Chapter 9

Capital Asset Pricing Model

9-1

Capital Asset Pricing Model (CAPM) • It is the 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 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 Continued • 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 9-5

Resulting Equilibrium Conditions Continued • 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 9-6

Figure 9.1 The Efficient Frontier and the Capital Market Line (CML) 9-7

Capital Market Line (CML)

E(r

c

) = r

f

+[(E(r

M

) – r

f

)/ σ

M

] σ

c

CML describes return and risk for portfolios on efficient frontier

9-8

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 • An individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio 9-9

Economic Intuition Behind the CAPM

Observation: portfolio is



M 2 Each investor holds the market portfolio and thus the risk of his/her



M 2 =



i



j W i W j



ij =



i



j W i W j Cov(r i , r j ) =



i W i {



j W j Cov(r i , r j )} =



i W i {W 1 Cov(r i , r 1 ) + W 2 Cov(r i , r 2 ) + …….. + W n Cov(r i , r n )} =



i W i {Cov(r i , W 1 r 1 ) + Cov(r i , W 2 r 2 ) + …….. + Cov(r i , W n r n )} =



i W i Cov(r i , W 1 r 1 + W 2 r 2 + …….. + W n r n ) =



i W i Cov(r i , r M ) Hence,



M 2 = W 1 Cov(r 1 , r M ) + W 2 Cov(r 2 , r M ) + …. + W n Cov(r n , r M ) Conclusion: Security i’s contribution to the risk of the market portfolio is measured by Cov(r i , r M ).

9-10

Note that 

M 2 = W 1 Cov(r 1 , r M ) + W 2 Cov(r 2 , r M ) + …. + W n Cov(r n , r M ).

Dividing both sides by  M 2 , we obtain

1 = W 1 Cov(r 1 , r M )/



M 2 + W 2 Cov(r 2 , r M )/



M 2 + …. + W n Cov(r n , r M )/



M 2 .

= W 1



1 + W 2



2 + .…. + W n



n .

Conclusion: The risk of the market portfolio can be viewed as the weighted sum of individual stock betas. Hence, for those who hold the market portfolio, the proper measure of risk for individual stocks is beta.

9-11

Implications: The risk premium for an individual security (portfolio) must be determined by its beta. Two stocks (portfolios) with the same beta should earn the same risk premium.

{E(r i ) - r f }/



i = {E(r j ) - r f }/



j = {E(r p ) - r f }/



p = {E(r M ) - r f }/



M {E(r i ) - r f }/



i ={E(r M ) - r f }/



M {E(r i ) - r f }/



i ={E(r M ) - r f }/1 E(r i ) - r f ={E(r M ) - r f }



i

E(r i ) = r f +{E(r M ) - r f }



i : Security Market Line (SML)

9-12

Figure 9.2 The Security Market Line 9-13

Figure 9.3 The SML and a Positive Alpha Stock 9-14

The Index Model and Realized Returns • To move from expected to realized returns—use the index model in excess return form:

R it = α i + β i R Mt + e it

where

R it = r it – r ft and R Mt = r Mt – r ft α i

= Jensen’s alpha for stock i • The index model beta coefficient turns out to be the same beta as that of the CAPM expected return-beta relationship 9-15

Figure 9.4 Estimates of Individual Mutual Fund Alphas, 1972-1991 9-16

The CAPM and Reality • Is the condition of zero alphas for all stocks as implied by the CAPM met – Not perfect but one of the best available • Is the CAPM testable – Proxies must be used for the market portfolio • CAPM is still considered the best available description of security pricing and is widely accepted 9-17

9-18

Econometrics and the Expected Return Beta Relationship • It is important to consider the econometric technique used for the model estimated • Statistical bias is easily introduced – Miller and Scholes paper demonstrated how econometric problems could lead one to reject the CAPM even if it were perfectly valid 9-19

Extensions of the CAPM • Zero-Beta Model – Helps to explain positive alphas on low beta stocks and negative alphas on high beta stocks 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

Liquidity and the CAPM • Liquidity • Illiquidity Premium • Research supports a premium for illiquidity.

– Amihud and Mendelson – Acharya and Pedersen 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

Figure 9.5 The Relationship Between Illiquidity and Average Returns 9-27