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Efficient Diversification I Covariance and Portfolio Risk Mean-variance Frontier Efficient Portfolio Frontier

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Investments 92 Some Empirical Evidence In 2000, 40% of stocks in Russell 3000 had returns of -20% or worse. Meanwhile, less than 12% of U.S. stock mutual funds had returns of -20% or below. Of the 2,397 U.S. stocks in existence throughout 1990s, 22% had negative returns. In contrast, 0.4% of U.S. equity mutual funds had negative returns.

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Investments 93 Diversification and Portfolio Risk “Don’t put all your eggs in one basket” Effect of portfolio diversification # of securities in the portfolio 5 10 15 20 Diversifiable risk, non-systematic risk, firm-specific risk, idiosyncratic risk Non-diversifiable risk, systematic risk, market risk

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Investments 94 Covariance and Correlation Covariance and correlation Degree of co-movement of two stocks Covariance: non-standardized measure Correlation coefficient: standardized measure r1r1 r2r2 r1r1 r2r2 r1r1 r2r2 0< 12 <1 -1< 12 <0 12 =0

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Investments 95 Covariance and Correlation Example: Two risky assets Calculating the covariance MeansStd. Dev.Cov.Corr.

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Investments 96 Diversification and Portfolio Risk A portfolio of two risky assets w 1 : % invested in bond w 2 : % invested in stock Expected return Variance

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Investments 97 Diversification and Portfolio Risk Example: Portfolio of two risky securities w in security 1, (1 – w) in security 2 Expected return (Mean): Variance What happens when w changes? Expected return decreases with increasing w How about variance ?..

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Investments 98 Mean-Variance Frontier w from 0 to 1 Security 1 Security 2 Mean-variance frontier GMVP GMVP: Global Minimum Variance Portfolio

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Investments 99 Mean-Variance Frontier Global Minimum Variance Port. (GMVP) A unique w Associated characteristics

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Investments 910 Efficient Portfolio Frontier 67% in Security 1 and 33% in Security 2, what’s so special? Efficient portfolio has 33% in 2 Inefficient Frontier Efficient Frontier GMVP w 1 =1 w 1 =.6733 w 1 =0 P

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Investments 911 Efficient Portfolio Frontier Portfolio “P” dominates Security 1 The same standard deviation The higher expected return How to find it? Since the portfolio has the same standard deviation as Security 1 Solve the quadratic equation w = 1 (Security 1) or w =.3465 (Portfolio P)

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Investments 912 Efficient Portfolio Frontier The effect of correlation Lower correlation means greater risk reduction If = +1.0, no risk reduction is possible

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Investments 913 Efficient Portfolio Frontier Efficient Portfolio of Many securities E[r p ]: Weighted average of n securities p 2 : Combination of all pair-wise covariance measures Construction of the efficient frontier is complicated Analytical solution without short-sale constraints Numerical solution with short-sale constraints Numerical solution General Features Optimal combination results in lowest risk for given return Efficient frontier describes optimal trade-off Portfolios on efficient frontier are dominant

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Investments 914 Efficient Frontier E[r] Efficientfrontier Globalminimumvarianceportfolio Minimumvariancefrontier Individualassets St. Dev.

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Investments 915 Wrap-up How to estimate portfolio return and risk? What is the mean-variance frontier? What is the efficient portfolio frontier? Why do portfolios on efficient frontier dominate other combinations?

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