
1 = 19% – [3% + 1.5(15% – 3%)] = 19% – 21% = –2%
2 = 16% – [3%+ 1.0(15% – 3%)] =
16% – 15% = 1%
Here, not only does the second investment adviser appear to be a
better stock selector, but the first adviser's selections appear valueless
(or worse).
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7.4 MULTIFACTOR MODELS AND THE CAPM
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Fama French Three-Factor Model
• In reality, the systematic risk is not from one source
• It is obvious that developing models that allow for
several systematic risks can provide better descriptions
of security returns
• In addition to the market risk premium , Fama and
French propose the size premium and the book-tomarket premium
– The size premium is constructed as the difference in returns
between small and large firms and is denoted by SMB (“small
minus big”)
– The book-to-market premium is calculated as the difference in
returns between firms with a high versus low B/M ratio, and is
denoted by HML (“high minus low”)
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Fama French Three-Factor Model
• The Fama and French three-factor model is
E (ri )  rf  iM [ E (rM )  rf )]  iHML E (rHML )  iSMB E (rSMB )
– rSMB is the return of a portfolio consisting of a long
position of $1 in a small-size-firm portfolio and a short
position of $1 in a large-size-firm portfolio
– rHML is the return of a portfolio consisting of a long
position of $1 in a higher-B/M (value stock) portfolio and
a short position of $1 in a lower-B/M (growth stock)
portfolio
– The roles of rSMB and rHML are to identify the average
reward compensating holders of the security i exposed to
the sources of risk for which they proxy
– Note that it is not necessary to calculate the excess
return for rSMB and rHML
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Fama-French (FF) 3 factor Model
rGM – rf =αGM + βM(rM – rf ) + βHMLrHML + βSMBrSMB + eGM
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Arbitrage Pricing Theory
• Arbitrage –Creation of riskless profits by trading
relative mispricing among securities
1. Constructing a zero-investment portfolio today and earn
a profit for certain in the future
2. 2. Or if there is a security priced differently in two
markets, a long position in the cheaper market financed
by a short position in the more expensive market will
lead to a profit as long as the position can be offset each
other in the future
• Since there is no risk for arbitrage, an investor will
create arbitrarily large positions to obtain large levels
of profit
– No arbitrage argument: in efficient markets, profitable
arbitrage opportunities will quickly disappear
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