Lecture 6: CAPM & APT

Lecture 6: CAPM & APT • The following topics are covered: • Deriving CAPM • Extensions of CAPM • Roll’s critique • APT L6: CAPM & APT

Assumptions for CAPM • Investors are risk-averse individuals who maximize the expected utility of their wealth • Investors are price takers and have homogeneous expectations about asset returns that have a joint normal distribution • When all individuals have homogeneous expectations, the market portfolio must be efficient • 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

Deriving 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

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

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

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

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

Roll (1977)’s Critiques • Roll (1977) : page 174 • We are looking at an efficient index, rather than the market portfolio. L6: CAPM & APT

APT • 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 Є=a random zero mean noise term for the ith asset • We create arbitrage portfolios using the above assets. I.e., • No wealth • Having no risk and earning no return on average L6: CAPM & APT

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

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

Lecture 6: CAPM & APT