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CHAPTER7 OptimalRiskyPortfolios 7 2 TheInvestmentDecision Top downprocesswith3steps Capitalallocationbetweentheriskyportfolioandrisk freeassetAssetallocationacrossbroadassetclassesSecurityselectionofindividualassetswithineachassetclass 7 3 DiversificationandPortfolioRisk MarketriskSystematicornondiversifiableFirm specificriskDiversifiableornonsystematic 7 4 Figure7 1PortfolioRiskasaFunctionoftheNumberofStocksinthePortfolio 7 5 Figure7 2PortfolioDiversification 7 6 CovarianceandCorrelation PortfolioriskdependsonthecorrelationbetweenthereturnsoftheassetsintheportfolioCovarianceandthecorrelationcoefficientprovideameasureofthewayreturnsoftwoassetsvary 7 7 Two SecurityPortfolio Return 7 8 Two SecurityPortfolio Risk 7 9 Two SecurityPortfolio Risk Anotherwaytoexpressvarianceoftheportfolio 7 10 D E Correlationcoefficientofreturns Cov rD rE DE D E D StandarddeviationofreturnsforSecurityD E StandarddeviationofreturnsforSecurityE Covariance 7 11 Rangeofvaluesfor 1 2 1 0 r 1 0 Ifr 1 0 thesecuritiesareperfectlypositivelycorrelatedIfr 1 0 thesecuritiesareperfectlynegativelycorrelated CorrelationCoefficients PossibleValues 7 12 CorrelationCoefficients When DE 1 thereisnodiversificationWhen DE 1 aperfecthedgeispossible 7 13 Table7 2ComputationofPortfolioVarianceFromtheCovarianceMatrix 7 14 Three AssetPortfolio 7 15 Figure7 3PortfolioExpectedReturnasaFunctionofInvestmentProportions 7 16 Figure7 4PortfolioStandardDeviationasaFunctionofInvestmentProportions 7 17 TheMinimumVariancePortfolio Theminimumvarianceportfolioistheportfoliocomposedoftheriskyassetsthathasthesmalleststandarddeviation theportfoliowithleastrisk Whencorrelationislessthan 1 theportfoliostandarddeviationmaybesmallerthanthatofeitheroftheindividualcomponentassets Whencorrelationis 1 thestandarddeviationoftheminimumvarianceportfolioiszero 7 18 Figure7 5PortfolioExpectedReturnasaFunctionofStandardDeviation 7 19 Theamountofpossibleriskreductionthroughdiversificationdependsonthecorrelation Theriskreductionpotentialincreasesasthecorrelationapproaches 1 Ifr 1 0 noriskreductionispossible Ifr 0 Pmaybelessthanthestandarddeviationofeithercomponentasset Ifr 1 0 arisklesshedgeispossible CorrelationEffects 7 20 Figure7 6TheOpportunitySetoftheDebtandEquityFundsandTwoFeasibleCALs 7 21 TheSharpeRatio MaximizetheslopeoftheCALforanypossibleportfolio P Theobjectivefunctionistheslope TheslopeisalsotheSharperatio 7 22 Figure7 7TheOpportunitySetoftheDebtandEquityFundswiththeOptimalCALandtheOptimalRiskyPortfolio 7 23 Figure7 8DeterminationoftheOptimalOverallPortfolio 7 24 Figure7 9TheProportionsoftheOptimalOverallPortfolio 7 25 MarkowitzPortfolioSelectionModel SecuritySelectionThefirststepistodeterminetherisk returnopportunitiesavailable Allportfoliosthatlieontheminimum variancefrontierfromtheglobalminimum varianceportfolioandupwardprovidethebestrisk returncombinations 7 26 Figure7 10TheMinimum VarianceFrontierofRiskyAssets 7 27 MarkowitzPortfolioSelectionModel WenowsearchfortheCALwiththehighestreward to variabilityratio 7 28 Figure7 11TheEfficientFrontierofRiskyAssetswiththeOptimalCAL 7 29 MarkowitzPortfolioSelectionModel EveryoneinvestsinP regardlessoftheirdegreeofriskaversion Moreriskaverseinvestorsputmoreintherisk freeasset LessriskaverseinvestorsputmoreinP 7 30 CapitalAllocationandtheSeparationProperty TheseparationpropertytellsusthattheportfoliochoiceproblemmaybeseparatedintotwoindependenttasksDeterminationoftheoptimalriskyportfolioispurelytechnical AllocationofthecompleteportfoliotoT billsversustheriskyportfoliodependsonpersonalpreference 7 31 Figure7 13CapitalAllocationLineswithVariousPortfoliosfromtheEfficientSet 7 32 ThePowerofDiversification Remember Ifwedefinetheaveragevarianceandaveragecovarianceofthesecuritiesas 7 33 ThePowerofDiversification Wecanthenexpressportfoliovarianceas 7 34 Table7 4RiskReductionofEquallyWeightedPortfoliosinCorrelatedandUncorrelatedUniverses 7 35 OptimalPortfoliosandNonnormalReturns Fat taileddistributionscanresultinextremevaluesofVaRandESandencouragesmallerallocationstotheriskyportfolio IfotherportfoliosprovidesufficientlybetterVaRandESvaluesthanthemean varianceefficientportfolio wemaypreferthesewhenfacedwithfat taileddistributions 7 36 RiskPoolingandtheInsurancePrinciple Riskpooling merginguncorrelated riskyprojectsasameanstoreducerisk increasesthescaleoftheriskyinvestmentbyaddingadditionaluncorrelatedassets Theinsuranceprinciple riskincreaseslessthanproportionallytothenumberofpoliciesinsuredwhenthepoliciesareuncorrelatedSharperatioincreases 7 37 RiskSharing Asriskyassetsareaddedtotheportfolio aportionofthepoolissoldtomaintainariskyportfoliooffixedsize Risksharingcombinedwithriskpoolingisthekeytotheinsuranceindustry Truediversificationmeansspreadingaportfoliooffixedsizeacrossmanyassets notmerelyaddingmoreriskybetstoanever growingriskyportfolio 7 38 InvestmentfortheLongRun LongTermStra
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