Optimal Risky Portfolios Chapter 8

Risk Reduction with Diversification St. Deviation Unique Risk Market Risk Number of Securities

Two-Security Portfolio: Return 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: Risk 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

Covariance Cov(r1r2) = 1,212 1,2 = Correlation coefficient of returns 1 = Standard deviation of returns for Security 1 2 = Standard deviation of returns for Security 2

Correlation Coefficients: Possible Values Range of values for 1,2 + 1.0 > r > -1.0 If r= 1.0, the securities would be perfectly positively correlated If r= - 1.0, the securities would be perfectly negatively correlated

Three-Security Portfolio rp = W1r1 + W2r2 + W3r3 2p = W1212 + W2212 + W3232 + 2W1W2 Cov(r1r2) + 2W1W3 Cov(r1r3) + 2W2W3 Cov(r2r3)

In General, For An N-Security Portfolio: rp = Weighted average of the n securities p2 = (Consider all pairwise covariance measures)

Two-Security Portfolio E(rp) = W1r1 + W2r2 p2 = w1212 + w2222 + 2W1W2 Cov(r1r2) p = [w1212 + w2222 + 2W1W2 Cov(r1r2)]1/2

Portfolios with Different Correlations 13%  = -1  = .3  = -1  = 1 %8 St. Dev 12% 20%

Correlation Effects The relationship depends on correlation coefficient. -1.0 <  < +1.0 The smaller the correlation, the greater the risk reduction potential. If r = +1.0, no risk reduction is possible.

Minimum-Variance Combination 2 E(r2) = .14 = .20 Sec 2 12 = .2 E(r1) = .10 = .15 Sec 1   1 r22 - Cov(r1r2) = W1 s2 s2 - 2Cov(r1r2) + 1 2 W2 = (1 - W1)

Minimum-Variance Combination:  = .2 W1 = (.2)2 - (.2)(.15)(.2) (.15)2 + (.2)2 - 2(.2)(.15)(.2) = .6733 W2 = (1 - .6733) = .3267

Risk and Return: Minimum Variance rp = .6733(.10) + .3267(.14) = .1131  = [(.6733)2(.15)2 + (.3267)2(.2)2 + p 1/2 2(.6733)(.3267)(.2)(.15)(.2)] s 1/2 = [.0171] = .1308 p

Minimum - Variance Combination:  = -.3 W1 = (.2)2 - (.2)(.15)(.2) (.15)2 + (.2)2 - 2(.2)(.15)(-.3) = .6087 W2 = (1 - .6087) = .3913

Risk and Return: Minimum Variance rp = .6087(.10) + .3913(.14) = .1157 s = [(.6087)2(.15)2 + (.3913)2(.2)2 + p 1/2 2(.6087)(.3913)(.2)(.15)(-.3)] s 1/2 = [.0102] = .1009 p

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.

Minimum-Variance Frontier of Risky Assets Efficient frontier Individual assets Global minimum variance portfolio Minimum variance frontier St. Dev.

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

Alternative CALs E(r) CAL (P) CAL (A) M M P P CAL (Global minimum variance) A A G F  P P&F M A&F

Portfolio Selection & Risk Aversion U’’ U’ U’’’ E(r) Efficient frontier of risky assets S P Q Less risk-averse investor More risk-averse investor St. Dev

Efficient Frontier with Lending & Borrowing CAL E(r) B Q P A rf F St. Dev