投资学第10版习题答案06分析解析

1、Copyright2014McGraw-HillEducation.Allrightsreserved.NoreproductionordistributionwithoutthepriorwrittenconsentofCopyright2014McGraw-HillEducation.Allrightsreserved.NoreproductionordistributionwithoutthepriorwrittenconsentofMcGraw-HillEducation.CHAPTER6:CAPITALALLOCATIONTORISKYASSETSPROBLEMSETS(e)Thef

2、irsttwoanswerchoicesareincorrectbecauseahighlyriskaverseinvestorwouldavoidportfolioswithhigherriskpremiumsandhigherstandarddeviations.Inaddition,higherorlowerSharperatiosarenotanindicationofaninvestorstoleranceforrisk.TheSharperatioissimplyatooltoabsolutelymeasurethereturnpremiumearnedperunitofrisk.

3、(b)Ahigherborrowingrateisaconsequenceoftheriskoftheborrowersdefault.Inperfectmarketswithnoadditionalcostofdefault,thisincrementwouldequalthevalueoftheborrowersoptiontodefault,andtheSharpemeasure,withappropriatetreatmentofthedefaultoption,wouldbethesame.However,inrealitytherearecoststodefaultsothatth

4、ispartoftheincrementlowerstheSharperatio.Also,noticethatanswer(c)isnotcorrectbecausedoublingtheexpectedreturnwithafixedrisk-freeratewillmorethandoubletheriskpremiumandtheSharperatio.Assumingnochangeinrisktolerance,thatis,anunchangedrisk-aversioncoefficient(A),higherperceivedvolatilityincreasestheden

5、ominatoroftheequationfortheoptimalinvestmentintheriskyportfolio(Equation6.7).Theproportioninvestedintheriskyportfoliowillthereforedecrease.a.Theexpectedcashflowis:(0.5x$70,000)+(0.5x200,000)=$135,000.Withariskpremiumof8%overtherisk-freerateof6%,therequiredrateofreturnis14%.Therefore,thepresentvalueo

6、ftheportfoliois:$135,000/1.14=$118,421Iftheportfolioispurchasedfor$118,421andprovidesanexpectedcashinflowof$135,000,thentheexpectedrateofreturnE(r)isasfollows:$118,421x1+E(r)=$135,000Therefore,E(r)=14%.Theportfoliopriceissettoequatetheexpectedrateofreturnwiththerequiredrateofreturn.Iftheriskpremiumo

7、verT-billsisnow12%,thentherequiredreturnis:6%+12%=18%Thepresentvalueoftheportfolioisnow:$135,000/1.18=$114,407Foragivenexpectedcashflow,portfoliosthatcommandgreaterrisk6-1premiumsmustsellatlowerprices.Theextradiscountfromexpectedvalueisapenaltyforrisk.WhenwespecifyutilitybyU=E(r)0.5Ao2,theutilitylev

8、elforT-billsis:0.07Theutilitylevelfortheriskyportfoliois:U=0.120.5xAx(0.18)2=0.120.0162xAInorderfortheriskyportfoliotobepreferredtobills,thefollowingmusthold:0.120.0162A0.07nA0.05/0.0162=3.09Amustbelessthan3.09fortheriskyportfoliotobepreferredtobills.PointsonthecurvearederivedbysolvingforE(r)inthefo

9、llowingequation:U=0.05=E(r)0.5AL=E(r)1.52ThevaluesofE(r),giventhevaluesofo2,aretherefore:GG2E(r)0.000.00000.050000.050.00250.053750.100.01000.065000.150.02250.083750.200.04000.110000.250.06250.14375Theboldlineinthegraphonthenextpage(labeledQ6,forQuestion6)depictstheindifferencecurve.Repeatingtheanal

10、ysisinProblem6,utilityisnow:U=E(r)0.5Ao2=E(r)2.0=0.05Theequal-utilitycombinationsofexpectedreturnandstandarddeviationarepresentedinthetablebelow.Theindifferencecurveistheupwardslopinglineinthegraphonthenextpage,labeledQ7(forQuestion7).GG2E(r)0.000.00000.05000.050.00250.05500.100.01000.07000.150.0225

11、0.09500.200.04000.13000.250.06250.17506-2TheindifferencecurveinProblem7differsfromthatinProblem6inslope.WhenAincreasesfrom3to4,theincreasedriskaversionresultsinagreaterslopefortheindifferencecurvesincemoreexpectedreturnisneededinordertocompensateforadditionalo.Thecoefficientofriskaversionforariskneu

12、tralinvestoriszero.Therefore,thecorrespondingutilityisequaltotheportfoliosexpectedreturn.Thecorrespondingindifferencecurveintheexpectedreturn-standarddeviationplaneisahorizontalline,labeledQ8inthegraphabove(seeProblem6).Arisklover,ratherthanpenalizingportfolioutilitytoaccountforrisk,derivesgreaterut

13、ilityasvarianceincreases.Thisamountstoanegativecoefficientofriskaversion.Thecorrespondingindifferencecurveisdownwardslopinginthegraphabove(seeProblem6),andislabeledQ9.6-310.Theportfolioexpectedreturnandvariancearecomputedasfollows:壯畳畳豊x(眈0%。2“。10.0.05%0.04002468000055550.813.011.4%=0.11416%=0.160.60

14、.05%0.04002468000055550.813.011.4%=0.11416%=0.160.613.09.8%=0.09812%=0.1213.08.2%=0.0828%=0.06.6%=0.0664%=0.040.013.05.0%=0.0500%=0.001.013.0%13.0%=0.13020%=0.200.02560.01440.00640.00160.0000ComputingutilityfromU=E(r)0.5xAo2=E(r)o2,wearriveatthevaluesinthecolumnlabeledU(A=2)inthefollowing

15、table:0.01.00.1300.200.04000.0900.070060.02560.0884.07560.60.1140.0980.120.01440.0836.076820.080.00640.0756.070.0660.00160.0644.06361.00.00.0500.040.000.00000.0500.0500WG2rGU(A=2)U(A=3)ThecolumnlabeledU(A=2)impliesthatinvestorswithA=2preferaportfoliothatisinvested100%in

16、themarketindextoanyoftheotherportfoliosinthetable.ThecolumnlabeledU(A=3)inthetableaboveiscomputedfrom:U=E(r)-0.5Ao2=E(r)-1.5o2Themoreriskaverseinvestorsprefertheportfoliothatisinvested40%inthemarket,ratherthanthe100%marketweightpreferredbyinvestorswithA=2.Expectedreturn=(0.7x18%)+(0.3x8%)=15%Standar

17、ddeviation=0.7x28%=19.6%Investmentproportions:30.0%inT-bills0.7x25%=17.5%inStockA0.7x32%=22.4%inStockB0.7x43%=30.1%inStockC15.Yourreward-to-volatilityratio:S=.18-.08.2815.Yourreward-to-volatilityratio:S=.18-.08.28二0.3571Clientsreward-to-volatilityratio:S.15-.08.196二0.35716-416.17.a.E(rC)=片+yxE(rp)-r

18、=8+yx(18-8)Iftheexpectedreturnfortheportfoliois16%,then:16%=8%+10%xyny二.16-.8二0.8.10Therefore,inordertohaveaportfoliowithexpectedrateofreturnequalto16%,theclientmustinvest80%oftotalfundsintheriskyportfolioand20%inT-bills.b.Clientsinvestmentproportions:20.0%inT-bills0.8x25%=20.0%inStockA0.8x32%=25.6%

19、inStockB0.8x43%=34.4%inStockCc.c=0.8x%=0.8x28%=22.4%6-518.a.oC=yx28%Ifyourclientprefersastandarddeviationofatmost18%,then:y=18/28=0.6429=64.29%investedintheriskyportfolio.b.E(rc)=.08+.1xy=.08+(0.6429x.1)二14.429%19.a.E(r)r19.a.E(r)rA02P0.180.08_0.103.5x0.2820.2744Therefore,theclientsoptimalproportion

20、sare:36.44%investedintheriskyportfolioand63.56%investedinT-bills.b.E(rC)=0.08+0.10 xy*=0.08+(0.3644x0.1)=0.1164or11.644%gc=0.3644x28=10.203%20.a.Iftheperiod1926-2012isassumedtoberepresentativeoffutureexpectedperformance,thenweusethefollowingdatatocomputethefractionallocatedtoequity:A=4,E(rM)-rf=8.10

21、%,oM=20.48%(weusethestandarddeviationoftheriskpremiumfromTable6.7).Theny*isgivenby:4X0.20484X0.20482Thatis,48.28%oftheportfolioshouldbeallocatedtoequityand51.72%shouldbeallocatedtoT-bills.b.Iftheperiod1968-1988isassumedtoberepresentativeoffutureexpectedperformance,thenweusethefollowingdatatocomputet

22、hefractionallocatedtoequity:A=4,E(rM)-rf=3.44%,oM=16.71%andy*isgivenby:E(r)rE(r)rMfAg2M0.03444x0.16712二0.3080Therefore,30.80%ofthecompleteportfolioshouldbeallocatedtoequityand69.20%shouldbeallocatedtoT-bills.c.Inpart(b),themarketriskpremiumisexpectedtobelowerthaninpart(a)andmarketriskishigher.Theref

23、ore,thereward-to-volatilityratioisexpectedtobelowerinpart(b),whichexplainsthegreaterproportioninvestedinT-bills.6-621.a.E(rC)=8%=5%+yx(11%5%)ny-.08.05-0.5C.11.05b.o=yxo=0.50 x15%=7.5%CPc.Thefirstclientismoreriskaverse,preferringinvestmentsthathavelessriskasevidencedbythelowerstandarddeviation.Johnso

24、nrequeststheportfoliostandarddeviationtoequalonehalfthemarketportfoliostandarddeviation.Themarketportfolioc二20%,whichimpliesMc二10%.TheinterceptoftheCMLequalsr二0.05andtheslopeoftheCMLPfequalstheSharperatioforthemarketportfolio(35%).ThereforeusingtheCML:E(r)rE(r)二r+mfc二0.05+0.35x0.10二0.085二8.5%PfcPMDa

25、ta:r.=5%,E(r)=13%,=25%,andrB=9%fMMfTheCMLandindifferencecurvesareasfollows:Forytobelessthan1.0(thattheinvestorisalender),riskaversion(A)mustbelargeenoughsuchthat:E(r)r0.130.05y二m匚二1.28Ao20.252MForytobegreaterthan1(theinvestorisaborrower),Amustbesmallenough:0.130.090.2520.130.090.252二0.64y二Mf1nAAo2M6

26、-7Forvaluesofriskaversionwithinthisrange,theclientwillneitherborrownorlendbutwillholdaportfoliocomposedonlyoftheoptimalriskyportfolio:y=1for0.64A1.2825.a.ThegraphforProblem23hastoberedrawnhere,with:Foraborrowingposition:A二0.890.1125.a.ThegraphforProblem23hastoberedrawnhere,with:Foraborrowingposition

27、:A二0.890.152Therefore,y=1for0.89A1,theborrowingrate,9%,istherelevantrisk-freerate.Thenwenoticethat,evenwithoutafee,theactivefundisinferiortothepassivefundbecause:11-09f13-09-=0.13-=0.16f=.004.15.25Morerisktolerantinvestors(whoaremoreinclinedtoborrow)willnotbeclientsofthefund.Wefindthatfisnegative:th

28、atis,youwouldneedtopayinvestorstochooseyouractivefund.Theseinvestorsdesirehigherrisk-higherreturncomplete6-8portfoliosandthusareintheborrowingrangeoftherelevantCAL.Inthisrange,thereward-to-variabilityratiooftheindex(thepassivefund)isbetterthanthatofthemanagedfund.13_0827.a.SlopeoftheCML=0.20Thediagr

29、amfollows.nurtKHdeeDxThediagramfollows.nurtKHdeeDxb.Myfundallowsaninvestortoachieveahighermeanforanygivenstandarddeviationthanwouldapassivestrategy,i.e.,ahigherexpectedreturnforanygivenlevelofrisk.28.a.With70%ofhismoneyinvestedinmyfundsportfolio,theclientsexpectedreturnis15%peryearwithastandarddevia

30、tionof19.6%peryear.Ifheshiftsthatmoneytothepassiveportfolio(whichhasanexpectedreturnof13%andstandarddeviationof25%),hisoverallexpectedreturnbecomes:E(rC)=rf+0.7xE(rM)rf=.08+0.7x(.13-.08)=.115,or11.5%CfMfThestandarddeviationofthecompleteportfoliousingthepassiveportfoliowouldbe:oc=0.7x=0.7x25%=17.5%Th

31、erefore,theshiftentailsadecreaseinmeanfrom15%to11.5%andadecreaseinstandarddeviationfrom19.6%to17.5%.Sincebothmeanreturnandstandarddeviationdecrease,itisnotyetclearwhetherthemoveisbeneficial.Thedisadvantageoftheshiftisthat,iftheclientiswillingtoacceptameanreturnonhistotalportfolioof11.5%,hecanachieve

32、itwithalowerstandarddeviationusingmyfundratherthanthepassiveportfolio.Toachieveatargetmeanof11.5%,wefirstwritethemeanofthecomplete6-9portfolioasafunctionoftheproportioninvestedinmyfund(y):E(rC)=.08+yx(.18-.08)=.08+.10 xyOurtargetis:E(rC)=11.5%.Therefore,theproportionthatmustbeinvestedinmyfundisdeter

33、minedasfollows:11508.115=.08+.10 xyny-0.35.10Thestandarddeviationofthisportfoliowouldbe:oc=yx28%=0.35x28%=9.8%Thus,byusingmyportfolio,thesame11.5%expectedreturncanbeachievedwithastandarddeviationofonly9.8%asopposedtothestandarddeviationof17.5%usingthepassiveportfolio.b.Thefeewouldreducethereward-to-

34、volatilityratio,i.e.,theslopeoftheCAL.Theclientwillbeindifferentbetweenmyfundandthepassiveportfolioiftheslopeoftheafter-feeCALandtheCMLareequal.Letfdenotethefee:.18.08f.10fSlopeofCALwithfee-.28.281308SlopeofCML(whichrequiresnofee)-去-0.20Settingtheseslopesequalwehave:1；8f-0.20nf0.044-4.4%peryear29.a.

35、Theformulafortheoptimalproportiontoinvestinthepassiveportfoliois:E(r)ry*-mfA02MSubstitutethefollowing:E(rM)=13%;=8%;oM=25%;A=3.5:013008y*-0.2286,or22.86%inthepassiveportfolio3.5x0.2526-10b.TheanswerhereisthesameastheanswertoProblem28(b).Thefeethatyoucanchargeaclientisthesameregardlessoftheassetalloc

36、ationmixoftheclientsportfolio.Youcanchargeafeethatwillequatethereward-to-volatilityratioofyourportfoliotothatofyourcompetition.CFAPROBLEMSUtilityforeachinvestment=E(r)0.5x4xo2Wechoosetheinvestmentwiththehighestutilityvalue,Investment3.InvestmentExpectedreturnE(r)StandarddeviationGUtilityU10.120.30-0

37、.060020.150.50-0.3500588518Wheninvestorsareriskneutral,thenA=0;theinvestmentwiththehighestutilityisInvestment4becauseithasthehighestexpectedreturn.(b)Indifferencecurve2becauseitistangenttotheCAL.PointE(0.6x$50,000)+0.4x($30,000)-$5,000=$13,000(b)Higherborrowingrateswillreduce

38、thetotalreturntotheportfolioandthisresultsinapartofthelinethathasalowerslope.Expectedreturnforequityfund=T-billrate+Riskpremium=6%+10%=16%Expectedrateofreturnoftheclientsportfolio=(0.6x16%)+(0.4x6%)=12%Expectedreturnoftheclientsportfolio=0.12x$100,000=$12,000(whichimpliesexpectedtotalwealthattheendo

39、ftheperiod=$112,000)Standarddeviationofclientsoverallportfolio=0.6x14%=8.4%6-119.Reward-to-volatilityratio=9.CHAPTER6:APPENDIXByyear-end,the$50,000investmentwillgrowto:$50,000 x1.06=$53,000Withoutinsurance,theprobabilitydistributionofend-of-yearwealthis:ProbabilityWealthNofire0.999$253,000Fire0.0015

40、3,000Forthisdistribution,expectedutilityiscomputedasfollows:EU(W)=0.999xln(253,000)+0.001xln(53,000)=12.439582Thecertaintyequivalentis:W=e12.439582=$252,604.85CEWithfireinsurance,atacostof$P,theinvestmentintherisk-freeassetis:$(50,000P)Year-endwealthwillbecertain(sinceyouarefullyinsured)andequalto:$

41、(50,000P)x1.06+$200,000SolveforPinthefollowingequation:$(50,000P)x1.06+$200,000=$252,604.85nP=$372.78Thisisthemostyouarewillingtopayforinsurance.Notethattheexpectedlossis“only”$200,soyouarewillingtopayasubstantialriskpremiumovertheexpectedvalueoflosses.Theprimaryreasonisthatthevalueofthehouseisalargeproportionofyourwealth.a.