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McGraw-Hill/Irwin © 2008 The McGraw-Hill Companies, Inc., All Rights Reserved. Efficient Diversification CHAPTER 6

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6-2 6.1 DIVERSIFICATION AND PORTFOLIO RISK

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6-3 Diversification and Portfolio Risk Market risk –Systematic or Nondiversifiable Firm-specific risk –Diversifiable or nonsystematic

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6-4 Figure 6.1 Portfolio Risk as a Function of the Number of Stocks

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6-5 Figure 6.2 Portfolio Risk as a Function of Number of Securities

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6-6 6.2 ASSET ALLOCATION WITH TWO RISKY ASSETS

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6-7 Covariance and Correlation Portfolio risk depends on the correlation between the returns of the assets in the portfolio Covariance and the correlation coefficient provide a measure of the returns on two assets to vary

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6-8 Two Asset Portfolio Return – Stock and Bond

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6-9 Covariance and Correlation Coefficient Covariance: Correlation Coefficient:

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6-10 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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6-11 Two Asset Portfolio Std. Dev. – Stock and Bond

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6-12 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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6-13 Three ‘Rules’ of Two-Risky-Asset Portfolios Rate of return on the portfolio: Expected rate of return on the portfolio:

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6-14 Three ‘Rules’ of Two-Risky-Asset Portfolios Variance of the rate of return on the portfolio:

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6-15 Numerical Text Example: Bond and Stock Returns (Page 169) 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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6-16 Numerical Text Example: Bond and Stock Returns (Page 169) Expected Return =.08 = 8% =.5(0.06) +.5 (0.10)=0.08 Standard Deviation =.1387=13.87% =[(.5) 2 (0.12) 2 + (.5) 2 (0.25) 2 + 2 (.5) (.5) (0.12) (0.25) (0)] ½ 2 (.5) (.5) (0.12) (0.25) (0)] ½ =[1.9225] 1/2 =.1387 = 13.87%

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6-17 Figure 6.3 Investment Opportunity Set for Stocks and Bonds

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6-18 Figure 6.4 Investment Opportunity Set for Stocks and Bonds with Various Correlations

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6-19 6.3 THE OPTIMAL RISKY PORTFOLIO WITH A RISK-FREE ASSET

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6-20 Extending to Include Riskless Asset The optimal combination becomes linear A single combination of risky and riskless assets will dominate

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6-21 Figure 6.5 Opportunity Set Using Stocks and Bonds and Two Capital Allocation Lines

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6-22 Dominant CAL with a Risk-Free Investment (F) CAL(O) dominates other lines -- it has the best risk/return or the largest slope Slope =

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6-23 Dominant CAL with a Risk-Free Investment (F) Regardless of risk preferences, combinations of O & F dominate

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6-24 Figure 6.6 Optimal Capital Allocation Line for Bonds, Stocks and T-Bills

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6-25 Optimal Allocation We want to maximize the Sharpe ratio – the ratio of the expected return to risk for the complete portfolio risk The complete portfolio is the portfolio that is a combination of one of our risky asset portfolios (in our portfolio choice set) and the risk free asset. The highest Sharpe ratio is for the CAL just tangent to the portfolio choice set

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6-26 Optimal Allocation Maximizing the Sharpe Ratio:

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6-27 Figure 6.7 The Complete Portfolio

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6-28 Figure 7.8 Determination of the Optimal Overall Portfolio

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6-29 Figure 6.8 The Complete Portfolio – Solution to the Asset Allocation Problem

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6-30 6.4 EFFICIENT DIVERSIFICATION WITH MANY RISKY ASSETS

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6-31 2 p = w 1 2 1 2 + w 2 2 1 2 + 2w 1 w 2 Cov(r 1, r 2 ) + w 3 2 3 2 Cov(r 1, r 3 ) + 2w 1 w 3 Cov(r 2, r 3 )+ 2w 2 w 3 Three-Security Portfolio

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6-32 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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6-33 Figure 6.9 Portfolios Constructed from Three Stocks A, B and C

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6-34 Figure 6.10 The Efficient Frontier of Risky Assets and Individual Assets

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6-35 Portfolio Choice Set – Minimum Variance Finding the minimum variance portfolio:

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6-36 Investment Opportunity Set for Bond and Stock Funds

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6-37 Optimal Allocation We want to maximize the Sharpe ratio – the ratio of the expected return to risk for the complete portfolio risk The complete portfolio is the portfolio that is a combination of one of our risky asset portfolios (in our portfolio choice set) and the risk free asset. The highest Sharpe ratio is for the CAL just tangent to the portfolio choice set

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6-38 6.5 A SINGLE-FACTOR ASSET MARKET

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6-39 Single Factor Model β i = index of a securities’ particular return to the factor M = unanticipated movement commonly related to security returns E i = unexpected event relevant only to this security Assumption: a broad market index like the S&P500 is the common factor

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6-40 Specification of a Single-Index Model of Security Returns Use the S&P 500 as a market proxy Excess return can now be stated as: –This specifies the both market and firm risk

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6-41 Figure 6.11 Scatter Diagram for Dell

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6-42 Figure 6.12 Various Scatter Diagrams

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6-43 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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6-44 Measuring Components of Risk i 2 = i 2 m 2 + 2 (e i ) where; i 2 = total variance of firm i i 2 m 2 = systematic variance 2 (e i ) = unsystematic variance

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6-45 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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6-46 Advantages of the Single Index Model Reduces the number of inputs for diversification Easier for security analysts to specialize

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6-47 6.6 RISK OF LONG-TERM INVESTMENTS

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6-48 Are Stock Returns Less Risky in the Long Run? Consider a 2-year investment Variance of the 2-year return is double of that of the one-year return and σ is higher by a multiple of the square root of 2

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6-49 Are Stock Returns Less Risky in the Long Run? Generalizing to an investment horizon of n years and then annualizing:

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6-50 The Fly in the ‘Time Diversification’ Ointment Annualized standard deviation is only appropriate for short-term portfolios Variance grows linearly with the number of years Standard deviation grows in proportion to

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6-51 The Fly in the ‘Time Diversification’ Ointment To compare investments in two different time periods: –Risk of the total (end of horizon) rate of return –Accounts for magnitudes and probabilities

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