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Efficient Diversification CHAPTER 6
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Diversification and Portfolio Risk Market risk –Systematic or Nondiversifiable Firm-specific risk –Diversifiable or nonsystematic
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Figure 6.1 Portfolio Risk as a Function of the Number of Stocks
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Figure 6.2 Portfolio Risk as a Function of Number of Securities
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Two Asset Portfolio Return – Stock and Bond
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Covariance 1,2 = Correlation coefficient of returns 1,2 = Correlation coefficient of returns Cov(r 1 r 2 ) = 1 2 1 = Standard deviation of returns for Security 1 2 = Standard deviation of returns for Security 2 1 = Standard deviation of returns for Security 1 2 = Standard deviation of returns for Security 2
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Correlation Coefficients: Possible Values If = 1.0, the securities would be perfectly positively correlated If = - 1.0, the securities would be perfectly negatively correlated Range of values for 1,2 -1.0 < < 1.0
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Two Asset Portfolio St Dev – Stock and Bond
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r p = Weighted average of the n securities r p = Weighted average of the n securities p 2 = (Consider all pair-wise covariance measures) p 2 = (Consider all pair-wise covariance measures) In General, For an n- Security Portfolio:
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Numerical Example: Bond and Stock Returns Bond = 6%Stock = 10% Standard Deviation Bond = 12%Stock = 25% Weights Bond =.5Stock =.5 Correlation Coefficient (Bonds and Stock) = 0
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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
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Figure 6.3 Investment Opportunity Set for Stock and Bonds
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Figure 6.4 Investment Opportunity Set for Stock and Bonds with Various Correlations
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Figure 6.3 Investment Opportunity Set for Bond and Stock Funds
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Extending to Include Riskless Asset The optimal combination becomes linear A single combination of risky and riskless assets will dominate
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Figure 6.5 Opportunity Set Using Stock and Bonds and Two Capital Allocation Lines
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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) / E(R P ) - R f ) / P E(R A ) - R f ) / Regardless of risk preferences combinations of O & F dominate
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Figure 6.6 Optimal Capital Allocation Line for Bonds, Stocks and T-Bills
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Figure 6.7 The Complete Portfolio
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Figure 6.8 The Complete Portfolio – Solution to the Asset Allocation Problem
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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
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Figure 6.9 Portfolios Constructed from Three Stocks A, B and C
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Figure 6.10 The Efficient Frontier of Risky Assets and Individual Assets
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Single Factor Model r i = E(R i ) + ß i F + e ß i = 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
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Single Index Model Risk Prem Market Risk Prem or Index Risk Prem or Index Risk Prem i = the stock’s expected return if the market’s excess return is zero market’s excess return is zero ß i (r m - r f ) = the component of return due to movements in the market index movements in the market index (r m - r f ) = 0 e i = firm specific component, not due to market movements movements e rrrr i fm i i fi
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Let: R i = (r i - r f ) R m = (r m - r f ) R m = (r m - r f ) Risk premium format R i = i + ß i (R m ) + e i Risk Premium Format
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Figure 6.11 Scatter Diagram for Dell
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Figure 6.12 Various Scatter Diagrams
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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
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Measuring Components of Risk i 2 = i 2 m 2 + 2 (e i ) where; i 2 = total variance i 2 m 2 = systematic variance 2 (e i ) = unsystematic variance
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Total Risk = Systematic Risk + Unsystematic Risk Systematic Risk/Total Risk = 2 ß i 2 m 2 / 2 = 2 i 2 m 2 / i 2 m 2 + 2 (e i ) = 2 Examining Percentage of Variance
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Advantages of the Single Index Model Reduces the number of inputs for diversification Easier for security analysts to specialize
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