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Index Models

Index Models. Diversification. Random selection The effect of diversification Markowitz diversification What information are needed? How to simplify the approach?. Linear Regression . Review Properties R-square Example spreadsheet. The Single Index Model. Advantages :

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Index Models

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  1. Index Models

  2. Diversification • Random selection • The effect of diversification • Markowitz diversification • What information are needed? • How to simplify the approach?

  3. Linear Regression • Review • Properties • R-square • Example • spreadsheet

  4. The Single Index Model Advantages: • Reduces the number of inputs for diversification • Easier for security analysts to specialize Drawback: • the simple dichotomy rules out important risk sources (such as industry events)

  5. Single Factor Model ßi = index of a security’s 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

  6. Single Index Model ai = stock’s expected return if market’s excess return is zero bi(rM-ri) = the component of return due to market movements ei = the component of return due to unexpected firm-specific events

  7. Let: Ri = (ri - rf) Risk premium format Rm = (rm - rf) Ri = αi + ßiRm + ei Risk Premium Format

  8. 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

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

  10. Examining Percentage of Variance Total Risk = Systematic +Unsystematic

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

  12. Index Model • Spreadsheet example

  13. St. Deviation Unique Risk s2(eP)=s2(e) / n bP2sM2 Market Risk No. of Securities Index Model and Diversification

  14. Industry Prediction of Beta • BMO Nesbitt Burns and Merrill Lynch examples • BMO NB uses returns not risk premiums • a has a different interpretation: a + rf (1-b) • Merill Lynch’s ‘adjusted b’ • Forecasting beta as a function of past beta • Forecasting beta as a function of firm size, growth, leverage etc.

  15. Tests of the Single Factor Model Tests of the expected return beta relationship • First Pass Regression • Estimate beta, average risk premiums and unsystematic risk • Second Pass: Using estimates from the first pass to determine if model is supported by the data • Most tests do not generally support the single factor model

  16. Return % Predicted Actual Beta Single Factor Test Results

  17. Roll’s Criticism on the Tests • The only testable hypothesis: the mean-variance efficiency of the market portfolio • All other implications are not independently testable • CAPM is not testable unless we use the true market portfolio • The benchmark error

  18. Measurement Error in Beta Statistical property: • If beta is measured with error in the first stage, • Second stage results will be biased in the direction the tests have supported • Test results could result from measurement error

  19. Conclusions on the Tests’ Results • Tests proved that CAPM seems qualitatively correct • Rates of return are linear and increase with beta • Returns are not affected by nonsystematic risk • But they do not entirely validate its quantitative predictions • The expected return-beta relationship is not fully consistent with empirical observation.

  20. 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 • Chen, Roll and Ross • Returns a function of several macroeconomic and bond market variables instead of market returns • Fama and French • Returns a function of size and book-to-market value as well as market returns

  21. Researchers’ Responses to Fama and French • Utilize better econometric techniques • Improve estimates of beta • Reconsider the theoretical sources and implications of the Fama and French-type results • Return to the single-index model, accounting for non-traded assets and cyclical behavior of betas

  22. Jaganathan and Wang Study (1996) • Included factors for cyclical behavior of betas and human capital • When these factors were included the results showed returns were a function of beta • Size is not an important factor when cyclical behavior and human capital are included

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