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

2. 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.

3. 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

4. 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

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

6. 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

7. 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 ?..

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

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

10. 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

11. 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)

12. Investments 912 Efficient Portfolio Frontier  The effect of correlation  Lower correlation means greater risk reduction  If  = +1.0, no risk reduction is possible

13. 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

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

15. 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?