Class 8
The Capital Asset Pricing
Model
Efficient Portfolios with
Multiple Assets
E[r]
Efficient
Frontier
Asset
Portfolios
Asset 1
of other
Portfolios of
assets
2 Asset 1 and Asset 2
Minimum-Variance
Portfolio
0
s
Utility in Risk-Return Space
Higher Utility
Expected
Return
Higher Utility
Investor B
Investor A
Standard
Deviation
Individual Asset Allocations
Expected
Return
Investor B
Investor A
Standard
Deviation
Introducing a Riskfree Asset
n
n
Suppose we introduce the opportunity to invest
in a riskfree asset. How does this alter investors’
portfolio choices?
The riskfree asset has a zero variance and a zero
covariance with every other asset (or portfolio).
 var(rf) = 0.
 cov(rf, rj) = 0 for all j.
n
What is the expected return and variance of a
portfolio consisting of a fraction af of the riskfree
asset and (1-af) of the risky asset (or portfolio)?
Introducing a Riskfree Asset
n
Expected Return
E rp  a f rf  (1  a f ) E rj
n
Variance and Standard Deviation
Var rp  s  (1  a f ) s
2
p
2
SD rp  s p  (1  a f )s j
2
j
Introducing a Riskfree Asset
Expected
Return
Asset j
E(rj)
rf
sj
Standard Deviation
Introducing a Riskless Asset
1
a=-
Expected
Return
a=0.5
E[rM ]
M
a=0
rf
a=1
0
M
Standard
Deviation
Individual Asset Allocations
Higher
Utility
Expected
Return
Higher
Utility
Investor B
M
Investor A
rf
Standard
Deviation
The Capital Market Line
Expected
Return
M
E [ rm ]
E [ rIBM ]
A
IBM
rf
14243 1444
424444
3
Systematic
Diversifiable
Risk
Risk
Standard
Deviation
The Capital Market Line
n
n
The CML gives the tradeoff between risk and
return for portfolios consisting of the riskfree
asset and the tangency portfolio M.
The equation of the CML is:
sp
E ( rp )  rf  [ E ( rM )  rf ]
sM
n
Portfolio M is the market portfolio.
Historical Returns, 1926-1993
Portfolio
Treasury Bills
Average
St. Deviation
Annual Rate of Annual
of Return
Rates of
(Nominal)
Return
3.8%
3.3%
Ave. Risk
Premium
over Treas.
Bills
Treasury Bonds
5.2%
8.6%
1.4%
Corporate Bonds
5.8%
8.5%
2.0%
Common Stocks
12.4%
20.6%
8.6%
Relationship Between Risk
and Return for Individual Assets
n
The expected rate of return on a risky asset can
be thought of as composed of two terms.
 The return on a riskfree security, like U.S. Treasury
bills.
 A risk premium to compensate investors for bearing
risk.
E(r) = rf + Risk x [Market Price of Risk]
The Capital Asset Pricing Model
n
The tradeoff between risk and return for the
market portfolio is:
E ( rM )  rf
s
n
2
M

This gives us the following relationship:
E ( rM )  rf  s 2M
The Capital Asset Pricing Model
n
This can be written explicitly as follows:
N
N
j 1
j 1
 x j [ E ( rj )  rf ]    x js jM
n
This implies that for each security j the
following relationship must hold:
E ( rj )  rf  s
jM
The Capital Asset Pricing Model
n
Using the definition of , we have the relationship
between risk and expected return for individual
stocks and portfolios. This is called the Security
Market Line.
E ( rj )  rf  [ E ( rM )  rf ] j
where
s jM
j  2
sM
The CAPM
Expected
Return
E [ rm ]
M
Security Market Line
rf
1
Beta
The Capital Asset Pricing Model
n
n
n
n
n
The appropriate measure of risk for an individual
stock is its beta.
Beta measures the stock’s sensitivity to market
risk factors. The higher the beta, the more
sensitive the stock is to market movements.
The average stock has a beta of 1.0.
Portfolio betas are weighted averages of the betas
for the individual stocks in the portfolio.
The market price of risk is [E(rM)-rf].
The CML and SML
E(r)
CML
E(r)
SML
M
M
E(rM)
E(rIBM)
IBM
rf
rf
sIBM,M/sM sM
sIBM
s
IBM
1.0

Review of Intuition for the CAPM
From economics/utility theory
Investors prefer high returns
Investors are risk averse
Diversify as much as possible
to reduce risk
In a diversified portfolio, the variance of the asset
is irrelevant. Only the covariance with the returns
of the existing portfolio are relevant
All investors spread their wealth over the
entire market
All investors assess individual assets according
to their contribution to the risk of the
market portfolio
The relevant measure of risk is the covariance of
the return of an asset with the return of
the market
Investors will demand a higher return for holding
assets whose returns are highly correlated
with those of the market
Using Regression Analysis
to Measure Betas
Rate of Return
on Stock A
Slope = Beta
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Jan 1995
Rate of Return
on the Market
Betas of Selected
Common Stocks
Stock
Beta
Stock
Beta
AT&T
0.96 Ford Motor
1.03
Boston Ed.
0.49 Home Depot
1.34
BM Squibb
0.92 McDonalds
1.06
Delta Airlines
1.31 Microsoft
1.20
Digital Equip.
1.23 Nynex
0.77
Dow Chem.
1.05 Polaroid
0.96
Exxon
0.46 Tandem
1.73
Merck
1.11 UAL
1.84
Betas based on 5 years of monthly returns through
mid-1993.
Estimating the Expected Rate of
Return on Equity
n
The SML gives us a way to estimate the
expected (or required) rate of return on equity.
E ( rj )  rf  [ E ( rM )  rf ] j
n
We need estimates of three things:
 Riskfree interest rate, rf.
 Market price of risk, [E(rM)-rf].
 Beta for the stock,j.
Estimating the Expected Rate of
Return on Equity
n
n
n
The riskfree rate can be estimated by the
current yield on one-year Treasury bills. As of
early 1996, one-year Treasury bills were
yielding about 5.0%.
The market price of risk can be estimated by
looking at the historical difference between the
return on stocks and the return on Treasury
bills. Earlier we saw this difference has
averaged about 8.6% since 1926.
The betas are estimated by regression analysis.
Estimating the Expected Rate of
Return on Equity
E(r) = 5.0% + (8.6%)
Stock
AT&T
Boston Ed.
BM Squibb
Delta Airlines
Digital Equip.
Dow Chem.
Exxon
Merck
E(r)
13.3%
9.2%
12.9%
16.3%
15.6%
14.0%
9.0%
14.5%
Stock
Ford Motor
Home Depot
McDonalds
Microsoft
Nynex
Polaroid
Tandem
UAL
E(r)
13.9%
16.5%
14.1%
15.3%
11.6%
13.3%
19.9%
20.8%
Example of Portfolio Betas and
Expected Returns
n
n
n
What is the beta and expected rate of return of
an equally-weighted portfolio consisting of
Exxon and Polaroid?
Portfolio Beta
 p  (1 / 2 )(.46)  (1 / 2 )(.96)
 p  0.71
Expected Rate of Return
E ( rp )  5.0%  ( 8.6%)(0.71)  111%
.
Example of Portfolio Betas and
Expected Returns
n
n
How would you construct a portfolio with the
same beta and expected return, but with the
lowest possible standard deviation?
Use the figure on the following page to locate
the equally-weighted portfolio of Exxon and
Polaroid. Also locate the minimum variance
portfolio with the same expected return.
Graphical Illustration
E(r)
E(r)
SML
CML
13.6%
M
M
11.1%
5.0%
5.0%
sM
s
0.71
1.0

Example
n
n
n
The S&P500 Index has a standard deviation of
about 12% per year.
Gold mining stocks have a standard deviation of
about 24% per year and a correlation with the
S&P500 of about r = 0.15.
If the yield on U.S. Treasury bills is 6% and the
market risk premium is [E(rM)-rf] = 7.0%, what
is the expected rate of return on gold mining
stocks?
Example
n
The beta for gold mining stocks is calculated as
follows:
s gM r gM s gs M .15(.24 )
Beta  2 

 0.30
2
sM
sM
.12
n
The expected rate of return on gold mining stocks
is:
E ( rg )  6.0%  ( 7 .0%)(0.30)  8.1%
Example
n
n
Question: What portfolio has the same expected
return as gold mining stocks, but the lowest
possible standard deviation?
Answer: A portfolio consisting of 70% invested
in U.S. Treasury bills and 30% invested in the
S&P500 Index.
Beta  (.7 )( 0)  (.3)(1.0)  0.30
E ( rp )  6.0%  ( 7 .0%)(0.30)  8.1%
Sd ( rp )  (.7 )( 0)  (.3)(12 .0%)  3.6%
Using the CAPM for
Project Evaluation
n
Suppose Microsoft is considering an expansion of
its current operations. The expansion will cost
$100 million today and is expected to generate a
net cash flow of $25 million per year for the next
20 years.
 What is the appropriate risk-adjusted discount rate for
the expansion project?
 What is the NPV of Microsoft’s investment project?
Microsoft’s Expansion Project
n
The risk-adjusted discount rate for the project, rp,
can be estimated by using Microsoft’s beta and the
CAPM.
d
rp  rf   p E rm  rf
i
b g
rp  0.05  12
. 0.086  15.3%
n
Thus, the NPV of the project is:
20
$25
NPV  
 $100  $53.92 million
t
.
)
t 1 (1153
Company Risk Versus
Project Risk
n
n
n
The company-wide discount rate is the
appropriate discount rate for evaluating
investment projects that have the same risk as
the firm as a whole.
For investment projects that have different risk
from the firm’s existing assets, the companywide discount rate is not the appropriate
discount rate.
In these cases, we must rely on industry betas
for estimates of project risk.
Company Risk versus
Project Risk
n
Suppose Microsoft is considering investing in
the development of a new airline.
 What is the risk of this investment?
 What is the appropriate risk-adjusted discount rate
for evaluating the project?
 Suppose the project offers a 17% rate of return. Is
the investment a good one for Microsoft?
Industry Asset Betas
Industry
Beta
Industry
Beta
Airlines
1.80 Agriculture
1.00
Electronics
1.60 Food
1.00
Consumer Durables
1.45 Liquor
0.90
Producer Goods
1.30 Banks
0.85
Chemicals
1.25 International Oils
0.85
Shipping
1.20 Tobacco
0.80
Steel
1.05 Telephone Utilities
0.75
Containers
1.05 Energy Utilities
0.60
Nonferrous Metals
1.00 Gold
0.35
Source: D. Mullins, “Does the Capital Asset Pricing Model
Work?,” Havard Business Review, vol. 60, pp. 105-114.
Company Risk versus
Project Risk
n
n
The project risk is closer to the risk of other
airlines than it is to the risk of Microsoft’s
software business.
The appropriate risk-adjusted discount rate for
the project depends upon the risk of the project.
If the average asset beta for airlines is 1.8, then
the project’s cost of capital is:
d
rp  rf   p E rm  rf
b g
i
rp  0.05  18
. 0.086  20.5%
Company Risk versus
Project Risk
Required
Return
SML
Project-specific
Discount Rate
Project IRR
A
Company-wide
Discount Rate
Company Beta
Project Beta

Summary
n
n
n
The risk of an investment project is given by the
project’s beta.
The Security Market Line provides an estimate
of an appropriate discount rate for the project
based upon the project’s beta.
This discount rate is used when computing the
project’s net present value.