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EssentialsofInvestments(BKM5thEd.)AnswerstoSelectedProblems–Lecture3Chapter5:6.c.Foreachportfolio:Utility=E(r)–(½×4×σ2)Wechoosetheportfoliowiththehighestutilityvalue.7.d.Whenaninvestorisriskneutral,A=0,sothattheportfoliowiththehighestutilityistheportfoliowiththehighestexpectedreturn.8.b.18.a)Expectedcashflow:0.5($50,000)+0.5($150,000)=$100,000HPR=(P1-P0)/P00.15=(100,000-P0)/P0P0=$86,956.52b)HPR=(100,000-86956.52)/86956.52=15%c)Arisk-premiumof15%,leadstoanexpectedreturnof15%+5%=20%.0.20=(100,000-P0)/P0P0=$83,333.00d)Thereisaninverserelationship:pricedecreasesastheriskpremiumincreases.Inordertoearnahigherrisk-premium(assumingthecashflowsstaythesame),youmustbeabletobuythesecurityatalowerprice.Theinvestorrequiringthe15%riskpremium(20%HPR)isrequiringalargerdiscountascompensationforrisk.19.a)E(RP)=0.3(7%)+0.7(17%)=14%σP=0.7(27%)=18.9%b)T-Bills=30.0%StockA=0.7(27%)=18.9%(Thetotalweightintheportfoliois70%,andtheStockB=0.7(33%)=23.1%portfolioconsistsof27%A,33%B,and40%C)StockC=0.7(40%)=28.0%TotalPortfolio=100%c)YourReward-to-variability=(RP-RF)/σP=(17%-7%)/27%=0.3704Client’sReward-to-variability=(14%-7%)/18.9%=0.37040.00%2.00%4.00%6.00%8.00%10.00%12.00%14.00%16.00%18.00%0.00%5.00%10.00%15.00%20.00%25.00%30.00%StandardDeviationExpectedReturnRisk-FreeRiskyPortfoliow=0.7d)Theslopeofthecapitalallocationlineequalsthereward-to-variabilityratio(0.3704).NotethatthisisthesameatanypointyouchooseontheCAL.20.a.Rule1:E(RC)=RF+y(RP-RF)0.15=0.07+y(0.17-0.07)y=0.80b.T-Bills=20.0%StockA=21.6%(0.8*27%)StockB=26.4%(0.8*33%)StockC=32.0%(0.8*40%)c.Rule2:σC=0.80(0.27)=0.216=21.6%21.a.Portfoliostandarddeviation=y×27%.Iftheclientwantsastandarddeviationof20%,theny=20/27=.7407=74.07%intheriskyportfolio.b.Rule2:E(R)=7+10y=7+.7407×10=7+7.407=14.407%22.a.SlopeoftheCML=13–725=.24Thediagramisshownbelow.024681012141618200102030σ(%)CAL(slope=.3704)CML(slope=.24)b.Myfundhasahigherreward-to-variability(orasteeperCAL).ThisallowsaninvestorinmyfundtoachieveahigherexpectedreturnforanygivenstandarddeviationthantheywouldearnonthepassiveS&Pfund.Inotherwords,myfundprovidesahigherreturnatanygivenlevelofrisk.**Note:Theanswerstotheconceptchecksareprovidedinthetextbookattheendofthechapter.Chapter6:2.a.–1.0Thelowerthecorrelation,thegreaterthebenefitsfromdiversification.Infact,ifwecanfindtwosecuritieswithacorrelationofnegativeone,wecancreateaperfecthedgeportfolio(i.e.,aportfoliocombinationwithzerorisk).6.Theparametersoftheopportunitysetare:E(rS)=15%,E(rB)=9%,σS=32%,σB=23%,ρ=0.15,rf=5.5%ThisgivesCov(rS,rB)=ρσSσB=.01104UsingRule1*andRule2*,youcancalculatethereturnandstandarddeviationoneachofthesixportfoliocombinations(in20%increments):WeightSWeightBE(R)Std.Dev.100%0%15.0%32.00%802013.8%26.6870.7529.2513.2524.57604012.6%22.50406011.4%20.1831.4268.5810.8919.94208010.2%20.3701009.0%23.0015.0%13.8%12.6%11.4%10.2%9.0%0.00%2.00%4.00%6.00%8.00%10.00%12.00%14.00%16.00%0.00%5.00%10.00%15.00%20.00%25.00%30.00%35.00%CALOptimalRiskyPortfolio(70.75%Stocks,29.25%Bonds)MinimumVariancePortfolio(31.42%Stocks,68.58%Bonds)CALOfthesesixpossibleportfolios,theminimumvarianceportfolioisthecombinationwith40%instocksand60%inbonds.Note:Thetrueminimumvarianceportfolioisactuallybetweenthe40%stockand20%stockchoices-theexactminimumvarianceportfoliohasaweightof31.42%instocksand68.58%inbonds.Theresultcanbefoundbyminimizingthevarianceformula,Rule2*,aswediscussedinclass.Theminimum-varianceportfolioproportionsare:)r,r(Cov2)r,r(Cov)S(wBS2B2SBS2BMin−σ+σ−σ=3142.0)01104.02(0529.01024.001104.00529.0=×−+−=wMin(B)=1–0.3142=0.6858PluggingtheseweightsintoRule1*andRule2*givestheexpectedreturnandstandarddeviationshowninthetableabove.7.Ifyouconsideronlythesixportfoliosyoucreatedat20%increments,theoptimalriskyportfolioistheportfoliothatincludes60%stocksand40%bonds.Thisportfoliohasanexpectedreturnof12.6%andastandarddeviationof22.5%.Youcanusethisportfoliotoanswerquestions8through10.(Note:Thetrueoptimalriskyportfolio(P)issomewherebetweenthe60%stockand80%stockchoices.Theportfoliowith70%stocksand30%bondsisclosetotheoptimalriskyportfolio.Thisportfoliohasanexpectedreturnof13.2%andastandarddeviationofapproximately24.41%.AlthoughIwillnotrequireyoutodothiscalculation,youcansolvefortheexactweightsintheoptimalportfoliousingtheformulainfootnote3onpage185ofthetext:theweightsare70.75%stockand29.25%bonds.Thisportfoliohasanexpectedreturnof13.25%andastandarddeviationof24.57%.)8.Thereward-to-variabilityratiooftheoptimalCAL(usingthe60/40portfolio)is:3156.05.225.56.12)(=−=−pfprrEσ9.a)TheequationfortheCAL(usingthe60/40portfolio)is:CCpfpfCrrErrEσσσ3156.0%5.5)()(+=−+=SettingE(rC)equalto12%yieldsastandarddeviationof20.6%.b.Themeanofthecompleteportfolioasafunctionoftheproportioninvestedintheriskyportfolio(y)is:E(rC)=(l−y)rf+yE(rP)=rf+y[E(rP)−rf]=5.5+y(12.6−5.5)SettingE(rC)=12%⇒y=0.9155(91.55%intheriskyportfolio)1−y=0.0845(8.45%inT-bills)Fromthecompositionoftheoptimalriskyportfolio:Proportionofstocksincompleteportfolio=0.9155×0.60=0.5493Proportionofbondsincompleteportfolio=0.9155×0.40=0.450710.Usingonlythestockandbondfundstoachieveameanof12