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CHAPTER 6

CHAPTER 6. Efficient Diversification. Diversification and Portfolio Risk. Market risk Systematic or Nondiversifiable Firm-specific risk Diversifiable or nonsystematic. Figure 6.1 Portfolio Risk as a Function of the Number of Stocks.

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CHAPTER 6

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  1. CHAPTER 6 Efficient Diversification

  2. Diversification and Portfolio Risk • Market risk • Systematic or Nondiversifiable • Firm-specific risk • Diversifiable or nonsystematic

  3. Figure 6.1 Portfolio Risk as a Function of the Number of Stocks

  4. Figure 6.2 Portfolio Risk as a Function of Number of Securities

  5. Two Asset Portfolio Return – Stock and Bond

  6. Covariance Cov(r1r2) = r1,2s1s2 r1,2 = Correlation coefficient of returns s1 = Standard deviation of returns for Security 1 s2 = Standard deviation of returns for Security 2

  7. Correlation Coefficients: Possible Values Range of values for r1,2 -1.0 <r < 1.0 If r = 1.0, the securities would be perfectly positively correlated If r = - 1.0, the securities would be perfectly negatively correlated

  8. Two Asset Portfolio St Dev – Stock and Bond

  9. In General, For an n-Security Portfolio: rp = Weighted average of the n securities sp2 = (Consider all pair-wise covariance measures)

  10. Numerical Example: Bond and Stock Returns Bond = 6% Stock = 10% Standard Deviation Bond = 12% Stock = 25% Weights Bond = .5 Stock = .5 Correlation Coefficient (Bonds and Stock) = 0

  11. Return and Risk for Example Return = 8% .5(6) + .5 (10) Standard Deviation = 13.87% [(.5)2 (12)2 + (.5)2 (25)2 + … 2 (.5) (.5) (12) (25) (0)] ½ [192.25] ½ = 13.87

  12. Figure 6.3 Investment Opportunity Set for Stock and Bonds

  13. Figure 6.4 Investment Opportunity Set for Stock and Bonds with Various Correlations

  14. Figure 6.3 Investment Opportunity Set for Bond and Stock Funds

  15. Extending to Include Riskless Asset • The optimal combination becomes linear • A single combination of risky and riskless assets will dominate

  16. Figure 6.5 Opportunity Set Using Stock and Bonds and Two Capital Allocation Lines

  17. Dominant CAL with a Risk-Free Investment (F) CAL(O) dominates other lines -- it has the best risk/return or the largest slope Slope = (E(R) - Rf) / s [ E(RP) - Rf) / s P] > [E(RA) - Rf) / sA] Regardless of risk preferences combinations of O & F dominate

  18. Figure 6.6 Optimal Capital Allocation Line for Bonds, Stocks and T-Bills

  19. Figure 6.7 The Complete Portfolio

  20. Figure 6.8 The Complete Portfolio – Solution to the Asset Allocation Problem

  21. Extending Concepts to All Securities • The optimal combinations result in lowest level of risk for a given return • The optimal trade-off is described as the efficient frontier • These portfolios are dominant

  22. Figure 6.9 Portfolios Constructed from Three Stocks A, B and C

  23. Figure 6.10 The Efficient Frontier of Risky Assets and Individual Assets

  24. 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

  25. Single Index Model ( ) ( ) b a e r r r r - = + - + i f m f i i i 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

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

  27. Figure 6.11 Scatter Diagram for Dell

  28. Figure 6.12 Various Scatter Diagrams

  29. 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

  30. Measuring Components of Risk si2 = bi2sm2 + s2(ei) where; si2 = total variance bi2sm2 = systematic variance s2(ei) = unsystematic variance

  31. Examining Percentage of Variance Total Risk = Systematic Risk + Unsystematic Risk Systematic Risk/Total Risk = r2 ßi2 sm2 / s2 = r2 bi2sm2 / bi2sm2 + s2(ei) = r2

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

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