# 2. Capital Asset pricing Model

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1 2. Capital Asset pricing Model Dr. Youchang Wu WS 2007 Asset Management Youchang Wu 1

2 Efficient frontier in the presence of a risk-free asset Asset Management Youchang Wu 2

3 Capital market line When a risk-free asset exists, i.e., when a capital market is introduced, the efficient frontier is linear. This linear frontier is called capital market line The capital market line touches the efficient frontier of the risky assets only at the tangency portfolio The optimal portfolio of any investor with mean-variance preference can be constructed using the risk-free asset and the tangency portfolio, which contains only risky assets (Two-fund separation) All investors with a mean-variance preference, independent of their risk attitudes, hold the same portfolio of risky assets. The risk attitude only affects the relative weights of the risk-free asset and the risky portfolio Asset Management Youchang Wu 3

4 Market price of risk The equation for the capital market line E( rt ) rf E ( r P ) = r f + σ ( r p σ ( r ) T The slope of the capital market line represents the best risk-return trade-off that t is available on the market Without the risk-free asset, the marginal rate of substitution between risk and return will be different across investors After introducing i the risk-free asset, it will be the same for all investors. Almost all investors benefit from introducing the risk-free asset (capital market) Reminiscent of the Fisher Separation Theorem? ) Asset Management Youchang Wu 4

5 Remaining questions What would be the tangency portfolio in equilibrium? CML describes the risk-return relation for all efficient portfolios, but what is the equilibrium relation between risk and return for inefficient portfolios or individual assets? What if the risk-free asset does not exist? Asset Management Youchang Wu 5

6 CAPM If everybody holds the same risky portfolio, then the risky portfolio must be the MARKET portfolio, i.e., the value-weighted portfolio of all securities (demand must equal supply in equilibrium!). It follows that the market portfolio is a frontier portfolio. According to Property III of portfolio frontier (discussed last time), we must have E( r ) = r f + β [ E( r ) r ], where β = COV( r m f, r m 2 ) / σ ( r This is the famous CAPM independently derived by Treynor(1961), Sharpe (1964), Lintner(1965) and Mossin (1966) E(r m )-r f is called market risk premium. An asset s risk premium is given by its beta time the market risk premium. m ) Asset Management Youchang Wu 6

7 Underlying assumptions Mean-variance preference Homogeneous beliefs among investors regarding the planning horizon and the distribution of security returns No market frictions: no transaction costs, no tax, no restriction on short selling, no information cost etc. Asset Management Youchang Wu 7

8 Intuition for CAPM Since everybody holds the market portfolio, the risk of an individual asset is characterized by its contribution to variance of the market portfolio instead of its own variance. The contribution of an individual security to the variance of market portfolio is determined by its covariance with the market portfolio n n 2 2 σ ( rm ) σ ( r m ) = wiw COV ( ri, r ) => = w i= 1 = 1 2COV The part of an asset ss risk that is correlated with the market portfolio, the systematic risk, cannot be diversified away; thus, investors need to be compensated for bearing it. The part of an asset ss risk that is not correlated with the market portfolio, the unsystematic risk, can be diversified away; thus, bearing unsystematic risk need not be rewarded, and therefore, an asset s unsystematic risk does not affect its risk premium. i ( r i, r m ) Asset Management Youchang Wu 8

9 CAPM vs Property III Property III is a mathematical property of portfolio frontier that holds for any return distribution. CAPM is an asset pricing relation derived using economic reasoning. It specifies how risk and return are related in equilibrium. Asset returns under CAPM are not exogenously given. They are endogenously determined d in market equilibrium. i If the CAPM does not hold, i.e., market portfolio is not mean-variance efficient, then demand does not equal supply for some assets. The prices of such assets as well as other assets must change, therefore the whole return distribution will change This process will continue until we converge to an return distribution that makes the market portfolio mean-variance efficient. Asset Management Youchang Wu 9

10 Security market line E(r) Capital market line E(r) Security market line E(r m ) E(r m ) r f r f σ β=1 β Asset Management Youchang Wu 10

11 SML vs CML Capital market line describes the efficient frontier in the presence of a risk-free asset. It specifies the equilibrium relation between return and total risk of efficient portfolio. Security market line describes the equilibrium relation between return and systematic risk for all assets or portfolios Asset Management Youchang Wu 11

12 Portfolio beta CAPM holds both for individual assets and portfolios of assets The beta of a portfolio is simply the weighted average beta of each individual assets in the portfolio, this is because COV( r p, r m ) = COV( n i= 1 w r, r i i m ) n i= 1 w COV( r where w i is the weight of asset i in portfolio p. Asset Management Youchang Wu 12 = i i, r m )

13 Zero-beta CAPM What if there is no risk-free asset? Each investor will hold a different frontier portfolio depending on his own risk attitude According to Property I of portfolio frontier, the aggregate of each investor s portfolio, i.e., the market portfolio, is also a frontier portfolio By Property III of portfolio frontier, it follows that the linear relation between expected return and beta with respect to the market portfolio must hold. This argument leads to the zero-beta CAPM (Black 1972): E r ) = E( r ) + β [ E( r ) E( r )] ( z m z where r z is the return of a zero-beta portfolio Asset Management Youchang Wu 13

14 Implications for investment CAPM solves three most important t issues in investment t simultaneously: Where to invest? Invest in the market portfolio and risk-free asset! How to value an asset? Estimated the beta of this asset and discount all the future cash flows generated by this asset using a discount rate given by CAPM! How to evaluate investment performance? Adust the performance by beta! Not surprisingly this is regarded as one of the greatest results in finance But how does it fit the real world? Numerous studies have tried to answer this question Asset Management Youchang Wu 14

15 Earlier empirical tests of CAPM Cross-sectional sectional test First estimate the beta by running the following time-serise regression r t = α + β r + ε Then run the following (out of sample) cross-sectional regression mt r = γ + γ β + γ CHAR + = 0 1 γ 2 where CHAR is a charateristic of stock unrelated to CAPM such firm size, idiosyncratic risk CAPM predicts that γ 0 =risk free rate, γ 1 =market risk premium, and γ 2 =0 Tests are usually based on portfolio returns to avoid measurement errors in estimated t betas. Supportive results are reported by Fama and MacBeth (1974) t e Asset Management Youchang Wu 15

16 Earlier empirical tests of CAPM (2) Time series test r t r = α + β ( r r ) + ε f mt f t Prediction of CAPM: α is zero for every stock or portfolio Black, Jensen and Scholes (1972) reect the standard CAPM in favor of the zero-beta CAPM Asset Management Youchang Wu 16

17 Roll s critique Roll (1977) argues that CAPM is inherently untestable The only economic prediction of CAPM is that the market portfolio is mean-variance efficient The linear return/beta relation can be found in any sample irrespective of how returns are determined in the market (All we need is to identify an index portfolio which is ex post efficient) All existing tests only tell us whether the market proxy used by researchers are efficient or not, they say nothing about the efficiency of the market portfolio itself A true market portflio should include all assets (human capital etc) and is unobservable, therefore CAPM is untestable Asset Management Youchang Wu 17

18 Roll s critique (2) Roll (1978) further argues that using CAPM to measure performance is problematic If performance is measure relative to an index that is ex post efficient, then from PIII of porftolio frontier, all portfolios will be on the security market line (no performance) If performance is measured relative to an ex post inefficient index, then any ranking of portfolio performance is possible depending on which inefficient index has been chosen Asset Management Youchang Wu 18

19 Findings from more recent tests The relation between beta and return is weak at best (Fama and French 1992) Size effect: small stocks outperform large stocks (Banz 1981,Fama and French 1992) Book-to-market k t effect: stocks with high h book-tomarket equity ratios outperform stocks with low book-to-market ratios (Fama and French 1992) Momentum effect: Stocks outperforming in the last 3-12 months tend to outperform in the following 3-12 months (Jegadeesh and Titman 1993). Asset Management Youchang Wu 19

20 Possible explanations for empirical shortcomings of CAPM Measurement errors Measurement errors in beta Measurement errors in expected return (survivorship bias) Behavioral biases of investors Small investors are subect to many behavioral biases Institutional investors are subect to agency issues Missing risk factors People do not ust hold bonds and stocks Therefore they do not ust care about an asset s covariance with a market index They also care about its covariance with other components (such as human capital) of their true portfolio This means we have to consider other risk factors as well Asset Management Youchang Wu 20

21 Summary CAPM is an elegant model that gives simple answers to several key issues in finance The real prediction of CAPM is that the market portfolio is mean-variance efficient. Empirical results should be interpreted with great caution in the light of Roll s critique Asset Management Youchang Wu 21

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