The Capital Asset Pricing Model
Global Financial Management
Campbell R. Harvey
Fuqua School of Business
Duke University
charvey@mail.duke.edu
http://www.duke.edu/~charvey
1
Overview



Utility and risk aversion
» Choosing efficient portfolios
Investing with a risk-free asset
» Borrowing and lending
» The markt portfolio
» The Capital Market Line (CML)
The Capital Asset Pricing Model (CAPM)
» The Security Market Line (SML)
» Beta
» Project analysis
2
Efficient Portfolios with Multiple Assets
Efficient
Frontier
Investors
prefer
E[r]
Asset
Portfolios
Asset 1
of other
Portfolios of
assets
2 Asset 1 and Asset 2
Minimum-Variance
Portfolio
0
s
3
Utility in Risk-Return Space
Indifference curves
25.00%
tau=0.5, Ubar=6%
tau=0.5, Ubar=8%
20.00%
tau=0.5, Ubar=10%
Investors
prefer
tau=0.25, Ubar=6%
tau=0.25, Ubar=8%
tau=0.5, Ubar=10%
Return
15.00%
10.00%
5.00%
0.00%
Risk
4
Individual Asset Allocations

Return
16.00%
14.00%
12.00%

x
10.00%
y
8.00%
Point x is the optimal
portfolio for the less risk
averse investor (red line)
Point y is the optimal
portfolio for the more risk
averse investor (black
line)
6.00%
4.00%
2.00%
Risk
20.00%
19.00%
18.00%
17.00%
16.00%
15.00%
14.00%
13.00%
12.00%
11.00%
10.00%
9.00%
8.00%
7.00%
6.00%
5.00%
0.00%
5
Introducing a Riskfree Asset



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 zero covariance with
every other asset (or portfolio).
» var(rf) = 0.
» cov(rf, rj) = 0 for all j.
What is the expected return and variance of a portfolio consisting
of a fraction (1-a) of the riskfree asset and a of the risky asset (or
portfolio)?
6
Risk and Return with a Riskfree asset

Expected Return
 
E rP   aE rj  1  a )r f

Variance and Standard Deviation
VarrP   s 2P  a 2 s 2j  s P  as j

Hence, the risk-return tradeoff is:
sP
ErP   r f 
Erj   rj
sj

)
7
Risk and Return with a Riskfree asset
Expected
Return

Asset j (a=1)
The line represents
all portfolios
depending on a
E(rj)
rf
Riskfree asset
(a=0)
0
sj
Standard Deviation
8
Investing with Borrowing and Lending
Expected
Return
a =2
a = 0.5
E[rM ]
M
a =1
rf
a=0
Lending
0
Borrowing
sM
Standard
Deviation
9
Optimal Investing With Borrowing
and Lending
25.00%
Return

20.00%
tau=0.5, Ubar=8%
tau=0.25, Ubar=6%
15.00%

Portfolio
10.00%
Y = optimal riskreturn tradeoff
for risk-averse
investor
X = optimal riskreturn tradeoff
for risk-tolerant
investor
X
5.00%
Y
Risk
0.00%
1
3
rf=4%
5
7
9
11
13
15
17
19
21
23
25
27
10
The Capital Market Line
Expected
Return
M
E [ rm ]
E [ rIBM ]
A
IBM
rf
Systematic
Risk
Diversifiable
Risk
Standard
Deviation
11
The Capital Market Line


The CML gives the tradeoff between risk and return for portfolios
consisting of the riskfree asset and the tangency portfolio M.
» Portfolio M is the market portfolio.
The equation of the CML is:
E ( rp )  rf  s p

E (rM )  rf
sM
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;
compensating investors for the time value of money.
» A risk premium to compensate investors for bearing risk.
E(r) = rf + Risk x [Market Price of Risk]
12
Everybody holds the Market



Everybody holds the tangency portfolio M
» If all hold the same portfolio, it must be the market!
Nobody can do better than holding the market
» If another asset existed which offers a better return for the
same risk, buy that!
Can’t be an equilibrium
Write the weight of asset j in the market portfolio as wj. Then we
have:
j N
E rM )   j 1 w j E rj  rf  rf
  )
Var rM )  i 1
i N
)
 j 1 wi w j Covrj , ri )
j N
» Simply use expressions for multi-asset case
13
All Risk-Return Tradeoffs are Equal

Hence, if you increase the weight of asset j in your portfolio
(relative to the market),
» Then expected returns increase by:
E rj )  rf
» Then the riskiness of the portfolio increases by:
i 1 wi Covrj , ri )  Covrj , rM )
N
» Hence, the return/risk gain is:
E  rj )  rf
Cov rj , rM )
» This must be the same for all assets
– Why?
14
All Assets are Equal

Suppose that for two assets A and B:
E rA )  rf
CovrA , rM )

E rB )  rf
CovrB , rM )
» Asset A offers a better return/risk ratio than asset B
– Buy A, sell B
– What if everybody does this?
» Hence, in equilibrium, all return/risk ratios must be equal for
all assets
E rA )  rf
CovrA , rM )

E rB )  rf
CovrB , rM )
15
The Capital Asset Pricing Model

If the risk-return tradeoff is the same for all assets, than it is the
one of the market:
E rA )  rf
CovrA , rM )

E rB )  rf
CovrB , rM )

E rM )  rf
Var rM )
This gives the relationship between risk and expected return for
individual stocks and portfolios.
» This is called the Security Market Line.
CovrA , rM )
E rA )  rf 
E rM )  rf  rf   A E rM )  rf
Var rM )

where
A 
)

)
Cov rA , rM )
Var  rM )
16
Capital Asset Pricing Model
A Graphical Illustration
Expected
Return
Expected
Market
Return
Expected
market risk
premium
Risk free
rate
0
Expected
return
=
0.5
Risk free
rate
+
1.0
Beta
factor
x
Beta
Expected market
risk premium
17
The Intuitive Argument For the
CAPM




Everybody holds the same portfolio, hence the market.
Portfolio-risk cannot be diversified.
Investors demand a premium on non-diversifiable risk only,
hence portfolio or market risk.
Beta measures the market risk, hence it is the correct measure
for non-diversifiable risk.
Conclusion:
In a market where investors can diversify by holding many
assets in their portfolio, they demand a risk premium
proportional to beta.
18
The SML and mispriced stocks

Suppose for a particular stock:
 )
E rj  r f 

) Er )  r
 M f
VarrM )
Remember the definition of expected returns:
 )
E rj 


Cov rj , rM

)
E Pj1  D1j  Pj0
Pj0
Then P0 falls, so that E(rj) increases until disequilibrium
vanishes and the equation holds!
19
The SML and mispriced stocks
Expected
Return

E(rM)
Y

X
Stock j is overvalued at X:
» price drops,
» expected return rises.
At Y, stock j would be
undervalued!
» expected return falls
» price increases
rf
j
=1
20
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

21
The Capital Asset Pricing Model





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].
22
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
Rate of Return
on the Market
x
Jan 1995
23
Calculating the beta of BA
Return on BA
40
30
20
10
0
-10
-20
Beta
-30
-40
-30
-20
-10
0
Beta is theslope of a regression line which best fits
the scatter of monthly returns on the share and on
the market index.
10
20
Return on the market index
24
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.
25
Beta and Standard Deviation
Risk of a
Share (Variance)
=
Market risk
of the share
Beta of
share
Risk of a
portfolio
x
+
Risk of
market
the portfolio
Beta of
Portfolio
x
Risk of
market
of the share
This is the major
element of a share's risk
Market risk of
=
Specific risk
Specific risk of
+
the portfolio
This is negligible
for a diversified portfolio
26
Testing the CAPM
Black, Jensen and Scholes
Average
Monthly
Return
Theoretical
Line
•
•
•
•
•
•
Fitted Line
•
•
•
Beta
27
Estimating the Expected Rate of
Return on Equity

The SML gives us a way to estimate the expected (or required) rate of
return on equity.
 )

E rj  r f   j ErM )  r f


We need estimates of three things:
» Riskfree interest rate, rf.
» Market price of risk, [E(rM)-rf].
» Beta for the stock,j.
28
Estimating the Expected Rate of
Return on Equity



The riskfree rate can be estimated by the current yield on one-year
Treasury bills.
» As of early 1997, 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.
» This difference has averaged about 8.6% since 1926.
The betas are estimated by regression analysis.
29
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%
30
Example of Portfolio Betas and
Expected Returns


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%
.


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.
31
Graphical Illustration
E(r)
E(r)
SML
CML
13.6%
M
M
11.1%
5.0%
5.0%
sM
s
0.71
1.0

32
Example

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?
33
Example

The beta for gold mining stocks is calculated as follows:
s gM r gM s g s M .15(.24)
 2 

 0.30
2
.
12
sM
sM

The expected rate of return on gold mining stocks is:
E (rg )  6.0%  ( 7.0%)(0.30)  7.1%


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%
34
Using the CAPM for
Project Evaluation

Suppose Microsoft is considering an expansion of its current
operations.
» The expansion will cost $100 million today
» 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?
35
Microsoft’s Expansion Project

The risk-adjusted discount rate for the project, rp, can be
estimated by using Microsoft’s beta and the CAPM.

rP  r f   E rm   r f

)
Thus, the NPV of the project is:
rP  0.05  1.2 *  0.086)
NPV  t 1
20
$25
 $100  $53.92 million
t
. )
1153
36
Company Risk Versus
Project Risk



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 company-wide discount rate is not the appropriate
discount rate.
In these cases, we must rely on industry betas for estimates of
project risk.
37
Company Risk versus
Project Risk

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?
38
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.
39
Company Risk versus
Project Risk


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:
rp  rf   p  E rm   rf )
rp  0.05  18
. 0.086)  20.5%
40
Company Risk versus
Project Risk
Required
Return
SML
Project-specific
Discount Rate
Project IRR
A
Company-wide
Discount Rate
Company Beta
Project Beta

41
Project Evaluation: Rules



The risk of an investment project is given by the project’s beta.
» Can be different from company’s beta
» Can often use industry as approximation
The Security Market Line provides an estimate of an appropriate
discount rate for the project based upon the project’s beta.
» Same company may use different discount rates for different
projects
This discount rate is used when computing the project’s net present
value.
42
Summary



Optimal investments depend on trading off risk and return
» Investors with higher risk tolerance invest more in risky
assets
» Only risk that can’t be diversified counts
If investors can borrow and lend, then everybody holds a
combination of two portfolios
» The market portfolio of all risky assets
» The riskless asset
– Covariance with the market portfolio counts
In equilibrium, all stocks must lie on the security market line
» Beta measures the amount of nondiversifiable risk
» Expected returns reflect only market risk
» Use these as required returns in project evaluation
43