11-1
McGraw-Hill/Irwin
Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.
Key Concepts and Skills
• Know how to calculate expected
returns
• Understand:
– The impact of diversification
– The systematic risk principle
– The security market line and the riskreturn trade-off
11-2
Chapter Outline
11.1 Expected Returns and Variances
11.2 Portfolios
11.3 Announcements, Surprises, and Expected
Returns
11.4 Risk: Systematic and Unsystematic
11.5 Diversification and Portfolio Risk
11.6 Systematic Risk and Beta
11.7 The Security Market Line
11.8 The SML and the Cost of Capital: A Preview
11-3
Expected Returns
• Expected returns are based on the
probabilities of possible outcomes
E( R ) 
n
p R
i 1
i
i
Where:
pi = the probability of state “i” occurring
Ri = the expected return on an asset in state i
Return to
Quick Quiz
11-4
Example: Expected Returns
State (i)
Recession
Neutral
Boom
E(R)
p(i)
0.25
0.50
0.25
1.00
E(R)
Stock A Stock B
E(Ra)
E(Rb)
-20%
30%
15%
15%
35%
-10%
25%
20%
n
E ( R )   pi R i
i 1
11-5
Example: Expected Returns
E(R)
State (i)
Recession
Neutral
Boom
E(R)
p(i)
0.25
0.50
0.25
1.00
Stock A
E(Ra) p(i) x E(Ra)
-20%
-5.0%
15%
7.5%
35%
8.8%
11.3%
Stock B
E(Rb)
p(i) x E(Rb)
30%
7.5%
15%
7.5%
-10%
-2.5%
12.5%
n
E ( R )   pi R i
i 1
11-6
Variance and Standard Deviation
• Variance and standard deviation measure
the volatility of returns
• Variance = Weighted average of squared
deviations
• Standard Deviation = Square root of variance
n
σ   p i ( R i  E ( R ))
2
2
i 1
Return to
Quick Quiz
11-7
Variance & Standard Deviation
p(i)
0.25
0.50
0.25
1.00
Expected Return
Variance
Standard Deviation
State (i)
Recession
Neutral
Boom
p(i)
0.25
0.50
0.25
1.00
Expected Return
Variance
Standard Deviation
State (i)
Recession
Neutral
Boom
E(R)
-20%
15%
35%
Stock A
DEV^2
10%
2%
6%
x p(i)
0.0244141
0.0112500
0.0141016
11.3%
0.0497656
22.3%
E(R)
30%
15%
-10%
Stock B
DEV^2
3%
2%
5%
x p(i)
0.0076563
0.0112500
0.0126563
12.5%
0.0316
17.8%
11-8
Portfolios
• Portfolio = collection of assets
• An asset’s risk and return impact how the
stock affects the risk and return of the
portfolio
• The risk-return trade-off for a portfolio is
measured by the portfolio expected
return and standard deviation, just as
with individual assets
11-9
Portfolio Expected Returns
• The expected return of a portfolio is the
weighted average of the expected
returns for each asset in the portfolio
• Weights (wj) = % of portfolio invested in
each asset
m
E ( RP )   w j E ( R j )
j 1
Return to
Quick Quiz
11-10
Example: Portfolio Weights
Asset
A
B
C
D
E
Dollars
Invested
$15,000
$8,600
$11,000
$9,800
$5,800
$50,200
% of Pf
w(j)
30%
17%
22%
20%
12%
100%
E( Rj )
12.5%
9.5%
10.0%
7.5%
8.5%
w(j) x
E( Rj )
3.735%
1.627%
2.191%
1.464%
0.982%
10.000%
11-11
Expected Portfolio Return
Alternative Method
A
1
2
3
4
5
6
7
State (i)
Recession
Neutral
Boom
E(R)
B
C
Stock V
w(j)
30%
p(i)
0.25
-20.0%
0.50
17.5%
0.25
35.0%
1.00
12.5%
D
Stock W
17%
18.0%
15.0%
-10.0%
9.5%
E
F
Stock X Stock Y
22%
20%
Expected Return
5.0%
-8.0%
10.0%
11.0%
15.0%
16.0%
10.0%
7.5%
G
Stock Z
12%
H
Portfolio
100%
4.0%
9.0%
12.0%
8.5%
-3%
13%
17%
10%
Steps:
5
1. Calculate expected portfolio return in each
state:
2. Apply the probabilities of each state to the
expected return of the portfolio in that state
E ( R P ,i )   w j E ( R j )
j 1
3
E ( R P )   p i E ( R P ,i )
i 1
3. Sum the result of step 2
Return to
Slide 11-15
11-12
Portfolio Risk
Variance & Standard Deviation
• Portfolio standard deviation is NOT
a weighted average of the standard
deviation of the component
securities’ risk
– If it were, there would be no benefit to
diversification.
11-13
Portfolio Variance
• Compute portfolio return for each state:
RP,i = w1R1,i + w2R2,i + … + wmRm,i
• Compute the overall expected portfolio
return using the same formula as for
an individual asset
• Compute the portfolio variance and
standard deviation using the same
formulas as for an individual asset
Return to
Quick Quiz
11-14
Portfolio Risk
Portfolio
State (i)
Recession
Neutral
Boom
E(R)
p(i)
0.25
0.50
0.25
1.00
E( R )
-3%
13%
17%
10%
Dev
Dev^2
x p(i)
-13%
3%
7%
0.01663
0.00416
0.00101
0.00050
0.00428
0.00107
VAR(Pf) 0.00573259
Std(Pf)
7.6%
1. Calculate Expected Portfolio Return in each state of the economy and
overall (Slide 11-12)
2. Compute deviation (DEV) of expected portfolio return in each state
from total expected portfolio return
3. Square deviations (DEV^2) found in step 2
4. Multiply squared deviations from Step 3 times the probability of each
state occurring (x p(i)).
5. The sum of the results from Step 4 = Portfolio Variance
11-15
Announcements, News and
Efficient markets
• Announcements and news contain both
expected and surprise components
• The surprise component affects stock prices
• Efficient markets result from investors trading
on unexpected news
– The easier it is to trade on surprises, the more
efficient markets should be
• Efficient markets involve random price
changes because we cannot predict surprises
11-16
Returns
• Total Return = Expected return +
unexpected return
R = E(R) + U
• Unexpected return (U) = Systematic
portion (m) + Unsystematic portion (ε)
• Total Return = Expected return
E(R)
+ Systematic portion
m
+ Unsystematic portion ε
= E(R) + m + ε
11-17
Systematic Risk
• Factors that affect a large number of
assets
• “Non-diversifiable risk”
• “Market risk”
• Examples: changes in GDP, inflation,
interest rates, etc.
Return to
Quick Quiz
11-18
Unsystematic Risk
• = Diversifiable risk
• Risk factors that affect a limited number of
assets
• Risk that can be eliminated by combining
assets into portfolios
• “Unique risk”
• “Asset-specific risk”
• Examples: labor strikes, part shortages,
etc.
Return to
Quick Quiz
11-19
The Principle of Diversification
• Diversification can substantially reduce
risk without an equivalent reduction in
expected returns
– Reduces the variability of returns
– Caused by the offset of worse-thanexpected returns from one asset by betterthan-expected returns from another
• Minimum level of risk that cannot be
diversified away = systematic portion
11-20
Standard Deviations of Annual Portfolio Returns
Table 11.7
11-21
Portfolio Conclusions
• As more stocks are added, each new
stock has a smaller risk-reducing impact
on the portfolio
 sp falls very slowly after about 40
stocks are included
– The lower limit for sp ≈ 20% = sM.
Forming well-diversified portfolios can
eliminate about half the risk of owning a
single stock.
11-22
Portfolio Diversification
Figure 11.1
11-23
Total Risk = Stand-alone 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
Total risk for a diversified portfolio is
essentially equivalent to the systematic risk
11-24
Systematic Risk Principle
• There is a reward for bearing risk
• There is no reward for bearing risk
unnecessarily
• The expected return (market required
return) on an asset depends only on that
asset’s systematic or market risk.
Return to
Quick Quiz
11-25
Market Risk for Individual Securities
• The contribution of a security to the
overall riskiness of a portfolio
• Relevant for stocks held in well-diversified
portfolios
• Measured by a stock’s beta coefficient
• For stock i, beta is:
i = (ri,M si) / sM = siM / sM2
• Measures the stock’s volatility relative to
the market
11-26
The Beta Coefficient
i = (ri,M si) / sM = siM / sM2
Where:
ρi,M = Correlation coefficient of this asset’s returns with
the market
σi = Standard deviation of the asset’s returns
σM = Standard deviation of the market’s returns
σM2 = Variance of the market’s returns
σiM = Covariance of the asset’s returns and the market
Slides describing covariance and correlation
11-27
Interpretation of beta
If  = 1.0, stock has average risk
If  > 1.0, stock is riskier than average
If  < 1.0, stock is less risky than average
Most stocks have betas in the range of 0.5
to 1.5
• Beta of the market = 1.0
• Beta of a T-Bill = 0
•
•
•
•
11-28
Beta Coefficients for Selected Companies
Table 11.8
11-29
Example: Work the Web
• Many sites provide betas for companies
• Yahoo! Finance provides beta, plus a lot
of other information under its profile link
• Click on the Web surfer to go to Yahoo!
Finance
– Enter a ticker symbol and get a basic quote
– Click on key statistics
– Beta is reported under stock price history
11-30
Quick Quiz:
Total vs. Systematic Risk
• Consider the following information:
Security C
Security K
Standard Deviation
20%
30%
Beta
1.25
0.95
• Which security has more total risk?
• Which security has more systematic risk?
• Which security should have the higher
expected return?
11-31
Beta and the Risk Premium
• Risk premium = E(R ) – Rf
• The higher the beta, the greater the risk
premium should be
• Can we define the relationship between
the risk premium and beta so that we can
estimate the expected return?
– YES!
11-32
SML and Equilibrium
Figure 11.4
11-33
Reward-to-Risk Ratio
• Reward-to-Risk Ratio:
E ( Ri )  R f
i
• = Slope of line on graph
• In equilibrium, ratio should be the same for all
assets
• When E(R) is plotted against β for all assets, the
result should be a straight line
11-34
Market Equilibrium
• In equilibrium, all assets and portfolios
must have the same reward-to-risk ratio
• Each ratio must equal the reward-to-risk
ratio for the market
E ( R A )  Rf E ( R M  Rf )

A
M
11-35
Security Market Line
• The security market line (SML) is the
representation of market equilibrium
• The slope of the SML = reward-to-risk
ratio:
(E(RM) – Rf) / M
• Slope = E(RM) – Rf = market risk premium
– Since  of the market is always 1.0
11-36
The SML and Required Return
• The Security Market Line (SML) is part of
the Capital Asset Pricing Model (CAPM)
E ( Ri )  Rf  E ( RM )  Rf  i
E ( Ri )  Rf  RPM  i
Rf = Risk-free rate (T-Bill or T-Bond)
RM = Market return ≈ S&P 500
RPM = Market risk premium = E(RM) – Rf
E(Ri) = “Required Return”
11-37
Capital Asset Pricing Model
• The capital asset pricing model (CAPM)
defines the relationship between risk and
return
E(RA) = Rf + (E(RM) – Rf)βA
• If an asset’s systematic risk () is known,
CAPM can be used to determine its
expected return
11-38
SML example
Expected vs Required Return
Stock
A
B
E(R)
14%
10%
Beta
1.3
0.8
Assume: Market Return =
Risk-free Rate =

Req R
13.4%
11.1%
Undervalued
Overvalued
12.0%
7.5%

E ( Ri )  R f  E ( RM )  R f  i
11-39
Factors Affecting Required Return
E( Ri )  Rf  E( RM )  Rf  i
• Rf measures the pure time value of
money
• RPM = (E(RM)-Rf) measures the
reward for bearing systematic risk
• i measures the amount of systematic
risk
11-40
Portfolio Beta
βp = Weighted average of the Betas of the
assets in the portfolio
Weights (wi) = % of portfolio invested in
asset i
n
 p  wi i
i 1
11-41
Quick Quiz
1. How do you compute the expected return
and standard deviation:
•
•
For an individual asset? (Slide 11-4 and Slide 11-7)
For a portfolio? (Slide 11-10 and Slide 11-14)
2. What is the difference between systematic
and unsystematic risk? (Slide 11-18 and Slide 1119)
3. What type of risk is relevant for determining
the expected return? (Slide 11-25)
11-42
Quick Quiz
4. Consider an asset with a beta of 1.2, a riskfree rate of 5%, and a market return of 13%.
–
What is the reward-to-risk ratio in equilibrium?
E ( R A )  Rf
A

E ( RM  Rf )
M

13%  5%
1.0
0.08  1.2  E ( R A ) R f  .096  .05  E ( R A )
E ( R A )  14.6%
–
What is the expected return on the asset?
•
E(R) = 5% + (13% - 5%)* 1.2 = 14.6%
11-43
Covariance of Returns
• Measures how much the returns on two
risky assets move together.
Cov(a , b)  s ab
s ab   Ra  E ( Ra )Rb  E ( Rb ) pi
i
i
i
11-44
Covariance vs. Variance of
Returns
Cov(a , b)  s ab
s ab   Ra  E ( Ra )Rb  E ( Rb ) pi
i
i
i
Var(a )  s aa  s
2
a
s   Ra  E ( Ra )Ra  E ( Ra ) pi
2
a
i
i
i
11-45
Correlation Coefficient
• Correlation Coefficient = ρ (rho)
• Scales covariance to [-1,+1]
– -1 = Perfectly negatively correlated
– 0 = Uncorrelated; not related
– +1 = Perfectly positively correlated
s ab
r ab 
s as b
11-46
Two-Stock Portfolios
• If r = -1.0
– Two stocks can be combined to form a
riskless portfolio
• If r = +1.0
– No risk reduction at all
• In general, stocks have r ≈ 0.65
– Risk is lowered but not eliminated
• Investors typically hold many stocks
11-47
Covariance & Correlation
Coefficient
Covariance
State (i)
Recession
Boom
p(i)
0.5
0.5
1.0
Expected Return
Standard Deviation
Covariance
Correlation Coefficient
Stock L
E(R)
Dev L
-20%
-45%
70%
45%
Stock U
E(R)
Dev U
30%
10%
10%
-10%
25%
45%
20%
10%
s ab   Ra  E ( Ra )Rb  E ( Rb ) pi
i
x p(i)
-0.0225
-0.0225
-4.50%
-1.00
Cov (a , b)  s ab
i
Dev*Dev
-4.5%
-4.5%
i
s ab
r ab 
s as b
11-48
s of n-Stock Portfolio
n
n
s    w i w js is j r ij
2
p
i 1 j 1
n
n
s    w i w js ij
2
p




s ab
r ab 
s as b
i 1 j 1
Subscripts denote stocks i and j
ri,j = Correlation between stocks i and j
σi and σj =Standard deviations of stocks i and j
σij = Covariance of stocks i and j
11-49
Portfolio Risk-n Risky Assets
n
n
s   wi w js ij
2
p
i 1 j 1
i
j
for n=2
1
1
w1w1s11 = w12s12
1
2
w1w2s12
2
1
w2w1s21
2
2
w2w2s22 = w22s22
sp2 = w12s12 + w22s22 + 2w1w2 s12
11-50
Portfolio Risk-2 Risky Assets
sp2 = w12s12 + w22s22 + 2w1w2 s12
i
j
for n=2
1
1
w1w1s11 = w12s12
= (.50)(.50)(45)(45)
1
2
w1w2s12
= (.50)(.50)(-.045)
2
1
w2w1s21
= (.50)(.50)(-.045)
2
2
w2w2s22 = w22s22
= (.50)(.50)(10)(10)
sp2 = w12s12 + w22s22 + 2w1w2 s12 = 0.030625
11-51
Portfolio Risk
Portfolio Variance & Standard Dev
Stock
PF %
L
50%
U
50%
Covariance
Portfolio Variance
Portfolio Standard Dev
n
σ
45%
10%
-4.50%
0.030625
17.50%
n
s   wi w js ij
2
p
i 1 j 1
Return to
Slideshow
11-52
Chapter 11
END