投资学第10版习题答案08.docx

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

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:

+（×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:

rirf=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:

2（eA）=582=3,364

2（eB）=712=5,041

2（eC）=602=3,600

2（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.

2（e）=[（–0.6142）2×3364]+[1.12652×5041]+[（–1.2181）2×3600]+[1.70582×3025]

=21,809.6

（e）=147.68%

TheleveredpositioninB[withhigh2（e）]overcomesthediversificationeffectandresultsinahighresidualstandarddeviation.Theoptimalriskyportfoliohasaproportionw*intheactiveportfolio,computedasfollows:

Thenegativepositionisjustifiedforthereasonstatedearlier.

Theadjustmentforbetais:

Sincew*isnegative,theresultisapositivepositioninstockswithpositivealphasandanegativepositioninstockswithnegativealphas.Thepositionintheindexportfoliois:

1–（–0.0486）=1.0486

c.TocalculatetheSharperatiofortheoptimalriskyportfolio,wecomputetheinformationratiofortheactiveportfolioandSharpe’smeasureforthemarketportfolio.Theinformationratiofortheactiveportfolioiscomputedasfollows:

A=

=–16.90/