Transcript Chapter 10

Chapter 11
Return and Risk: The Capital Asset Pricing
Model
McGraw-Hill/Irwin
Copyright © 2010 by the McGraw-Hill Companies, Inc. All rights reserved.
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

11-1
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 for Two Assets
11.5 The Efficient Set for Many Assets
11.6 Diversification
11.7 Riskless Borrowing and Lending
11.8 Market Equilibrium
11.9 Relationship between Risk and Expected Return
(CAPM)
11-2
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)
11-3
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
Rate of Return
Probability Stock Fund Bond Fund
33.3%
-7%
17%
33.3%
12%
7%
33.3%
28%
-3%
11-4
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
Rate of
Return
17%
7%
-3%
7.00%
0.0067
8.2%
Fund
Squared
Deviation
0.0100
0.0000
0.0100
11-5
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%
11-6
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
11-7
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
11-8
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
11-9
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.
11-10
Correlation

Cov(a, b)
 a b
 .0117

 0.998
(.143)(.082)
11-11
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.
11-12
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%)
11-13
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%)
11-14
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.
11-15
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.
11-16
11.4 The Efficient Set for Two Assets
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%
Portfolo Risk and Return Combinations
Portfolio
Return
% in stocks
12.0%
11.0%
10.0%
9.0%
8.0%
7.0%
6.0%
5.0%
0.0%
100%
stocks
100%
bonds
5.0%
10.0%
15.0%
20.0%
Portfolio Risk (standard deviation)
We can consider other
portfolio weights besides
50% in stocks and 50% in
bonds.
11-17
% 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%
Portfolio Return
The Efficient Set for Two Assets
Portfolo Risk and Return Combinations
12.0%
11.0%
100%
stocks
10.0%
9.0%
8.0%
7.0%
100%
bonds
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)
Note that some portfolios are
“better” than others. They have
higher returns for the same level of
risk or less.
11-18
retur
n
Portfolios with Various Correlations
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
11-19
return
11.5 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.
11-20
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.
11-21
Announcements, Surprises, and Expected
Returns

The return on any security consists of two parts.



First, the expected returns
Second, the unexpected or risky returns
A way to write the return on a stock in the
coming month is:
R  R U
where
R is the expected part of the return
U is the unexpected part of the return
11-22
Announcements, Surprises, and Expected
Returns



Any announcement can be broken down into two
parts, the anticipated (or expected) part and the
surprise (or innovation):
 Announcement = Expected part + Surprise.
The expected part of any announcement is the
part of the information the market uses to form
the expectation, R, of the return on the stock.
The surprise is the news that influences the
unanticipated return on the stock, U.
11-23
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.

11-24
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
11-25
Risk: Systematic and Unsystematic





A systematic risk is any risk that affects a large
number of assets, each to a greater or lesser degree.
An unsystematic risk is a risk that specifically affects
a single asset or small group of assets.
Unsystematic risk can be diversified away.
Examples of systematic risk include uncertainty
about general economic conditions, such as GNP,
interest rates or inflation.
On the other hand, announcements specific to a
single company are examples of unsystematic risk.
11-26
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.

11-27
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.
11-28
return
11.7 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.
11-29
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.
11-30
return
11.8 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.
11-31
return
Market Equilibrium
100%
stocks
Balanced
fund
rf
100%
bonds

Where the investor chooses along the Capital Market
Line depends on her risk tolerance. The big point is that
all investors have the same CML.
11-32
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 
Cov( Ri , RM )
 ( RM )
2
11-33
Security Returns
Estimating b with Regression
Slope = bi
Return on
market %
Ri = a i + biRm + ei
11-34
The Formula for Beta
 ( Ri )
bi 


2
 ( RM )
 ( RM )
Cov( Ri , RM )
Clearly, your estimate of beta will
depend upon your choice of a proxy
for the market portfolio.
11-35
11.9 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.
11-36
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
11-37
Expected return
Relationship Between Risk & Return
Ri  RF  βi  ( R M  RF )
RM
RF
1.0
b
11-38
Expected
return
Relationship Between Risk & Return
13.5%
3%
1.5
β i  1.5
RF  3%
b
R M  10%
R i  3%  1.5  (10%  3%)  13.5%
11-39
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?
11-40