Irwin/McGraw-Hill 10-1 ? The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Single Index and Multifactor Models Chapter 6 Irwin/McGraw-Hill 10-2 ? The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus ? Reduces the number of inputs for diversification. ? Portfolio of 50 assets ? ---50 expected returns; 50 variances; ? 1225 covariance. ? ---too difficult a task. Advantages of the Single Index Model Irwin/McGraw-Hill 10-3 ? The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus (r i -r f )= i + ? i (r m -r f )+ e i α Risk Premium Market Risk Premium or Index Risk Premium i = the stock’s expected return if the market’s excess return is zero ? i (r m -r f ) = the component of return due to movements in the market index E(r m -r f )= 0 e i = firm specific component, not due to market movements, E(ei)=0 ?α Single Index Model Irwin/McGraw-Hill 10-4 ? The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Let: R i = (r i -r f ) R m = (r m -r f ) Risk premium format R i = α i + ? i (R m )+ e i Risk Premium Format Irwin/McGraw-Hill 10-5 ? The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Security Characteristic Line Excess Returns (i) SCL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Excess returns on market index R i = α i + ? i R m + e i Irwin/McGraw-Hill 10-6 ? The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Jan. Feb. . . Dec Mean Std Dev 5.41 -3.44 . . 2.43 -.60 4.97 7.24 .93 . . 3.90 1.75 3.32 Excess Mkt. Ret. Excess GM Ret. Using the Text Example from Table 10-1 Irwin/McGraw-Hill 10-7 ? The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 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) r GM -r f = + ?(r m -r f ) α α Regression Results Irwin/McGraw-Hill 10-8 ? The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus ? Market or systematic risk: risk related to the macro economic factor or market index ? Unsystematic or firm specific risk: risk not related to the macro factor or market index ? Total risk = Systematic + Unsystematic Components of Risk Irwin/McGraw-Hill 10-9 ? The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus σ i 2 = β i 2 σ m 2 + σ 2 (e i ) where; σ i 2 = total variance β i 2 σ m 2 = systematic variance σ 2 (e i ) = unsystematic variance Measuring Components of Risk Irwin/McGraw-Hill 10-10 ? The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Total Risk = Systematic Risk + Unsystematic Risk Systematic Risk/Total Risk = 2 ? i 2 σ m 2 / σ 2 = 2 β i 2 σ m 2 / (β i 2 σ m 2 + σ 2 (e i ) )= 2 Examining Percentage of Variance R R R Irwin/McGraw-Hill 10-11 ? The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Index model and portfolio construction ? n assets ? n expected returns ? n betas ? )( 2 i en σ 2 M σ Irwin/McGraw-Hill 10-12 ? The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Index Model and Diversification )( 1 1 1 2222 1 1 1 PMP P N i iP N i iP N i iP PMPPP e e N e N N eRR σσβσ αα ββ βα += = = = ++= ∑ ∑ ∑ = = = Irwin/McGraw-Hill 10-13 ? The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Risk Reduction with Diversification Number of Securities variance Market Risk Unique Risk σ 2 (e P )=σ 2 (e) / n β P 2 σ M 2 Irwin/McGraw-Hill 10-14 ? The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Industry Prediction of Beta ? Merrill Lynch Example - Use returns not risk premiums ?α has a different interpretation ?α = α + r f (1-β) ? Forecasting beta as a function of past beta ? Forecasting beta as a function of firm size, growth, leverage etc. Irwin/McGraw-Hill 10-15 ? The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Multifactor Models ? Use factors in addition to market return - Examples include industrial production, expected inflation etc. - Estimate a beta for each factor using multiple regression ? Fama and French - Returns a function of size and book-to-market value as well as market returns