This chapter contains additional material on the “art” of selecting reasonable parameter values for portfolio construction, and a discussion of what can go wrong when inputs are derived solely from recent historical experience. | Chapter 8 Optimal Risky Portfolios Risk Reduction with Diversification Number of Securities St. Deviation Market Risk Unique Risk rp = W1r1 + W2r2 W1 = Proportion of funds in Security 1 W2 = Proportion of funds in Security 2 r1 = Expected return on Security 1 r2 = Expected return on Security 2 Two-Security Portfolio: Return p2 = w12 12 + w22 22 + 2W1W2 Cov(r1r2) 12 = Variance of Security 1 22 = Variance of Security 2 Cov(r1r2) = Covariance of returns for Security 1 and Security 2 Two-Security Portfolio: Risk 1,2 = Correlation coefficient of returns Cov(r1r2) = 1,2 1 2 1 = Standard deviation of returns for Security 1 2 = Standard deviation of returns for Security 2 Covariance Range of values for 1,2 + > r > If r= , the securities would be perfectly positively correlated If r= - , the securities would be perfectly negatively correlated Correlation Coefficients: Possible Values 2p = W12 12 + W22 12 + 2W1W2 rp = W1r1 + W2r2 + W3r3 Cov(r1r2) + W32 32 Cov(r1r3) + 2W1W3 Cov(r2r3) + 2W2W3 Three-Security Portfolio rp = Weighted average of the n securities p2 = (Consider all pairwise covariance measures) In General, For An N-Security Portfolio: E(rp) = W1r1 + W2r2 p2 = w12 12 + w22 22 + 2W1W2 Cov(r1r2) p = [w12 12 + w22 22 + 2W1W2 Cov(r1r2)]1/2 Two-Security Portfolio Portfolios with Different Correlations = 1 13% %8 E(r) St. Dev 12% 20% = .3 = -1 = -1 The relationship depends on correlation coefficient. < < + The smaller the correlation, the greater the risk reduction potential. If r = +, no risk reduction is possible. Correlation Effects 1 1 2 r22 - Cov(r1r2) W1 = + - 2Cov(r1r2) W2 = (1 - W1) s2 2 E(r2) = .14 = .20 Sec 2 12 = .2 E(r1) = .10 = .15 Sec 1 s 2 Minimum-Variance Combination W1 = (.2)2 - (.2)(.15)(.2) (.15)2 + (.2)2 - 2(.2)(.15)(.2) W1 = .6733 W2 = (1 - .6733) = .3267 Minimum-Variance Combination: = .2 rp = .6733(.10) + .3267(.14) = .1131 p = [(.6733)2(.15)2 + (.3267)2(.2)2 + 2(.6733)(.3267)(.2)(.15)(.2)] 1/2 p = [.0171] 1/2 = .1308 s Risk and Return: Minimum Variance W1 = (.2)2 - (.2)(.15)(.2) (.15)2 + (.2)2 - 2(.2)(.15)() W1 = .6087 W2 = (1 - .6087) = .3913 Minimum - Variance Combination: = rp = .6087(.10) + .3913(.14) = .1157 p = [(.6087)2(.15)2 + (.3913)2(.2)2 + 2(.6087)(.3913)(.2)(.15)()] 1/2 p = [.0102] 1/2 = .1009 s s Risk and Return: Minimum Variance 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. Extending Concepts to All Securities Minimum-Variance Frontier of Risky Assets E(r) Efficient frontier Global minimum variance portfolio Minimum variance frontier Individual assets St. Dev. The optimal combination becomes linear. A single combination of risky and riskless assets will dominate. Extending to Include Riskless Asset Alternative CALs M E(r) CAL (Global minimum variance) CAL (A) CAL (P) P A F P P&F A&F M A G P M Portfolio Selection & Risk Aversion E(r) Efficient frontier of risky assets More risk-averse investor U’’’ U’’ U’ Q P S St. Dev Less risk-averse investor Efficient Frontier with Lending & Borrowing E(r) F rf A P Q B CAL St. Dev
