Chapter 11
Return and Risk: The Capital Asset Pricing
Model (CAPM)
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Key Concepts and Skills
Know how to calculate expected returns
 Know how to calculate covariances,
correlations, and betas
 Understand the impact of diversification
 Understand the systematic risk principle
 Understand the security market line
 Understand the risk-return tradeoff
 Be able to use the Capital Asset Pricing Model

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Chapter Outline
11.1 Individual Securities
11.2 Expected Return, Variance, and Covariance
11.3 The Return and Risk for Portfolios
11.4 The Efficient Set
11.5 Riskless Borrowing and Lending
11.6 Announcements, Surprises, and Expected Return
11.7 Risk: Systematic and Unsystematic
11.8 Diversification and Portfolio Risk
11.9 Market Equilibrium
11.10 Relationship between Risk and Expected Return (CAPM)
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11.1 Individual Securities

The characteristics of individual securities
that are of interest are the:



Expected Return
Variance and Standard Deviation
Covariance and Correlation (to another security
or index)
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11.2 Expected Return, Variance, and Covariance
Consider the following two risky asset world.
There is a 1/3 chance of each state of the
economy, and the only assets are a stock
fund and a bond fund.
Scenario
Recession
Normal
Boom
McGraw-Hill/Irwin
Rate of Return
Probability Stock Fund Bond Fund
33.3%
-7%
17%
33.3%
12%
7%
33.3%
28%
-3%
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Expected Return
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
McGraw-Hill/Irwin
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond
Rate of
Return
17%
7%
-3%
7.00%
0.0067
8.2%
Fund
Squared
Deviation
0.0100
0.0000
0.0100
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Expected Return
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
0.0100
7%
0.0000
-3%
0.0100
7.00%
0.0067
8.2%
E (rS )  1  (7%)  1  (12%)  1  (28%)
3
3
3
E (rS )  11%
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Variance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
0.0100
7%
0.0000
-3%
0.0100
7.00%
0.0067
8.2%
(7%  11%)  .0324
2
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Variance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
0.0100
7%
0.0000
-3%
0.0100
7.00%
0.0067
8.2%
1
.0205  (.0324  .0001  .0289)
3
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Standard Deviation
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
0.0100
7%
0.0000
-3%
0.0100
7.00%
0.0067
8.2%
14.3%  0.0205
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Covariance
Scenario
Recession
Normal
Boom
Sum
Covariance
Stock
Bond
Deviation Deviation
-18%
10%
1%
0%
17%
-10%
Product
-0.0180
0.0000
-0.0170
Weighted
-0.0060
0.0000
-0.0057
-0.0117
-0.0117
Deviation compares return in each state to the expected return.
Weighted takes the product of the deviations multiplied by the
probability of that state.
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Correlation

Cov(a, b)
 a b
 .0117

 0.998
(.143)(.082)
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11.3 The Return and Risk for Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
0.0100
7%
0.0000
-3%
0.0100
7.00%
0.0067
8.2%
Note that stocks have a higher expected return than bonds
and higher risk. Let us turn now to the risk-return tradeoff
of a portfolio that is 50% invested in bonds and 50%
invested in stocks.
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Portfolios
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.0016
0.0000
0.0012
9.0%
0.0010
3.08%
The rate of return on the portfolio is a weighted average of
the returns on the stocks and bonds in the portfolio:
rP  wB rB  wS rS
5%  50%  (7%)  50%  (17%)
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Portfolios
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.0016
0.0000
0.0012
9.0%
0.0010
3.08%
The expected rate of return on the portfolio is a weighted
average of the expected returns on the securities in the
portfolio.
E (rP )  wB E (rB )  wS E (rS )
9%  50%  (11%)  50%  (7%)
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Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.0016
0.0000
0.0012
9.0%
0.0010
3.08%
The variance of the rate of return on the two risky assets
portfolio is
σ P2  (wB σ B )2  (wS σ S )2  2(wB σ B )(wS σ S )ρ BS
where BS is the correlation coefficient between the returns
on the stock and bond funds.
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Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.0016
0.0000
0.0012
9.0%
0.0010
3.08%
Observe the decrease in risk that diversification offers.
An equally weighted portfolio (50% in stocks and 50%
in bonds) has less risk than either stocks or bonds held
in isolation.
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% in stocks
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50.00%
55%
60%
65%
70%
75%
80%
85%
90%
95%
McGraw-Hill/Irwin
100%
Risk
Return
8.2%
7.0%
5.9%
4.8%
3.7%
2.6%
1.4%
0.4%
0.9%
2.0%
3.08%
4.2%
5.3%
6.4%
7.6%
8.7%
9.8%
10.9%
12.1%
13.2%
14.3%
7.0%
7.2%
7.4%
7.6%
7.8%
8.0%
8.2%
8.4%
8.6%
8.8%
9.00%
9.2%
9.4%
9.6%
9.8%
10.0%
10.2%
10.4%
10.6%
10.8%
11.0%
Portfolio Return
11.4 The Efficient Set
Portfolo Risk and Return Combinations
100%
stocks
12.0%
11.0%
10.0%
9.0%
8.0%
100%
bonds
7.0%
6.0%
5.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Portfolio Risk (standard deviation)
We can consider other
portfolio weights besides
50% in stocks and 50% in
bonds …
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% in stocks
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
McGraw-Hill/Irwin
100%
Risk
Return
8.2%
7.0%
5.9%
4.8%
3.7%
2.6%
1.4%
0.4%
0.9%
2.0%
3.1%
4.2%
5.3%
6.4%
7.6%
8.7%
9.8%
10.9%
12.1%
13.2%
14.3%
7.0%
7.2%
7.4%
7.6%
7.8%
8.0%
8.2%
8.4%
8.6%
8.8%
9.0%
9.2%
9.4%
9.6%
9.8%
10.0%
10.2%
10.4%
10.6%
10.8%
11.0%
Portfolio Return
The Efficient Set
Portfolo Risk and Return Combinations
12.0%
11.0%
10.0%
100%
stocks
9.0%
8.0%
7.0%
6.0%
100%
bonds
5.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Portfolio Risk (standard deviation)
Note that some portfolios are
“better” than others. They have
higher returns for the same level of
risk or less.
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Portfolios with Various Correlations
return

100%
stocks
 = -1.0
100%
bonds

 = 1.0
 = 0.2


McGraw-Hill/Irwin
Relationship depends
on correlation
coefficient
-1.0 <  < +1.0
If  = +1.0, no risk
reduction is possible
If  = –1.0, complete
risk reduction is
possible
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return
The Efficient Set for Many Securities
Individual Assets
P
Consider a world with many risky assets; we can
still identify the opportunity set of risk-return
combinations of various portfolios.
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return
The Efficient Set for Many Securities
minimum
variance
portfolio
Individual Assets
P
The section of the opportunity set above the
minimum variance portfolio is the efficient
frontier.
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return
Optimal Portfolio with a Risk-Free Asset
100%
stocks
rf
100%
bonds

In addition to stocks and bonds, consider a world
that also has risk-free securities like T-bills.
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return
11.5 Riskless Borrowing and Lending
100%
stocks
Balanced
fund
rf
100%
bonds

Now investors can allocate their money across
the T-bills and a balanced mutual fund.
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return
Riskless Borrowing and Lending
rf
P
With a risk-free asset available and the efficient
frontier identified, we choose the capital
allocation line with the steepest slope.
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Expected vs. Unexpected Returns
Realized returns are generally not equal to
expected returns.
 There is the expected component and the
unexpected component.

At any point in time, the unexpected return can be
either positive or negative.
 Over time, the average of the unexpected
component is zero.

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11.6 Announcements and News
Announcements and news contain both an
expected component and a surprise
component.
 It is the surprise component that affects a
stock’s price and, therefore, its return.
 This is very obvious when we watch how stock
prices move when an unexpected
announcement is made or earnings are
different than anticipated

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11.7 Risk: Systematic
Risk factors that affect a large number of
assets
 Also known as non-diversifiable risk or market
risk
 Includes such things as changes in GDP,
inflation, interest rates, etc.

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Risk: Unsystematic
Risk factors that affect a limited number of
assets
 Also known as unique risk and asset-specific
risk
 Includes such things as labor strikes, part
shortages, etc.

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Returns
Total Return = expected return + unexpected
return
 Unexpected return = systematic portion +
unsystematic portion
 Therefore, total return can be expressed as
follows:


Total Return = expected return + systematic
portion + unsystematic portion
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11.8 Diversification and Portfolio Risk
Diversification can substantially reduce the
variability of returns without an equivalent
reduction in expected returns.
 This reduction in risk arises because worse
than expected returns from one asset are offset
by better than expected returns from another.
 However, there is a minimum level of risk that
cannot be diversified away, and that is the
systematic portion.

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Portfolio Risk and Number of Stocks

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In a large portfolio the variance terms are effectively
diversified away, but the covariance terms are not.
Diversifiable Risk;
Nonsystematic Risk;
Firm Specific Risk;
Unique Risk
Portfolio risk
Nondiversifiable risk;
Systematic Risk;
Market Risk
n
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Diversifiable Risk
The risk that can be eliminated by combining
assets into a portfolio
 Often considered the same as unsystematic,
unique, or asset-specific risk
 If we hold only one asset, or assets in the same
industry, then we are exposing ourselves to
risk that we could diversify away.

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Total Risk
Total risk = systematic risk + unsystematic risk
 The standard deviation of returns is a measure
of total risk.
 For well-diversified portfolios, unsystematic
risk is very small.
 Consequently, the total risk for a diversified
portfolio is essentially equivalent to the
systematic risk.

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return
11.9 Market Equilibrium
M
rf
P
With the capital allocation line identified, all investors choose a
point along the line—some combination of the risk-free asset
and the market portfolio M. In a world with homogeneous
expectations, M is the same
for all investors.
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return
Market Equilibrium
100%
stocks
Balanced
fund
rf
100%
bonds

Just where the investor chooses along the Capital Market
Line depends on his risk tolerance. The big point is that
all investors have the same CML.
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Risk When Holding the Market Portfolio


Researchers have shown that the best measure
of the risk of a security in a large portfolio is
the beta (b)of the security.
Beta measures the responsiveness of a
security to movements in the market portfolio
(i.e., systematic risk).
bi 
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Cov( Ri , RM )
 ( RM )
2
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Security Returns
Estimating b with Regression
Slope = bi
Return on
market %
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Ri = a i + biRm + ei
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The Formula for Beta
bi 
Cov( Ri , RM )
 ( RM )
2
Clearly, your estimate of beta will
depend upon your choice of a proxy
for the market portfolio.
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11.10 Risk and Return (CAPM)

Expected Return on the Market:
R M  RF  Market Risk Premium
• Expected return on an individual security:
Ri  RF  βi  ( R M  RF )
Market Risk Premium
This applies to individual securities held within welldiversified portfolios.
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Expected Return on a Security

This formula is called the Capital Asset
Pricing Model (CAPM):
Ri  RF  βi  ( R M  RF )
Expected
return on
a security
RiskBeta of the
=
+
×
free rate
security
Market risk
premium
• Assume bi = 0, then the expected return is RF.
• Assume bi = 1, then Ri  R M
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Expected return
Relationship Between Risk & Return
Ri  RF  βi  ( R M  RF )
RM
RF
1.0
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b
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Expected
return
Relationship Between Risk & Return
13.5%
3%
βi  1.5
RF  3%
1.5
b
R M  10%
R i  3%  1.5  (10%  3%)  13.5%
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Quick Quiz




How do you compute the expected return and
standard deviation for an individual asset? For a
portfolio?
What is the difference between systematic and
unsystematic risk?
What type of risk is relevant for determining the
expected return?
Consider an asset with a beta of 1.2, a risk-free rate of
5%, and a market return of 13%.
 What is the expected return on the asset?
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