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(rc) = rf +[(E(rM) – rf)/σM]σc
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 M2
Each investor holds the market portfolio and thus the risk of his/her
M2 = i jWiWjij = i jWiWjCov(ri, rj) = iWi {jWjCov(ri, rj)}
= iWi {W1Cov(ri, r1) + W2Cov(ri, r2) + …….. + WnCov(ri, rn)}
= iWi {Cov(ri, W1r1) + Cov(ri, W2r2) + …….. + Cov(ri, Wnrn)}
= iWi Cov(ri, W1r1 + W2r2 + …….. + Wnrn) = iWi Cov(ri, rM)
Hence, M2 = W1 Cov(r1, rM) + W2 Cov(r2, rM) + …. + Wn Cov(rn, rM)
Conclusion: Security i’s contribution to the risk of the market portfolio is measured by
Cov(ri, rM).
9-10
Note that M2 = W1 Cov(r1, rM) + W2 Cov(r2, rM) + …. + Wn Cov(rn, rM).
Dividing both sides by M2, we obtain
1 = W1 Cov(r1, rM)/M2 + W2 Cov(r2, rM)/M2 + …. + Wn Cov(rn, rM)/M2.
= W1 1 + W2 2 + .…. + Wn 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(ri) - rf}/i = {E(rj) - rf}/j = {E(rp) - rf}/p= {E(rM) - rf}/M
{E(ri) - rf}/i ={E(rM) - rf}/M
{E(ri) - rf}/i ={E(rM) - rf}/1
E(ri) - rf ={E(rM) - rf}i
E(ri) = rf +{E(rM) - rf}i : Security Market Line (SML)
9-12
Figure 9.2 The Security Market Line
9-13
Figure 9.3 The SML and a PositiveAlpha Stock
9-14
The Index Model and Realized Returns
• To move from expected to realized returns—use
the index model in excess return form:
Rit = αi + βi RMt + eit
where Rit = rit – rft and RMt = rMt – rft
αi = 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 ReturnBeta 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 (ri )  E (rQ )  E (rP )  E (rQ )

Cov(ri , rP )  Cov(rP , rQ )
 P2  Cov(rP , rQ )
9-22
Efficient Portfolios and Zero
Companions
E(r)
Q
P
E[rz (Q)]
E[rz (P)]
Z(Q)
Z(P)

9-23
Zero Beta Market Model

E (ri )  E (rZ ( M ) )  E (rM )  E (rZ ( M ) )

Cov(ri , rM )
 M2
CAPM with E(rz (m)) replacing rf
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 (ri )  rf   i E (ri )  rf  f (ci )
f (ci) = liquidity premium for security i
f (ci) increases at a decreasing rate
9-26
Figure 9.5 The Relationship Between
Illiquidity and Average Returns
9-27