Tài chính doanh nghiệp - Chapter 8: Optimal risky portfolios

Chapter 8Optimal Risky PortfoliosRisk Reduction with DiversificationNumber of SecuritiesSt. DeviationMarket RiskUnique Riskrp = W1r1 + W2r2W1 = Proportion of funds in Security 1W2 = Proportion of funds in Security 2r1 = Expected return on Security 1r2 = Expected return on Security 2Two-Security Portfolio: Returnp2 = w1212 + w2222 + 2W1W2 Cov(r1r2)12 = Variance of Security 122 = Variance of Security 2Cov(r1r2) = Covariance of returns for Security 1 and Security 2Two-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 2CovarianceRange of values for 1,2+ 1.0 > r > -1.0If r= 1.0, the securities would be perfectly positively correlatedIf r= - 1.0, the securities would be perfectly negatively correlatedCorrelation Coefficients: Possible Values2p = W1212+ W2212+ 2W1W2rp = W1r1 + W2r2 + W3r3 Cov(r1r2)+ W3232 Cov(r1r3)+ 2W1W3 Cov(r2r3)+ 2W2W3Three-Security Portfoliorp = 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)]1/2Two-Security PortfolioPortfolios with Different Correlations = 113%%8E(r)St. Dev12%20% = .3 = -1 = -1The relationship depends on correlation coefficient.-1.0 <  < +1.0The smaller the correlation, the greater the risk reduction potential.If r = +1.0, no risk reduction is possible.Correlation Effects112 r22 - Cov(r1r2)W1=+ - 2Cov(r1r2)W2= (1 - W1)s22E(r2) = .14= .20Sec 212= .2E(r1) = .10= .15Sec 1s2Minimum-Variance CombinationW1=(.2)2 - (.2)(.15)(.2)(.15)2 + (.2)2 - 2(.2)(.15)(.2)W1= .6733W2= (1 - .6733) = .3267Minimum-Variance Combination:  = .2rp = .6733(.10) + .3267(.14) = .1131p= [(.6733)2(.15)2 + (.3267)2(.2)2 +2(.6733)(.3267)(.2)(.15)(.2)]1/2p= [.0171]1/2= .1308sRisk and Return: Minimum VarianceW1=(.2)2 - (.2)(.15)(.2)(.15)2 + (.2)2 - 2(.2)(.15)(-.3)W1= .6087W2= (1 - .6087) = .3913Minimum - Variance Combination:  = -.3rp = .6087(.10) + .3913(.14) = .1157p= [(.6087)2(.15)2 + (.3913)2(.2)2 +2(.6087)(.3913)(.2)(.15)(-.3)]1/2p= [.0102]1/2= .1009ssRisk and Return: Minimum VarianceThe 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 SecuritiesMinimum-Variance Frontier of Risky AssetsE(r)EfficientfrontierGlobalminimumvarianceportfolioMinimumvariancefrontierIndividualassetsSt. Dev.The optimal combination becomes linear.A single combination of risky and riskless assets will dominate.Extending to Include Riskless AssetAlternative CALsME(r)CAL (Globalminimum variance)CAL (A)CAL (P)PAFPP&FA&FMAGPMPortfolio Selection & Risk AversionE(r)Efficientfrontier ofrisky assetsMorerisk-averseinvestorU’’’U’’U’QPSSt. DevLessrisk-averseinvestorEfficient Frontier with Lending & BorrowingE(r)FrfAPQBCALSt. Dev