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Capital Asset Pricing Model

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1 Capital Asset Pricing Model
Chapter 9 Capital Asset Pricing Model

2 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 2

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

4 Assumptions Continued
Information is costless and available to all investors Investors are rational mean-variance optimizers There are homogeneous expectations

5 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

6 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

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

8 Capital Market Line (CML)
E(rc) = rf +[(E(rM) – rf)/σM]σc CML describes return and risk for portfolios on efficient frontier

9 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

10 Economic Intuition Behind the CAPM
Observation: Each investor holds the market portfolio and thus the risk of his/her portfolio is M2 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).

11 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.

12 E(ri) = rf +{E(rM) - rf}i : Security Market Line (SML)
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)

13 Figure 9.2 The Security Market Line

14 Figure 9.3 The SML and a Positive-Alpha Stock

15 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

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

17 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


19 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

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

21 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.

22 Black’s Zero Beta Model Formulation

23 Efficient Portfolios and Zero Companions
Q P E[rz (Q)] Z(Q) Z(P) E[rz (P)] s

24 Zero Beta Market Model CAPM with E(rz (m)) replacing rf

25 Liquidity and the CAPM Liquidity Illiquidity Premium
Research supports a premium for illiquidity. Amihud and Mendelson Acharya and Pedersen

26 CAPM with a Liquidity Premium
f (ci) = liquidity premium for security i f (ci) increases at a decreasing rate

27 Figure 9.5 The Relationship Between Illiquidity and Average Returns

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