博迪第八版投资学第十章课后习题答案Word格式.doc
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博迪第八版投资学第十章课后习题答案Word格式.doc
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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´
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(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,wecanchooseweightssuchthatbP=bCbutwithexpectedreturnhigherthanthatofPortfolioC.Hence,combiningPwithashortpositioninCwillcreateanarbitrageportfoliowithzeroinvestment,zerobeta,andpositiverateofreturn.
b. Theargumentinpart(a)leadstothepropositionthatthecoefficientofb2mustbezeroinordertoprecludearbitrageopportunities.
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´
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,wenotconfuseriskfactorsthatareimportanttoaparticularinvestorwithfactorsthatareimportanttoinvestorsingeneral;
onlythelatterarelikelytocommandariskpremiuminthecapitalmarkets.
13. Themaximumresidualvarianceistiedtothenumberofsecurities(n)intheportfoliobecause,asweincreasethenumberofsecurities,wearemorelikelytoencountersecuritieswithlargerresidualvariances.Thestartingpointistodeterminethepracticallimitontheportfolioresidualstandarddeviation,s(eP),thatstillqualifiesasa‘well-diversifiedportfolio.’Areasonableapproachistocompares2(eP)tothemarketvariance,orequivalently,tocompares(eP)tothemarketstandarddeviation.Supposewedonotallows(eP)toexceedpsM,wherepisasmalldecimalfraction,forexample,0.05;
then,thesmallerthevaluewechooseforp,themorestringentourcriterionfordefininghowdiversifieda‘well-diversified’portfoliomustbe.
Nowconstructaportfolioofnsecuritieswithweightsw1,w2,…,wn,sothatSwi=1.Theportfolioresidualvarianceis:
s2(eP)=Sw12s2(ei)
Tomeetourpracticaldefinitionofsufficientlydiversified,werequirethisresidualvariancetobelessthan(psM)2.Asureandsimplewaytoproceedistoassumetheworst,thatis,assumethattheresidualvarianceofeachsecurityisthehighestpossiblevalueallowedundertheassumptionsoftheproblem:
s2(ei)=ns2M
Inthatcase:
s2(eP)=Swi2nsM2
Nowapplytheconstraint:
Swi2nsM2≤(psM)2
Thisrequiresthat:
nSwi2≤p2
Or,equivalently,that:
Swi2≤p2/n
Arelativelyeasywaytogenerateasetofwell-diversifiedportfoliosistouseportfolioweightsthatfollowageometricprogression,sincethecomputationsthenbecomerelativelystraightforward.Choosew1andacommonfactorqforthegeometricprogressionsuchthatq<
1.Therefore,theweightoneachstockisafractionqoftheweightonthepreviousstockintheseries.Thenthesumofntermsis:
Swi=w1(1–qn)/(1–q)=1
or:
w1=(1–q)/(1–qn)
Thesumofthensquaredweightsissimilarlyobtainedfromw12andacommongeometricprogressionfactorofq2.Therefore:
Swi2=w12(1–q2n)/(1–q2)
Substitutingforw1fromabove,weobtain:
Swi2=[(1–q)2/(1–qn)2]×
[(1–q2n)/(1–q2)]
Forsufficientdiversification,wechooseqsothat:
Forexample,continuetoassumethatp=0.05andn=1,000.Ifwechoose
q=0.9
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