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Lecture 6: CAPM & APT

Lecture 6: CAPM & APT. The following topics are covered: CAPM CAPM extensions Critiques APT. CAPM: Assumptions. Investors are risk-averse individuals who maximize the expected utility of their wealth

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Lecture 6: CAPM & APT

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  1. Lecture 6: CAPM & APT • The following topics are covered: • CAPM • CAPM extensions • Critiques • APT L6: CAPM & APT

  2. CAPM: Assumptions • Investors are risk-averse individuals who maximize the expected utility of their wealth • Investors are price takers and they have homogeneous expectations about asset returns that have a joint normal distribution (thus market portfolio is efficient – page 148) • There exists a risk-free asset such that investors may borrow or lend unlimited amount at a risk-free rate. • The quantities of assets are fixed. Also all assets are marketable and perfectly divisible. • Asset markets are frictionless. Information is costless and simultaneously available to all investors. • There are no market imperfections such as taxes, regulations, or restriction on short selling. L6: CAPM & APT

  3. Derivation of CAPM • If market portfolio exists, the prices of all assets must adjust until all are held by investors. There is no excess demand. • The equilibrium proportion of each asset in the market portfolio is • (6.1) • A portfolio consists of a% invested in risky asset I and (1-a)% in the market portfolio will have the following mean and standard deviation: • (6.2) • (6.3) • A portfolio consists of a% invested in risky asset I and (1-a)% in the market portfolio will have the following mean and standard deviation: • Find expected value and standard deviation of with respect to the percentage of the portfolio as follows. L6: CAPM & APT

  4. Derivation of CAPM • Evaluating the two equations where a=0: • The slope of the risk-return trade-off: • Recall that the slope of the market line is: ; • Equating the above two slopes: L6: CAPM & APT

  5. Extensions of CAPM • No riskless assets • Forming a portfolio with a% in the market portfolio and (1-a)% in the minimum-variance zero-beta portfolio. • The mean and standard deviation of the portfolio are: • The partial derivatives where a=1 are: • ; • ; • Taking the ratio of these partials and evaluating where a=1: • Further, this line must pass through the point and the intercept is . The equation of the line must be: L6: CAPM & APT

  6. Extensions of CAPM • The existence of nonmarketable assets • E.g., human capital; page 162 • The model in continuous time • Inter-temporal CAPM • The existence of heterogeneous expectations and taxes L6: CAPM & APT

  7. Empirical tests of CAPM • Test form -- equation 6.36 • the intercept should not be significantly different from zero • There should be one factor explaining return • The relationship should be linear in beta • Coefficient on beta is risk premium • Test results – page 167 • Summary of the literature. L6: CAPM & APT

  8. Roll (1977)’s Critiques • Roll’s (1977) critiques (page 174) • The efficacy of CAPM tests is conditional on the efficiency of the market portfolio. • As long as the test involves an efficient index, we are fine. • The index turns out to be ex post efficient, if every asset is falling on the security market line. L6: CAPM & APT

  9. Arbitrage Pricing Theory • Assuming that the rate of return on any security is a linear function of k factors: Where Ri and E(Ri) are the random and expected rates on the ith asset Bik = the sensitivity of the ith asset’s return to the kth factor Fk=the mean zero kth factor common to the returns of all assets εi=a random zero mean noise term for the ith asset • We create arbitrage portfolios using the above assets. • No wealth -- arbitrage portfolio • Having no risk and earning no return on average L6: CAPM & APT

  10. Deriving APT • Return of the arbitrage portfolio: • To obtain a riskless arbitrage portfolio, one needs to eliminate both diversifiable and nondiversifiable risks. I.e., L6: CAPM & APT

  11. Deriving APT As: How does E(Ri) look like? -- a linear combination of the sensitivities L6: CAPM & APT

  12. APT • There exists a set of k+1 coefficients, such that, • (6.57) • If there is a riskless asset with a riskless rate of return Rf, then b0k =0 and Rf = • (6.58) • In equilibrium, all assets must fall on the arbitrage pricing line. L6: CAPM & APT

  13. APT vs. CAPM • APT makes no assumption about empirical distribution of asset returns • No assumption of individual’s utility function • More than 1 factor • It is for any subset of securities • No special role for the market portfolio in APT. • Can be easily extended to a multiperiod framework. L6: CAPM & APT

  14. Example • Page 182 • Empirical tests • Gehr (1975) • Reinganum (1981) • Conner and Korajczyk (1993) L6: CAPM & APT

  15. FF 3-factor Model • http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/f-f_factors.html L6: CAPM & APT

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