9-1

Chapter 9

Capital Asset Pricing Model

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• 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

Capital Asset Pricing Model (CAPM)

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• Individual investors are price takers

• Single-period investment horizon

• Investments are limited to traded financial assets

• No taxes and transaction costs

Assumptions

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• Information is costless and available to all investors

• Investors are rational mean-variance optimizers

• There are homogeneous expectations

Assumptions Continued

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• 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

Resulting Equilibrium Conditions

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• 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

Resulting Equilibrium Conditions Continued

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

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Capital Market Line (CML)

E(rc) = rf +[(E(rM) – rf)/σM]σc

CML describes return and risk for portfolios on efficient frontier

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• 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

Return and Risk For Individual Securities

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Economic Intuition Behind the CAPM

Observation: Each investor holds the market portfolio and thus the risk of his/her portfolio is M

2

M2 = i jWiWjij = 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).

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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)/M

2 + …. + 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.

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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)

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Figure 9.2 The Security Market Line

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Figure 9.3 The SML and a Positive-Alpha Stock

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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

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

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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

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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

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Extensions of the CAPM

• Zero-Beta Model

– Helps to explain positive alphas on low beta stocks and negative alphas on high beta stocks

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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.

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Black’s Zero Beta Model Formulation

),(

),(),()()()()(

2QPP

QPPiQPQi rrCov

rrCovrrCovrErErErE

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Efficient Portfolios and Zero Companions

Q

P

Z(Q)Z(P)

E[rz (Q)]

E[rz (P)]

E(r)

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Zero Beta Market Model

2)()(

),()()()()(

M

MiMZMMZi

rrCovrErErErE

CAPM with E(rz (m)) replacing rf

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Liquidity and the CAPM

• Liquidity

• Illiquidity Premium

• Research supports a premium for illiquidity.

– Amihud and Mendelson

– Acharya and Pedersen

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CAPM with a Liquidity Premium

)()()( ifiifi cfrrErrE

f (ci) = liquidity premium for security i

f (ci) increases at a decreasing rate

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Figure 9.5 The Relationship Between Illiquidity and Average Returns