博迪第八版投资学第十章课后习题答案.docx
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博迪第八版投资学第十章课后习题答案.docx
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博迪第八版投资学第十章课后习题答案
CHAPTER10:
ARBITRAGEPRICINGTHEORY
ANDMULTIFACTORMODELSOFRISKANDRETURN
PROBLEMSETS
1.Therevisedestimateoftheexpectedrateofreturnonthestockwouldbetheoldestimateplusthesumoftheproductsoftheunexpectedchangeineachfactortimestherespectivesensitivitycoefficient:
revisedestimate=12%+[(12%)+(0.53%)]=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+P1[E(r1)rf]+P2[E(r2)–rf]
Weneedtofindtheriskpremium(RP)foreachofthetwofactors:
RP1=[E(r1)rf]andRP2=[E(r2)rf]
Inordertodoso,wesolvethefollowingsystemoftwoequationswithtwounknowns:
31=6+(1.5RP1)+(2.0RP2)
27=6+(2.2RP1)+[(–0.2)RP2]
Thesolutiontothissetofequationsis:
RP1=10%andRP2=5%
Thus,theexpectedreturn-betarelationshipis:
E(rP)=6%+(P110%)+(P25%)
5.TheexpectedreturnforPortfolioFequalstherisk-freeratesinceitsbetaequals0.
ForPortfolioA,theratioofriskpremiumtobetais:
(126)/1.2=5
ForPortfolioE,theratioislowerat:
(8–6)/0.6=3.33
Thisimpliesthatanarbitrageopportunityexists.Forinstance,youcancreateaPortfolioGwithbetaequalto0.6(thesameasE’s)bycombiningPortfolioAandPortfolioFinequalweights.TheexpectedreturnandbetaforPortfolioGarethen:
E(rG)=(0.512%)+(0.56%)=9%
G=(0.51.2)+(0.50)=0.6
ComparingPortfolioGtoPortfolioE,Ghasthesamebetaandhigherreturn.Therefore,anarbitrageopportunityexistsbybuyingPortfolioGandsellinganequalamountofPortfolioE.Theprofitforthisarbitragewillbe:
rG–rE=[9%+(0.6F)][8%+(0.6F)]=1%
Thatis,1%ofthefunds(longorshort)ineachportfolio.
6.Substitutingtheportfolioreturnsandbetasintheexpectedreturn-betarelationship,weobtaintwoequationswithtwounknowns,therisk-freerate(rf)andthefactorriskpremium(RP):
12=rf+(1.2RP)
9=rf+(0.8RP)
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+(1.0RM)]$1,000,000[(–0.02)+(1.0RM)]
=$1,000,0000.04=$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[(100,0000.30)2]=18,000,000,000
Thestandarddeviationofdollarreturnsis$134,164.
b.Ifn=50stocks(25stockslongand25stocksshort),theinvestorwillhavea$40,000positionineachstock,andthevarianceofdollarreturnsis:
50[(40,0000.30)2]=7,200,000,000
Thestandarddeviationofdollarreturnsis$84,853.
Similarly,ifn=100stocks(50stockslongand50stocksshort),theinvestorwillhavea$20,000positionineachstock,andthevarianceofdollarreturnsis:
100[(20,0000.30)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-diversifiedportfolioofeachtypewillhaveonlysystematicrisksincethenon-systematicriskwillapproachzerowithlargen.Themeanwillequalthatoftheindividual(identical)stocks.
c.Thereisnoarbitrageopportunitybecausethewell-diversifiedportfoliosallplotonthesecuritymarketline(SML).Becausetheyarefairlypriced,thereisnoarbitrage.
9.a.Alongpositioninaportfolio(P)comprisedofPortfoliosAandBwillofferanexpectedreturn-betatradeofflyingonastraightlinebetweenpointsAandB.Therefore,wecanchooseweightssuchthatP=CbutwithexpectedreturnhigherthanthatofPortfolioC.Hence,combiningPwithashortpositioninCwillcreateanarbitrageportfoliowithzeroinvestment,zerobeta,andpositiverateofreturn.
b.Theargumentinpart(a)leadstothepropositionthatthecoefficientof2mustbezeroinordertoprecludearbitrageopportunities.
10.a.E(r)=6+(1.26)+(0.58)+(0.33)=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+(16)+(0.52)+(0.754)=16%
Accordingtotheequationforthereturnonthestock,theactuallyexpectedreturnonthestockis15%(becausetheexpectedsurprisesonallfactorsarezerobydefinition).Becausetheactuallyexpectedreturnbasedonriskislessthantheequilibriumreturn,weconcludethatthestockisoverpriced.
12.Thefirsttwofactorsseempromisingwithrespecttothelikelyimpactonthefirm’scostofcapital.Botharemacrofactorsthatwouldelicithedgingdemandsacrossbroadsectorsofinvestors.Thethirdfactor,whileimportanttoPorkProducts,isapoorchoiceforamultifactorSMLbecausethepriceofhogsisofminorimportancetomostinvestorsandisthereforehighlyunlikelytobeapricedriskfactor.Betterchoiceswouldfocusonvariablesthatinvestorsinaggregatemightfindmoreimportanttotheirwelfare.Examplesinclude:
inflationuncertainty,short-terminterest-raterisk,energypricerisk,orexchangeraterisk.Theimportantpointhereisthat,inspecifyingamultifactorSML,wenotconfuseriskfactorsthatareimportanttoaparticularinvestorwithfactorsthatareimportanttoinvestorsingeneral;onlythelatterarelikelytocommandariskpremiuminthecapitalmarkets.
13.Themaximumresidualvarianceistiedtothenumberofsecurities(n)intheportfoliobecause,asweincreasethenumberofsecurities,wearemorelikelytoencountersecuritieswithlargerresidualvariances.Thestartingpointistodeterminethepracticallimitontheportfolioresidualstandarddeviation,(eP),thatstillqualifiesasa‘well-diversifiedportfolio.’Areasonableapproachistocompare2(eP)tothemarketvariance,orequivalently,tocompare(eP)tothemarketstandarddeviation.Supposewedonotallow(eP)toexceedpM,wherepisasmalldecimalfraction,forexample,0.05;then,thesmallerthevaluewechooseforp,themorestringentourcriterionfordefininghowdiversifieda‘well-diversified’portfoliomustbe.
Nowconstructaportfolioofnsecuritieswithweightsw1,w2,…,wn,sothatwi=1.Theportfolioresidualvarianceis:
2(eP)=w122(ei)
Tomeetourpracticaldefinitionofsufficientlydiversified,werequirethisresidualvariancetobelessthan(pM)2.Asureandsimplewaytoproceedistoassumetheworst,thatis,assumethattheresidualvarianceofeachsecurityisthehighestpossiblevalueallowedundertheassumptionsoftheproblem:
2(ei)=n2M
Inthatcase:
2(eP)=wi2nM2
Nowapplytheconstraint:
wi2nM2≤(pM)2
Thisrequiresthat:
nwi2≤p2
Or,equivalently,that:
wi2≤p2/n
Arelativelyeasywaytogenerateasetofwell-diversifiedportfoliosistouseportfolioweightsthatfollowageometricprogression,sincethecomputationsthenbecomerelativelystraightforward.Choosew1andacommonfactorqforthegeometricprogressionsuchthatq<1.Therefore,theweightoneachstockisafractionqoftheweightonthepreviousstockintheseries.Thenthesumofntermsis:
wi=w1(1–qn)/(1–q)=1
or:
w1=(1–q)/(1–qn)
Thesumofthensquaredweightsissimilarlyobtainedfromw12andacommongeometricprogressionfactorofq2.Therefore:
wi2=w12(1–q2n)/(1–q2)
Substitutingforw1fromabove,weobtain:
wi2=[(1–q)2/(1–qn)2]×[(1–q2n)/(1–q2)]
Forsufficientdiversification,wechooseqsothat:
wi2≤p2/n
Forexample,continuetoassumethatp=0.05andn=1,000.Ifwechoose
q=0.9973,thenwewillsatisfytherequiredcondition.Atthisvalueforq:
w1=0.0029andwn=0.0029×0.
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