Single Index and Multifactor Models

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Presentation transcript:

Single Index and Multifactor Models 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 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.

a (ri - rf) = i + ßi(rm - rf) + ei Single Index Model 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

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

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

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

  rGM - rf = + ß(rm - rf) ß Estimated coefficient 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)

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

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

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

Index Model and Diversification

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

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.

Use factors in addition to market return 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.