CorporateFinanceFifthEditionChapter11OptimalPortfolioChoiceandtheCapitalAssetPricingModelCopyright?2020,2017,2014PearsonEducation,Inc.

AllRightsReservedChapterOutline(1of2)11.1

TheExpectedReturnofaPortfolio11.2

TheVolatilityofaTwo-StockPortfolio11.3

TheVolatilityofaLargePortfolio11.4

RiskVersusReturn:ChoosinganEfficientPortfolio11.5

Risk-FreeSavingandBorrowing11.6

TheEfficientPortfolioandRequiredReturnsChapterOutline(2of2)11.7TheCapitalAssetPricingModel11.8DeterminingtheRiskPremiumAppendixLearningObjectives(1of7)Givenaportfolioofstocks,includingtheholdingsineachstockandtheexpectedreturnineachstock,computethefollowing:Portfolioweightofeachstock(Eq.11.1)Expectedreturnontheportfolio(Eq.11.3)Covarianceofeachpairofstocksintheportfolio(Eq.11.5)Correlationcoefficientofeachpairofstocksintheportfolio(Eq.11.6)Varianceoftheportfolio(Eq.11.8)StandarddeviationoftheportfolioLearningObjectives(2of7)Computethevarianceofanequallyweightedportfolio,usingEq.11.12.Describethecontributionofeachsecuritytotheportfolio.UsethedefinitionofanefficientportfoliofromChapter10todescribetheefficientfrontier.Explainhowanindividualinvestorwillchoosefromthesetofefficientportfolios.LearningObjectives(3of7)Describewhatismeantbyashortsale,andillustratehowshortsellingextendsthesetofpossibleportfolios.Explaintheeffectofcombiningarisk-freeassetwithaportfolioofriskyassets,andcomputetheexpectedreturnandvolatilityforthatcombination.LearningObjectives(4of7)Illustratewhytherisk-returncombinationsoftherisk-freeinvestmentandariskyportfoliolieonastraightline.DefinetheSharperatio,andexplainhowithelpsidentifytheportfoliowiththehighestpossibleexpectedreturnforanylevelofvolatility,andhowthisinformationcanbeusedtoidentifythetangency(efficient)portfolio.LearningObjectives(5of7)Calculatethebetaofinvestmentwithaportfolio.Usethebetaofasecurity,expectedreturnonaportfolio,andtherisk-freeratetodecidewhetherbuyingsharesofthatsecuritywillimprovetheperformanceoftheportfolio.Explainwhytheexpectedreturnmustequaltherequiredreturn.LearningObjectives

(6of7)Usetherisk-freerate,theexpectedreturnontheefficient(tangency)portfolio,andthebetaofasecuritywiththeefficientportfoliotocalculatetheriskpremiumforaninvestment.ListthethreemainassumptionsunderlyingtheCapitalAssetPricingModel.LearningObjectives

(7of7)ExplainwhytheC

A

P

Mimpliesthatthemarketportfolioofallriskysecuritiesistheefficientportfolio.Compareandcontrastthecapitalmarketlinewiththesecuritymarketline.Definebetaforanindividualstockandforaportfolio.11.1TheExpectedReturnofaPortfolio(1of3)PortfolioWeightsThefractionofthetotalinvestmentintheportfolioheldineachindividualinvestmentintheportfolioTheportfolioweightsmustaddupto1.00or100%.11.1TheExpectedReturnofaPortfolio(2of3)Thenthereturnontheportfolio,istheweightedaverageofthereturnsontheinvestmentsintheportfolio,wheretheweightscorrespondtoportfolioweights.TextbookExample11.1(1of2)CalculatingPortfolioReturnsProblemSupposeyoubuy200sharesofDolbyLaboratoriesat$30pershareofCoca-Colastockat$40pershare.IfDolby’ssharepricegoesupto$36andCoca-Cola’sfallsto$38,whatisthenewvalueoftheportfolio,andwhatreturndiditearn?ShowthatEq.11.2holds.Afterthepricechange,whataretheportfolioweights?TextbookExample11.1(2of2)SolutionThenewofvalueoftheportfolioisforagainof$1000ora10%returnonyour$10,000investment.Dolby’sreturnwasandCoca-Cola’swasGiventheinitialportfolioweightsof60%Dolbyand40%Coca-Cola,wecanalsocomputetheportfolio’sreturnfromEq.11.2Afterthepricechange,thenewportfolioweightsareWithouttrading,theweightsincreaseforthosestockswhosereturnsexceedtheportfolio’sreturn.AlternativeExample11.1(1of4)ProblemSupposeyoubuy500sharesofFordat$11pershareand100sharesofCitigroupstockat$28pershare.IfFord’ssharepricegoesupto$13andCitigroup’srisesto$40,whatisthenewvalueoftheportfolio,andwhatreturndiditearn?ShowthatEq.11.2holds.Afterthepricechange,whatarethenewportfolioweights?AlternativeExample11.1(2of4)SolutionTheinitialvalueoftheportfolioisThenewvalueoftheportfolioisforagainof$2,200ora26.5%returnonyour$8,300investment.Ford’sreturnwasandCitigroup’swasAlternativeExample11.1(3of4)SolutionGiventheinitialportfolioweightsofforFordandforCitigroup,wecanalsocomputetheportfolio’sreturnfromEq.11.2:Thus,Eq.11.2holds.AlternativeExample11.1

(4of4)SolutionAfterthepricechange,thenewportfolioweightsareforFordandforCitigroup.11.1TheExpectedReturnofaPortfolio(3of3)Theexpectedreturnofaportfolioistheweightedaverageoftheexpectedreturnsoftheinvestmentswithinit.TextbookExample11.2(1of2)PortfolioExpectedReturnProblemSupposeyouinvest$10,000inFordstock,and$30,000inTycoInternationalstock,youexpectareturnof10%forTyco.Whatisyourportfolio’sexpectedreturn?TextbookExample11.2(2of2)SolutionYouinvested$40,000intotal,soyourportfolioweightsareinFordandTherefore,yourportfolio’sexpectedreturnisAlternativeExample11.2(1of2)ProblemAssumeyourportfolioconsistsof$25,000ofIntelstockand$35,000ofA

T

POilandGas.Yourexpectedreturnis18%forInteland25%forA

T

POilandGas.Whatistheexpectedreturnforyourportfolio?AlternativeExample11.2(2of2)SolutionTotalPortfolioPortfolioWeightsIntel:A

T

P:ExpectedReturn

11.2TheVolatilityofaTwo-StockPortfolio(1of4)CombiningRisksTable11.1

ReturnsforThreeStocks,andPortfoliosofPairsofStocks11.2TheVolatilityofaTwo-StockPortfolio(2of4)CombiningRisksWhilethethreestocksintheprevioustablehavethesamevolatilityandaveragereturn,thepatternoftheirreturnsdiffers.Forexample,whentheairlinestocksperformedwell,theoilstocktendedtodopoorly,andwhentheairlinesdidpoorly,theoilstocktendedtodowell.11.2TheVolatilityofaTwo-StockPortfolio(3of4)CombiningRisksConsidertheportfoliowhichconsistsofequalinvestmentsinWestAirandTexOil.Theaveragereturnoftheportfolioisequaltotheaveragereturnofthetwostocks.However,thevolatilityof5.1%ismuchlessthanthevolatilityofthetwoindividualstocks.11.2TheVolatilityofaTwo-StockPortfolio(4of4)CombiningRisksBycombiningstocksintoaportfolio,wereduceriskthroughdiversification.Theamountofriskthatiseliminatedinaportfoliodependsonthedegreetowhichthestocksfacecommonrisksandtheirpricesmovetogether.DeterminingCovarianceandCorrelation(1of3)Tofindtheriskofaportfolio,onemustknowthedegreetowhichthestocks’returnsmovetogether.DeterminingCovarianceandCorrelation

(2of3)CovarianceTheexpectedproductofthedeviationsoftworeturnsfromtheirmeans.CovariancebetweenReturnsEstimateoftheCovariancefromHistoricalDataIfthecovarianceispositive,thetworeturnstendtomovetogether.Ifthecovarianceisnegative,thetworeturnstendtomoveinoppositedirections.DeterminingCovarianceandCorrelation(3of3)CorrelationAmeasureofthecommonrisksharedbystocksthatdoesnotdependontheirvolatilityThecorrelationbetweentwostockswillalwaysbebetweenFigure11.1CorrelationTextbookExample11.3(1of2)TheCovarianceandCorrelationofaStockwithItselfProblemWhatarethecovarianceandthecorrelationofastock’sreturnwithitself?TextbookExample11.3(2of2)SolutionLetRbethestock’sreturn.Formthedefinitionofthecovariance,wherethelastEq.followsfromthedefinitionofthevariance.Thatis,thecovarianceofastockwithitselfissimplyitsvariance.Then,WherethelastEq.followsfromthedefinitionofthestandarddeviation.Thatis,astock’sreturnisperfectlycorrelatedwithitself,asitalwaysmovestogetherwithitselfinperfectsynchrony.Table11.2ComputingtheCovarianceandCorrelationBetweenPairsofStocksTable11.3HistoricalAnnualVolatilitiesandCorrelationsforSelectedStocksBlankMicrosoftHPAlaskaAirSouthwestAirlinesFordMotorKelloggGeneralMillsVolatility(StandardDeviation)Correlationwith32%32%36%36%31%47%19%17%Microsoft1.000.400.180.220.270.040.10HP0.401.000.270.340.270.100.06AlaskaAir0.180.271.000.400.150.150.20SouthwestAirlines0.220.340.401.000.300.150.21FordMotor0.270.270.150.301.000.170.08Kellogg0.040.100.150.150.171.000.55GeneralMills0.100.060.200.210.080.551.00TextbookExample11.4(1of2)ComputingtheCovarianceandCorrelationProblemUsingthedatainTable11.1,WhatarethecorrelationbetweenNorthAirandWestAir?BetweenWestAirandTexOil?TextbookExample11.4(2of2)SolutionGiventhereturnsinTable11.1,wedeductthemeanreturn(10%)fromeachandcomputetheproductofthesedeviationsbetweenthepairsofstocks.Wethensumthemanddividebytocomputethecovariance,asinTable11.2.Fromthetable,weseethatNorthAirandWestAirhaveapositivecovariance,indicatingatendencytomovetogether,whereasWestAirandTexOilhaveanegativecovariance,indicatingatendencytomoveoppositely.Wecanassessthestrengthofthesetendenciesfromcorrelation,obtainedbydividingthecovariancebythestandarddeviationofeachstock(13.4%).ThecorrelationforNorthAirandWestAiris62.4%;thecorrelationforWestAirandTexOilisTextbookExample11.5(1of2)ComputingtheCovariancefromtheCorrelationProblemUsingthedatafromTable11.3,whatisthecovariancebetweenMicrosoftandHP?TextbookExample11.5(2of2)SolutionWecanrewriteEq.11.6tosolveforthecovarianceAlternativeExample11.5(1of2)ProblemUsingthedatafromTable11.3,whatisthecovariancebetweenGeneralMillsandFord?AlternativeExample11.5(2of2)SolutionComputingaPortfolio’sVarianceandVolatilityForatwosecurityportfolio,TheVarianceofaTwo-StockPortfolioTextbookExample11.6(1of2)ComputingtheVolatilityofaTwo-StockPortfolioProblemUsingthedatafromTable11.3,whatisthevolatilityofaportfoliowithequalamountsinvestedinMicrosoftandHewett-Packardstock?WhatisthevolatilityofaportfoliowithequalamountsinvestedinMicrosoftandAlaskaAirstock?TextbookExample11.6(2of2)SolutionWithportfolioweightsof50%eachinMicrosoftandHewlett-Packardstock,fromEq.11.9,theportfolio’svarianceisThevolatilityisthereforeAlternativeExample11.6(1of2)ProblemContinuingwithAlternativeExample11.2:Assumetheannualstandarddeviationofreturnsis43%forInteland68%forATPOilandGas.IfthecorrelationbetweenIntelandATPis0.49,whatisthestandarddeviationofyourportfolio?AlternativeExample11.6(2of2)Solution11.3TheVolatilityofaLargePortfolioThevarianceofaportfolioisequaltotheweightedaveragecovarianceofeachstockwiththeportfolio:whichreducestoDiversificationwithanEquallyWeightedPortfolioEquallyWeightedPortfolioAportfolioinwhichthesameamountisinvestedineachstockVarianceofanEquallyWeightedPortfolioofnStocksFigure11.2VolatilityofanEquallyWeightedPortfolioVersustheNumberofStocksTextbookExample11.7(1of2)DiversificationusingDifferentTypesofStocksProblemStockwithinasingleindustrytendtohavehighercorrelationthanstocksindifferentindustries.Likewise,stocksindifferentcountrieshavelowercorrelationonaveragethanstockswithintheUnitedStates.Whatisthevolatilityofaverylargeportfolioofstackswithinanindustryinwhichthestockshaveavolatilityof40%andacorrelationof60%?Whatisthevolatilityofaverylargeportfolioofinternationalstackswithavolatilityof40%andacorrelationof10%?TextbookExample11.7(2of2)SolutionFromEq.11.12,thevolatilityoftheindustryportfolioasisgivenbyThisvolatilityishigherthanwhenusingstocksfromdifferentindustriesasinFigure11.2.Combiningstocksfromthesameindustrythataremorehighlycorrelatedthereforeprovideslessdiversification.Wecanachievesuperiordiversificationusinginternationalstocks.Inthiscase,TextbookExample11.8(1of2)VolatilitywhenRisksAreIndependentProblemWhatisthevolatilityofanequallyweightedaverageofnindependentrisks?TextbookExample11.8(2of2)SolutionIfrisksareindependent,theyareuncorrelatedandtheircovarianceiszero.UsingEq.11.12,thevolatilityofanequallyweightedportfoliooftherisksisThisresultcoincideswithEq.10.8,whichweusedearliertoevaluateindependentrisks.Notethatasthevolatilitygoesto0—thatis,averylargeportfoliowillhavenorisk.Inthiscase,wecaneliminateallrisksbecausethereisnocommonrisk.DiversificationwithGeneralPortfoliosForaportfoliowitharbitraryweights,thestandarddeviationiscalculatedasfollows:VolatilityofaPortfoliowithArbitraryWeightsUnlessallofthestocksinaportfoliohaveaperfectpositivecorrelationof+1withoneanother,theriskoftheportfoliowillbelowerthantheweightedaveragevolatilityoftheindividualstocks:11.4RiskVersusReturn:ChoosinganEfficientPortfolio

(1of3)EfficientPortfolioswithTwoStocksIdentifyingInefficientPortfoliosInaninefficientportfolio,itispossibletofindanotherportfoliothatisbetterintermsofbothexpectedreturnandvolatility.IdentifyingEfficientPortfoliosRecallfromChapter10,inanefficientportfoliothereisnowaytoreducethevolatilityoftheportfoliowithoutloweringitsexpectedreturn.11.4RiskVersusReturn:ChoosinganEfficientPortfolio(2of3)EfficientPortfolioswithTwoStocksConsideraportfolioofIntelandCoca-Cola.Table11.4ExpectedReturnsandVolatilityforDifferentPortfoliosofTwoStocksFigure11.3VolatilityVersusExpectedReturnforPortfoliosofIntelandCoca-ColaStock11.4RiskVersusReturn:ChoosinganEfficientPortfolio(3of3)EfficientPortfolioswithTwoStocksConsiderinvesting100%inCoca-Colastock.Asshowninonthepreviousslide,otherportfolios—suchastheportfoliowith20%inIntelstockand80%inCoca-Colastock—maketheinvestorbetteroffintwoways:Ithasahigherexpectedreturn,andithaslowervolatility.Asaresult,investingsolelyinCoca-Colastockisinefficient.TextbookExample11.9(1of2)ImprovingReturnswithanEfficientportfolioProblemSallyFersonhasinvested100%ofhermoneyincoca–colastockandisseekinginvestmentadvice.Shewouldliketoearnthehighestexpectedreturnpossiblewithoutinincreasinghervolatility.Whichportfoliowouldyourecommend?TextbookExample11.9(2of2)SolutionInFigure11.3,wecanseethatSallycaninvestupto40%inIntelstockwithoutincreasinghervolatility.Becausestockhasahigherexpectedreturnthancoca-colastock,shewillearnhigherexpectedreturnsbyputtingmoremoneyinIntelstock.Therefore,youshouldrecommendthatsallyput40%ofhermoneyinIntelstock,leaving60%incoca-colastock.Thisportfoliohasthesamevolatilityof25%,butanexpectedreturnof14%ratherthanthe6%shehasnow.AlternativeExample11.9(1of2)ProblemUsingFigure11.3,whatcombinationofIntelandCoca-Colaprovidestheworstrisk/returntrade-off?Whatcombinationprovidesthelowestamountofrisk?Thegreatestamountofrisk?AlternativeExample11.9(2of2)SolutionTheworstrisk/returntrade-offis100%inCoca-Colabecauseyoucanearnahigherreturnwiththesameamountofriskorthesamereturnwithaloweramountofriskbymovingtoamoreefficientportfolio.20%inInteland80%inCoca-Colaprovidesthelowestrisk,while100%inInteloffersthehighestamountofrisk.TheEffectofCorrelationCorrelationhasnoeffectontheexpectedreturnofaportfolio.However,thevolatilityoftheportfoliowilldifferdependingonthecorrelation.Thelowerthecorrelation,thelowerthevolatilitywecanobtain.Asthecorrelationdecreases,thevolatilityoftheportfoliofalls.Thecurveshowingtheportfolioswillbendtothelefttoagreaterdegreeasshownonthenextslide.Figure11.4EffectonVolatilityandExpectedReturnofChangingtheCorrelationBetweenIntelandCoca-ColaStockShortSalesLongPositionApositiveinvestmentinasecurityShortPositionAnegativeinvestmentinasecurityInashortsale,yousellastockthatyoudonotownandthenbuythatstockbackinthefuture.Shortsellingisanadvantageousstrategyifyouexpectastockpricetodeclineinthefuture.TextbookExample11.10(1of3)ExpectedReturnandVolatilitywithaShortSaleProblemSupposeyouhave$20,000incashtoinvest.Youdecidetoshortsell$10,000worthofCoca-Colastockandinvesttheproceedsfromyourshortsale,plusyour$20,000,inIntel.Whatistheexpectedreturnandvolatilityofyourportfolio?TextbookExample11.10(2of3)SolutionWecanthinkofourshortsaleasanegativeinvestmentof$10,000inCoca-Colastock.Inaddition,weinvested+$30,000inIntelstock,foratotalnetinvestmentofcash.ThecorrespondingportfolioweightsareNotethattheportfolioweightsstilladdupto100%.Usingtheseportfolioweights,wecancalculatetheexpectedreturnandvolatilityoftheportfoliousingEq.11.3andEq.11.8asbefore:TextbookExample11.10(3of3)Notethatinthiscase,shortsellingincreasestheexpectedreturnofyourportfolio,butalsoitsvolatility,abovethoseoftheindividualstocks.Figure11.5PortfoliosofIntelandCoca-ColaAllowingforShortSalesEfficientPortfolioswithManyStocksConsideraddingBoreIndustriestothetwo-stockportfolio:BlankBlankBlankBlankCorrelationwithBlankStockExpectedReturnVolatilityIntelCoca-ColaBoreInd.Intel26%50%1.00.00.0Coca-Cola6%25%0.01.00.0BoreIndustries2%25%0.00.01.0AlthoughBorehasalowerreturnandthesamevolatilityasCoca-Cola,itstillmaybebeneficialtoaddBoretotheportfolioforthediversificationbenefits.Figure11.6ExpectedReturnandVolatilityforSelectedPortfoliosofIntel,Coca-Cola,andBoreIndustriesStocksFigure11.7TheVolatilityandExpectedReturnforAllPortfoliosofIntel,Coca-Cola,andBoreStockRiskVersusReturn:ManyStocksTheefficientportfolios,thoseofferingthehighestpossibleexpectedreturnforagivenlevelofvolatility,arethoseonthenorthwestedgeoftheshadedregion,whichiscalledtheefficientfrontierforthesethreestocks.Inthiscase,noneofthestocks,onitsown,isontheefficientfrontier,soitwouldnotbeefficienttoputallourmoneyinasinglestock.Figure11.8EfficientFrontierwithThreeStocksVersusTenStocks11.5Risk-FreeSavingandBorrowingRiskcanalsobereducedbyinvestingaportionofaportfolioinarisk-freeinvestment,likeT-Bills.However,doingsowilllikelyreducetheexpectedreturn.Ontheotherhand,anaggressiveinvestorwhoisseekinghighexpectedreturnsmightdecidetoborrowmoneytoinvestevenmoreinthestockmarket.InvestinginRisk-FreeSecurities(1of2)Consideranarbitraryriskyportfolioandtheeffectonriskandreturnofputtingafractionofthemoneyintheportfolio,whileleavingtheremainingfractioninrisk-freeTreasurybills.Theexpectedreturnwouldbe InvestinginRisk-FreeSecurities(2of2)Thestandarddeviationoftheportfoliowouldbecalculatedasfollows:Note:Thestandarddeviationisonlyafractionofthevolatilityoftheriskyportfolio,basedontheamountinvestedintheriskyportfolio.Figure11.9TheRisk–ReturnCombinationsfromCombiningaRisk-FreeInvestmentandaRiskyPortfolioBorrowingandBuyingStocksonMarginBuyingStocksonMarginBorrowingmoneytoinvestinastockAportfoliothatconsistsofashortpositionintherisk-freeinvestmentisknownasaleveredportfolio.Margininvestingisariskyinvestmentstrategy.TextbookExample11.11(1of2)MarginInvestingProblemSupposeyouhave$10,000incash,andyoudecidetoborrowanother$10,000ata5%interestrateinordertoinvest$20,000inportfolioQwhichhasa10%expectedreturnandvolatilityofyourinvestment?WhatisyourrealizedreturnifQgoesup30%overtheyear?WhatifQfallsby10%?TextbookExample11.11(2of2)SolutionYouhavedoubledyourinvestmentinQusingmargin,soFromEq.11.16,weseethatyouhaveincreasedbothyourexpectedreturnsandyourriskrelativetotheportfolioQ:IfQgoesup30%,yourinvestmentwillbeworth$26,000,butyouwilloweonyourloan,foranetpayoffof$15,500ora55%returnonyour$10,000initialinvestment.IfQdropsby10%,youareleftwithandyourreturnis?25%.Thustheuseofmargindoubledtherangeofyourreturnsversuscorrespondingtothedoublingofthevolatilityoftheportfolio.IdentifyingtheTangentPortfolio(1of4)Toearnthehighestpossibleexpectedreturnforanylevelofvolatilitywemustfindtheportfoliothatgeneratesthesteepestpossiblelinewhencombinedwiththerisk-freeinvestment.IdentifyingtheTangentPortfolio(2of4)SharpeRatioMeasurestheratioofreward-to-volatilityprovidedbyaportfolioTheportfoliowiththehighestSharperatioistheportfoliowherethelinewiththerisk-freeinvestmentistangenttotheefficientfrontierofriskyinvestments.Theportfoliothatgeneratesthistangentlineisknownasthetangentportfolio.Figure11.10TheTangentorEfficientPortfolioIdentifyingtheTangentPortfolio(3of4)Combinationsoftherisk-freeassetandthetangentportfolioprovidethebestriskandreturntrade-offavailabletoaninvestor.Thismeansthatthetangentportfolioisefficientandthatallefficientportfoliosarecombinationsoftherisk-freeinvestmentandthetangentportfolio.Everyinvestorshouldinvestinthetangentportfolioindependentofhisorhertasteforrisk.IdentifyingtheTangentPortfolio(4of4)Aninvestor’spreferenceswilldetermineonlyhowmuchtoinvestinthetangentportfolioversustherisk-freeinvestment.Conservativeinvestorswillinvestasmallamountinthetangentportfolio.Aggressiveinvestorswillinvestmoreinthetangentportfolio.Bothtypesofinvestorswillchoosetoholdthesameportfolioofriskyassets,thetangentportfolio,whichistheefficientportfolio.TextbookExample11.12(1of2)OptimalPortfolioChoiceProblemYouruncleasksforinvestmentadvice.Currently,hehas$100,000investedinportfoliopinFigure11.10,whichhasanexpectedreturnof10.5%andavolatilityof8%.Supposetherisk-freerateis5%,andthetangentportfoliohasanexpectedreturnof18.5%andavolatilityof13%.Tomaximizehisexpectedreturnwithoutincreasinghisvolatility,whichportfoliowouldyourecommend?Ifyourunclepreferstokeephisexpectedreturnthesamebutminimizehisrisk,whichportfoliowouldyourecommend?TextbookExample11.12(2of2)SolutionIneithercasethebestportfoliosarecombinationoftherisk-freeinvestmentandthetangentportfolio.T,usingEq.11.16,theexpectedreturnandvolatilityareSo,tomaintainthevolatilityat8%,Inthiscase,youruncleshouldinvest$61,500inthetangentportfolio,andtheremaining$38,500intherisk-freeinvestment.Hisexpectedreturnwillthenbe13.3%,thehighestpossiblegivenhislevelofrisk.Alternatively,tokeeptheexpectedreturnequaltothecurrentvalueof10.5%,xmustsatisfyNowyouruncleshouldinvest$40,700inthetangentportfolioand$59,300intherisk-freeinvestment,loweringhisvolatilityleveltothelowestpossiblegivenhisexpectedreturn.11.6TheEfficientPortfolioandRequiredReturns(1of5)PortfolioImprovement:BetaandtheRequiredReturnAssumethereisaportfolioofriskysecurities,PTodeterminewhetherPhasthehighestpossibleSharperatio,considerwhetheritsSharperatiocouldberaisedbyaddingmoreofsomeinvestmentitotheportfolio.Thecontributionofinvestmentitothevolatilityoftheportfoliodependsontheriskthatihasincommonwiththeportfolio,whichismeasuredbyi’svolatilitymultipliedbyitscorrelationwithP.11.6TheEfficientPortfolioandRequiredReturns(2of5)PortfolioImprovement:BetaandtheRequiredReturnIfyouweretopurchasemoreofinvestmentibyborrowing,youwouldearntheexpectedreturnofiminustherisk-freereturn.ThusaddingitotheportfolioPwillimproveourSharperatioif11.6TheEfficientPortfolioandRequiredReturns(3of5)PortfolioImprovement:BetaandtheRequiredReturnBetaofPortfolioiwithPortfolioP11.6TheEfficientPortfolioandRequiredReturns(4of5)PortfolioImprovement:BetaandtheRequiredReturnIncreasingtheamountinvestediniwillincreasetheSharperatioofportfolioPifitsexpectedreturnexceedstherequiredreturnwhichisgivenby11.6TheEfficientPortfolioandRequiredReturns(5of5)PortfolioImprovement:BetaandtheRequiredReturnRequiredReturnofiTheexpectedreturnthatisnecessarytocompensatefortheriskinvestmentiwillcontributetotheportfolio.TextbookExample11.13(1of2)TheRequiredReturnofaNewInvestmentProblemYouarecurrentlyinvestedintheomegaFund,abroad-basedfundwithanexpectedreturnof15%andavolatilityof20%,aswellasinrick-freeTreasuriespaying3%.Yourbrokersuggeststhatyouaddarealestatefundtoyourportfolio.Therealestatefundhasanexpectedreturnof9%,avolatilityof35%,andacorrelationof0.10withtheomegafund.Willaddingtherealestatefundimproveyourportfolio?TextbookExample11.13(2of2)SolutionLetbethereturnoftherealestatefundandbethereturnoftheOmegafund.FormEq.11.19,thebetaoftherealestatefundwiththeOmegaFundisWecanthenuseEq.11.