博迪第八版投资学第十章课后习题答案.doc
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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
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