投资学10版习题答案10.docx

投资学10版习题答案10.docx

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投资学10版习题答案10

CHAPTER10:

ARBITRAGEPRICINGTHEORYANDMULTIFACTORMODELSOFRISKANDRETURN

PROBLEMSETS

1.Therevisedestimateoftheexpectedrateofreturnonthestockwouldbetheoldestimateplusthesumoftheproductsoftheunexpectedchangeineachfactortimestherespectivesensitivitycoefficient:

Revisedestimate=12%+[（1×2%）+（0.5×3%）]=15.5%

NotethattheIPestimateiscomputedas:

1×（5%-3%）,andtheIRestimateiscomputedas:

0.5×（8%-5%）.

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

3.Anypatternofreturnscanbeexplainedifwearefreetochooseanindefinitelylargenumberofexplanatoryfactors.Ifatheoryofassetpricingistohavevalue,itmustexplainreturnsusingareasonablylimitednumberofexplanatoryvariables（i.e.,systematicfactorssuchasunemploymentlevels,GDP,andoilprices）.

4.Equation10.11applieshere:

E（rp）=rf+βP1[E（r1）rf]+βP2[E（r2）–rf]

Weneedtofindtheriskpremium（RP）foreachofthetwofactors:

RP1=[E（r1）rf]andRP2=[E（r2）rf]

Inordertodoso,wesolvethefollowingsystemoftwoequationswithtwounknowns:

.31=.06+（1.5×RP1）+（2.0×RP2）

.27=.06+（2.2×RP1）+[（–0.2）×RP2]

Thesolutiontothissetofequationsis

RP1=10%andRP2=5%

Thus,theexpectedreturn-betarelationshipis

E（rP）=6%+（βP1×10%）+（βP2×5%）

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%

βG=（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.Shortinganequallyweightedportfolioofthetennegative-alphastocksandinvestingtheproceedsinanequally-weightedportfolioofthe10positive-alphastockseliminatesthemarketexposureandcreatesazero-investmentportfolio.DenotingthesystematicmarketfactorasRM,theexpecteddollarreturnis（notingthattheexpectationofnonsystematicrisk,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（from$134,164to$60,000）.

8.a.

b.Ifthereareaninfinitenumberofassetswithidenticalcharacteristics,thenawell-diversifiedportfolioofeachtypewillhaveonlysystematicrisksincethenonsystematicriskwillapproachzerowithlargen.Eachvarianceissimplyβ2×marketvariance:

Themeanwillequalthatoftheindividual（identical）stocks.

c.Thereisnoarbitrageopportunitybecausethewell-diversifiedportfoliosallplotonthesecuritymarketline（SML）.Becausetheyarefairlypriced,thereisnoarbitrage.

9.a.Alongpositioninaportfolio（P）composedofportfoliosAandBwillofferanexpectedreturn-betatrade-offlyingonastraightlinebetweenpointsAandB.Therefore,wecanchooseweightssuchthatβP=βCbutwithexpectedreturnhigherthanthatofportfolioC.Hence,combiningPwithashortpositioninCwillcreateanarbitrageportfoliowithzeroinvestment,zerobeta,andpositiverateofreturn.

b.Theargumentinpart（a）leadstothepropositionthatthecoefficientofβ2mustbezeroinordertoprecludearbitrageopportunities.

10.a.E（r）=6%+（1.2×6%）+（0.5×8%）+（0.3×3%）=18.1%

b.Surprisesinthemacroeconomicfactorswillresultinsurprisesinthereturnofthestock:

Unexpectedreturnfrommacrofactors=

[1.2×（4%–5%）]+[0.5×（6%–3%）]+[0.3×（0%–2%）]=–0.3%

E（r）=18.1%−0.3%=17.8%

11.TheAPTrequired（i.e.,equilibrium）rateofreturnonthestockbasedonrfandthefactorbetasis

RequiredE（r）=6%+（1×6%）+（0.5×2%）+（0.75×4%）=16%

Accordingtotheequationforthereturnonthestock,theactuallyexpectedreturnonthestockis15%（becausetheexpectedsurprisesonallfactorsarezerobydefinition）.Becausetheactuallyexpectedreturnbasedonriskislessthantheequilibriumreturn,weconcludethatthestockisoverpriced.

12.Thefirsttwofactorsseempromisingwithrespecttothelikelyimpactonthefirm’scostofcapital.Botharemacrofactorsthatwouldelicithedgingdemandsacrossbroadsectorsofinvestors.Thethirdfactor,whileimportanttoPorkProducts,isapoorchoiceforamultifactorSMLbecausethepriceofhogsisofminorimportancetomostinvestorsandisthereforehighlyunlikelytobeapricedriskfactor.Betterchoiceswouldfocusonvariablesthatinvestorsinaggregatemightfindmoreimportanttotheirwelfare.Examplesinclude:

inflationuncertainty,short-terminterest-raterisk,energypricerisk,orexchangeraterisk.Theimportantpointhereisthat,inspecifyingamultifactorSML,wenotconfuseriskfactorsthatareimportantto

aparticularinvestorwithfactorsthatareimportanttoinvestorsingeneral;onlythelatterarelikelytocommandariskpremiuminthecapitalmarkets.

13.Theformulais

14.If

andbasedonthesensitivitiestorealGDP（0.75）andinflation（1.25）,McCrackenwouldcalculatetheexpectedreturnfortheOrbLargeCapFundtobe:

Therefore,Kwon’sfundamentalanalysisestimateiscongruentwithMcCracken’sAPTestimate.IfweassumethatbothKwonandMcCracken’sestimatesonthereturnofOrb’sLargeCapFundareaccurate,thennoarbitrageprofitispossible.

15.Inordertoeliminateinflation,thefollowingthreeequationsmustbesolvedsimultaneously,wheretheGDPsensitivitywillequal1inthefirstequation,inflationsensitivitywillequal0inthesecondequationandthesumoftheweightsmustequal1inthethirdequation.

Here,xrepresentsOrb’sHighGrowthFund,yrepresentsLargeCapFundandzrepresentsUtilityFund.Usingalgebraicmanipulationwillyieldwx=wy=1.6andwz=-2.2.

16.Sinceretireeslivingoffasteadyincomewouldbehurtbyinflation,thisportfoliowouldnotbeappropriateforthem.Retireeswouldwantaportfoliowithareturnpositivelycorrelatedwithinflationtopreservevalue,andlesscorrelatedwiththevariablegrowthofGDP.Thus,Stilesiswrong.McCrackeniscorrectinthatsupplysidemacroeconomicpoliciesaregenerallydesignedtoincreaseoutputataminimumofinflationarypressure.IncreasedoutputwouldmeanhigherGDP,whichinturnwouldincreasereturnsofafundpositivelycorrelatedwithGDP.

17.Themaximumresidualvarianceistiedtothenumberofsecurities（n）intheportfoliobecause,asweincreasethenumberofsecurities,wearemorelikelytoencountersecuritieswithlargerresidualvariances.Thestartingpointistodeterminethepracticallimitontheportfolioresidualstandarddeviation,（eP）,thatstillqualifiesasawell-diversifiedportfolio.Areasonableapproachistocompare

2（eP）tothemarketvariance,orequivalently,tocompare（eP）tothemarketstandarddeviation.Supposewedonotallow（eP）toexceedpM,wherepisasmalldecimalfraction,forexample,0.05;then,thesmallerthevaluewechooseforp,themorestringe