Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.
McGraw-Hill/Irwin

Capital Asset Pricing Model (CAPM)
It is the equilibrium model that is a centerpiece of modern financial theory.
It gives us a precise prediction of the relationship between risk and return.
Gives us a benchmark to make comparisons
Gives us a way to evaluate non traded assets such as investment projects (use in Capital
Budgeting)
Derived using principles of diversification with simplified assumptions.
Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development.
9-2

Assumptions
Individual investors are price takers.
Single-period investment horizon.
Investments are limited to traded financial assets.
No taxes and transaction costs.
9-3

Assumptions (cont’d)
Information is costless and available to all investors.
Investors are rational mean-variance optimizers.
There are homogeneous expectations.
9-4

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. (What would happen if this were not true?)
9-5

Capital Market Line
E(r)
E(r
M
) r f
The CML is the CAL between the risk free asset and the Market Portfolio
CML
M
 m

9-6

Derivation of the CAPM
9-7

Recall the optimal choice equation y
* 
E ( r p
)

0 .
01 A
 r f
2 p
Given that on balance, for every $1 borrower there must be $1 lent, we can plug y=1 into this equation and compute the market risk premium – it depends on variance of the market portfolio, and overall average risk aversion
9-8

Resulting Equilibrium Conditions (cont’d)
Risk premium on the 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. i.e. the beta
9-9

Mutual Fund Theorem
The passive investment strategy is efficient
Another way of stating the “separation theorem” of last chapter
Portfolio selection has two components
First the technological to create a mutual fund of the market portfolio (index fund)
Second to optimize to an individual, based on risk tolerance, how much of the risk free each person should hold
9-10

Slope and Market Risk Premium
M = Market portfolio r f
E(r
M
) - r f
=
=
Risk free rate
Market risk premium
E(r
M
) - r f

M
= Market reward to variability
= Slope of the CAPM
9-11

Return and Risk For Individual Securities
The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio (i.e. the beta).
An individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio.
 i
 cov(
 r i
2
M
, r
M
)
  i , M 
 i
M
9-12

Security Market Line (Graph of CAPM)
E ( r i
)

( E ( r
M
)
 r f
)
 i
 r f
E(r)
SML
E(r
M
) r f


M
= 1.0
9-13

SML Relationships

= [COV(r i
,r m
)] /
 m
2
Slope SML = E(r m
) - r f
= market risk premium
SML = r f
+

[E(r m
) - r f
]
Beta m
= [Cov (r m
,r m
)] /
 m
2
=
 m
2 /
 m
2 = 1
9-14

Sample Calculations for SML
E(r m
) - r f
= .08 (market risk premium) r f
= .03
 x
= 1.25
E(r x
) = 0.03 + 1.25(0.08) = .13 or 13%
 y
= .6
E(r y
) = 0.03 + 0.6(0.08) = .078 or 7.8%
These show, the expected return is a function of beta
9-15

Graph of Sample Calculations
E(r)
R x
=13%
R m
=11%
R y
=7.8%
3%
SML
.
08

.6
 y
1.0
1.25
 x
9-16

Disequilibrium Example
Suppose a security with a
 of 1.25 is offering expected return of 15%.
According to SML, it should be 13%.
This security has an alpha of 2%
Under-priced: offering too high of a rate of return for its level of risk.
The price must rise, so that the expected return falls to 13% to regain equilibrium
9-17

Disequilibrium Example (illustrate alpha)
“alpha” is the vertical distance from a point to the line
E(r)
SML
15%
R m
=11% r f
=3%

1.0
1.25
9-18

Adjustments to CAPM
Black’s zero beta version if there is no risk free
Adjustments to consider liquidity
(require a reward for bearing illiquidity)
Market to Book ratio’s seem to have empirical value (No theoretical basis)
9-19

9-20

Black’s Zero Beta Model
Absence of a risk-free asset
Combinations of portfolios on the efficient frontier are efficient.
All frontier portfolios have companion portfolios that are uncorrelated.
Returns on individual assets can be expressed as linear combinations of efficient portfolios.
9-21

Black’s Zero Beta Model Formulation
E ( r i
)

E ( r
Q
)


E ( r
P
)

E ( r
Q
)
 Cov ( r i

2
P
, r
P

)

Cov ( r
P
, r
Q
)
Cov ( r
P
, r
Q
)
9-22

Efficient Portfolios and Zero Companions
E(r)
E[r z (Q)
]
E[r z (P)
]
Q
P
Z(Q)
Z(P)

9-23

Zero Beta Market Model
E ( r i
)

E ( r
Z ( M )
)


E ( r
M
)

E ( r
Z ( M )
)

Cov (
 r i
2
M
, r
M
)
CAPM with E(r z (m)
) replacing r f
9-24

CAPM & Liquidity
Liquidity
Illiquidity Premium
Research supports a premium for illiquidity.
Amihud and Mendelson
9-25

CAPM with a Liquidity Premium
E ( r i
)
 r f
  i

E ( r i
)
 r f

 f ( c i
) f (c i
) = liquidity premium for security i f (c i
) increases at a decreasing rate
9-26

Liquidity and Average Returns
Average monthly return(%)
Bid-ask spread (%)
9-27