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Chapter 10

Chapter 10. Single Index and Multifactor Models. Advantages of the Single Index Model. Reduces the number of inputs for diversification Easier for security analysts to specialize. Single Factor Model. r i = E(R i ) + ß i F + e ß i = index of a securities’ particular return to the factor

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Chapter 10

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  1. Chapter 10 Single Indexand Multifactor Models 10-1

  2. Advantages of the Single Index Model • Reduces the number of inputs for diversification • Easier for security analysts to specialize 10-2

  3. Single Factor 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 10-3

  4. (ri - rf)= i + ßi(rm - rf)+ ei Single Index Model Risk Prem Market Risk Prem or Index Risk Prem  = the stock’s expected return if the market’s excess return is zero i (rm - rf)= 0 ßi(rm - rf)= the component of return due to movements in the market index ei = firm specific component, not due to market movements 10-4

  5. Let: Ri = (ri - rf) Risk premium format Rm = (rm - rf) Ri = i + ßi(Rm)+ ei Risk Premium Format 10-5

  6. Excess Returns (i) SCL . . . . . . . . . . . . . . . . . . . . . . . . . . Excess returns on market index . . . . . . . . . . . . . . . . . . . . . . . . Ri =  i + ßiRm + ei Security Characteristic Line 10-6

  7. Using the Text Example from Table 10-1 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 10-7

  8. 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) 10-8

  9. Components of Risk • 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 10-9

  10. Measuring Components of Risk i2 = i2m2 + 2(ei) where; i2 = total variance i2m2 = systematic variance 2(ei) = unsystematic variance 10-10

  11. Examining Percentage of Variance Total Risk = Systematic Risk + Unsystematic Risk Systematic Risk/Total Risk = 2 ßi2 m2/ 2 = 2 i2m2/i2m2 + 2(ei) = 2 10-11

  12. Index Model and Diversification 10-12

  13. Risk Reduction with Diversification St. Deviation Unique Risk s2(eP)=s2(e) / n bP2sM2 Market Risk Number of Securities 10-13

  14. 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. 10-14

  15. 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 10-15

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