投资学第10版习题答案06.docx
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投资学第10版习题答案06
CHAPTER6:
CAPITALALLOCATIONTORISKYASSETS
PROBLEMSETS
1.（e）Thefirsttwoanswerchoicesareincorrectbecauseahighlyriskaverseinvestorwouldavoidportfolioswithhigherriskpremiumsandhigherstandarddeviations.Inaddition,higherorlowerSharperatiosarenotanindicationofaninvestor'stoleranceforrisk.TheSharperatioissimplyatooltoabsolutelymeasurethereturnpremiumearnedperunitofrisk.
2.（b）Ahigherborrowingrateisaconsequenceoftheriskoftheborrowers’default.Inperfectmarketswithnoadditionalcostofdefault,thisincrementwouldequalthevalueoftheborrower’soptiontodefault,andtheSharpemeasure,withappropriatetreatmentofthedefaultoption,wouldbethesame.However,inrealitytherearecoststodefaultsothatthispartoftheincrementlowerstheSharperatio.Also,noticethatanswer（c）isnotcorrectbecausedoublingtheexpectedreturnwithafixedrisk-freeratewillmorethandoubletheriskpremiumandtheSharperatio.
3.Assumingnochangeinrisktolerance,thatis,anunchangedrisk-aversioncoefficient（A）,higherperceivedvolatilityincreasesthedenominatoroftheequationfortheoptimalinvestmentintheriskyportfolio（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$118,421andprovidesanexpectedcashinflowof$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,portfoliosthatcommandgreaterriskpremiumsmustsellatlowerprices.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>0.07⇒A<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:
σ
σ2
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）.
σ
σ2
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）
σPortfolio
（3）×20%
σ2Portfolio
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
σPortfolio
σ2Portfolio
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×（18-8）
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）=0.08+0.10×y*=0.08+（0.3644×0.1）=0.1164or11.644%
σC=0.3644×28=10.203%
20.a.Iftheperiod1926–2012isassumedtoberepresentativeoffutureexpectedperformance,thenweusethefollowingdatatocomputethefractionallocatedtoequity:
A=4,E（rM）−rf=8.10%,σM=20.48%（weusethestandarddeviationoftheriskpremiumfromTable6.7）.Theny*isgivenby:
Thatis,48.28%oftheportfolioshouldbeallocatedtoequityand51.72%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,preferringinvestmentsthathavelessriskasevidencedbythelowerstandarddeviation.
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,theclientwillneitherborrownorlendbutwillholdaportfoliocomposedonlyoftheoptimalriskyportfolio:
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:
.11–.09–f
=0.13<
.13–.09
=0.16→f=–.004
.15
.25
Morerisktolerantinvestors（whoaremoreinclinedtoborrow）willnotbeclientsofthefund.Wefindthatfisnegative:
thatis,youwouldneedtopayinv
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