Department of Banking and Finance
SPRING 2007-08
Capital Asset Pricing and Arbitrage Pricing Theory
by
Asst. Prof. Sami Fethi
Ch 7: CAPM and APT
Capital Asset Pricing Model (CAPM)

The asset pricing models aim to use the concepts of
portfolio valuation and market equilibrium in order
to determine the market price for risk and
appropriate measure of risk for a single asset.
 Capital Asset Pricing Model (CAPM) has an
observation that the returns on a financial asset
increase with the risk. CAPM concerns two types of
risk namely unsystematic and systematic risks. The
central principle of the CAPM is that, systematic
risk, as measured by beta, is the only factor affecting
the level of return.
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Ch 7: CAPM and APT
Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model (CAPM) was
developed independently by Sharpe (1964),
Lintner (1965) and Mossin (1966) as a financial
model of the relation of risk to expected return for
the practical world of finance.
 CAPM is originally depending on the mean
variance theory which was demonstrated by
Markowitz’s portfolio selection model (1952)
aiming higher average returns with lower risk.
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Ch 7: CAPM and APT
Capital Asset Pricing Model (CAPM)

Equilibrium model that underlies all modern financial
theory
 Derived using principles of diversification with
simplified assumptions
 Markowitz, Sharpe, Lintner and Mossin are
researchers credited with its development
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Ch 7: CAPM and APT
Capital Asset Pricing Model (CAPM)

From the point that most of the investors think that
the variance or standard deviation of their
portfolio’s return will enable them to quantify the
risk, portfolio selection model is used in order to
find the efficient portfolios and secondly to
generate an equation that relates the risk of an
asset to its expected return.
 The reason for this is that portfolios are expected
to have maximum return given the variance of
future returns. Therefore, Mean Variance Analysis
is one of the tools for achieving higher average
returns with lower risk and as a second tool
Capital Asset Pricing Model’s main parameters are
depending on mean and variance of returns.
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Ch 7: CAPM and APT
Capital Asset Pricing Model (CAPM)

Moreover, CAPM requires that in the equilibrium
the market portfolio must be an efficient portfolio.
One way to establish its efficiency is to argue that
if investors have homogenous expectations, the set
of optimal portfolios they would face would be
using the same values of expected returns,
variances and co variances.
 Therefore, the efficiency of the market portfolio
and the CAPM are joint hypothesis and it is not
possible to test the validity of one without the other
(Roll, 1977).
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Ch 7: CAPM and APT
CAPM Assumptions-Summary







Individual investors are price takers
Single-period investment horizon
Investments are limited to traded financial assets
No taxes, and transaction costs
Information is costless and available to all
investors
Investors are rational mean-variance optimizers
Homogeneous expectations
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Ch 7: CAPM and APT
Assumptions

Asset markets are frictionless and information
liquidity is high.
 All investors are price takers; so that, they are not able
to influence the market price by their actions.
 All investors have homogenous expectations about
asset returns and what the uncertain future holds for
them.
 All investors are risk averse and they operate in the
market rationally and perceive utility in terms of
expected return.
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Assumptions (cont.)
Ch 7: CAPM and APT

All investors are operating in perfect markets which
enables them to operate without tax payments on
returns and without cost of transactions entailed in
trading securities.
 All securities are highly divisible for instance they
can be traded in small parcels (Elton and Gruber,
1995, p.294).
 All investors can lend and borrow unlimited
amount of funds at the risk-free rate of return.
 All investors have single period investment time
horizon in means of different expectations from
their investments leads them to operate for short or
long term returns from their investments.
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Ch 7: CAPM and APT
Resulting Equilibrium Conditions

All investors will hold the same portfolio for risky
assets – market portfolio
 Market portfolio contains all securities and the
proportion of each security is its market value as a
percentage of total market value
 Risk premium on the market depends on the
average risk aversion of all market participants
 Risk premium on an individual security is a
function of its covariance with the market
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Ch 7: CAPM and APT
Capital Market Line
E(r)
CML
E(rM)
M
rf
sm
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s
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Ch 7: CAPM and APT
Capital Market Line

If a fully diversified investor is able to
invest in the market portfolio and lend or
borrow at the risk free rate of return, the
alternative risk and return relationships can
be generally placed around a market line
which is called the Capital Market Line
(CML).
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Ch 7: CAPM and APT
Capital Market Line

CML: E(rp)= rF+ λσp

E(rp): Expected return on portfolio
 rF : Return on the risk free asset
λ
: Market price risk
 σp : Market portfolio risk

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Ch 7: CAPM and APT
Slope and Market Risk Premium
M
rf
E(rM) - rf
=
=
=
Market portfolio
Risk free rate
Market risk premium
E(rM) - rf
=
Market price of risk
=
Slope of the CAPM
sM
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Ch 7: CAPM and APT
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
 Individual security’s risk premium is a
function of the covariance of returns with
the assets that make up the market
portfolio
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Ch 7: CAPM and APT
Security Market Line

The SML shows the relationship between
risk measured by beta and expected return.
The model states that the stock’s expected
return is equal to the risk-free rate plus a
risk premium obtained by the price of the
risk multiplied by the quantity of the risk.
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Ch 7: CAPM and APT
Security Market Line
E(r)
SML
E(rM)
rf
ß M = 1.0
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ß
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Ch 7: CAPM and APT
SML Relationships
b = [COV(ri,rm)] / sm2
Slope SML = E(rm) - rf
= market risk premium
SML = rf + b[E(rm) - rf]
SML: E(rS)=rF+ λσSpS,M
(σSpS,M) is the market price of risk
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Ch 7: CAPM and APT
Sample Calculations for SML
E(rm) - rf = .08 rf = .03
bx = 1.25
E(rx) = .03 + 1.25(.08) = .13 or 13%
by = .6
e(ry) = .03 + .6(.08) = .078 or 7.8%
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Ch 7: CAPM and APT
Graph of Sample Calculations
E(r)
SML
Rx=13%
Rm=11%
Ry=7.8%
3%
.08
.6 1.0 1.25
ßy ßm ßx
Investment Management
ß
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© 2005, Sami Fethi, EMU, All Right Reserved.
Ch 7: CAPM and APT
Disequilibrium Example

Suppose a security with a beta of 1.25 is
offering expected return of 15%
 According to SML, it should be 13%
 Underpriced: offering too high of a rate of
return for its level of risk
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Ch 7: CAPM and APT
Disequilibrium Example
E(r)
SML
15%
Rm=11%
rf=3%
1.0 1.25
ß
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Ch 7: CAPM and APT
Security Characteristic Line
Excess Returns (i)
SCL
.
.
.
.
.
.
. . .
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
Excess returns
.
.
.
on market index
.
.
.
.
.
.
.
.
.
. . . .
.
.
.
.
. . . . . R =. a + ß R + e
i
i
i
m
i
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Ch 7: CAPM and APT
Using the Text Example p. 231, Table 7.5
Jan.
Feb.
.
.
Dec
Mean
Std Dev
Excess
GM Ret.
5.41
-3.44
.
.
2.43
-.60
4.97
Excess
Mkt. Ret.
7.24
.93
.
.
3.90
1.75
3.32
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Ch 7: CAPM and APT
Regression Results:
rGM - rf = a + ß(rm - rf)
a
ß
Estimated coefficient -2.590 1.1357
Std error of estimate (1.547) (0.309)
Variance of residuals = 12.601
Std dev of residuals = 3.550
R-SQR = 0.575
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Ch 7: CAPM and APT
Arbitrage Pricing Theory


Arbitrage Pricing Theory was developed by Stephen Ross
(1976). His theory begins with an analysis of how investors
construct efficient portfolios and offers a new approach for
explaining the asset prices and states that the return on any
risky asset is a linear combination of various
macroeconomic factors that are not explained by this theory
namely.
Similar to CAPM it assumes that investors are fully
diversified and the systematic risk is an influencing factor in
the long run. However, unlike CAPM model APT specifies
a simple linear relationship between asset returns and the
associated factors because each share or portfolio may have
a different set of risk factors and a different degree of
sensitivity to each of them.
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The Assumptions of APT
Ch 7: CAPM and APT
Capital asset returns’ properties are consistent
with a linear structure of the factors. The returns
can be described by a factor model.
 Either there are no arbitrage opportunities in the
capital markets or the markets have perfect
competition.
 The number of the economic securities are either
inestimable or so large that the law of large
number can be applied that makes it possible to
form portfolios that diversify the firm specific
risk of individual stocks.
 Lastly, the number of the factors can be estimated
by the investor or known in advance (K. C. John
Wei, 1988)

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The Model of APT






Ch 7: CAPM and APT
k
Ri= E( Ri )+ ∑ δj βij+ εi
j=1
where,
R i
: The single period expected rate on security i , i
=1,2….,n
δj
: The zero mean j factor common to the all assets
under consideration
βij
: The sensitivity of security i’s returns to the
fluctuations in the j th common factor portfolio
εi
: A random of i th security that constructed to
have a mean of zero.
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Ch 7: CAPM and APT
Arbitrage Pricing Theory-briefly
• Arbitrage
- arises if an investor can
construct a zero investment portfolio with a
sure profit
 Since no investment is required, an investor
can create large positions to secure large
levels of profit
 In efficient markets, profitable arbitrage
opportunities will quickly disappear
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Ch 7: CAPM and APT
Arbitrage Example
Current
Stock Price$
A
10
B
10
C
10
D
10
Expected
Return%
25.0
20.0
32.5
22.5
Standard
Dev.%
29.58
33.91
48.15
8.58
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Ch 7: CAPM and APT
Arbitrage Portfolio
Mean
Return
Portfolio
A,B,C
D
Stan.
Dev.
25.83
6.40
22.25
8.58
Correlation
Of Returns
0.94
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Ch 7: CAPM and APT
Arbitrage Action and Returns
E. Ret.
* P
* D
St.Dev.
Short 3 shares of D and buy 1 of A, B & C to form
P
You earn a higher rate on the investment than
you pay on the short sale
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Ch 7: CAPM and APT
APT and CAPM Compared

APT applies to well diversified portfolios and not
necessarily to individual stocks
 With APT it is possible for some individual stocks
to be mispriced - not lie on the SML
 APT is more general in that it gets to an expected
return and beta relationship without the
assumption of the market portfolio
 APT can be extended to multifactor models
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Ch 7: CAPM and APT
Example-market risk





Suppose the risk free rate is 5%, the average investor
has a risk-aversion coefficient of A* is 2, and the st.
dev. Of the market portfolio is 20%.
A) Calculate the market risk premium.
B) Find the expected rate of return on the market.
C) Calculate the market risk premium as the riskaversion coefficient of A* increases from 2 to 3.
D) Find the expected rate of return on the market
referring to part c.
34
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Ch 7: CAPM and APT
Answer-market risk
A) E(rm-rf)=A*σ2m
 Market Risk Premium =2(0.20)2=0.08
 B) E(rm) = rf +Eq. Risk prem

= 0.05+0.08=0.13 or 13%
 C) Market Risk Premium =3(0.20)2=0.12
 D) E(rm) = rf +Eq. Risk prem

= 0.05+0.12=0.17 or 17%

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Ch 7: CAPM and APT
Example-risk premium

Suppose an av. Excess return over Treasury
bill of 8% with a st. dev. Of 20%.

A) Calculate coefficient of risk-aversion of
the av. investor.

B) Calculate the market risk premium as the
risk-aversion coefficient is 3.5
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Ch 7: CAPM and APT
Answer-risk premium

A) A*= E(rm-rf)/ σ2m =0.085/0.202=2.1

B) E(rm)-rf =A*σ2m =3.5(0.20)2=0.14 or 14%
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Ch 7: CAPM and APT
Example-EROR

Suppose the risk premium of the market portfolio
is 9%, and the estimated beta is 1.3. The risk
premium for stock is 1.3 times the market risk
premium.
 A) Calculate expected ROR if T-bill rate is 5%.
 B) Calculate ROR and the risk premium if the
estimated of the beta is 1.2 for the company.
 C) Find the company’s risk premium if market risk
premium and beta are 8% and 1.3 respectively.
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Ch 7: CAPM and APT
Answer-EROR

A) E(rc)=rf+βc[E(rm-rf)]

=5%+1.3(9%)=16.7%
 B) E(rc)=rf+βc[E(rm-rf)]

=5%+1.2(9%)=15.8%

1.2(9%)=10.8% risk premium
 C)
1.3(8%)=10.4%
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Ch 7: CAPM and APT
Example-Portfolio beta and risk premium
Asset
Beta
Risk
prem.
Portfolio  Consider the following
Weight
X
1.2
9%
0.5
Y
0.8
6
0.3
Z
0.0
0
0.2
Port.
0.84

1.0


portfolio:
A) Calculate the risk
premium on each portfolio
B) Calculate the total
portfolio if Market risk
premium is 7.5%.
40
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Ch 7: CAPM and APT
Answer-Portfolio beta and risk premium

A) (9%) (0.5)=4.5

(6%) (0.3)=1.8

=6.3%
 B) 0.84(7.5)=6.3%
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Ch 7: CAPM and APT
Example-risk premium

Suppose the risk premium of the market portfolio
is 8%, with a st. dev. Of 22%.
A) Calculate portfolio’s beta.
 B) Calculate the risk premium of the portfolio
referring to a portfolio invested 25% in x motor
company with beta 0f 1.15 and 75% in y motor
company with a beta of 1.25.

42
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Ch 7: CAPM and APT
Answer-risk premium
A) βy= 1.25, βx= 1.15
 βp=wy βy+ wx βx
 =0.75(1.25)+0.25(1.15)=1.225
 B) E(rp)-rf=βp[E(rm)-rf]

=1.225(8%)=9.8%

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Ch 7: CAPM and APT
Example-SML

Suppose the return on the market is
expected to be 14%, a stock has a beta of
1.2, and the T-bill rate is 6%.
 A) Calculate the expected return of the SML

B) If the return is 17%, calculate alpha of the
stock
44
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Ch 7: CAPM and APT
Answer-SML
E(r)
17%


Stock
15.6%
14%
A) E(rp)=rf+β[E(rm)-rf]
=6+1.2(14-6)=15.6%
SML
α=17-15.6=1.4
M
6%
1.0 1.2
ß
45
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Ch 7: CAPM and APT
Example-SML






Stock xyz has an expected return of 12% and risk of
beta is 1.5. Stock ABC is expected to return 13% with
a beta of 1.5. The market expected return is 11% and
rf=5%.
A) Based on CAPM, which stock is a better buy?
B) What is the alpha of each stock?
C) Plot the relevant SML of the two stocks
D) rf is 8% ER on the market portfolio is 16%, and
estimated beta is 1.3, what is the required ROR on the
project?
E) If the IRR of the project is 19%, what is the project
alpha?
46
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Ch 7: CAPM and APT
Answer-SML
A and B) α=E(r)-{rf+β[E(rm)-rf]}

αXYZ= 12-{5+1.0[11-5]}=1

– UNDERVALUED

αABC= 13-{5+1.5[11-5]}= -1
– OVERVALUED
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Ch 7: CAPM and APT
Answer-SML-C
E(r)
14%
Stock
13%
12%
SML
13-14=-1
α=
ABC
α xyz
=13-12=1
5%
1.0 1.5
ß
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Ch 7: CAPM and APT
Answer-SML

D) E(r)={rf+β[E(rm)-rf]}

= 8+1.3[16-8]=18.4%
 E) If the IRR of the project is 19%, it is
desirable. However, any project with an
IRR by using similar beta is less than 18.4%
should be rejected.
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Ch 7: CAPM and APT
Example-SML

Consider the following table:
Market
Return
Aggressive
stock
Defensive
stock
5%
2%
3.5%
20
32
14
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Ch 7: CAPM and APT
Example-SML cont..

A) What are the betas of the two stock?
 B) What is the E(ROR) on each stock if Market
return is equally likely to be 5% or 20%?
 C) If T-bill rate is 8% and Market return is equally
likely to be 5% or 20%, draw SML for the
economy?
 D) Plot the two securities on the SML graph and
show the alphas
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Ch 7: CAPM and APT
Answer-SML
A) βA=2-32/5-20=2 βB=3.5-14/5-20=0.7
 B) E(rA)={rf+β[E(rm)-rf]}

=0.5(2%+32%)=17%

=0.5(3.5%+14%)=8.75%

52
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Ch 7: CAPM and APT
The End
Thanks
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