投资学第10版习题答案08

Chapter8-IndexModels8-1Copyright©2014McGraw-HillEducation.Allrightsreserved.NoreproductionordistributionwithoutthepriorwrittenconsentofMcGraw-HillEducation.CHAPTER8:INDEXMODELSPROBLEMSETS1.Theadvantageoftheindexmodel,comparedtotheMarkowitzprocedure,isthevastlyreducednumberofestimatesrequired.Inaddition,thelargenumberofestimatesrequiredfortheMarkowitzprocedurecanresultinlargeaggregateestimationerrorswhenimplementingtheprocedure.Thedisadvantageoftheindexmodelarisesfromthemodel’sassumptionthatreturnresidualsareuncorrelated.Thisassumptionwillbeincorrectiftheindexusedomitsasignificantriskfactor.2.Thetrade-offentailedindepartingfrompureindexinginfavorofanactivelymanagedportfolioisbetweentheprobability(orthepossibility)ofsuperiorperformanceagainstthecertaintyofadditionalmanagementfees.3.Theanswertothisquestioncanbeseenfromtheformulasforw0(equation8.20)andw*(equation8.21).Otherthingsheldequal,w0issmallerthegreatertheresidualvarianceofacandidateassetforinclusionintheportfolio.Further,weseethatregardlessofbeta,whenw0decreases,sodoesw*.Therefore,otherthingsequal,thegreatertheresidualvarianceofanasset,thesmalleritspositionintheoptimalriskyportfolio.Thatis,increasedfirm-specificriskreducestheextenttowhichanactiveinvestorwillbewillingtodepartfromanindexedportfolio.4.Thetotalriskpremiumequals:+(×Marketriskpremium).Wecallalphaanonmarketreturnpremiumbecauseitistheportionofthereturnpremiumthatisindependentofmarketperformance.TheSharperatioindicatesthatahigheralphamakesasecuritymoredesirable.Alpha,thenumeratoroftheSharperatio,isafixednumberthatisnotaffectedbythestandarddeviationofreturns,thedenominatoroftheSharperatio.Hence,anincreaseinalphaincreasestheSharperatio.Sincetheportfolioalphaistheportfolio-weightedaverageofthesecurities’alphas,then,holdingallotherparametersfixed,anincreaseinasecurity’salpharesultsinanincreaseintheportfolioSharperatio.Chapter8-IndexModels8-2Copyright©2014McGraw-HillEducation.Allrightsreserved.NoreproductionordistributionwithoutthepriorwrittenconsentofMcGraw-HillEducation.5.a.Tooptimizethisportfolioonewouldneed:n=60estimatesofmeansn=60estimatesofvariances770,122nnestimatesofcovariancesTherefore,intotal:890,1232nnestimatesb.Inasingleindexmodel:rirf=αi+βi(rM–rf)+eiEquivalently,usingexcessreturns:Ri=αi+βiRM+eiThevarianceoftherateofreturncanbedecomposedintothecomponents:(l)Thevarianceduetothecommonmarketfactor:22Mi(2)Thevarianceduetofirmspecificunanticipatedevents:)(σ2ieInthismodel:σββ),(CovjijirrThenumberofparameterestimatesis:n=60estimatesofthemeanE(ri)n=60estimatesofthesensitivitycoefficientβin=60estimatesofthefirm-specificvarianceσ2(ei)1estimateofthemarketmeanE(rM)1estimateofthemarketvariance2MTherefore,intotal,182estimates.Thesingleindexmodelreducesthetotalnumberofrequiredestimatesfrom1,890to182.Ingeneral,thenumberofparameterestimatesisreducedfrom:)23(to232nnn6.a.Thestandarddeviationofeachindividualstockisgivenby:2/1222)](β[σiMiieSinceβA=0.8,βB=1.2,σ(eA)=30%,σ(eB)=40%,andσM=22%,weget:σA=(0.82×222+302)1/2=34.78%σB=(1.22×222+402)1/2=47.93%Chapter8-IndexModels8-3Copyright©2014McGraw-HillEducation.Allrightsreserved.NoreproductionordistributionwithoutthepriorwrittenconsentofMcGraw-HillEducation.b.Theexpectedrateofreturnonaportfolioistheweightedaverageoftheexpectedreturnsoftheindividualsecurities:E(rP)=wA×E(rA)+wB×E(rB)+wf×rfE(rP)=(0.30×13%)+(0.45×18%)+(0.25×8%)=14%Thebetaofaportfolioissimilarlyaweightedaverageofthebetasoftheindividualsecurities:βP=wA×βA+wB×βB+wf×βfβP=(0.30×0.8)+(0.45×1.2)+(0.25×0.0)=0.78Thevarianceofthisportfoliois:)(σβσ2222PMPPewhere22σβMPisthesystematiccomponentand)(2Peisthenonsystematiccomponent.Sincetheresiduals(ei)areuncorrelated,thenonsystematicvarianceis:2222222()()()()PAABBffewewewe=(0.302×302)+(0.452×402)+(0.252×0)=405whereσ2(eA)andσ2(eB)arethefirm-specific(nonsystematic)variancesofStocksAandB,andσ2(ef),thenonsystematicvarianceofT-bills,iszero.Theresidualstandarddeviationoftheportfolioisthus:σ(eP)=(405)1/2=20.12%Thetotalvarianceoftheportfolioisthen:47.699405)2278.0(σ222PThetotalstandarddeviationis26.45%.7.a.Thetwofiguresdepictthestocks’securitycharacteristiclines(SCL).StockAhashigherfirm-specificriskbecausethedeviationsoftheobservationsfromtheSCLarelargerforStockAthanforStockB.DeviationsaremeasuredbytheverticaldistanceofeachobservationfromtheSCL.b.BetaistheslopeoftheSCL,whichisthemeasureofsystematicrisk.TheSCLforStockBissteeper;henceStockB’ssystematicriskisgreater.Chapter8-IndexModels8-4Copyright©2014McGraw-HillEducation.Allrightsreserved.NoreproductionordistributionwithoutthepriorwrittenconsentofMcGraw-HillEducation.c.TheR2(orsquaredcorrelationcoefficient)oftheSCListheratiooftheexplainedvarianceofthestock’sreturntototalvariance,andthetotalvarianceisthesumoftheexplainedvarianceplustheunexplainedvariance(thestock’sresidualvariance):)(σσβσβ222222iMiMieRSincetheexplainedvarianceforStockBisgreaterthanforStockA(theexplainedvarianceis22σβMB,whichisgreatersinceitsbetaishigher),anditsresidualvariance2()Beissmaller,itsR2ishigherthanStockA’s.d