Capital Asset Pricing and Arbitrage Pricing Theory Chapter 8 Capital Asset Pricing and Arbitrage Pricing Theory

Capital Asset Pricing Model (CAPM) 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

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

Assumptions (cont.) Information is costless and available to all investors Investors are rational mean-variance optimizers Homogeneous expectations

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

Resulting Equilibrium Conditions (cont.) 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

Capital Market Line E(r) CML M E(rM) rf s sm

Slope and Market Risk Premium M = Market portfolio rf = Risk free rate E(rM) - rf = Market risk premium E(rM) - rf = Market price of risk = Slope of the CAPM s M

Expected Return and Risk on Individual Securities The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio Individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio

Security Market Line E(r) SML E(rM) rf ß ß = 1.0 M

SML Relationships b = [COV(ri,rm)] / sm2 Slope SML = E(rm) - rf = market risk premium SML = rf + b[E(rm) - rf]

Sample Calculations for SML E(rm) - rf = .08 rf = .03 bx = 1.25 E(rx) = .03 + 1.25(.08) = .13 or 13% by = .6 e(ry) = .03 + .6(.08) = .078 or 7.8%

Graph of Sample Calculations E(r) SML Rx=13% .08 Rm=11% Ry=7.8% 3% ß .6 1.0 1.25 ß ß ß y m x

Disequilibrium Example E(r) SML 15% Rm=11% rf=3% ß 1.0 1.25

Disequilibrium Example Suppose a security with a b of 1.25 is offering expected return of 15% According to SML, it should be 13% Underpriced: offering too high of a rate of return for its level of risk

Security Characteristic Line Excess Returns (i) SCL . . . . . . . . . . . . . . . . . . . . . . . . . . . Excess returns on market index . . . . . . . . . . . . . . . . . . . . . . . Ri = a i + ßiRm + ei

Using the Text Example p. 245, Table 8.5: Excess GM Ret. Excess Mkt. Ret. Jan. Feb. . Dec Mean Std Dev 5.41 -3.44 . 2.43 -.60 4.97 7.24 .93 . 3.90 1.75 3.32

a a Regression Results: rGM - rf = + ß(rm - rf) ß Estimated coefficient Std error of estimate Variance of residuals = 12.601 Std dev of residuals = 3.550 R-SQR = 0.575 -2.590 (1.547) 1.1357 (0.309)

Arbitrage Pricing Theory Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit Since no investment is required, an investor can create large positions to secure large levels of profit In efficient markets, profitable arbitrage opportunities will quickly disappear

Arbitrage Example from Text pp. 255-257 Current Expected Standard Stock Price$ Return% Dev.% A 10 25.0 29.58 B 10 20.0 33.91 C 10 32.5 48.15 D 10 22.5 8.58

Arbitrage Portfolio Mean Stan. Correlation Return Dev. Of Returns A,B,C 25.83 6.40 0.94 D 22.25 8.58

Arbitrage Action and Returns E. Ret. * P * D St.Dev. Short 3 shares of D and buy 1 of A, B & C to form P You earn a higher rate on the investment than you pay on the short sale

APT and CAPM Compared APT applies to well diversified portfolios and not necessarily to individual stocks With APT it is possible for some individual stocks to be mispriced - not lie on the SML APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio APT can be extended to multifactor models