INVESTMENTS-投资学-(博迪BODIE--KANE--MARCUS)Chap007-Opt

CHAPTER7OptimalRiskyPortfoliosTheInvestmentDecisionTop-downprocesswith3steps:Capitalallocationbetweentheriskyportfolioandrisk-freeassetAssetallocationacrossbroadassetclassesSecurityselectionofindividualassetswithineachassetclassDiversificationandPortfolioRiskMarketriskSystematicornondiversifiableFirm-specificriskDiversifiableornonsystematicFigure7.1PortfolioRiskasaFunctionoftheNumberofStocksinthePortfolioFigure7.2PortfolioDiversificationCovarianceandCorrelationPortfolioriskdependsonthecorrelationbetweenthereturnsoftheassetsintheportfolioCovarianceandthecorrelationcoefficientprovideameasureofthewayreturnsoftwoassetsvaryTwo-SecurityPortfolio:Return=VarianceofSecurityD=VarianceofSecurityE=CovarianceofreturnsforSecurityDandSecurityETwo-SecurityPortfolio:RiskTwo-SecurityPortfolio:RiskAnotherwaytoexpressvarianceoftheportfolio:D,E=CorrelationcoefficientofreturnsCov(rD,rE)=DEDED=StandarddeviationofreturnsforSecurityDE=StandarddeviationofreturnsforSecurityECovarianceRangeofvaluesfor1,2+1.0>r>-1.0Ifr=1.0,thesecuritiesareperfectlypositivelycorrelatedIfr=-1.0,thesecuritiesareperfectlynegativelycorrelatedCorrelationCoefficients:PossibleValuesCorrelationCoefficientsWhenρDE=1,thereisnodiversificationWhenρDE=-1,aperfecthedgeispossibleTable7.2ComputationofPortfolioVarianceFromtheCovarianceMatrixThree-AssetPortfolioFigure7.3PortfolioExpectedReturnasaFunctionofInvestmentProportionsFigure7.4PortfolioStandardDeviationasaFunctionofInvestmentProportionsTheMinimumVariancePortfolioTheminimumvarianceportfolioistheportfoliocomposedoftheriskyassetsthathasthesmalleststandarddeviation,theportfoliowithleastrisk.Whencorrelationislessthan+1,theportfoliostandarddeviationmaybesmallerthanthatofeitheroftheindividualcomponentassets.Whencorrelationis-1,thestandarddeviationoftheminimumvarianceportfolioiszero.Figure7.5PortfolioExpectedReturnasaFunctionofStandardDeviationTheamountofpossibleriskreductionthroughdiversificationdependsonthecorrelation.Theriskreductionpotentialincreasesasthecorrelationapproaches-1.Ifr=+1.0,noriskreductionispossible.Ifr=0,σPmaybelessthanthestandarddeviationofeithercomponentasset.Ifr=-1.0,arisklesshedgeispossible.CorrelationEffectsFigure7.6TheOpportunitySetoftheDebtandEquityFundsandTwoFeasibleCALsTheSharpeRatioMaximizetheslopeoftheCALforanypossibleportfolio,P.Theobjectivefunctionistheslope:TheslopeisalsotheSharperatio.Figure7.7TheOpportunitySetoftheDebtandEquityFundswiththeOptimalCALandtheOptimalRiskyPortfolioFigure7.8DeterminationoftheOptimalOverallPortfolioFigure7.9TheProportionsoftheOptimalOverallPortfolioMarkowitzPortfolioSelectionModelSecuritySelectionThefirststepistodeterminetherisk-returnopportunitiesavailable.Allportfoliosthatlieontheminimum-variancefrontierfromtheglobalminimum-varianceportfolioandupwardprovidethebestrisk-returncombinationsFigure7.10TheMinimum-VarianceFrontierofRiskyAssetsMarkowitzPortfolioSelectionModelWenowsearchfortheCALwiththehighestreward-to-variabilityratioFigure7.11TheEfficientFrontierofRiskyAssetswiththeOptimalCALMarkowitzPortfolioSelectionModelEveryoneinvestsinP,regardlessoftheirdegreeofriskaversion.Moreriskaverseinvestorsputmoreintherisk-freeasset.LessriskaverseinvestorsputmoreinP.CapitalAllocationandtheSeparationPropertyTheseparationpropertytellsusthattheportfoliochoiceproblemmaybeseparatedintotwoindependenttasksDeterminationoftheoptimalriskyportfolioispurelytechnical.AllocationofthecompleteportfoliotoT-billsversustheriskyportfoliodependsonpersonalpreference.Figure7.13CapitalAllocationLineswithVariousPortfoliosfromtheEfficientSetThePowerofDiversificationRemember:Ifwedefinetheaveragevarianceandaveragecovarianceofthesecuritiesas:ThePowerofDiversificationWecanthenexpressportfoliovarianceas:Table7.4RiskReductionofEquallyWeightedPortfoliosinCorrelatedandUncorrelatedUniversesOptimalPortfoliosandNonnormalReturnsFat-taileddistributionscanresultinextremevaluesofVaRandESandencouragesmallerallocationstotheriskyportfolio.IfotherportfoliosprovidesufficientlybetterVaRandESvaluesthanthemean-varianceefficientportfolio,wemaypreferthesewhenfacedwithfat-taileddistributions.RiskPoolingandtheInsurancePrincipleRiskpooling:merginguncorrelated,riskyprojectsasameanstoreducerisk.increasesthescaleoftheriskyinvestmentbyaddingadditionaluncorrelatedassets.Theinsuranceprinciple:riskincreaseslessthanproportionallytothenumberofpoliciesinsuredwhenthepoliciesareuncorrelatedSharperatioincreasesRiskSharingAsriskyassetsareaddedtotheportfolio,aportionofthepoolissoldtomaintainariskyportfoliooffixedsize.Risksharingcombinedwithriskpoolingisthekeytotheinsuranceindustry.Truediversificationmeansspreadingaportfoliooffixedsizeacrossmanyassets,notmerelyaddingmoreriskybetstoanever-growingriskyportfolio.InvestmentfortheLongRunLongTermStrategyInvestintheriskyportfoliofor2years.Long-termstrategyisriskier.Riskcanbereducedbysellingsomeoftheriskyassetsinyear2.“Timediversification”isnottruediversification.ShortTermStrategyInvestintheriskyportfoliofor1yearandintherisk-freeassetforthesecondyear.