投资学第10版习题答案08.docx
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投资学第10版习题答案08.docx
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投资学第10版习题答案08
投资学第10版习题答案08
CHAPTER8:
INDEXMODELS
PROBLEMSETS
1.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:
α+(β×Marketriskpremium).Wecallalphaanonmarketreturnpremiumbecauseitistheportionofthereturnpremiumthatisindependentofmarketperformance.
TheSharperatioindicatesthatahigheralphamakesasecuritymoredesirable.Alpha,thenumeratoroftheSharperatio,isafixednumberthatisnotaffectedbythestandarddeviationofreturns,thedenominatoroftheSharperatio.Hence,anincreaseinalphaincreasestheSharperatio.Sincetheportfolioalphaistheportfolio-weightedaverageofthesecurities’alphas,then,holdingallotherparametersfixed,anincreaseinasecurity’salpharesultsinanincreaseintheportfolioSharperatio.
5.a.Tooptimizethisportfolioonewouldneed:
n=60estimatesofmeans
n=60estimatesofvariances
estimatesofcovariances
Therefore,intotal:
estimates
b.Inasingleindexmodel:
ri-rf=αi+βi(rM–rf)+ei
Equivalently,usingexcessreturns:
Ri=αi+βiRM+ei
Thevarianceoftherateofreturncanbedecomposedintothecomponents:
(l)Thevarianceduetothecommonmarketfactor:
(2)Thevarianceduetofirmspecificunanticipatedevents:
Inthismodel:
Thenumberofparameterestimatesis:
n=60estimatesofthemeanE(ri)
n=60estimatesofthesensitivitycoefficientβi
n=60estimatesofthefirm-specificvarianceσ2(ei)
1estimateofthemarketmeanE(rM)
1estimateofthemarketvariance
Therefore,intotal,182estimates.
Thesingleindexmodelreducesthetotalnumberofrequiredestimatesfrom1,890to182.Ingeneral,thenumberofparameterestimatesisreducedfrom:
6.a.Thestandarddeviationofeachindividualstockisgivenby:
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%
b.Theexpectedrateofreturnonaportfolioistheweightedaverageoftheexpectedreturnsoftheindividualsecurities:
E(rP)=wA×E(rA)+wB×E(rB)+wf×rf
E(rP)=(0.30×13%)+(0.45×18%)+(0.25×8%)=14%
Thebetaofaportfolioissimilarlyaweightedaverageofthebetasoftheindividualsecurities:
βP=wA×βA+wB×βB+wf×βf
βP=(0.30×0.8)+(0.45×1.2)+(0.25×0.0)=0.78
Thevarianceofthisportfoliois:
where
isthesystematiccomponentand
isthenonsystematiccomponent.Sincetheresiduals(ei)areuncorrelated,thenonsystematicvarianceis:
=(0.302×302)+(0.452×402)+(0.252×0)=405
whereσ2(eA)andσ2(eB)arethefirm-specific(nonsystematic)variancesofStocksAandB,andσ2(ef),thenonsystematicvarianceofT-bills,iszero.Theresidualstandarddeviationoftheportfolioisthus:
σ(eP)=(405)1/2=20.12%
Thetotalvarianceoftheportfolioisthen:
Thetotalstandarddeviationis26.45%.
7.a.Thetwofiguresdepictthestocks’securitycharacteristiclines(SCL).StockAhashigherfirm-specificriskbecausethedeviationsoftheobservationsfromtheSCLarelargerforStockAthanforStockB.DeviationsaremeasuredbytheverticaldistanceofeachobservationfromtheSCL.
b.BetaistheslopeoftheSCL,whichisthemeasureofsystematicrisk.TheSCLforStockBissteeper;henceStockB’ssystematicriskisgreater.
c.
TheR2(orsquaredcorrelationcoefficient)oftheSCListheratiooftheexplainedvarianceofthestock’sreturntototalvariance,andthetotalvarianceisthesumoftheexplainedvarianceplustheunexplainedvariance(thestock’sresidualvariance):
SincetheexplainedvarianceforStockBisgreaterthanforStockA(theexplainedvarianceis
whichisgreatersinceitsbetaishigher),anditsresidualvariance
issmaller,itsR2ishigherthanStockA’s.
d.AlphaistheinterceptoftheSCLwiththeexpectedreturnaxis.StockAhasasmallpositivealphawhereasStockBhasanegativealpha;hence,StockA’salphaislarger.
e.ThecorrelationcoefficientissimplythesquarerootofR2,soStockB’scorrelationwiththemarketishigher.
8.a.Firm-specificriskismeasuredbytheresidualstandarddeviation.Thus,stockAhasmorefirm-specificrisk:
10.3%>9.1%
b.Marketriskismeasuredbybeta,theslopecoefficientoftheregression.Ahasalargerbetacoefficient:
1.2>0.8
c.R2measuresthefractionoftotalvarianceofreturnexplainedbythemarketreturn.A’sR2islargerthanB’s:
0.576>0.436
d.RewritingtheSCLequationintermsoftotalreturn(r)ratherthanexcessreturn(R):
Theinterceptisnowequalto:
Sincerf=6%,theinterceptwouldbe:
9.ThestandarddeviationofeachstockcanbederivedfromthefollowingequationforR2:
Therefore:
ForstockB:
10.ThesystematicriskforAis:
Thefirm-specificriskofA(theresidualvariance)isthedifferencebetweenA’stotalriskanditssystematicrisk:
980–196=784
ThesystematicriskforBis:
B’sfirm-specificrisk(residualvariance)is:
4,800–576=4,224
11.ThecovariancebetweenthereturnsofAandBis(sincetheresidualsareassumedtobeuncorrelated):
ThecorrelationcoefficientbetweenthereturnsofAandBis:
12.NotethatthecorrelationisthesquarerootofR2:
13.ForportfolioPwecancompute:
σP=[(0.62×980)+(0.42×4800)+(2×0.4×0.6×336)]1/2=[1282.08]1/2=35.81%
βP=(0.6×0.7)+(0.4×1.2)=0.90
Cov(rP,rM)=βP
=0.90×400=360
Thissameresultcanalsobeattainedusingthecovariancesoftheindividualstockswiththemarket:
Cov(rP,rM)=Cov(0.6rA+0.4rB,rM)=0.6×Cov(rA,rM)+0.4×Cov(rB,rM)
=(0.6×280)+(0.4×480)=360
14.NotethatthevarianceofT-billsiszero,andthecovarianceofT-billswithanyassetiszero.Therefore,forportfolioQ:
15.a.BetaBooksadjustsbetabytakingthesampleestimateofbetaandaveragingitwith1.0,usingtheweightsof2/3and1/3,asfollows:
adjustedbeta=[(2/3)×1.24]+[(1/3)×1.0]=1.16
b.Ifyouuseyourcurrentestimateofbetatobeβt–1=1.24,then
βt=0.3+(0.7×1.24)=1.168
16.ForStockA:
ForstockB:
StockAwouldbeagoodadditiontoawell-diversifiedportfolio.AshortpositioninStockBmaybedesirable.
17.a.
Alpha(α)
Expectedexcessreturn
αi=ri–[rf+βi×(rM–rf)]
E(ri)–rf
αA=20%–[8%+1.3×(16%–8%)]=1.6%
20%–8%=12%
αB=18%–[8%+1.8×(16%–8%)]=–4.4%
18%–8%=10%
αC=17%–[8%+0.7×(16%–8%)]=3.4%
17%–8%=9%
αD=12%–[8%+1.0×(16%–8%)]=–4.0%
12%–8%=4%
StocksAandChavepositivealphas,whereasstocksBandDhavenegativealphas.
Theresidualvariancesare:
σ2(eA)=582=3,364
σ2(eB)=712=5,041
σ2(eC)=602=3,600
σ2(eD)=552=3,025
b.Toconstructtheoptimalriskyportfolio,wefirstdeterminetheoptimalactiveportfolio.UsingtheTreynor-Blacktechnique,weconstructtheactiveportfolio:
A
0.000476
–0.6142
B
–0.000873
1.1265
C
0.000944
–1.2181
D
–0.001322
1.7058
Total
–0.000775
1.0000
Beunconcernedwiththenegativeweightsofthepositiveαstocks—theentireactivepositionwillbenegative,returningeverythingtogoodorder.
Withtheseweights,theforecastfortheactiveportfoliois:
α=[–0.6142×1.6]+[1.1265×(–4.4)]–[1.2181×3.4]+[1.7058×(–4.0)]
=–16.90%
β=[–0.6142×1.3]+[1.1265×1.8]–[1.2181×0.70]+[1.7058×1]=2.08
Thehighbeta(higherthananyindividualbeta)resultsfromtheshortpositionsintherelativelylowbetastocksandthelongpositionsintherelativelyhighbetastocks.
σ2(e)=[(–0.6142)2×3364]+[1.12652×5041]+[(–1.2181)2×3600]+[1.70582×3025]
=21,809.6
σ(e)=147.68%
TheleveredpositioninB[withhighσ2(e)]overcomesthediversificationeffectandresultsinahighresidualstandarddeviation.Theoptimalriskyportfoliohasaproportionw*intheactiveportfolio,computedasfollows:
Thenegativepositionisjustifiedforthereasonstatedearlier.
Theadjustmentforbetais:
Sincew*isnegative,theresultisapositivepositioninstockswithpositivealphasandanegativepositioninstockswithnegativealphas.Thepositionintheindexportfoliois:
1–(–0.0486)=1.0486
c.TocalculatetheSharperatiofortheoptimalriskyportfolio,wecomputetheinformationratiofortheactiveportfolioandSharpe’smeasureforthemarketportfolio.Theinformationratiofortheactiveportfolioiscomputedasfollows:
A=
=–16.90/147.68=–0.1144
A2=0.0
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