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Chapter 10

Index Models. Chapter 10. 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. Index Models Chapter 10

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

  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.

  4. Single Index Model a (ri - rf)= i + ßi(rm - rf)+ ei Risk Prem Market Risk Prem or Index Risk Prem a = 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

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

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

  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

  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)

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

  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

  12. Index Model and Diversification

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

  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.

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