博迪第八版投资学第十章课后习题答案.doc

博迪第八版投资学第十章课后习题答案.doc

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Chapter10-ArbitragePricingTheoryandMultifactorModelsofRiskandReturn

CHAPTER10:

ARBITRAGEPRICINGTHEORY

ANDMULTIFACTORMODELSOFRISKANDRETURN

PROBLEMSETS

1. Therevisedestimateoftheexpectedrateofreturnonthestockwouldbetheoldestimateplusthesumoftheproductsoftheunexpectedchangeineachfactortimestherespectivesensitivitycoefficient:

revisedestimate=12%+[（1´2%）+（0.5´3%）]=15.5%

2. TheAPTfactorsmustcorrelatewithmajorsourcesofuncertainty,i.e.,sourcesofuncertaintythatareofconcerntomanyinvestors.Researchersshouldinvestigatefactorsthatcorrelatewithuncertaintyinconsumptionandinvestmentopportunities.GDP,theinflationrate,andinterestratesareamongthefactorsthatcanbeexpectedtodetermineriskpremiums.Inparticular,industrialproduction（IP）isagoodindicatorofchangesinthebusinesscycle.Thus,IPisacandidateforafactorthatishighlycorrelatedwithuncertaintiesthathavetodowithinvestmentandconsumptionopportunitiesintheeconomy.

3. Anypatternofreturnscanbe“explained”ifwearefreetochooseanindefinitelylargenumberofexplanatoryfactors.Ifatheoryofassetpricingistohavevalue,itmustexplainreturnsusingareasonablylimitednumberofexplanatoryvariables（i.e.,systematicfactors）.

4. Equation10.9applieshere:

E（rp）=rf+bP1[E（r1）-rf]+bP2[E（r2）–rf]

Weneedtofindtheriskpremium（RP）foreachofthetwofactors:

RP1=[E（r1）-rf]andRP2=[E（r2）-rf]

Inordertodoso,wesolvethefollowingsystemoftwoequationswithtwounknowns:

31=6+（1.5´RP1）+（2.0´RP2）

27=6+（2.2´RP1）+[（–0.2）´RP2]

Thesolutiontothissetofequationsis:

RP1=10%andRP2=5%

Thus,theexpectedreturn-betarelationshipis:

E（rP）=6%+（bP1´10%）+（bP2´5%）

5. TheexpectedreturnforPortfolioFequalstherisk-freeratesinceitsbetaequals0.

ForPortfolioA,theratioofriskpremiumtobetais:

（12-6）/1.2=5

ForPortfolioE,theratioislowerat:

（8–6）/0.6=3.33

Thisimpliesthatanarbitrageopportunityexists.Forinstance,youcancreateaPortfolioGwithbetaequalto0.6（thesameasE’s）bycombiningPortfolioAandPortfolioFinequalweights.TheexpectedreturnandbetaforPortfolioGarethen:

E（rG）=（0.5´12%）+（0.5´6%）=9%

bG=（0.5´1.2）+（0.5´0）=0.6

ComparingPortfolioGtoPortfolioE,Ghasthesamebetaandhigherreturn.Therefore,anarbitrageopportunityexistsbybuyingPortfolioGandsellinganequalamountofPortfolioE.Theprofitforthisarbitragewillbe:

rG–rE=[9%+（0.6´F）]-[8%+（0.6´F）]=1%

Thatis,1%ofthefunds（longorshort）ineachportfolio.

6. Substitutingtheportfolioreturnsandbetasintheexpectedreturn-betarelationship,weobtaintwoequationswithtwounknowns,therisk-freerate（rf）andthefactorriskpremium（RP）:

12=rf+（1.2´RP）

9=rf+（0.8´RP）

Solvingtheseequations,weobtain:

rf=3%andRP=7.5%

7. a. Shortinganequally-weightedportfolioofthetennegative-alphastocksandinvestingtheproceedsinanequally-weightedportfolioofthetenpositive-alphastockseliminatesthemarketexposureandcreatesazero-investmentportfolio.DenotingthesystematicmarketfactorasRM,theexpecteddollarreturnis（notingthattheexpectationofnon-systematicrisk,e,iszero）:

$1,000,000´[0.02+（1.0´RM）]-$1,000,000´[（–0.02）+（1.0´RM）]

=$1,000,000´0.04=$40,000

Thesensitivityofthepayoffofthisportfoliotothemarketfactoriszerobecausetheexposuresofthepositivealphaandnegativealphastockscancelout.（NoticethatthetermsinvolvingRMsumtozero.）Thus,thesystematiccomponentoftotalriskisalsozero.Thevarianceoftheanalyst’sprofitisnotzero,however,sincethisportfolioisnotwelldiversified.

Forn=20stocks（i.e.,long10stocksandshort10stocks）theinvestorwillhavea$100,000position（eitherlongorshort）ineachstock.Netmarketexposureiszero,butfirm-specificriskhasnotbeenfullydiversified.Thevarianceofdollarreturnsfromthepositionsinthe20stocksis:

20´[（100,000´0.30）2]=18,000,000,000

Thestandarddeviationofdollarreturnsis$134,164.

b. Ifn=50stocks（25stockslongand25stocksshort）,theinvestorwillhavea$40,000positionineachstock,andthevarianceofdollarreturnsis:

50´[（40,000´0.30）2]=7,200,000,000

Thestandarddeviationofdollarreturnsis$84,853.

Similarly,ifn=100stocks（50stockslongand50stocksshort）,theinvestorwillhavea$20,000positionineachstock,andthevarianceofdollarreturnsis:

100´[（20,000´0.30）2]=3,600,000,000

Thestandarddeviationofdollarreturnsis$60,000.

Noticethat,whenthenumberofstocksincreasesbyafactorof5（i.e.,from20to100）,standarddeviationdecreasesbyafactorof=2.23607（f