Chapter 7
Capital Asset
Pricing and
Arbitrage
Pricing Theory
McGraw-Hill/Irwin
Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
Capital Asset Pricing Model
(CAPM)
•
•
Equilibrium model that underlies all modern financial theory:
What should be the “appropriate” level of return commensurate
with a given amount of “risk” for an individual security
Derived using principles of diversification, with other simplifying
assumptions
•
7-2
Simplifying Assumptions
•
Individual investors are price takers
•
Single-period investment horizon
•
Investments are limited to traded financial assets
•
No taxes and no transaction costs
7-3
Simplifying Assumptions (cont.)
•
Information is costless and available to all investors
•
Investors are rational mean-variance optimizers
•
Homogeneous expectations
7-4
Resulting Equilibrium Conditions
• All investors will hold the same portfolio for risky
assets; the “market portfolio” and CML
•
Market portfolio contains all securities and the
proportion of each security is its market value as a
percentage of total market value
•
Equilibrium security prices or “risk-appropriate”
level of return is determined according to CAPM
7-5
Capital Market Line
E(r)
E(rM)
M = The value weighted “Market”
Portfolio of all risky assets. All
investors will hold the same portfolio
for risky securities
M
CML
Efficient
Frontier
rf
sm
s
7-6
Slope and Market Risk Premium
M = Market portfolio
Risk
free
rate
rf =
Excess
return
on
the
E(rM) - rf =
E(rM) - rf
sM
{
=
=
market portfolio
Optimal Market price of risk
Slope of the CML
Capital Market Line
E(r)
E(r)
E(rM)
→
M = The value weighted “Market”
Market”
Portfolio of all risky assets.
M
CML
rf
sm
s
7-7
Expected Return and Risk on
Individual Securities
• The risk premium on individual securities is a function
of the individual security’s
contribution to the risk of THE market portfolio
__________________________________________
• What type of individual security risk will matter,
systematic or unsystematic risk?
• An individual security’s total risk (s2i) can be
partitioned into systematic and unsystematic risk:
s2i = bi2 sM2 + s2(ei)
M = market portfolio of all risky securities
7-8
Expected Return and Risk on
Individual Securities
• Individual security’s contribution to the risk of the
market portfolio is a function of the __________
covariance of
the stock’s returns with the market portfolio’s returns
and is measured by BETA
With respect to an individual security, systematic
risk can be measured by bi = [COV(ri,rM)] / s2M
7-9
Individual Stocks: Security Market Line
Slope SML
= [E(rM) – rf ]
= price of risk for market
E(r)
Equation of the SML (=CAPM)
E(ri) = rf + [E(rM) - rf] bi
SML
E(rM)
rf
ß M = 1.0
ß
7-10
Sample Calculations for SML
Equation of the SML
E(ri) = rf + [E(rM) - rf] bi
E(rm) - rf = 0.08
rf = 0.03
Return per unit of systematic risk = 8% & the risk free return = 3%
bx = 1.25
E(rx) =
0.03 + (0.08)*1.25 = 0.13 or 13%
by = 0.6
E(ry) =
0.03 + (0.08)*0.6 = 0.078 or 7.8%
If b = 1?
If b = 0?
7-11
Graph of Sample Calculations
E(r)
SML
Rx=13%
RM=11%
Ry=7.8%
3%
0.08
If the CAPM is correct, only β
risk matters in determining the
risk premium for a given slope
of the SML.
0.6 1.0 1.25
ßy ßM ßx
ß
7-12
Disequilibrium Example
E(r)
SML
15%
Rm=11%
13%
rf=3%
1.0 1.25
ß
Suppose a security with a b of
____
1.25 is offering an expected
15%
return of ____
According to the SML, the E(r)
13%
should be _____
E(r) = 0.03 + 1.25(.08) = 13%
Is the security under or overpriced?
Underpriced: It is offering a higher rate of return for its level of risk
The difference between the return required for the risk level as measured
by the CAPM in this case and the actual return is called the stock’s _____
alpha
denoted by 
__
What is the __
 in this case?  = +2% Positive  is good, negative  is bad
+  gives the buyer a + abnormal return
7-13
More on Alpha and Beta
E(rM) = 14%
βS = 1.5
rf
= 5%
Required return = rf + [E(rM) – rf] βS
= 5 + [14 – 5]*1.5 = 18.5%
If you project that the stock will actually provide a return of
17% what is the implied alpha?
____,
 = 17% - 18.5% = -1.5%
A stock with a negative alpha plots below the SML
& gives the buyer a negative abnormal return
7-14
Portfolio Betas
βP = Wi βi
If you put half your money in a stock with a beta of 1.5
___ and
30% of your money in a stock with a beta of 0.9
____
___and the
rest in T-bills, what is the portfolio beta?
βP = 0.50(1.5) + 0.30(0.9) + 0.20(0) = 1.02
• All portfolio beta expected return combinations
should also fall on the SML.
7-15
Measuring Beta
• Concept:
We need to estimate the relationship between the
security and the “Market” portfolio.
• Method
Can calculate the Security Characteristic Line or SCL
using historical time series excess returns of the
security, and a proxy for the Market portfolio (DJI, S&P, etc).
7-16
Security Characteristic Line (SCL)
Excess Returns (i)
Dispersion of the points
around the line
measures
______________.
unsystematic risk
SCL
Slope = b
.
.
.
.
.
.
. . .
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
Excess returns
.
.
.
on market index
.
.
.
= 
.
.
.
.
.
.
. . . .
.
.
.
.
. . . . . R =.  + ß R + e
What should
 equal?
i
i
i
M
i
7-17
GM Excess Returns May 00 to April 05
“True” b is between 0.81 and 1.74!
If rf = 5% and rm – rf = 6%, then we
would predict GM’s return (rGM) to be
5% + (6%)*1.276 = 12.66%
-0.0143
0.01108
0.5858
(Adjusted) = 33.18%
8.57%
1.276
0.2318
7-18
Adjusted Betas
Calculated betas are adjusted to account for the empirical
finding that betas different from 1
_ tend to move toward 1
_ over
time.
A firm with a beta __
>1 will tend to have a ___________________
lower beta (closer to 1)
in the future. A firm with a beta ___
< 1 will tend to have a
____________________
higher beta (closer to 1) in the future.
Adjusted β = 2/3 (Calculated β) + 1/3 (1)
= 2/3 (1.276) + 1/3 (1)
= 1.184
7-19
7.3 The CAPM and the Real
World
7-20
Evaluating the CAPM
• The CAPM is “false” based on the
validity of its assumptions
____________________________.
•
–
The CAPM could still be a useful predictor of expected
returns. That is an empirical question.
Huge measurability problems because the market
portfolio is unobservable.
Conclusion: As a theory the CAPM is untestable.
7-21
Evaluating the CAPM
practicality of the CAPM is testable.
• However, the __________
Betas are ___________
not as useful at predicting returns as other
measurable factors may be.
• More advanced versions of the CAPM that do a
estimating the market portfolio are
better job at ___________________________
useful at predicting stock returns.
Still widely used and well understood.
7-22
Evaluating the CAPM
– The principles
_________ we learn from the CAPM are still
entirely valid.
• Investors should diversify.
• Systematic risk is the risk that matters.
• A well diversified risky portfolio can be suitable
for a wide range of investors.
– The risky portfolio would have to be adjusted for tax
and liquidity differences.
– Differences in risk tolerances can be handled by
changing the asset allocation decisions in the
complete portfolio.
7-23
7.4 Multifactor Models and
the CAPM
7-24
Fama-French (FF) 3 Factor Model
Fama and French noted that stocks of ____________
smaller firms and
stocks of firms with a high
_________________
book to market have had
higher stock returns than predicted by single factor
models.
– Problem: Empirical model without a theory
7-25
Fama-French (FF) 3 factor Model
FF proposed a 3 factor model of stock returns as follows:
• rM – rf = Market index excess return
• Ratio of ______________________________________
book value of equity to market value of equity
measured with a variable called HML
____:
– HML:
High minus low or difference in returns between
firms with a high versus a low book to market ratio.
size variable measured by the SMB
• Firm
_______________
____ variable
– SMB:
Small minus big or the difference in returns between
small and large firms.
7-26
Fama-French (FF) 3 factor Model
rGM – rf =αGM + βM(rM – rf ) + βHMLrHML + βSMBrSMB + eGM
7-27
Fama-French (FF) 3 factor Model
rGM – rf =αGM + βM(rM – rf ) + βHMLrHML + βSMBrSMB + eGM
If rf = 5%, rm – rf = 6%, & return on HML portfolio will
be 5%, then we would predict GM’s return (rGM) to be
5% + -2.62% + 1.2029(6%) + 0.6923(5%) = 13.06%
0.6454
(Adjusted) = 38.52%
8.22%
-0.0262*
1.2029*
0.6923*
0.3646
0.0116
0.2411
0.2749
0.3327
Compared to single factor model:
Better Adjusted R2; lower βM higher
E(r), but negative alpha.
7-28
Arbitrage Pricing Theory (APT)
• Arbitrage:
Arises if an investor can construct a zero
investment portfolio with a sure profit
• Zero investment:
Since no net investment outlay is
required, an investor can create
arbitrarily large positions to secure
large levels of profit
• Efficient markets:
With efficient markets, profitable
arbitrage opportunities will quickly
disappear
7-29
Simple Arbitrage Example
If all of these stocks cost
___
$8 today are there any
arbitrage opportunities?
Portfolio
Short
C
Buy (A+B) / 2
•
•
Cost
8
8
Final Outcome
9
10
The A&B combo dominates portfolio C, but costs the same.
Arbitrage opportunity: Buy A&B combo and short C, $0 net
investment, sure gain of $1
The opportunity should not persist in competitive capital markets.
•
7-30
Arbitrage Pricing Example
1.3
6% and a well diversified portfolio P has a beta of ___
Suppose Rf = ___
and an alpha of ___.
1% Another well diversified portfolio Q has a beta
0.9 and an alpha of 2%
of ___
___.
If we construct a portfolio of P and Q with the following weights:
WP = -2.25and WQ = 3.25;
Then βp = (-2.25 x 1.3) + (3.25 x 0.9) = 0
WP = - β Q / (β P - β Q)
WQ = β P / (β P - β Q)
Note: Σ W = 1
What should αp = 6%?
(-2.25 x 1%) + (3.25 x 2%) = 1.25%
αp = 1.25% means an investor will earn rf 1.25% on portfolio PQ.
In theory one could short this portfolio and pay 1.25%, and invest in
the riskless asset and earn 6%, netting the 4.75% difference.
Arbitrage should eliminate the portfolio alpha quickly.
7-31
Arbitrage Pricing Model
The result: For a well diversified portfolio
RS is the excess return on
Rp = βpRS (Excess returns)
a portfolio with a beta of 1
(rp – rf) = βp(rS – rf)
relative to systematic
factor “S”
and for an individual security
(ri – rf) = βi(rS – rf) + ei
Advantage of the APT over the CAPM:
• No particular role for the “Market Portfolio,” which can’t be
measured anyway
• Easily extended to multiple systematic factors, for example
– => (ri – rf) = βp,1(r1,i – rf) + βp,2(r2,i – rf) + βp,3(r3,i – rf) + ei
7-32
APT and CAPM Cont.
APT employs fewer restrictive assumptions
APT does NOT specify the systematic factors
Chen, Roll and Ross (1986) suggest:
 Industrial production
 Yield curve
 Default spreads
 Inflation
7-33
Selected Problems
7-34
Problem 1
CAPM: E(ri) = rf + β(E(rM)-rf)
a.
CAPM: E(ri) = 5% + β(14% -5%)
–
–
–
–
E(rX)
X
E(rY)
Y
=
=
=
=
5% + 0.8(14% – 5%) = 12.2%
14% – 12.2% = 1.8%
5% + 1.5(14% – 5%) = 18.5%
17% – 18.5% = –1.5%
7-35
Problem 1
X = 1.8%
Y = -1.5%
b. Which stock?
b. Which stock?
ii. Held alone:
i. Well diversified:
Relevant Risk Measure?
Relevant Risk
s
Measure? β: CAPM Model
Best Choice?
Calculate Sharpe
Stock X with the
ratios
Best Choice? positive alpha
7-36
Problem 1
b. (continued) Sharpe RatiosSharpe Ratio  E(r)σ r
f
ii. Held Alone:
Sharpe Ratio X = (0.14 – 0.05)/0.36 = 0.25
Better Sharpe Ratio Y = (0.17 – 0.05)/0.25 = 0.48
Sharpe Ratio Index = (0.14 – 0.05)/0.15 = 0.60
7-37
Problem 2
E(rP) = rf + b[E(rM) – rf]
20% = 5% + b(15% – 5%)
b = 15/10 = 1.5
7-38
Problem 3
E(rP) = rf + b[E(rM) – rf]
13% = 7% + β(8%) or β = 0.75
E(rp) when double the beta: E(rP) = 7% + 1.5(8%) or E(rP) = 19%
If the stock pays a constant dividend in perpetuity, then we know
from the original data that the dividend (D) must satisfy the
equation for a perpetuity:
so the Dividend = $40 x 0.13 = $5.20
Price = Dividend / E(r)
$40 = Dividend / 0.13
At the new discount rate of 19%, the stock would be worth:
$5.20 / 0.19 = $27.37
7-39
Problem 4
a. False. b = 0 implies E(r) = rf , not zero.
a. Depends on what one means by ‘volatility.’ If one means the s then
this statement is false. Investors require a risk premium for bearing
systematic (i.e., market or undiversifiable) risk.
b. False. You should invest 0.75 of your portfolio in the market portfolio,
which has β = 1, and the remainder in T-bills. Then:
•
bP = (0.75 x 1) + (0.25 x 0) = 0.75
7-40
Problems 5 & 6
9.
Not possible. Portfolio A has a higher beta than Portfolio B, but the
expected return for Portfolio A is lower.
10. Possible.
Portfolio A's lower expected rate of return can be paired with a higher
standard deviation, as long as Portfolio A's beta is lower than that of
Portfolio B.
7-41
Problem 7
Sharpe Ratio 
E(r)  rf
σ
Calculate Sharpe ratios for both portfolios:
Sharpe M 
.18  .10
 0.33
.24
Sharpe A 
.16  .10
 0.5
.12
Not possible. The reward-to-variability ratio for Portfolio A is better
than that of the market, which is not possible according to the CAPM,
since the CAPM predicts that the market portfolio is the portfolio with
the highest return per unit of risk.
7-42
Problem 8
8.
Need to calculate Sharpe ratios?
Not possible. Portfolio A clearly dominates the market
portfolio. It has a lower standard deviation with a higher
expected return.
7-43
Problem 9
9.
Given the data, the SML is:
E(r) = 10% + b(18% – 10%)
A portfolio with beta of 1.5 should have an expected return of:
E(r) = 10% + 1.5(18% – 10%) = 22%
Not Possible: The expected return for Portfolio A is 16% so that
Portfolio A plots below the SML (i.e., has an  = –6%), and hence is an
overpriced portfolio. This is inconsistent with the CAPM.
7-44
Problem 10
E(r) = 10% + b(18% – 10%)
The SML is the same as in the prior problem. Here, the required
expected return for Portfolio A is:
10% + (0.9  8%) = 17.2%
Not Possible: The required return is higher than 16%. Portfolio A is
overpriced, with  = –1.2%.
7-45
Problem 11
Sharpe A = (16% - 10%) / 22% = .27
Sharpe M = (18% - 10%) / 24% = .33
Possible: Portfolio A's ratio of risk premium to standard
deviation is less attractive than the market's. This situation
is consistent with the CAPM. The market portfolio should
provide the highest reward-to-variability ratio.
7-46
Problem 12
Since the stock's beta is equal to 1.0, its expected rate of return
the market return, or 18%
should be equal to ______________________.
E(r) =
D1  P1  P0
P0
9  P1  100
0.18 =
100
In Equilibriu m :
or P1 = $109
D1  P1  P0
 rf  β(E(r M )  rf )
P0
7-47
Problem 13
a.
b.
r1 = 19%; r2 = 16%; b1 = 1.5; b2 = 1.0
We can’t tell which adviser did the better job selecting stocks because
we can’t calculate either the alpha or the return per unit of risk.
CAPM: ri = 6% + β(14%-6%)
r1 = 19%; r2 = 16%; b1 = 1.5; b2 = 1.0, rf = 6%; rM = 14%
1 = 19% – [6% + 1.5(14% – 6%)] = 19% – 18% = 1%
2 = 16% – [6% + 1.0(14% – 6%)] = 16% – 14% = 2%
The second adviser did the better job selecting stocks (bigger + alpha)
7-48
Problem 14
a.
b.
McKay should borrow funds and invest those funds proportionally in Murray’s
existing portfolio (i.e., buy more risky assets on margin). In addition to
increased expected return, the alternative portfolio on the capital market line
(CML) will also have increased variability (risk), which is caused by the higher
proportion of risky assets in the total portfolio.
McKay should substitute low beta stocks for high beta stocks in order to
reduce the overall beta of York’s portfolio. Because York does not permit
borrowing or lending, McKay cannot reduce risk by selling equities and using
the proceeds to buy risk free assets (i.e., by lending part of the portfolio).
7-50
Problem 15
i.
ii.
Since the beta for Portfolio F is zero, the expected return for Portfolio
F equals the risk-free rate.
For Portfolio A, the ratio of risk premium to beta is: (10% - 4%)/1 = 6%
The ratio for Portfolio E is:
(9% - 4%)/(2/3) = 7.5%
Create Portfolio P by buying Portfolio E and shorting F in the
proportions to give βp = βA = 1, the same beta as A.
βp =Wi βi
1 = WE(βE) + (1-WE)(βF); WE = 1 / (2/3) or WE = 1.5 and WF = (1-WE) = -.5
E(rp) = 1.5(9) + -0.5(4) = 11.5%, Buying Portfolio P and shorting A
creates an arbitrage opportunity since
both have β = 1
p,-A = 11.5% - 10% = 1.5%
7-51
Problem 16
E(IP) = 4% &
E(IR) = 6%;
E(rstock) = 14%
βIP = 1.0
&
βIR = 0.4
Actual IP = 5%, so unexpected ΔIP = 1%
Actual IR = 7%, so unexpected ΔIR = 1%
The revised estimate of the expected rate of return of the stock would
be the old estimate plus the sum of the unexpected changes in the
factors times the sensitivity coefficients, as follows:
E(rstock) + Δ due to unexpected Δ Factors
Revised estimate = 14% + [(1  1%) + (0.4  1%)] = 15.4%
7-52