投資學(xué)課后習(xí)題答案 Chap008

CHAPTERSINDEXMODELS

CHAPTER8:INDEXMODELS

PROBLEMSETS

1.Theadvantageoftheindexmodel,comparedtotheMarkowitzprocedure,isthe

vastlyreducednumberofestimatesrequired.Inaddition,thelargenumberof

estimatesrequiredfortheMarkowitzprocedurecanresultinlargeaggregate

estimationerrorswhenimplementingtheprocedure.Thedisadvantageoftheindex

modelarisesfromthemodel'sassumptionthatreturnresidualsareuncorrelated.

Thisassumptionwillbeincorrectiftheindexusedomitsasignificantriskfactor.

2.Thetrade-offentailedindepartingfrompureindexinginfavorofanactively

managedportfolioisbetweentheprobability(orthepossibility)ofsuperior

performanceagainstthecertaintyofadditionalmanagementfees.

3.Theanswertothisquestioncanbeseenfromtheformulasforw°(equation8.20)

andw*(equation8.21).Otherthingsheldequal,w°issmallerthegreaterthe

residualvarianceofacandidateassetforinclusionintheportfolio.Further,wesee

thatregardlessofbeta,whenw°decreases,sodoesw*.Therefore,otherthings

equal,thegreatertheresidualvarianceofanasset,thesmalleritspositioninthe

optimalriskyportfolio.Thatis,increasedfirm-specificriskreducestheextentto

whichanactiveinvestorwillbewillingtodepartfromanindexedportfolio.

4.Thetotalriskpremiumequals:a+(「xmarketriskpremium).Wecallalphaa

“nonmarket“returnpremiumbecauseitistheportionofthereturnpremiumthatis

independentofmarketperformance.

TheSharperatioindicatesthatahigheralphamakesasecuritymoredesirable.

Alpha,thenumeratoroftheSharperatio,isafixednumberthatisnotaffectedby

thestandarddeviationofreturns,thedenominatoroftheSharperatio.Hence,an

increaseinalphaincreasestheSharperatio.Sincetheportfolioalphaisthe

portfolio-weightedaverageofthesecurities?alphas,then,holdingallother

parametersfixed,anincreaseinasecurity'salpharesultsinanincreaseinthe

portfolioSharperatio.

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5.a.Tooptimizethisportfolioonewouldneed:

n=60estimatesofmeans

n=60estimatesofvariances

n2-n.

---=1,770estimatesofcovariances

Therefore,intotal:--~~~=1,890estimates

b.Inasingleindexmodel:n-rf=ai+pi(rM—rf)+ei

Equivalently,usingexcessreturns:Ri=ai+0iRM+ei

Thevarianceoftherateofreturncanbedecomposedintothecomponents:

(1)Thevarianceduetothecommonmarketfactor:

(2)Thevarianceduetofirmspecificunanticipatedevents:(52(6；)

Inthismodel:Cov&jj)=艮BQ

Thenumberofparameterestimatesis:

n=60estimatesofthemeanE(u)

n=60estimatesofthesensitivitycoefficient0i

n=60estimatesofthefirm-specificvariancea2(ej)

1estimateofthemarketmeanE(TM)

1estimateofthemarketvariance

Therefore,intotal,182estimates.

Thesingleindexmodelreducesthetotalnumberofrequiredestimatesfrom

1,890to182.Ingeneral,thenumberofparameterestimatesisreducedfrom:

n2+3n

to(3n+2)

2

6.a.Thestandarddeviationofeachindividualstockisgivenby:

2l/2

6=[俾％+o(ei)]

Since0A=0.8,PB=1.2,o(eA)=30%,O(CB)=40%,andOM=22%,weget:

GA=(0.82x222+302嚴=34.78%

OB=(1.22X222+402)1/2=47.93%

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b.Theexpectedrateofreturnonaportfolioistheweightedaverageofthe

expectedreturnsoftheindividualsecurities:

E(rp)=WAxE(FA)+WBXE(FB)+WfXrf

E(FP)=(0.30x13%)+(0.45x18%)+(0.25x8%)=14%

Thebetaofaportfolioissimilarlyaweightedaverageofthebetasofthe

individualsecurities:

PP=WAxpA+WBX0B+WfX0(-

PP=(0.30x0.8)+(0.45x1.2)+(0.25x0.0)=0.78

Thevarianceofthisportfoliois:

2

whereisthesystematiccomponentanda(ep)isthenonsystematic

component.Sincetheresiduals(ei)areuncorrelated,thenon-systematic

varianceis:

X/⑸)+X2X4(%)

/(%)=戌伺a(eB)+用

=(0.302x302)+(0.452x402)+(0.252x())=405

whereO2(CA)andO2(CB)arethefirm-specific(nonsystematic)variancesof

StocksAandB,ando2(ef),thenonsystematicvarianceofT-bills,iszero.

Theresidualstandarddeviationoftheportfolioisthus:

,/2

a(ep)=(405)=20.12%

Thetotalvarianceoftheportfolioisthen:

Qp=(0.782X222)+405=699.47change699.47to697.3

Thetotalstandarddeviationis26.41%.

7.a.Thetwofiguresdepictthestocks9securitycharacteristiclines(SCL).Stock

Ahashigherfirm-specificriskbecausethedeviationsoftheobservations

fromtheSCLarelargerforStockAthanforStockB.Deviationsare

measuredbytheverticaldistanceofeachobservationfromtheSCL.

b.BetaistheslopeoftheSCL,whichisthemeasureofsystematicrisk.The

SCLforStockBissteeper;henceStockB'ssystematicriskisgreater.

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c.TheR2(orsquaredcorrelationcoefficient)oftheSCListheratioofthe

explainedvarianceofthestock'sreturntototalvariance,andthetotal

varianceisthesumoftheexplainedvarianceplustheunexplainedvariance

(thestock9sresidualvariance):

R2

SincetheexplainedvarianceforStockBisgreaterthanforStockA(the

explainedvariance,whichisgreatersinceitsbetaishigher),andits

22

residualvariancecr(eff)issmaller,itsRishigherthanStockA's.

d.AlphaistheinterceptoftheSCLwiththeexpectedreturnaxis.StockAhasa

smallpositivealphawhereasStockBhasanegativealpha;hence,StockA's

alphaislarger.

e.ThecorrelationcoefficientissimplythesquarerootofR2,soStockB's

correlationwiththemarketishigher.

8.a.Firm-specificriskismeasuredbytheresidualstandarddeviation.Thus,stock

Ahasmorefirm-specificrisk:10.3%>9.1%

b.Marketriskismeasuredbybeta,theslopecoefficientoftheregression.Ahas

alargerbetacoefficient:1.2>0.8

c.R2measuresthefractionoftotalvarianceofreturnexplainedbythemarket

return.A'sR2islargerthanB's:0.576>0.436

d.RewritingtheSCLequationintermsoftotalreturn(r)ratherthanexcess

return(R):

rA-rf=a+/x(3一。)二

G=a+「x(l-0+￡x%

Theinterceptisnowequalto:

a+0x(1-7?)=1%+ox(1-1.2)

Sincerf=6%,theinterceptwouldbe:1%+6%(1-1.2)=1%-1.2%=-0.2%

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9.Thestandarddeviationofeachstockcanbederivedfromthefollowing

equationforR2:

PHExplainedvariance

R；

2:—-Totalvariance

Therefore:

oA=31.30%

ForstockB:

卡=4,800

%69.28%

10.ThesystematicriskforAis:

禺xq；=0.702x202=196

Thefirm-specificriskofA(theresidualvariance)isthedifferencebetween

A'stotalriskanditssystematicrisk:

980-196=784

ThesystematicriskforBis:

區(qū)xbj=1.202X202=576

B'sfirm-specificrisk(residualvariance)is:

4800-576=4224

11.ThecovariancebetweenthereturnsofAandBis(sincetheresidualsareassumed

tobeuncorrelated):

Cov(rA,rB)=pAPBo^=0.70x1.20x400=336

ThecorrelationcoefficientbetweenthereturnsofAandBis:

pAB=5(53)=——336——=0155

AB

oAoB31.30x69.28

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GHAPIERSINDEXMODELS

12.NotethatthecorrelationisthesquarerootofR2:p=

=0.20,/2x31.30x20=280

Cov(rArM)=pc8

=0.1212x69.28x20=480

Cov(rBrM)=

13.ForportfolioPwecancompute:

op=[(0.62x980)+(0.42x4800)+(2x0.4x0.6x336)]1/2=[1282.08]1/2=35.81%

pp=(0.6x0.7)+(0.4x1.2)=0.90

0/)=0：-限j=1282.08-(0.902X400)=958.08

Cov(rp,rM)=pp。:=0.90x400=360

Thissameresultcanalsobeattainedusingthecovariancesoftheindividualstocks

withthemarket:

Cov(rp,rM)=Cov(0.6rA+0.4FB,FM)=0.6xCov(tA,FM)+0.4xCOV(FBJM)

=(0.6x280)+(0.4x480)=360

14.NotethatthevarianceofT-billsiszero,andthecovarianceofT-billswithanyasset

iszero.Therefore,forportfolioQ:

BQ=[wjbb+w/+2xwpxwMxCov(rp,rM)]

=[(0.52x1,282.08)+(0.32x400)+(2x().5x().3x36())[=21.55%

=(0.5x0.90)+(0.3xl)+(0.20x0)=0.75

PQ=wp/3p+wM0M

22

a(eQ)=《一后￡=464.52-(0.75x400)=239.52

Cov(rQ,rM)-=0.75X400=300

15.a.BetaBooksadjustsbetabytakingthesampleestimateofbetaandaveragingit

with1.0,usingtheweightsof2/3and1/3,asfollows:

adjustedbeta=[(2/3)x1.24]+[(1/3)x1.0]=1.16

b.IfyouuseyourcurrentestimateofbetatobePt_|=1.24,then

pt=0.3+(0.7x1.24)=1.168

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16.ForStockA:

aA=rA-[rf+/?4x{rM-/y)]=.11-[.06+0.8x(.12-.06)]=0.2%

ForstockB:

aB="—1號+0B義CM.0)]=.14-[.06+1.5x(.12-.06)]=-1%

StockAwouldbeagoodadditiontoawell-diversifiedportfolio.Ashortposition

inStockBmaybedesirable.

17.a.

Alpha(a)Expectedexcessreturn

aj=rj-[rf+.x(FM-rf)]E(n)-rf

aA=20%-[8%+1.3x(16%-8%)]=1.6%20%-8%=12%

aB=18%-[8%+1.8x(16%-8%)]=-4.4%18%-8%=10%

ac=17%-[8%+0.7x(16%-8%)]=3.4%17%-8%=9%

aD=12%-[8%+1.0x(16%-8%)]=-4.0%12%-8%=4%

StocksAandChavepositivealphas,whereasstocksBandDhave

negativealphas.

Theresidualvariancesare:

O2(CA)=582=3,364

Q2(CB)=712=5,041

c2(ec)=602=3,600

Q2(CD)=552=3,025

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b.Toconstructtheoptimalriskyportfolio,wefirstdeterminetheoptimalactive

portfolio.UsingtheTreynor-Blacktechnique,weconstructtheactiveportfolio:

aa/(J??

o2(e)Sa/n2(e)

A0.000476-0.6142

B-0.0008731.1265

c0.000944-1.2181

D-0.0013221.7058

Total-0.0007751.0000

Beunconcernedwiththenegativeweightsofthepositiveastocks—theentire

activepositionwillbenegative,returningeverythingtogoodorder.

Withtheseweights,theforecastfortheactiveportfoliois:

a=[-0.6142x1.6]+[1.1265x(-4.4)]-[1.2181x3.4]+[1.7058x(-4.0)]

=-16.90%

P=[-0.6142x1.3]+[1.1265x1.8]-[1.2181x0.70]+[1.7058xl]=2.08

Thehighbeta(higherthananyindividualbeta)resultsfromtheshort

positionsintherelativelylowbetastocksandthelongpositionsinthe

relativelyhighbetastocks.

c2(e)=[(-0.6142)2x3364]+[1.12652x5041]+[(-1.2181)2x3600]+[1.70582x3025]

=21,809.6

a(e)=147.68%

TheleveredpositioninB[withhigho2(e)]overcomesthediversification

effect,andresultsinahighresidualstandarddeviation.Theoptimalrisky

portfoliohasaproportionw'intheactiveportfolio,computedasfollows:

a/a2(e)-.1690/21,809.6

%=--------r=-0.05124

.08/23?

Thenegativepositionisjustifiedforthereasonstatedearlier.

Theadjustmentforbetais:

-0.05124

=-0.0486

1+(1-P)wo1+(1-2.08)(-0.05124)

Sincew*isnegative,theresultisapositivepositioninstockswithpositive

alphasandanegativepositioninstockswithnegativealphas.Thepositionin

theindexportfoliois:

1-(-0.0486)=1.0486

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c.TocalculateSharpe'smeasurefortheoptimalriskyportfolio,wecomputethe

informationratiofortheactiveportfolioandSharpe'smeasureforthemarket

portfolio.Theinformationratiofortheactiveportfolioiscomputedasfollows:

nr

A=——=-16.90/147.68=-0.1144

cr(e)

A2=0.0131

Hence,thesquareofSharpe'smeasure(S)oftheoptimizedriskyportfoliois:

S2=S；+A2=怎]+0.0131=0.1341

S=0.3662

Comparethistothemarket'sSharpemeasure:

SM=8/23=0.3478->Adifferenceof:0.0184

Theonly-moderateimprovementinperformanceresultsfromonlyasmall

positiontakenintheactiveportfolioAbecauseofitslargeresidualvariance.

d.Tocalculatethemakeupofthecompleteportfolio,firstcomputethebeta,the

meanexcessreturnandthevarianceoftheoptimalriskyportfolio:

0P=WM+(WAxPA)=1.0486+[(-0.0486)x2.08]=0.95

E(RP)=aP+PPE(RM)=[(-0.0486)x(-16.90%)]+(0.95x8%)=8.42%

222

Qp=+a(ep)=(0.95x23)+((-0.0486)x21,809.6)=528.94

%=23.00%

SinceA=2.8,theoptimalpositioninthisportfoliois:

8.42

=0.5685

0.01x2.8x528.94

Incontrast,withapassivestrategy:

y=------------------=0.5401-Adifferenceof:0.0284

0.01x2.8x232

Thefinalpositionsare(MmayincludesomeofstocksAthroughD):

Bills1-0.5685=43.15%

M0.5685x1.0486=59.61%

A0.5685x(-0.0486)x(-0.6142)=1.70%

B0.5685x(-0.0486)x1.1265=-3.11%

c0.5685x(-0.0486)x(-1.2181)=3.37%

D0.5685x(-0.0486)x1.7058=-4.71%

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(subjecttoroundingerror)100.00%

18.a.Ifamanagerisnotallowedtosellshorthewillnotincludestockswithnegative

alphasinhisportfolio,sohewillconsideronlyAandC:

a

A?(e)a/a2(e)

o2(e)

Sa/n2(e)

A1.63,3640.0004760.3352

c3.43,6000.0009440.6648

0.00142()1.0000

Theforecastfortheactiveportfoliois:

a=(0.3352x1.6)+(0.6648x3.4)=2.80%

p=(0.3352x1.3)+(0.6648x0.7)=0.90

c2(e)=(0.33522x3,364)+(0.66482x3,600)=1,969.03

o(e)=44.37%

Theweightintheactiveportfoliois:

a/o2(e)2.80/1,969.03，

w()=--------------=-----------；-----=0n.0n9Q40n

E(RM)/O：8/232

Adjustingforbeta:

0.094

=0.0931

1+(1-P)wo1+[(1-0.90)x0.094]

Theinformationratiooftheactiveportfoliois:

2.80

=0.0631

b(e)44.37

Hence,thesquareofSharpe'smeasureis:

s2+0.06312=0.1250

Therefore:S=0.3535

Themarket9sSharpemeasureis:SM=0.3478

Whenshortsalesareallowed(Problem17),themanager'sSharpemeasureis

higher(0.3662).ThereductionintheSharpemeasureisthecostoftheshort

salerestriction.

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Thecharacteristicsoftheoptimalriskyportfolioare:

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Mx4=(1—0.0931)+(0.0931x0.9)=0.99

E0)=ctp+/3PxE(RM)=(0.093lx2.8%)+(0.99x8%)=8.18%

222

b：=區(qū)XG+(7(ep)=(0.99X23)+(0.0931x1969.03)=535.54

=23.14%

WithA=2.8,theoptimalpositioninthisportfoliois:

8.18

y==0.5455

0.01x2.8x535.54

Thefinalpositionsineachassetare:

Bills1-0.5455=45.45%

M0.5455x(1-0.0931)=49.47%

A0.5455x0.0931x0.3352=1.70%

C0.5455x0,0931x0.6648=3.38%

100.00%

b.Themeanandvarianceoftheoptimizedcompleteportfoliosinthe

unconstrainedandshort-salesconstrainedcases,andforthepassivestrategyare:

E(Rc)Qc

Unconstrained0.5685x8.42%=4.790.56852x528.94=170.95

Constrained0.5455x8.18%=4.460.54552x535.54=159.36

Passive0.5401x8.00%=4.320.54012x529.00=154.31

Theutilitylevelsbelowarecomputedusingtheformula:E(rc)-0.005Aa1

Unconstrained8%+4.79%-(0.005x2,8x170.95)=10.40%

Constrained8%+4.46%-(0.005x2.8x159.36)=10.23%

Passive8%+4.32%-(0.005x2.8x154.31)=10.16%

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19.Allalphasarereducedto0.3timestheirvaluesintheoriginalcase.Therefore,the

relativeweightsofeachsecurityintheactiveportfolioareunchanged,butthealpha

oftheactiveportfolioisonly0.3timesitspreviousvalue:0.3x-16.90%=-5.07%

Theinvestorwilltakeasmallerpositionintheactiveportfolio.Theoptimalrisky

portfoliohasaproportionwintheactiveportfolioasfollows:

alcr(e)-0.0507/21,809.6

------------------------=—U.U1/

2

EG-iy)1bli().08/23

Thenegativepositionisjustifiedforthereasongivenearlier.

Theadjustmentforbetais:

-0.01537_______

=-0.0151

1+(1—P)w°1+[(1-2.08)x(-0.01537)]

Sincew*isnegative,theresultisapositivepositioninstockswithpositivealphas

andanegativepositioninstockswithnegativealphas.Thepositionintheindex

portfoliois:1-(-0.0151)=1.0151

TocalculateSharpe'smeasurefbrtheoptimalriskyportfoliowecomputethe

informationratiofbrtheactiveportfolioandSharpe'smeasurefbrthemarketportfolio.

Theinformationratiooftheactiveportfoliois().3timesitspreviousvalue:

2

A==Z12Z_=_O,O343andA=0.00118

b(e)147.68

Hence,thesquareofSharpe'smeasureoftheoptimizedriskyportfoliois:

S2=S2M+A2=(8%/23%)2+0.00118=0.1222

S=0.3495

8%

Comparethistothemarket9sSharpemeasure:SM=-----=0.3478

23%

Thedifferenceis:0.0017

Notethatthereductionoftheforecastalphasbyafactorof0.3reducedthesquared

informationratioandtheimprovementinthesquaredSharperatiobyafactorof:

0.32=0.09

20.Ifeachofthealphaforecastsisdoubled,thenthealphaoftheactiveportfoliowill

alsodouble.Otherthingsequal,theinformationratio(IR)oftheactiveportfolio

alsodoubles.ThesquareoftheSharperatiofortheoptimizedportfolio(S-square)

equalsthesquareoftheSharperatiofbrthemarketindex(SM-square)plusthe

squareoftheinformationratio.Sincetheinformationratiohasdoubled,itssquare

quadruples.Therefore:S-square=SM-square+(4xIR)

ComparedtothepreviousS-square,thedifferenceis:3IR

Nowyoucanembarkonthecalculationstoverifythisresult.

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CFAPROBLEMS

1.Theregressionresultsprovidequantitativemeasuresofreturnandriskbasedon

monthlyreturnsoverthefive-yearperiod.

PforABCwas0.60,considerablylessthantheaveragestock's0of1.0.This

indicatesthat,whentheS&P500roseorfellby1percentagepoint,