Chap006 Investments 9th edition by Bodie.docx
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Chap006 Investments 9th edition by Bodie.docx
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Chap006Investments9theditionbyBodie
CHAPTER6:
RISKAVERSIONAND
CAPITALALLOCATIONTORISKYASSETS
PROBLEMSETS
1.(e)
2.(b)Ahigherborrowingrateisaconsequenceoftheriskoftheborrowers’default.Inperfectmarketswithnoadditionalcostofdefault,thisincrementwouldequalthevalueoftheborrower’soptiontodefault,andtheSharpemeasure,withappropriatetreatmentofthedefaultoption,wouldbethesame.However,inrealitytherearecoststodefaultsothatthispartoftheincrementlowerstheSharperatio.Also,noticethatanswer(c)isnotcorrectbecausedoublingtheexpectedreturnwithafixedrisk-freeratewillmorethandoubletheriskpremiumandtheSharperatio.
3.Assumingnochangeinrisktolerance,thatis,anunchangedriskaversioncoefficient(A),thenhigherperceivedvolatilityincreasesthedenominatoroftheequationfortheoptimalinvestmentintheriskyportfolio(Equation6.7).Theproportioninvestedintheriskyportfoliowillthereforedecrease.
4.a.Theexpectedcashflowis:
(0.5×$70,000)+(0.5×200,000)=$135,000
Withariskpremiumof8%overtherisk-freerateof6%,therequiredrateofreturnis14%.Therefore,thepresentvalueoftheportfoliois:
$135,000/1.14=$118,421
b.Iftheportfolioispurchasedfor$118,421,andprovidesanexpectedcashinflowof$135,000,thentheexpectedrateofreturn[E(r)]isasfollows:
$118,421×[1+E(r)]=$135,000
Therefore,E(r)=14%.Theportfoliopriceissettoequatetheexpectedrateofreturnwiththerequiredrateofreturn.
c.IftheriskpremiumoverT-billsisnow12%,thentherequiredreturnis:
6%+12%=18%
Thepresentvalueoftheportfolioisnow:
$135,000/1.18=$114,407
d.Foragivenexpectedcashflow,portfoliosthatcommandgreaterriskpremiamustsellatlowerprices.Theextradiscountfromexpectedvalueisapenaltyforrisk.
5.WhenwespecifyutilitybyU=E(r)–0.5Aσ2,theutilitylevelforT-billsis:
0.07
Theutilitylevelfortheriskyportfoliois:
U=0.12–0.5×A×(0.18)2=0.12–0.0162×A
Inorderfortheriskyportfoliotobepreferredtobills,thefollowingmusthold:
0.12–0.0162A>0.07A<0.05/0.0162=3.09
Amustbelessthan3.09fortheriskyportfoliotobepreferredtobills.
6.PointsonthecurvearederivedbysolvingforE(r)inthefollowingequation:
U=0.05=E(r)–0.5Aσ2=E(r)–1.5σ2
ThevaluesofE(r),giventhevaluesofσ2,aretherefore:
2
E(r)
0.00
0.0000
0.05000
0.05
0.0025
0.05375
0.10
0.0100
0.06500
0.15
0.0225
0.08375
0.20
0.0400
0.11000
0.25
0.0625
0.14375
Theboldlineinthegraphonthenextpage(labeledQ6,forQuestion6)depictstheindifferencecurve.
7.RepeatingtheanalysisinProblem6,utilityisnow:
U=E(r)–0.5Aσ2=E(r)–2.0σ2=0.05
Theequal-utilitycombinationsofexpectedreturnandstandarddeviationarepresentedinthetablebelow.Theindifferencecurveistheupwardslopinglineinthegraphonthenextpage,labeledQ7(forQuestion7).
2
E(r)
0.00
0.0000
0.0500
0.05
0.0025
0.0550
0.10
0.0100
0.0700
0.15
0.0225
0.0950
0.20
0.0400
0.1300
0.25
0.0625
0.1750
TheindifferencecurveinProblem7differsfromthatinProblem6inslope.WhenAincreasesfrom3to4,theincreasedriskaversionresultsinagreaterslopefortheindifferencecurvesincemoreexpectedreturnisneededinordertocompensateforadditionalσ.
8.Thecoefficientofriskaversionforariskneutralinvestoriszero.Therefore,thecorrespondingutilityisequaltotheportfolio’sexpectedreturn.Thecorrespondingindifferencecurveintheexpectedreturn-standarddeviationplaneisahorizontalline,labeledQ8inthegraphabove(seeProblem6).
9.Arisklover,ratherthanpenalizingportfolioutilitytoaccountforrisk,derivesgreaterutilityasvarianceincreases.Thisamountstoanegativecoefficientofriskaversion.Thecorrespondingindifferencecurveisdownwardslopinginthegraphabove(seeProblem6),andislabeledQ9.
10.Theportfolioexpectedreturnandvariancearecomputedasfollows:
(1)
WBills
(2)
rBills
(3)
WIndex
(4)
rIndex
rPortfolio
(1)×
(2)+(3)×(4)
Portfolio
(3)×20%
2Portfolio
0.0
5%
1.0
13.0%
13.0%=0.130
20%=0.20
0.0400
0.2
5%
0.8
13.0%
11.4%=0.114
16%=0.16
0.0256
0.4
5%
0.6
13.0%
9.8%=0.098
12%=0.12
0.0144
0.6
5%
0.4
13.0%
8.2%=0.082
8%=0.08
0.0064
0.8
5%
0.2
13.0%
6.6%=0.066
4%=0.04
0.0016
1.0
5%
0.0
13.0%
5.0%=0.050
0%=0.00
0.0000
11.ComputingutilityfromU=E(r)–0.5×Aσ2=E(r)–σ2,wearriveatthevaluesinthecolumnlabeledU(A=2)inthefollowingtable:
WBills
WIndex
rPortfolio
Portfolio
2Portfolio
U(A=2)
U(A=3)
0.0
1.0
0.130
0.20
0.0400
0.0900
.0700
0.2
0.8
0.114
0.16
0.0256
0.0884
.0756
0.4
0.6
0.098
0.12
0.0144
0.0836
.0764
0.6
0.4
0.082
0.08
0.0064
0.0756
.0724
0.8
0.2
0.066
0.04
0.0016
0.0644
.0636
1.0
0.0
0.050
0.00
0.0000
0.0500
.0500
ThecolumnlabeledU(A=2)impliesthatinvestorswithA=2preferaportfoliothatisinvested100%inthemarketindextoanyoftheotherportfoliosinthetable.
12.ThecolumnlabeledU(A=3)inthetableaboveiscomputedfrom:
U=E(r)–0.5Aσ2=E(r)–1.5σ2
Themoreriskaverseinvestorsprefertheportfoliothatisinvested40%inthemarket,ratherthanthe100%marketweightpreferredbyinvestorswithA=2.
13.Expectedreturn=(0.7×18%)+(0.3×8%)=15%
Standarddeviation=0.7×28%=19.6%
14.
Investmentproportions:
30.0%inT-bills
0.7×25%=
17.5%inStockA
0.7×32%=
22.4%inStockB
0.7×43%=
30.1%inStockC
15.Yourreward-to-volatilityratio:
Client'sreward-to-volatilityratio:
16.
17.a.E(rC)=rf+y×[E(rP)–rf]=8+y×(188)
Iftheexpectedreturnfortheportfoliois16%,then:
16%=8%+10%×y
Therefore,inordertohaveaportfoliowithexpectedrateofreturnequalto16%,theclientmustinvest80%oftotalfundsintheriskyportfolioand20%inT-bills.
b.
Client’sinvestmentproportions:
20.0%inT-bills
0.8×25%=
20.0%inStockA
0.8×32%=
25.6%inStockB
0.8×43%=
34.4%inStockC
c.σC=0.8×σP=0.8×28%=22.4%
18.a.σC=y×28%
Ifyourclientprefersastandarddeviationofatmost18%,then:
y=18/28=0.6429=64.29%investedintheriskyportfolio
b.
19.a.y*
Therefore,theclient’soptimalproportionsare:
36.44%investedintheriskyportfolioand63.56%investedinT-bills.
b.E(rC)=8+10×y*=8+(0.3644×10)=11.644%
C=0.3644×28=10.203%
20.a.Iftheperiod1926-2009isassumedtoberepresentativeoffutureexpectedperformance,thenweusethefollowingdatatocomputethefractionallocatedtoequity:
A=4,E(rM)−rf=7.93%,σM=20.81%(weusethestandarddeviationoftheriskpremiumfromTable6.7).Theny*isgivenby:
Thatis,45.78%oftheportfolioshouldbeallocatedtoequityand54.22%shouldbeallocatedtoT-bills.
b.Iftheperiod1968-1988isassumedtoberepresentativeoffutureexpectedperformance,thenweusethefollowingdatatocomputethefractionallocatedtoequity:
A=4,E(rM)−rf=3.44%,σM=16.71%andy*isgivenby:
Therefore,30.80%ofthecompleteportfolioshouldbeallocatedtoequityand69.20%shouldbeallocatedtoT-bills.
c.Inpart(b),themarketriskpremiumisexpectedtobelowerthaninpart(a)andmarketriskishigher.Therefore,thereward-to-volatilityratioisexpectedtobelowerinpart(b),whichexplainsthegreaterproportioninvestedinT-bills.
21.a.E(rC)=8%=5%+y×(11%–5%)
b.σC=y×σP=0.50×15%=7.5%
c.Thefirstclientismoreriskaverse,allowingasmallerstandarddeviation.
22.Johnsonrequeststheportfoliostandarddeviationtoequalonehalfthemarketportfoliostandarddeviation.Themarketportfolio
whichimplies
.TheinterceptoftheCMLequals
andtheslopeoftheCMLequalstheSharperatioforthemarketportfolio(35%).ThereforeusingtheCML:
23.Data:
rf=5%,E(rM)=13%,σM=25%,and
=9%
TheCMLandindifferencecurvesareasfollows:
24.Forytobelessthan1.0(thattheinvestorisalender),riskaversion(A)mustbelargeenoughsuchthat:
Forytobegreaterthan1(theinvestorisaborrower),Amustbesmallenough:
Forvaluesofriskaversionwithinthisrange,theclientwillneitherborrownorlend,butwillholdaportfoliocomprisedonlyoftheoptimalriskyportfolio:
y=1for0.64≤A≤1.28
25.a.ThegraphforProblem23hastoberedrawnhere,with:
E(rP)=11%andσP=15%
b.Foralendingposition:
Foraborrowingposition:
Therefore,y=1for0.89≤A≤2.67
26.Themaximumfeasiblefee,denotedf,dependsonthereward-to-variabilityratio.
Fory<1,thelendingrate,5%,isviewedastherelevantrisk-freerate,andwesolveforfasfollows:
Fory>1,theborrowingrate,9%,istherelevantrisk-freerate.Thenwenoticethat,evenwithoutafee,theactivefundisinferiortothepassivefundbecause:
`
Morerisktolerantinvestors(whoaremoreinclinedtoborrow)willnotbeclientsof
thefund.Wefindthatfisnegative:
thatis,youwouldneedtopayinvestorstochooseyouractivefund.Theseinvestorsdesirehigherrisk-higherreturncompleteportfoliosandthusareintheborrowingrangeoftherelevantCAL.Inthisrange,thereward-to-variabilityratiooftheindex(thepassivefund)isbetterthanthatofthemanagedfund.
27.a.SlopeoftheCML
Thediagramfollows.
b.Myfundallowsaninvestortoachieveahighermeanforanygivenstandarddeviatio
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