Chapter 7 Capital Asset Pricing and Arbitrage Pricing Theory

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Assumptions Individual investors are price takers Single-period investment horizon Investments are limited to traded financial assets No taxes, and transaction costs

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Assumptions (cont.) Information is costless and available to all investors Investors are rational mean-variance optimizers Homogeneous expectations

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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 Resulting Equilibrium Conditions (cont.)

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. E(r) E(r M ) rfrfrfrf M CML mmmm Capital Market Line 

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. M=Market portfolio r f =Risk free rate E(r M ) - r f =Market risk premium E(r M ) - r f =Market price of risk =Slope of the CAPM Slope and Market Risk Premium M 

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. E(r) E(r M ) rfrfrfrf SML M ß ß = 1.0 Security Market Line

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. SML Relationships  = [COV(r i,r m )] /  m 2 Slope SML =E(r m ) - r f =market risk premium SML = r f +  [E(r m ) - r f ]

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Sample Calculations for SML E(r m ) - r f =.08r f =.03  x = 1.25 E(r x ) = (.08) =.13 or 13%  y =.6 e(r y ) = (.08) =.078 or 7.8%

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. E(r) R x =13% SML m m ß ß 1.0 R m =11% R y =7.8% 3% x x ß 1.25 y y ß.6.08 Graph of Sample Calculations

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. E(r) 15% SML ß 1.0 R m =11% r f =3% 1.25 Disequilibrium Example

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Disequilibrium Example Suppose a security with a  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

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Security Characteristic Line Excess Returns (i) SCL Excess returns on market index R i =  i + ß i R m + e i

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Using the Text Example p. 231, Table 7.5 Jan.Feb...DecMean Std Dev Excess Mkt. Ret. Excess GM Ret.

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Estimated coefficient Std error of estimate Variance of residuals = Std dev of residuals = R-SQR = ß ß (1.547)1.1357(0.309) r GM - r f = + ß(r m - r f ) Regression Results:  

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Arbitrage Example from Text pp Current ExpectedStandard Stock Price$ Return% Dev.% A B C D

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Arbitrage Portfolio Mean Stan. Correlation ReturnDev. Of Returns Portfolio A,B,C D

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Arbitrage Action and Returns E. Ret. St.Dev. * P * D 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

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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