solnch8optimalriskypf.docx
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solnch8optimalriskypf
CHAPTER8:
OPTIMALRISKYPORTFOLIOS
1.Theparametersoftheopportunitysetare:
E(rS)=20%,E(rB)=12%,σS=30%,σB=15%,ρ=.10
Fromthestandarddeviationsandthecorrelationcoefficientwegeneratethecovariancematrix[notethatCov(rS,rB)=ρσSσB]:
BondsStocks
Bonds22545
Stocks45900
Theminimum-varianceportfolioisfoundbyapplyingtheformula:
wMin(S)=
=
=.1739
wMin(B)=1.1739=.8261
Theminimumvarianceportfoliomeanandstandarddeviationare:
E(rMin)=.1739⨯20+.8261⨯12=13.39%
σMin=[w
+w
+2wSwBCov(rS,rB)]1/2
=[(.17392⨯900)+(.82612⨯225)+(2⨯.1739⨯.8261⨯45)]1/2=13.92%
2.
%instocks
%inbonds
Exp.return
Std.Dev
0.00%
100.00%
12.00
15.00
17.39%
82.61%
13.39
13.92
minimumvariance
20.00%
80.00%
13.60
13.94
40.00%
60.00%
15.20
15.70
45.16%
54.84%
15.61
16.54
tangencyportfolio
60.00%
40.00%
16.80
19.53
80.00%
20.00%
18.40
24.48
100.00%
0.00%
20.00
30.00
3.
E(r)σ
Thegraphapproximatesthepoints:
Min.VariancePortf.13.4%13.9%
TangencyPortfolio15.616.5
4.Theproportionofstocksintheoptimalriskyportfolioisgivenby:
wS=
=
=.4516
wB=.5484
Themeanandstandarddeviationoftheoptimalriskyportfolioare:
E(rp)=.4516⨯20+.5484⨯12=15.61%
σp=[(.45162⨯900)+(.54842⨯225)+(2⨯.4516⨯.5484⨯45)]1/2=16.54%
5.Thereward-to-variabilityratiooftheoptimalCALis:
=
=.4601
6.a.Ifyourequireyourportfoliotoyieldameanreturnof14%youcanfindthecorrespondingstandarddeviationfromtheoptimalCAL.TheformulaforthisCALis:
E(rC)=rf+
σC=8+.4601σC
SettingE(rC)equalto14%wefindthatthestandarddeviationoftheoptimalportfoliois13.04%.
b.TofindtheproportioninvestedinT-billswerememberthatthemeanofthecompleteportfolio,14%,isanaverageoftheT-billrateandtheoptimalcombinationofstocksandbonds,P.Letybetheproportioninthisportfolio.ThemeanofanyportfolioalongtheoptimalCALis:
E(rC)=(l-y)rf+yE(rp)=rf+y[E(rp)-rf]=8+y(15.61-8)
SettingE(rC)=14%wefind:
y=.7884,and1-y=.2116,theproportioninT-bills.
Tofindtheproportionsinvestedineachofthefundswemultiply.7884bytheproportionsofthestocksandbondsintheoptimalriskyportfolio:
Proportionofstocksincompleteportfolio=.7884⨯.4516=.3560
Proportionofbondsincompleteportfolio=.7884⨯.5484=.4324
7.Usingonlythestockandbondfundstoachieveaportfoliomeanof14%wemustfindtheappropriateproportioninthestockfund,wS,andwB=1-wSinthebondfund.Theportfoliomeanwillbe:
14=20wS+12(1-wS)=12+8wSwS=.25
Sotheproportionswillbe25%instocksand75%inbonds.Thestandarddeviationofthisportfoliowillbe:
σp=(.252⨯900+.752⨯225+2⨯.25⨯.75⨯45)1/2=14.13%
Thisisconsiderablylargerthanthestandarddeviationof13.04%achievedusingT-billsandtheoptimalportfolio.
8.Withnoopportunitytoborrowyouwishtoconstructaportfoliowithameanof24%.Sincethisexceedsthemeanonstocksof20%,youwillhavetogoshortonbonds,whichhaveameanof12%,andusetheproceedstobuyadditionalstock.ThegraphicalrepresentationofyourriskyportfolioispointQonthefollowinggraph:
PointQisthestock/bondcombinationwithmeanof24%.LetwSbetheproportionofstocksand1-wSbetheproportionofbondsrequiredtoachievethe24%mean.Then:
24=20⨯wS+12⨯(1-wS)=12+8wS
wS=1.50,and1-wS=-.50
Therefore,youwouldhavetosellshortanamountofbondsequalto.50ofyourtotalfunds,andinvest1.50timesyourtotalfundsinstocks.Thestandarddeviationofthisportfoliowouldbe:
σQ=[1.502⨯900+(-.50)2⨯225+2⨯(1.50)⨯(-.50)⨯45]1/2=44.87%
Ifyouwereallowedtoborrowattherisk-freerateof8%,thewaytoachievethetargetexpectedreturnof24%wouldbetoinvestmorethan100%ofyourfundsintheoptimalriskyportfolio,movingoutalongtheCALtotherightofP,uptoR,onthefollowinggraph.
RisthepointontheoptimalCALwhichhasthemeanof24%.UsingtheformulafortheoptimalCALwecanfindthecorrespondingstandarddeviation:
E(rC)=8+.4601σC=24
SettingE(rC)=24,weget:
σC=34.78%,whichisconsiderablylessthanthe44.87%standarddeviationyouwouldgetwithouttheabilitytoborrowattherisk-freerateof8%.
WhatistheportfoliocompositionofpointRontheoptimalCAL?
ThemeanofanyportfolioalongthisCALis:
E(rC)=rf+y[E(rP)-rf]
whereyistheproportioninvestedintheoptimalriskyportfolioPandrPisthemeanofthatportfolio,whichis15.61%.
24=8+y(15.61-8)
y=2.1025
Thismeansthatforevery$1ofyourownfundsinvestedinportfolioP,youwouldborrowanadditional$1.1025andinvestitalsoinportfolioP.
9.a.
Eventhoughgoldseemsdominatedbystocks,itstillmightbeanattractiveassettoholdasapartofaportfolio.Ifthecorrelationbetweengoldandstocksissufficientlylow,itwillbeheldasanelementinaportfolio--theoptimaltangencyportfolio.
b.Ifgoldhadacorrelationcoefficientwithstocksof+1,itwouldnotbeheld.TheoptimalCALwouldbecomprisedofbillsandstocksonly.Sincethesetofrisk/returncombinationsofstocksandgoldwouldplotasastraightlinewithanegativeslope(seethefollowinggraph),itwouldbedominatedbythestocksportfolio.Ofcourse,thissituationcouldnotpersist.Ifnoonedesiredgold,itspricewouldfallanditsexpectedrateofreturnwouldincreaseuntilitbecameasufficientlyattractiveassettohold.
10.SinceAandBareperfectlynegativelycorrelated,arisk-freeportfoliocanbecreatedanditsrateofreturninequilibriumwillbetherisk-freerate.Tofindtheproportionsofthisportfolio(withwAinvestedinAandwB=1–wAinB),setthestandarddeviationequaltozero.Withperfectnegativecorrelation,theportfoliostandarddeviationreducesto
σP=Absolutevalue[wAσA-wBσB]
0=5wA-10(1–wA)
wA=.6667
Theexpectedrateofreturnonthisrisk-freeportfoliois:
E(r)=.6667⨯10+.3333⨯15=11.67%
Therefore,therisk-freeratemustalsobe11.67%.
11.False.Iftheborrowingandlendingratesarenotidentical,thendependingonthetastesoftheindividuals(thatis,theshapeoftheirindifferencecurves),borrowersandlenderscouldhavedifferentoptimalriskyportfolios.
12.False.Theportfoliostandarddeviationequalstheweightedaverageofthecomponent-assetstandarddeviationsonlyinthespecialcasethatallassetsareperfectlypositivelycorrelated.Otherwise,astheformulaforportfoliostandarddeviationshows,theportfoliostandarddeviationislessthantheweightedaverageofthecomponent-assetstandarddeviations.Theportfoliovariancewillbeaweightedsumoftheelementsinthecovariancematrix,withtheproductsoftheportfolioproportionsasweights.
13.Theprobabilitydistributionis:
ProbabilityRateofReturn
.7100%
.3–50%
Mean=.7⨯100+.3⨯(-50)=55%
Variance=.7⨯(100-55)2+.3⨯(-50-55)2=4725
Standarddeviation=47251/2=68.74%
14.σP=30=yσ=40y
y=.75
E(rp)=12+.75(30-12)=25.5%
15.a.Restrictingtheportfolioto20stocksratherthan40-50willincreasetheriskoftheportfolio,butpossiblynotbymuch.If,forinstance,the50stocksinauniversehadthesamestandarddeviation,σ,andthecorrelationsbetweeneachpairwereidenticalwithcorrelationcoefficientρ(sothatthecovariancebetweeneachpairwouldbeρσ2),thevarianceofanequallyweightedportfoliowouldbe(seeAppendixA,equation8A.4),
σ
=
σ2+
ρσ2
Theeffectofthereductioninnonthesecondtermwouldberelativelysmall(since49/50iscloseto19/20andρσ2issmallerthanσ2),butthedenominatorofthefirsttermwouldbe20insteadof50.Forexample,ifσ=45%andρ=.2,thenthestandarddeviationwith50stockswouldbe20.91%,andwouldriseto22.05%whenonly20stocksareheld.Suchanincreasemightbeacceptableiftheexpectedreturnisincreasedsufficiently.
b.Hennessycouldcontaintheincreaseinriskbymakingsurethathemaintainsreasonablediversificationamongthe20stocksthatremaininhisportfolio.Thisentailsmaintainingalowcorrelationamongtheremainingstocks.Forexample,inpart(a),withρ=.2,theincreaseinportfolioriskwasminimal.Asapracticalmatter,thismeansthatHennessywouldneedtospreadhisportfolioamongmanyindustries;concentratingonjustafewwouldresultinhighercorrelationamongtheincludedstocks.
16.Riskreductionbenefitsfromdiversificationarenotalinearfunctionofthenumberofissuesintheportfolio.Rather,theincrementalbenefitsfromadditionaldiversificationaremostimportantwhenyouareleastdiversified.RestrictingHenneseyto10insteadof20issueswouldincreasetheriskofhisportfoliobyagreateramountthanwouldreducingthesizeoftheportfoliofrom30to20stocks.Inourexample,restrictingthenumberofstocksto10willincreasethestandarddeviationto23.81%.Theincreaseinstandarddeviationof1.76%fromgivingup10of20stocksisgreaterthantheincreaseof1.14%fromgivingup30stockswhenstartingwith50.
17.Thepointiswelltakenbecausethecommitteeshouldbeconcernedwiththevolatilityoftheentireportfolio.SinceHennessey'sportfolioisonlyoneofsixwell-diversifiedportfoliosandismallerthantheaverage,theconcentrationinfewerissuescouldhaveaminimaleffectonthediversificationofthetotalfund.Hence,unleashingHennessey
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