投资学第10版习题答案06.docx

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

投资学第10版习题答案06（总13页）

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（Equation.Theproportioninvestedintheriskyportfoliowillthereforedecrease.

4.a.Theexpectedcashflowis:

×$70,000）+×200,000）=$135,000.

Withariskpremiumof8%overtherisk-freerateof6%,therequiredrateofreturnis14%.Therefore,thepresentvalueoftheportfoliois:

$135,000/=$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/=$114,407

d.Foragivenexpectedcashflow,portfoliosthatcommandgreaterriskpremiumsmustsellatlowerprices.Theextradiscountfromexpectedvalueisapenaltyforrisk.

5.WhenwespecifyutilitybyU=E（r）–σ2,theutilitylevelforT-billsis:

Theutilitylevelfortheriskyportfoliois:

U=–×A×2=–×A

Inorderfortheriskyportfoliotobepreferredtobills,thefollowingmusthold:

–>A<=

Amustbelessthanfortheriskyportfoliotobepreferredtobills.

6.PointsonthecurvearederivedbysolvingforE（r）inthefollowingequation:

U==E（r）–σ2=E（r）–σ2

ThevaluesofE（r）,giventhevaluesofσ2,aretherefore:

2

E（r）

Theboldlineinthegraphonthenextpage（labeledQ6,forQuestion6）depictstheindifferencecurve.

7.RepeatingtheanalysisinProblem6,utilityisnow:

U=E（r）–σ2=E（r）–σ2=

Theequal-utilitycombinationsofexpectedreturnandstandarddeviationarepresentedinthetablebelow.Theindifferencecurveistheupwardslopinglineinthegraphonthenextpage,labeledQ7（forQuestion7）.

2

E（r）

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

5%

%

%=

20%=

5

%=

16%=

5

%=

12%=

5

%=

8%=

5

%=

4%=

5

%=

0%=

11.ComputingutilityfromU=E（r）–×Aσ2=E（r）–σ2,wearriveatthevaluesinthecolumnlabeledU（A=2）inthefollowingtable:

WBills

WIndex

rPortfolio

Portfolio

2Portfolio

U（A=2）

U（A=3）

.0700

.0756

.0764

.0724

.0636

.0500

ThecolumnlabeledU（A=2）impliesthatinvestorswithA=2preferaportfoliothatisinvested100%inthemarketindextoanyoftheotherportfoliosinthetable.

12.ThecolumnlabeledU（A=3）inthetableaboveiscomputedfrom:

U=E（r）–σ2=E（r）–σ2

Themoreriskaverseinvestorsprefertheportfoliothatisinvested40%inthemarket,ratherthanthe100%marketweightpreferredbyinvestorswithA=2.

13.Expectedreturn=×18%）+×8%）=15%

Standarddeviation=×28%=%

14.

Investmentproportions:

%inT-bills

×25%=

%inStockA

×32%=

%inStockB

×43%=

%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:

%inT-bills

×25%=

%inStockA

×32%=

%inStockB

×43%=

%inStockC

c.σC=×σP=×28%=%

18.a.σC=y×28%

Ifyourclientprefersastandarddeviationofatmost18%,then:

y=18/28==%investedintheriskyportfolio.

b.

19.a.y*

Therefore,theclient’soptimalproportionsare:

%investedintheriskyportfolioand%investedinT-bills.

b.E（rC）=+×y*=+×=or%

C=×28=%

20.a.Iftheperiod1926–2012isassumedtoberepresentativeoffutureexpectedperformance,thenweusethefollowingdatatocomputethefractionallocatedtoequity:

A=4,E（rM）−rf=%,σM=%（weusethestandarddeviationoftheriskpremiumfromTable.Theny*isgivenby:

Thatis,%oftheportfolioshouldbeallocatedtoequityand%shouldbeallocatedtoT-bills.

b.Iftheperiod1968–1988isassumedtoberepresentativeoffutureexpectedperformance,thenweusethefollowingdatatocomputethefractionallocatedtoequity:

A=4,E（rM）−rf=%,σM=%andy*isgivenby:

Therefore,%ofthecompleteportfolioshouldbeallocatedtoequityand%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=×15%=%

c.Thefirstclientismoreriskaverse,preferringinvestmentsthathavelessriskasevidencedbythelowerstandarddeviation.

22.Johnsonrequeststheportfoliostandarddeviationtoequalonehalfthemarketportfoliostandarddeviation.Themarketportfolio

whichimplies

.TheinterceptoftheCMLequals

andtheslopeoftheCMLequalstheSharperatioforthemarketportfolio（35%）.ThereforeusingtheCML:

23.Data:

rf=5%,E（rM）=13%,σM=25%,and

=9%

TheCMLandindifferencecurvesareasfollows:

24.Forytobelessthan（thattheinvestorisalender）,riskaversion（A）mustbelargeenoughsuchthat:

Forytobegreaterthan1（theinvestorisaborrower）,Amustbesmallenough:

Forvaluesofriskaversionwithinthisrange,theclientwillneitherborrownorlendbutwillholdaportfoliocomposedonlyoftheoptimalriskyportfolio:

y=1for≤A≤

25.a.ThegraphforProblem23hastoberedrawnhere,with:

E（rP）=11%andσP=15%

b.Foralendingposition:

Foraborrowingposition:

Therefore,y=1for≤A≤

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

 =<

.13–.09

=→f=–.004

.15

.25

  Morerisktolerantinvestors（whoaremoreinclinedtoborrow）willnotbeclientsofthefund.Wefindthatfisnegative:

thatis,youwouldneedtopayinvestorstochooseyouractivefund.Theseinvestorsdesirehigherrisk–higherreturncompleteportfoliosandthusareintheborrowingrangeoftherelevantCAL.Inthisrange,thereward-to-variabilityratiooftheindex（thepassivefund）isbetterthanthatofthemanagedfund.

27.a.SlopeoftheCML

Thediagramfollows.

b.Myfundallowsaninvestortoachieveahighermeanforanygivenstandarddeviationthanwouldapassivestrategy,.,ahigherexpectedreturnforanygivenlevelofrisk.

28.a.With70%ofhismoneyinvestedinmyfund’sportfolio,theclient’sexpectedreturnis15%peryearwithastandarddeviationof%peryear.Ifheshiftsthatmoneytothepassiveportfolio（whichhasanexpectedreturnof13%andstandarddeviationof25%）,hisoverallexpectedreturnbecomes:

E（rC）=rf+×[E（rM）−rf]=.08+[×（.13–.08）]=.115,or%

Thestandarddevia