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CHAPTER6RiskAversionandCapitalAllocationtoRiskyAssets6-2AllocationtoRiskyAssetsInvestorswillavoidriskunlessthereisareward.Theutilitymodelgivestheoptimalallocationbetweenariskyportfolioandarisk-freeasset.6-3RiskandRiskAversionSpeculationTakingconsiderableriskforacommensurategainPartieshaveheterogeneousexpectations6-4RiskandRiskAversionGambleBetorwageronanuncertainoutcomeforenjoymentPartiesassignthesameprobabilitiestothepossibleoutcomes6-5RiskAversionandUtilityValuesInvestorsarewillingtoconsider:risk-freeassetsspeculativepositionswithpositiveriskpremiumsPortfolioattractivenessincreaseswithexpectedreturnanddecreaseswithrisk.Whathappenswhenreturnincreaseswithrisk?6-6Table6.1AvailableRiskyPortfolios

(Risk-freeRate=5%)Eachportfolioreceivesautilityscoretoassesstheinvestor’srisk/returntradeoff6-7UtilityFunctionU=utilityE(r)=expectedreturnontheassetorportfolioA=coefficientofriskaversions2=varianceofreturns?=ascalingfactor

6-8Table6.2UtilityScoresofAlternativePortfoliosforInvestorswithVaryingDegreeofRiskAversion6-9Figure6.1TheTrade-offbetweenRiskandReturnofAPotentialInvestmentPortfolio,P6-10Figure6.2TheIndifferenceCurve6-11Mean-Variance(M-V)CriterionPortfolioAdominatesportfolioBif:and6-12EstimatingRiskAversionUsequestionnairesObserveindividuals’decisionswhenconfrontedwithriskObservehowmuchpeoplearewillingtopaytoavoidrisk6-13Investor’sWillingnesstoPayforCatastropheInsurancePayvdollarsperdollarofinsuredproperty,ascertaintyequivalentrateofreturn:6-14CapitalAllocationAcrossRiskyandRisk-FreePortfolios

AssetAllocation:Isaveryimportantpartofportfolioconstruction.Referstothechoiceamongbroadassetclasses.ControllingRisk:Simplestway:Manipulatethefractionoftheportfolioinvestedinrisk-freeassetsversustheportioninvestedintheriskyassets6-15BasicAssetAllocationTotalMarketValue$300,000Risk-freemoneymarketfund$90,000Equities$113,400Bonds(long-term)$96,600Totalriskassets$210,0006-16BasicAssetAllocationLety=weightoftheriskyportfolio,P,inthe

completeportfolio;(1-y)=weightofrisk-freeassets:6-17TheRisk-FreeAssetOnlythegovernmentcanissuedefault-freebonds.Risk-freeinrealtermsonlyifpriceindexedandmaturityequaltoinvestor’sholdingperiod.T-billsviewedastherisk-freeassetMoneymarketfundsalsoconsideredrisk-freeinpractice6-18Figure6.3SpreadBetween3-Month

CDandT-billRates

6-19It’spossibletocreateacompleteportfoliobysplittinginvestmentfundsbetweensafeandriskyassets.Lety=portionallocatedtotheriskyportfolio,P(1-y)=portiontobeinvestedinrisk-freeasset,F.PortfoliosofOneRiskyAssetandaRisk-FreeAsset6-20rf=7%

f=0%E(rp)=15%

p=22%y=%inp(1-y)=%inrfExampleUsingChapter6.4Numbers6-21Example(Ctd.)Theexpectedreturnonthecompleteportfolioistherisk-freerateplustheweightofPtimestheriskpremiumofP6-22Example(Ctd.)TheriskofthecompleteportfolioistheweightofPtimestheriskofP:6-23Example(Ctd.)Rearrangeandsubstitutey=sC/sP:6-24Figure6.4TheInvestmentOpportunitySet

6-25Lendatrf=7%andborrowatrf=9%Lendingrangeslope=8/22=0.36Borrowingrangeslope=6/22=0.27CALkinksatPCapitalAllocationLinewithLeverage6-26Figure6.5TheOpportunitySetwithDifferentialBorrowingandLendingRates6-27RiskToleranceandAssetAllocationTheinvestormustchooseoneoptimalportfolio,C,fromthesetoffeasiblechoicesExpectedreturnofthecompleteportfolio:Variance:6-28Table6.4UtilityLevelsforVariousPositionsinRiskyAssets(y)foranInvestorwithRiskAversionA=46-29Figure6.6UtilityasaFunctionofAllocationtotheRiskyAsset,y6-30OptimalyHowtoderivetheoptimalytoachievethehighestutilitylevel?Writetheproblemalgebraically

s.t.6-31Table6.5SpreadsheetCalculationsof

IndifferenceCurves6-32Figure6.7IndifferenceCurvesfor

U=.05andU=.09withA=2andA=4

6-33Figure6.8FindingtheOptimalCompletePortfolioUsingIndifferenceCurves

6-34Table6.6ExpectedReturnsonFourIndifferenceCurvesandtheCAL6-35PassiveStrategies:TheCapitalMarketLineThepassivestrategyavoidsanydirectorindirectsecurityanalysisSupplyanddemandforcesmaymakesuchastrategyareasonablechoiceformanyinvestors6-36PassiveStrategies:TheCapitalMarketLineAnaturalcandidateforapassivelyheldriskyassetwouldbeawell-diversifiedportfolioofcommonstockssuchastheS&P500.Thecapitalmarketline(CML)isthecapitalallocationlineformedfrom1-monthT-billsandabroadindexofcommonstocks(e.g.theS&P500).6-37PassiveStrategies:TheCapitalMarketLineTheCMLisgivenbyastrategythatinvolvesinvestmentintwopassiveportfolios:virtuallyrisk-freeshort-termT-bills(oramoneymarketfund)afund

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