INVESTMENTS Optimal Risky Portfolios Chapter 4 INVESTMENTS Risk Reduction with Diversification Number of Securities St. Deviation Market Risk Unique Risk INVESTMENTS r p = W 1 r 1 +W 2 r 2 W 1 = Proportion of funds in Security 1 W 2 = Proportion of funds in Security 2 r 1 = Expected return on Security 1 r 2 = Expected return on Security 2 1= ∑ = n 1i i w Two-Security Portfolio: Return INVESTMENTS 　σ p 2 = w 1 2 σ 1 2 + w 2 2 σ 2 2 + 2W 1 W 2 Cov(r 1 r 2 ) 　σ 1 2 = Variance of Security 1 　σ 2 2 = Variance of Security 2 Cov(r 1 r 2 ) = Covariance of returns for Security 1 and Security 2 Two-Security Portfolio: Risk INVESTMENTS 　ρ 1,2 = Correlation coefficient of returns Cov(r 1 r 2 ) = 　ρ ??2 σ 1 σ 2 　σ 1 = Standard deviation of returns for Security 1 　σ 2 = Standard deviation of returns for Security 2 Covariance INVESTMENTS Range of values for 　ρ 1,2 + 1.0 > ρ　???> ?-1.0 If ρ　?= 1.0, the securities would be perfectly positively correlated If ρ　?= - 1.0, the securities would be perfectly negatively correlated Correlation Coefficients: Possible Values INVESTMENTS 　σ 2 p = W 1 σ 1 2 + 2W 1 W 2 r p = W 1 r 1 +W 2 r 2 + W 3 r 3 Cov(r 1 r 2 ) + W 3 2 σ 3 2 Cov(r 1 r 3 )+ 2W 1 W 3 Cov(r 2 r 3 )+ 2W 2 W 3 Three-Security Portfolio 2 22 σW+ INVESTMENTS r p = Weighted average of the n securities 　σ p 2 = (Consider all pairwise covariance measures) In General, For an n-Security Portfolio: INVESTMENTS E(r p ) = W 1 r 1 +W 2 r 2 σ p 2 = w 1 2 σ 1 2 + w 2 2 σ 2 2 + 2W 1 W 2 Cov(r 1 r 2 ) σ p = [w 1 2 σ 1 2 + w 2 2 σ 2 2 + 2W 1 W 2 Cov(r 1 r 2 )] 1/2 Two-Security Portfolio INVESTMENTS Two-Security Portfolios with Different Correlations 　ρ = 1 13% E(r) St. Dev 12% 20% ρ = .3 ρ = -1 ρ = -1 8% INVESTMENTS ? Relationship depends on correlation coefficient ? -1.0 < ρ < +1.0 ? The smaller the correlation, the greater the risk reduction potential ? If ρ = +1.0, no risk reduction is possible Portfolio Risk/Return Two Securities: Correlation Effects INVESTMENTS 1 - Cov(r 1 r 2 ) W 1 = + - 2Cov(r 1 r 2 ) W 2 = (1 - W 1 ) 2 E(r 2 ) = .14 = .20Sec 2 12 = .2 E(r 1 ) = .10 = .15Sec 1 σ　 σ　 σ　 Minimum-Variance Combination 2 2 σ 2 1 σ 2 2 σ INVESTMENTS W 1 = (.2) 2 - (.2)(.15)(.2) (.15) 2 + (.2) 2 - 2(.2)(.15)(.2) W 1 = .6733 W 2 = (1 - .6733) = .3267 Minimum-Variance Combination: ρ = .2 INVESTMENTS r p = .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 σ　 　 σ　 　 Minimum -Variance: Return and Risk with ρ = .2 INVESTMENTS W 1 = (.2) 2 - (.2)(.15)(.2) (.15) 2 + (.2) 2 - 2(.2)(.15)(-.3) W 1 = .6087 W 2 = (1 - .6087) = .3913 Minimum -Variance Combination: ρ = -.3 INVESTMENTS r p = .6087(.10) + .3913(.14) = .1157 p = [(.6087) 2 (.15) 2 + (.3913) 2 (.2) 2 + 2(.6087)(.3913)(.2)(.15)(-.3)] 1/2 p = [.0102] 1/2 = .1009 σ　 　 　σ 　 Minimum -Variance: Return and Risk with ρ = -.3 INVESTMENTS ? 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 INVESTMENTS The Minimum-Variance Frontier of Risky Assets E(r) Efficient frontier Global minimum variance portfolio Minimum variance frontier Individual assets St. Dev. INVESTMENTS ? The optimal combination becomes linear ? A single combination of risky and riskless assets will dominate Extending to Include Riskless Asset INVESTMENTS Alternative CALs M E(r) CAL (Global minimum variance) CAL (A) CAL (P) P A F PP&F A&F M A G P M 　 INVESTMENTS 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 INVESTMENTS Efficient Frontier with Lending & Borrowing E(r) F r f A P Q B CAL St. Dev