CHAPTER
10
McGraw-Hill/Irwin
Return and Risk
The Capital Asset
Pricing Model (CAPM)
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
Slide 2
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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Slide 3
Chapter Outline
10.1 Individual Securities
10.2 Expected Return, Variance, and Covariance
10.3 The Return and Risk for Portfolios
10.4 The Efficient Set for Two Assets
10.5 The Efficient Set for Many Assets
10.6 Diversification: An Example
10.7 Riskless Borrowing and Lending
10.8 Market Equilibrium
10.9 Relationship between Risk and Expected
Return (CAPM)
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Slide 4
10.1 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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Slide 5
10.2 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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Slide 6
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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Slide 7
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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Slide 8
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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Slide 9
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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Slide 10
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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Slide 11
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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Slide 12
Correlation
Cov(a, b)

 a b
 .0117

 0.998
(.143)(. 082)
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Slide 13
10.3 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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Slide 14
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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Slide 15
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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Slide 16
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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Slide 17
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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Slide 18
% in stocks
Risk
Return
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50.00%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
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%
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Portfolio Return
10.4 The Efficient Set for Two Assets
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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Slide 19
% in stocks
Risk
Return
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
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%
McGraw-Hill/Irwin
Portfolio Return
The Efficient Set for Two Assets
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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return
Portfolios with Various
Correlations
Slide 20
100%
stocks
 = -1.0
100%
bonds
 = 1.0
 = 0.2

• 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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Slide 21
return
10.5 The Efficient Set for Many Securities
Individual Assets
P
Consider a world with many risky assets; we
can still identify the opportunity set of riskreturn combinations of various portfolios.
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Slide 22
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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Slide 23
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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Slide 24
Portfolio Risk and Number of Stocks

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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Slide 25
Systematic Risk
• 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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Slide 26
Unsystematic (Diversifiable) Risk
• 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.
• The risk that can be eliminated by combining
assets into a portfolio
• 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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Slide 27
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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Slide 28
return
Optimal Portfolio with a RiskFree 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
10.7 Riskless Borrowing and
Lending
Slide 29
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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Slide 30
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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Slide 31
return
10.8 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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Slide 32
return
Market Equilibrium
100%
stocks
Balanced
fund
rf
100%
bonds

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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Slide 33
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 
McGraw-Hill/Irwin
Cov( Ri , RM )
 ( RM )
2
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Security Returns
Estimating b with Regression
Slide 34
Slope = bi
Return on
market %
Ri = a i + biRm + ei
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Slide 35
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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Slide 36
10.9 Relationship between Risk and
Expected 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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Slide 37
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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Slide 38
Expected return
Relationship Between Risk &
Return
Ri  RF  βi  ( R M  RF )
RM
RF
1.0
McGraw-Hill/Irwin
b
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Expected
return
Relationship Between Risk &
Return
Slide 39
13.5%
3%
1.5
β i  1.5
RF  3%
b
R M  10%
R i  3%  1.5  (10%  3%)  13.5%
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Slide 40
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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