Chapter 7
Capital Asset
Pricing and
Arbitrage
Pricing Theory
McGraw-Hill/Irwin
Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
7.1 The Capital Asset
Pricing Model
7-2
Capital Asset Pricing Model
(CAPM)
•
Equilibrium model that underlies all modern financial theory
•
Derived using principles of diversification, but with other
simplifying assumptions
Markowitz, Sharpe, Lintner and Mossin are researchers
credited with its development
•
7-3
Simplifying Assumptions
•
•
Individual investors are price takers
Single-period investment horizon
Investments are limited to traded financial assets
•
No taxes and no transaction costs
•
7-4
Simplifying Assumptions (cont.)
•
•
Information is costless and available to all investors
Investors are rational mean-variance optimizers
Homogeneous expectations
•
7-5
Resulting Equilibrium Conditions
• All investors will hold the same portfolio for risky
assets; the “market portfolio”
•
Market portfolio contains all securities and the
proportion of each security is its market value as a
percentage of total market value
• Market price of risk or return per unit of risk depends
on the average risk aversion of all market participants
7-6
Capital Market Line
E(r)
E(rM)
M = The value weighted “Market”
Portfolio of all risky assets.
M
CML
Efficient
Frontier
rf
sm
s
7-7
Known Tangency Portfolio of CML
• Equilibrium conditions: All investors will hold the
__________________________________________
same portfolio for risky assets; the “market portfolio”
Capital Market Line
E(r)
E(rM)
M = The value weighted “Market”
Market”
Portfolio of all risky assets.
M
CML
Pricing of individual securities
is therefore related to the risk
that individual securities
have when they are included
in the market portfolio.
rf
sm
s
7-8
Slope and Market Risk Premium
M =
rf =
E(rM) - rf =
E(rM) - rf
sM
{
=
=
Market portfolio
Risk free rate
Excess return on the
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
7-9
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
7-10
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
7-11
Individual Stocks: Security Market Line
Slope SML
= (E(rM) – rf )/ βM
= price of risk for market
E(r)
Equation of the SML (CAPM)
E(ri) = rf + bi[E(rM) - rf]
SML
E(rM)
rf
ß M = 1.0
ß
7-12
Sample Calculations for SML
Equation of the SML
E(ri) = rf + bi[E(rM) - rf]
E(rm) - rf =
.08
rf =
.03
Return per unit of systematic risk = 8% & the return due to the TVM = 3%
bx = 1.25
E(rx) =
0.03 + 1.25(.08) = .13 or 13%
by = .6
E(ry) =
0.03 + 0.6(0.08) = 0.078 or 7.8%
If b = 1?
If b = 0?
7-13
Graph of Sample Calculations
E(r)
SML
Rx=13%
RM=11%
Ry=7.8%
3%
.08
If the CAPM is correct, only β
risk matters in determining the
risk premium for a given slope
of the SML.
.6 1.0 1.25
ßy ßM ßx
ß
7-14
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 too high of a 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
7-15
More on Alpha and Beta
E(rM) = 14%
βS = 1.5
rf
= 5%
Required return = rf + β S [E(rM) – rf]
= 5 + 1.5 [14 – 5] = 18.5%
If you believe 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
7-16
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.
• All (E(ri) – rf) / βi should be the same for all
stocks.
7-17
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 unfortunately, a proxy for the Market
portfolio.
7-18
7.2 The CAPM and Index
Models
7-19
Security Characteristic Line (SCL)
Excess Returns (i)
Dispersion of the points
around the line
measures
______________.
unsystematic
risk
SCL
Slope = b
.
.
.
.
.
.
.
.
.
The statistic is
.
.
called s
.
.
.
.
.
.
.
.
.
. .
.
.
Excess returns
.
.
.
on market index
.
.
.
= 
.
.
.
.
.
.
. . . .
.
.
.
.
. . . . . R =.  + ß R + e
e
What should
 equal?
i
i
i
M
i
7-20
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% + 1.276(6%) = 12.66%
-0.0143
0.01108
0.5858
(Adjusted) = 33.18%
8.57%
1.276
0.2318
7-21
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
7-22
7.3 The CAPM and the Real
World
7-23
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.
7-24
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.
7-25
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.
Even if the CAPM is “false,” the markets can still be
– “efficient.”
7-26
7.4 Multifactor Models and
the CAPM
7-27
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
– Will the variables continue to have predictive power?
7-28
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.
7-29
Fama-French (FF) 3 factor Model
rGM – rf =αGM + βM(rM – rf ) + βHMLrHML + βSMBrSMB + eGM
7-30
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.
7-31
7.5 Factor Models and the
Arbitrage Pricing Theory
7-32
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
7-33
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.
•
7-34
Arbitrage Pricing Example
1.3
6% and a well diversified portfolio P has a beta of ___
Suppose Rf = ___
and an alpha of ___
2% when regressed against a systematic factor S.
Another well diversified portfolio Q has a beta of ___
0.9 and an alpha of
1%
___.
If we construct a portfolio of P and Q with the following weights:
WP = -2.25 and WQ = 3.25;
WP = - β Q / (β P - β Q)
Then βp = (-2.25 x 1.3) + (3.25 x 0.9) = 0
WQ = β P / (β P - β Q)
αp = (-2.25 x 2%) + (3.25 x 1%) = - 1.25%
Note: Σ W = 1
0
What should αp = ___
αp = -1.25% means an investor will earn rf – 1.25% or 4.75% on
portfolio PQ.
In theory one could short this portfolio and pay 4.75%, and invest in
the riskless asset and earn 6%, netting the 1.25% difference.
Arbitrage should eliminate the negative portfolio alpha quickly.
7-35
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,i – rf) = βp(rS,i – rf)
relative to systematic
factor “S”
and for an individual security
(rp,i – rf) = βp(rS,i – 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
– (rp,i – rf) = βp,1(r1,i – rf) + βp,2(r2,i – rf) + βp,3(r3,i – rf) + ei
7-36
APT and CAPM Cont.
APT applies to well diversified portfolios but not necessarily to all
individual stocks
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
In order to actually use this you would
need to find 4 well diversified portfolios:
Portfolio 1 would have to have a beta of
1 wrt Δ Industrial Production and betas
of zero on the other 3 factors.
Portfolio 2 would have to have a beta of
1 wrt Δ Yield curve and betas of zeros
wrt the other 3 factors. …
(rp,i – rf) = βp,1(r1,i – rf) + βp,2(r2,i – rf) + βp,3(r3,i – rf) + βp,4(r4,i – rf) + ei
7-37