This chapter a simplifying assumption that at once eases our computational burden and offers significant new insights into the nature of systematic risk versus firm-specific risk. This abstraction is the notion of an “index model,” specifying the process by which security returns are generated. | Chapter 10 Index Models Reduces the number of inputs for diversification. Easier for security analysts to specialize. Advantages of the Single Index Model ri = E(Ri) + ßiF + e ßi = index of a securities’ particular return to the factor F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns Assumption: a broad market index like the S&P500 is the common factor. Single Factor Model (ri - rf) = i + ßi(rm - rf) + ei a Risk Prem Market Risk Prem or Index Risk Prem i = the stock’s expected return if the market’s excess return is zero ßi(rm - rf) = the component of return due to movements in the market index (rm - rf) = 0 ei = firm specific component, not due to market movements a Single Index Model Let: Ri = (ri - rf) Rm = (rm - rf) Risk premium format Ri = i + ßi(Rm) + ei Risk Premium Format Security Characteristic Line Excess Returns (i) SCL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Excess returns on market index Ri = i + ßiRm + ei . . . Jan. Feb. . . Dec Mean Std Dev . . .93 . . Excess Mkt. Ret. Excess GM Ret. Using the Text Example from Table 10-1 Estimated coefficient Std error of estimate Variance of residuals = Std dev of residuals = R-SQR = ß () () rGM - rf = + ß(rm - rf) Regression Results 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 i2 = i2 m2 + 2(ei) where; i2 = total variance i2 m2 = systematic variance 2(ei) = unsystematic variance Measuring Components of Risk Total Risk = Systematic Risk + Unsystematic Risk Systematic Risk/Total Risk = 2 ßi2 m2 / 2 = 2 i2 m2 / i2 m2 + 2(ei) = 2 Examining Percentage of Variance Index Model and Diversification Risk Reduction with Diversification Number of Securities St. Deviation Market Risk Unique Risk s2(eP)=s2(e) / n bP2sM2 Industry Prediction of Beta Merrill Lynch Example Use returns not risk premiums a has a different interpretation a = a + rf (1-b) Forecasting beta as a function of past beta Forecasting beta as a function of firm size, growth, leverage etc. | Chapter 10 Index Models Reduces the number of inputs for diversification. Easier for security analysts to specialize. Advantages of the Single Index Model ri = E(Ri) + ßiF + e ßi = index of a securities’ particular return to the factor F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns Assumption: a broad market index like the S&P500 is the common factor. Single Factor Model (ri - rf) = i + ßi(rm - rf) + ei a Risk Prem Market Risk Prem or Index Risk Prem i = the stock’s expected return if the market’s excess return is zero ßi(rm - rf) = the component of return due to movements in the market index (rm - rf) = 0 ei = firm specific component, not due to market movements a Single Index Model Let: Ri = (ri - rf) Rm = (rm - rf) Risk premium format Ri = i + ßi(Rm) + ei Risk Premium Format Security Characteristic Line Excess Returns (i) SCL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
