solution-to-第三章-习题

1.Youmanageariskyportfoliowithanexpectedrateofreturnof18%andastandarddeviationof28%.TheT-billrateis8%.(1)Yourclientchoosestoinvest70%ofaportfolioinyourfundand30%inaT-billmoneymarketfund.Whatistheexpectedvalueandstandarddeviationoftherateofreturnonhisportfolio?(2)SupposethatyourriskyportfolioincludesthefollowinginvestmentsinthegivenproportionstockA25%;stockB32%;stockC43%.Whataretheinvestmentproportionsofyourclient’soverallportfolio,includingthepositioninT-bill?(3)Whatistherewardtovariabilityratio(S)ofyourriskyportfolio?Yourclient’s(4)DrawtheCALofyourportfolioonanexpectedreturn-standarddeviationdiagram,whatistheslopeoftheCAL?Showthepositionofyourclientonyourfund’sCal(5)Supposethatyourclientdecidestoinvestinyourportfolioaproportionyofthetotalinvestmentbudgetsothattheoverallportfoliowillhaveanexpectedrateofreturnof16%.Whatistheproportiony?Whatareyourclient’sinvestmentproportionsinyourthreestocksandtheT-billfund?Whatisthestandarddeviationoftherateofreturnonyourclient’sportfolio?(6)Supposethatyourclientpreferstoinvestinyourfundaproportionythatmaximizestheexpectedreturnonthecompleteportfoliosubjecttotheconstraintthatthecompleteportfolio’sstandarddeviationwillnotexceed18%.Whatistheinvestmentproportiony?Whatistheexpectedrateofreturnonthecompleteportfolio?(7)Yourclient’sdegreeofriskaversionisA=3.5.Whatproportionyofthetotalinvestmentshouldbeinvestedinyourfund?Whatistheexpectedvalueandstandarddeviationoftherateofreturnonyourclient’soptimizedportfolio?Solution:(1)Expectedreturn=0.3*8%+0.7*18%=15%standardvariance=0.7*28%=19.6%(2)Proportion:T-bill30%A:0.7*25%=17.5%;B:0.7*32=22.4%;C:0.7*43%=30.1%(3)Myrewardtovariabilityration=(18-8)/28=0.3571;client’s=(15-8)/19.6=0.3571(4)(5)theexpectedreturnofportfolio=yyrrrfpf108*)(Iftheexpectedreturnofportfolio=16%theny=0.8whichmeans80%mustinvestedinriskfundand20%inT-bill,thecompletelyproportionas:T-bill20%;A0.8*25%=20%,B:0.8*32%=25.6%;C:0.8*43%=34.4%Standardvariance=0.8*28%=22.4%(6)becausethestandardvariance=y*28%,ifitlessthan18%,theny=18/28=64.29%andexpectedreturn=8+10y=14.429%(7)pfpArrEy**001.0)(*=(18-8)/(0.01*3.5*28*28)=0.3644Expectedreturnofoptimal=8+10y=11.644%Standardvariance=0.3644*28=10.20%2.Apensionfundmanagerisconsideringthreemutualfunds.Thefirstisastockfund,thesecondisalong-termgovernmentandcorporatebondfund,andthethirdisaT-billmoneymarketfundthatyieldsarateof8%.Theprobabilitydistributionoftheriskyfundsisasfollows:ExpectedreturnStandarddeviationStockfund(S)20%30%Bondfund(B)12%15%Thecorrelationbetweenthefundreturnsis0.1.1)Whataretheinvestmentproportionsintheminimum-varianceportfolioofthetworiskyfunds,andwhatistheexpectedvalueandstandarddeviationofitsrateofreturn?2)Tabulateanddrawtheinvestmentopportunitysetoftworiskyfunds.Useinvestmentproportionsforthestockfundsofzeroto100%inincrementsof20%.3)Drawatangentfromtherisk-freeratetotheopportunityset.Solvenumericallyfortheproportionsofeachassetandfortheexpectedreturnandstandarddeviationoftheoptimalriskyportfolio.4)Yourequirethatyourportfolioyieldanexpectedreturnof14%a)Whatisthestandarddeviationofyouroptimalportfolio?b)WhatistheproportioninvestedintheT-billfundandeachoftworiskyfunds?5)Ifyouweretouseonlythetworiskyfunds,andstillrequireanexpectedreturnof14%,whatmustbetheinvestmentproportionsofyourportfolio?CompareitsstandarddeviationtothatoftheoptimizedportfolioinProblem4).Whatdoyouconclude?Solution:(1)8261.0,1739.0,min45),cov(,225,900,1),cov(222222222bspsbbssbbsbbsspwwSBwwSBExpectedreturn=0.1739*20+0.8261*12=13.39%Standardvariance=13.92%(2)Stock(%)Bond(%)ExpectedreturnVariance0100121517.3982.6113.3913.92208013.613.94406015.215.7045.1654.8415.6116.54604016.8019.53802018.4024.4810002030(3)MaxslopeofCalthenproportionofstock=0.4516andbond=0.5484Expectedreturnofoptimalportfolio=0.4516*20+0.5484*12=15.61Standardvariance=16.54(4)A:because%54.16%,61.15)()()(ppcpfpfcrErrErrENowrequiredreturn=14%,thenstandardvariance=13.04%B:because%54.16)(%,14)()()1()(pcpfcrErnowEryEryrEtheny=0.7884,1-y=0.2116,theproportionofstock=0.7884*0.4516=0.3560;bond’sproportion=0.7884*0.5484=0.4324(5)Ifthecompleteportfoliojustincludestworiskysecurities:then%13.14,75.0,25.0),1%(12%20%14pbssscomparestandardvariancewiththatofquestion4,obviouslyitisnottheoptimalportfolio