投资学_第七版_英文版_(Zvi Bodie)_课后答案 140页

'课后答案网（http://www.khdaw.com）CHAPTER2:ASSETCLASSESANDFINANCIALINSTRUMENTS1.(d)2.Theequivalenttaxableyieldis:6.75%/(1−0.34)=10.23%3.(a)Writingacallentailsunlimitedpotentiallossesasthestockpricerises.4.a.Youwouldhavetopaytheaskedpriceof:94:30=94.9375%ofpar=$949.375b.Thecouponrateis3.625%implyingcouponpaymentsof$36.25annuallyor,moreprecisely,$18.125semiannually.c.Currentyield=(Annualcouponincome/price)=$36.25/$949.375=0.0382=3.82%5.Preferredstockislikelong-termdebtinthatittypicallypromisesafixedpaymenteachyear.Inthisway,itisaperpetuity.Preferredstockisalsolikelong-termdebtinthatitdoesnotgivetheholdervotingrightsinthefirm.Preferredstockislikeequityinthatthefirmisundernocontractualobligationtomakethepreferredstockdividendpayments.Failuretomakepaymentsdoesnotsetoffcorporatebankruptcy.Withrespecttothepriorityofclaimstotheassetsofthefirmintheeventofcorporatebankruptcy,preferredstockhasahigherprioritythancommonequitybutalowerprioritythanbonds.6.Moneymarketsecuritiesarecalled“cashequivalents”becauseoftheirgreatliquidity.Thepricesofmoneymarketsecuritiesareverystable,andtheycanbeconvertedtocash(i.e.,sold)onveryshortnoticeandwithverylowtransactioncosts.7.P=$10,000/1.02=$9,803.922- 课后答案网（http://www.khdaw.com）8.Thetotalbefore-taxincomeis$4.Afterthe70%exclusionforpreferredstockdividends,thetaxableincomeis:0.30×$4=$1.20Therefore,taxesare:0.30×$1.20=$0.36After-taxincomeis:$4.00–$0.36=$3.64Rateofreturnis:$3.64/$40.00=9.10%9.a.GeneralDynamicsclosedtodayat$112.65,whichwas$1.56higherthanyesterday’sprice.Yesterday’sclosingpricewas:$111.09b.Youcouldbuy:$5,000/$112.65=44.4sharesc.Yourannualdividendincomewouldbe:44.4×$1.60=$71.04d.Theprice-to-earningsratiois16andthepriceis$112.65.Therefore:$112.65/Earningspershare=16⇒Earningspershare=$7.0410.a.Att=0,thevalueoftheindexis:(90+50+100)/3=80Att=1,thevalueoftheindexis:(95+45+110)/3=83.333Therateofreturnis:(83.333/80)−1=4.17%b.Intheabsenceofasplit,StockCwouldsellfor110,sothevalueoftheindexwouldbe:250/3=83.333Afterthesplit,StockCsellsfor55.Therefore,weneedtofindthedivisor(d)suchthat:83.333=(95+45+55)/d⇒d=2.340c.Thereturniszero.Theindexremainsunchangedbecausethereturnforeachstockseparatelyequalszero.11.a.Totalmarketvalueatt=0is:($9,000+$10,000+$20,000)=$39,000Totalmarketvalueatt=1is:($9,500+$9,000+$22,000)=$40,500Rateofreturn=($40,500/$39,000)–1=3.85%b.Thereturnoneachstockisasfollows:rA=(95/90)–1=0.0556rB=(45/50)–1=–0.10rC=(110/100)–1=0.10Theequally-weightedaverageis:[0.0556+(-0.10)+0.10]/3=0.0185=1.85%2- 课后答案网（http://www.khdaw.com）12.Theafter-taxyieldonthecorporatebondsis:0.09×(1–0.30)=0.0630=6.30%Therefore,municipalsmustofferatleast6.30%yields.13.a.Thetaxablebond.Withazerotaxbracket,theafter-taxyieldforthetaxablebondisthesameasthebefore-taxyield(5%),whichisgreaterthantheyieldonthemunicipalbond.b.Thetaxablebond.Theafter-taxyieldforthetaxablebondis:0.05×(1–0.10)=4.5%c.Youareindifferent.Theafter-taxyieldforthetaxablebondis:0.05×(1–0.20)=4.0%Theafter-taxyieldisthesameasthatofthemunicipalbond.d.Themunicipalbondoffersthehigherafter-taxyieldforinvestorsintaxbracketsabove20%.14.Equation(2.5)showsthattheequivalenttaxableyieldis:r=rm/(1–t)a.4.00%b.4.44%c.5.00%d.5.71%15.Iftheafter-taxyieldsareequal,then:0.056=0.08×(1–t)Thisimpliesthatt=0.30=30%.16.a.Thehighercouponbond.b.Thecallwiththelowerexerciseprice.c.Theputonthelowerpricedstock.2- 课后答案网（http://www.khdaw.com）17.a.Youboughtthecontractwhenthefuturespricewas1687.00(seeFigure2.12).Thecontractclosesatapriceof1700,whichis13.0higherthantheoriginalfuturesprice.Thecontractmultiplieris$100.Therefore,theprofitwillbe:13.0×$100=$1,300b.Openinterestis47,990contracts.18.a.Sincethestockpriceexceedstheexerciseprice,youwillexercisethecall.Thepayoffontheoptionwillbe:$37−$35=$2Theoptionoriginallycost$1.97,sotheprofitis:$2.00−$1.97=$0.03Rateofreturn=$0.03/$1.97=0.0152=1.52%b.Iftheexercisepricewere$32.50,youwouldexercisethecall.Thepayoffontheoptionwillbe:$37.00−$32.50=$4.50Theoptionoriginallycost$3.70sotheprofitis:$4.50−$3.70=$0.80Rateofreturn=$0.80/$3.70=0.2162=21.62%c.Iftheputhasanexercisepriceof$35,youwouldnotexerciseforanystockpriceof$35orabove.Thelossontheputwouldbetheinitialcost,$1.30.19.Thereisalwaysapossibilitythattheoptionwillbein-the-moneyatsometimepriortoexpiration.Investorswillpaysomethingforthispossibilityofapositivepayoff.20.ValueofcallatexpirationInitialCostProfita.08-8b.08-8c.08-8d.1082e.20812ValueofputatexpirationInitialCostProfita.20128b.1012-2c.012-12d.012-12e.012-122- 课后答案网（http://www.khdaw.com）21.Aputoptionconveystherighttoselltheunderlyingassetattheexerciseprice.Ashortpositioninafuturescontractcarriesanobligationtoselltheunderlyingassetatthefuturesprice.22.Acalloptionconveystherighttobuytheunderlyingassetattheexerciseprice.Alongpositioninafuturescontractcarriesanobligationtobuytheunderlyingassetatthefuturesprice.23.Thespreadwillwiden.Deteriorationoftheeconomyincreasescreditrisk,thatis,thelikelihoodofdefault.Investorswilldemandagreaterpremiumondebtsecuritiessubjecttodefaultrisk.24.Onthedaywetriedthisexperiment,36ofthe50stocksmetthiscriterion,leadingustoconcludethatreturnsonstockinvestmentscanbequitevolatile.CHAPTER3:HOWSECURITIESARETRADED1.a.Inadditiontotheexplicitfeesof$70,000,FBNappearstohavepaidanimplicitpriceinunderpricingoftheIPO.Theunderpricingis$3pershare,oratotalof$300,000,implyingtotalcostsof$370,000.b.No.Theunderwritersdonotcapturethepartofthecostscorrespondingtotheunderpricing.Theunderpricingmaybearationalmarketingstrategy.Withoutit,theunderwriterswouldneedtospendmoreresourcesinordertoplacetheissuewiththepublic.Theunderwriterswouldthenneedtochargehigherexplicitfeestotheissuingfirm.Theissuingfirmmaybejustaswelloffpayingtheimplicitissuancecostrepresentedbytheunderpricing.2.a.Inprinciple,potentiallossesareunbounded,growingdirectlywithincreasesinthepriceofHarrierGroup.b.Ifthestop-buyordercanbefilledat£17.50,themaximumpossiblelosspershareis£1.50.IfthepriceofHarrierGroupsharesgoesabove£17.50,thenthestop-buyorderwouldbeexecuted,limitingthelossesfromtheshortsale.3.ThebrokerisinstructedtoattempttosellyourMarks&SpencerstockassoonastheMarks&Spencerstocktradesatabidpriceof£7.03orless.Here,thebrokerwillattempttoexecute,butmaynotbeabletosellat£7.03,sincethebid2- 课后答案网（http://www.khdaw.com）priceisnow£7.00.Thepriceatwhichyousellmaybemoreorlessthan£7.03becausethestop-lossbecomesamarketordertosellatcurrentmarketprices.4.a.€55.50b.€55.25c.Thetradewillnotbeexecutedbecausethebidpriceislowerthanthepricespecifiedinthelimitsellorder.d.Thetradewillnotbeexecutedbecausetheaskedpriceisgreaterthanthepricespecifiedinthelimitbuyorder.2- 课后答案网（http://www.khdaw.com）5.a.Inaspecialistmarket,therecanbepriceimprovementinthetwomarketorders.Brokersforeachofthemarketorders(i.e.,thebuyandthesellorders)canagreetoexecuteatradeinsidethequotedspread.Forexample,theycantradeat€55.37,thusimprovingthepriceforbothcustomersby€0.12or€0.13relativetothequotedbidandaskedprices.Thebuyergetsthestockfor€0.13lessthanthequotedaskedprice,andthesellerreceives€0.12moreforthestockthanthequotedbidprice.b.Whereasthelimitordertobuyat€55.30wouldnotbeexecutedinadealermarket(sincetheaskedpriceis€55.50),itcouldbeexecutedinaspecialistmarket.Abrokerforanothercustomerwithanordertosellatmarketwouldviewthelimitbuyorderasthebestbidprice;thetwobrokerscouldagreetothetradeandbringittothespecialist,whowouldthenexecutethetrade.6.TheSuperDotsystemexpeditestheflowofordersfromexchangememberstothespecialists.Itallowsmemberstosendcomputerizedordersdirectlytotheflooroftheexchange,whichallowsthenearlysimultaneoussaleofeachstockinalargeportfolio.Thiscapabilityisnecessaryforprogramtrading.7.a.Youbuy200sharesofTelecomfor€8,000.Thesesharesincreaseinvalueby10%,or€800.Youpayinterestof:0.08×€4,000=€320Therateofreturnwillbe:800−320=0.12=12%4,000b.Thevalueofthe200sharesis200P.Equityis(200P–€4,000).Youwillreceiveamargincallwhen:200P−4,000=0.30⇒whenP=€28.57orlower200P8.a.Initialmarginis50%of€4,000or€2,000.b.Totalassetsare€6,000(€4,000fromthesaleofthestockand€2,000putupformargin).Liabilitiesare100P.Therefore,networthis(€6,000–100P).Amargincallwillbeissuedwhen:6,000−100P=0.30⇒whenP=€46.15orhigher100P2- 课后答案网（http://www.khdaw.com）9.Thetotalcostofthepurchaseis:$40×500=$20,000Youborrow$5,000fromyourbroker,andinvest$15,000ofyourownfunds.Yourmarginaccountstartsoutwithnetworthof$15,000.a.(i)Networthincreasesto:($44×500)–$5,000=$17,000Percentagegain=$2,000/$15,000=0.1333=13.33%(ii)Withpriceunchanged,networthisunchanged.Percentagegain=zero(iii)Networthfallsto($36×500)–$5,000=$13,000Percentagegain=(–$2,000/$15,000)=–0.1333=–13.33%Therelationshipbetweenthepercentagereturnandthepercentagechangeinthepriceofthestockisgivenby:Totalinvestment%return=%changeinprice×=%changeinprice×1.333Investor"sinitialequityForexample,whenthestockpricerisesfrom$40to$44,thepercentagechangeinpriceis10%,whilethepercentagegainfortheinvestoris:$20,000%return=10%×=13.33%$15,000b.Thevalueofthe500sharesis500P.Equityis(500P–$5,000).Youwillreceiveamargincallwhen:500P−$5,000=0.25⇒whenP=$13.33orlower500Pc.Thevalueofthe500sharesis500P.Butnowyouhaveborrowed$10,000insteadof$5,000.Therefore,equityis(500P–$10,000).Youwillreceiveamargincallwhen:500P−$10,000=0.25⇒whenP=$26.67500PWithlessequityintheaccount,youarefarmorevulnerabletoamargincall.2- 课后答案网（http://www.khdaw.com）d.Bytheendoftheyear,theamountoftheloanowedtothebrokergrowsto:$5,000×1.08=$5,400Theequityinyouraccountis(500P–$5,400).Initialequitywas$15,000.Therefore,yourrateofreturnafteroneyearisasfollows:(500×$44)−$5,400−$15,000(i)=0.1067=10.67%$15,000(500×$40)−$5,400−$15,000(ii)=–0.0267=–2.67%$15,000(500×$36)−$5,400−$15,000(iii)=–0.1600=–16.00%$15,000TherelationshipbetweenthepercentagereturnandthepercentagechangeinthepriceofIntelisgivenby:%return=⎛Totalinvestment⎞⎛Fundsborrowed⎞⎜⎜%changeinprice×⎟⎟−⎜⎜8%×⎟⎟⎝Investor"sinitialequity⎠⎝Investor"sinitialequity⎠Forexample,whenthestockpricerisesfrom$40to$44,thepercentagechangeinpriceis10%,whilethepercentagegainfortheinvestoris:⎛$20,000⎞⎛$5,000⎞⎜10%×⎟−⎜8%×⎟=10.67%⎝$15,000⎠⎝$15,000⎠e.Thevalueofthe500sharesis500P.Equityis(500P–$5,400).Youwillreceiveamargincallwhen:500P−$5,400=0.25⇒whenP=$14.40orlower500P10.a.Thegainorlossontheshortpositionis:(–500×∆P)Investedfunds=$15,000Therefore:rateofreturn=(–500×∆P)/15,000Therateofreturnineachofthethreescenariosis:(i)rateofreturn=(–500×$4)/$15,000=–0.1333=–13.33%(ii)rateofreturn=(–500×$0)/$15,000=0%(iii)rateofreturn=[–500×(–$4)]/$15,000=+0.1333=+13.33%2- 课后答案网（http://www.khdaw.com）b.Totalassetsinthemarginaccountare:$20,000(fromthesaleofthestock)+$15,000(theinitialmargin)=$35,000Liabilitiesare500P.Amargincallwillbeissuedwhen:$35,000−500P=0.25⇒whenP=$56orhigher500Pc.Witha$1dividend,theshortpositionmustnowpayontheborrowedshares:($1/share×500shares)=$500.Rateofreturnisnow:[(–500×∆P)–500]/15,000(i)rateofreturn=[(–500×$4)–$500]/$15,000=–0.1667=–16.67%(ii)rateofreturn=[(–500×$0)–$500]/$15,000=–0.0333=–3.33%(iii)rateofreturn=[(–500)×(–$4)–$500]/$15,000=+0.1000=+10.00%Totalassetsare$35,000,andliabilitiesare(500P+500).Amargincallwillbeissuedwhen:35,000−500P−500=0.25⇒whenP=$55.20orhigher500P11.a.Thestockispurchasedfor:300×¥4,000=¥1,200,000Theamountborrowedis¥400,000.Therefore,theinvestorputupequity,ormargin,of¥800,000.b.Ifthesharepricefallsto¥3,000,thenthevalueofthestockfallsto¥900,000.Bytheendoftheyear,theamountoftheloanowedtothebrokergrowsto:¥400,000×1.08=¥432,000Therefore,theremainingmarginintheinvestor’saccountis:¥900,000−¥432,000=¥468,000Thepercentagemarginisnow:¥468,000/¥900,000=0.52=52%Therefore,theinvestorwillnotreceiveamargincall.c.Therateofreturnontheinvestmentovertheyearis:(Endingequityintheaccount−Initialequity)/Initialequity=(¥468,000−¥800,000)/¥800,000=−0.415=−41.5%2- 课后答案网（http://www.khdaw.com）12.a.Theinitialmarginwas:0.50×1,000×¥4,000=¥2,000,000AsaresultoftheincreaseinthestockpriceOldEconomyTradersloses:¥1,000×1,000=¥1,000,000Therefore,margindecreasesby¥1,000,000.Moreover,OldEconomyTradersmustpaythedividendof¥200persharetothelenderoftheshares,sothatthemarginintheaccountdecreasesbyanadditional¥200,000.Therefore,theremainingmarginis:¥2,000,000–¥1,000,000–¥200,000=¥800,000b.Thepercentagemarginis:¥800,000/¥5,000,000=0.16=16%Sotherewillbeamargincall.c.Theequityintheaccountdecreasedfrom¥2,000,000to¥800,000inoneyear,forarateofreturnof:(−¥1,200,000/¥2,000,000)=−0.60=−60%13.Muchofwhatthespecialistdoes(e.g.,crossingordersandmaintainingthelimitorderbook)canbeaccomplishedbyacomputerizedsystem.Infact,someexchangesuseanautomatedsystemfornighttrading.Amoredifficultissuetoresolveiswhetherthemorediscretionaryactivitiesofspecialistsinvolvingtradingfortheirownaccounts(e.g.,maintaininganorderlymarket)canbereplicatedbyacomputersystem.14.a.Thebuyorderwillbefilledatthebestlimit-sellorderprice:$50.25b.Thenextmarketbuyorderwillbefilledatthenext-bestlimit-sellorderprice:$51.50c.Youwouldwanttoincreaseyourinventory.Thereisconsiderablebuyingdemandatpricesjustbelow$50,indicatingthatdownsideriskislimited.Incontrast,limitsellordersaresparse,indicatingthatamoderatebuyordercouldresultinasubstantialpriceincrease.15.a.Youwillnotreceiveamargincall.Youborrowed$20,000andwithanother$20,000ofyourownequityyoubought1,000sharesofDisneyat$40pershare.At$35pershare,themarketvalueofthestockis$35,000,yourequityis$15,000,andthepercentagemarginis:$15,000/$35,000=42.9%Yourpercentagemarginexceedstherequiredmaintenancemargin.b.Youwillreceiveamargincallwhen:2- 课后答案网（http://www.khdaw.com）1,000P−$20,000=0.35⇒whenP=$30.77orlower1,000P2- 课后答案网（http://www.khdaw.com）16.Thedealersetsthebidandaskedprice.Spreadsshouldbehigheroninactivelytradedstocksandloweronactivelytradedstocks.17.Answerstothisproblemwillvary.18.a.Overshortperiodsoftime,thepriceofanexchangemembershipgenerallyincreaseswithincreasesintradingactivity.Thismakessensebecausetradingcommissionsdependontradingvolume.b.Thepriceofanexchangemembershiphasrisenfarlessinpercentagetermsthantradingvolume.Thissuggeststhatthecommissionschargedtotraderson"typical"tradeshavefallenovertime.19.Theproceedsfromtheshortsale(netofcommission)were:(C$14×100)–C$50=C$1,350AdividendpaymentofC$200waswithdrawnfromtheaccount.CoveringtheshortsaleatC$9persharecostyou(includingcommission):C$900+C$50=C$950Therefore,thevalueofyouraccountisequaltothenetprofitonthetransaction:C$1350–C$200–C$950=C$200Notethatyourprofit(C$200)equals(100shares×profitpershareofC$2).Yournetproceedspersharewas:C$14sellingpriceofstock–C$9repurchasepriceofstock–C$2dividendpershare–C$12trades×C$0.50commissionpershareC$220.(d)Thebrokerwillsell,atcurrentmarketprice,afterthefirsttransactionat$55orless.21.(b)22.(d)CHAPTER4:MUTUALFUNDSANDOTHERINVESTMENTCOMPANIES2- 课后答案网（http://www.khdaw.com）1.a.Unitinvestmenttrusts:diversificationfromlarge-scaleinvesting,lowertransactioncostsassociatedwithlarge-scaletrading,lowmanagementfees,predictableportfoliocomposition,guaranteedlowportfolioturnoverrate.b.Open-endfunds:diversificationfromlarge-scaleinvesting,lowertransactioncostsassociatedwithlarge-scaletrading,professionalmanagementthatmaybeabletotakeadvantageofbuyorsellopportunitiesastheyarise,recordkeeping.c.Individualstocksandbonds:Nomanagementfee,realizationofcapitalgainsorlossescanbecoordinatedwithinvestor’spersonaltaxsituation,portfoliocanbedesignedtoinvestor’sspecificriskprofile.2.Balancedfundskeeprelativelystableproportionsoffundsinvestedineachassetclass.Theyaremeantasconvenientinstrumentstoprovideparticipationinarangeofassetclasses.Life-cyclefundsarebalancedfundswhoseassetmixgenerallydependsontheageoftheinvestor.Aggressivelife-cyclefunds,withlargerinvestmentsinequities,aremarketedtoyoungerinvestors,whileconservativelife-cyclefunds,withlargerinvestmentsinfixed-incomesecurities,aredesignedforolderinvestors.Assetallocationfunds,incontrast,mayvarytheproportionsinvestedineachassetclassbylargeamountsaspredictionsofrelativeperformanceacrossclassesvary.Assetallocationfundsthereforeengageinmoreaggressivemarkettiming.3.Open-endfundsareobligatedtoredeeminvestor"ssharesatnetassetvalue,andthusmustkeepcashorcash-equivalentsecuritiesonhandinordertomeetpotentialredemptions.Closed-endfundsdonotneedthecashreservesbecausetherearenoredemptionsforclosed-endfunds.Investorsinclosed-endfundsselltheirshareswhentheywishtocashout.4.Theunitinvestmenttrustshouldhaveloweroperatingexpenses.Becausetheinvestmenttrustportfolioisfixedoncethetrustisestablished,itdoesnothavetopayportfoliomanagerstoconstantlymonitorandrebalancetheportfolioasperceivedneedsoropportunitieschange.Becausetheportfolioisfixed,theunitinvestmenttrustalsoincursvirtuallynotradingcosts.2- 课后答案网（http://www.khdaw.com）5.Theofferingpriceincludesa6%front-endload,orsalescommission,meaningthateverydollarpaidresultsinonly€0.94goingtowardpurchaseofshares.Therefore:NAV12.50Offeringprice===€13.301−load1−0.066.NAV=offeringprice×(1–load)=SFr14.50×0.95=SFr13.787.StockValueheldbyfundA$7,000,000B12,000,000C8,000,000D15,000,000Total$42,000,000$42,000,000−$30,000Netassetvalue==$10.494,000,0008.Valueofstockssoldandreplaced=$15,000,000$15,000,000Turnoverrate==0.357=35.7%$42,000,000200,000,000,000−3,000,000,0009.a.NAV==¥3,94050,000,000Price−NAV3,600−3,940b.Premium(ordiscount)===–0.086=-8.6%NAV3,940Thefundsellsatan8.6%discountfromNAV.10.Rateofreturn=NAV1−NAV0+distributions$12.10−$12.50+$1.50==0.088=8.8%NAV$12.5002- 课后答案网（http://www.khdaw.com）11.a.Start-of-yearprice:P0=£12.00×1.02=£12.24End-of-yearprice:P1=£12.10×0.93=£11.25AlthoughNAVincreasedby£0.10,thepriceofthefunddecreasedby£0.99.P1−P0+Distributions11.25−12.24+1.50Rateofreturn===0.042=4.2%P12.240b.AninvestorholdingthesameportfolioasthemanagerwouldhaveearnedarateofreturnbasedontheincreaseintheNAVoftheportfolio:Rateofreturn=NAV1−NAV0+distributions$12.10−$12.00+$1.50==0.133=13.3%NAV$12.00012.a.Empiricalresearchindicatesthatpastperformanceofmutualfundsisnothighlypredictiveoffutureperformance,especiallyforbetter-performingfunds.Whiletheremaybesometendencyforthefundtobeanaboveaverageperformernextyear,itisunlikelytoonceagainbeatop10%performer.b.Ontheotherhand,theevidenceismoresuggestiveofatendencyforpoorperformancetopersist.Thistendencyisprobablyrelatedtofundcostsandturnoverrates.Thusifthefundisamongthepoorestperformers,investorswouldbeconcernedthatthepoorperformancewillpersist.13.NAV0=€200,000,000/10,000,000=€20Dividendspershare=€2,000,000/10,000,000=€0.20NAV1isbasedonthe8%pricegain,lessthe1%12b-1fee:NAV1=€20×1.08×(1–0.01)=€21.38421.384−20+0.20Rateofreturn==0.0792=7.92%2014.Theexcessofpurchasesoversalesmustbeduetonewinflowsintothefund.Therefore,$400millionofstockpreviouslyheldbythefundwasreplacedbynewholdings.Soturnoveris:$400/$2,200=0.182=18.2%15.Feespaidtoinvestmentmanagerswere:0.007×$2.2billion=$15.4million2- 课后答案网（http://www.khdaw.com）Sincethetotalexpenseratiowas1.1%andthemanagementfeewas0.7%,weconcludethat0.4%mustbeforotherexpenses.Therefore,otheradministrativeexpenseswere:0.004×$2.2billion=$8.8million2- 课后答案网（http://www.khdaw.com）16.Asaninitialapproximation,yourreturnequalsthereturnonthesharesminusthetotaloftheexpenseratioandpurchasecosts:12%−1.2%−4%=6.8%Buttheprecisereturnislessthanthisbecausethe4%loadispaidupfront,notattheendoftheyear.Topurchasetheshares,youwouldhavehadtoinvest:$20,000/(1−0.04)=$20,833Thesharesincreaseinvaluefrom$20,000to:$20,000×(1.12−0.012)=$22,160Therateofreturnis:($22,160−$20,833)/$20,833=6.37%17.a.Aftertwoyears,eachdollarinvestedinafundwitha4%loadandaportfolioreturnequaltorwillgrowto:$0.96×(1+r–0.005)2EachdollarinvestedinthebankCDwillgrowto:$1×1.062Ifthemutualfundistobethebetterinvestment,thentheportfolioreturn(r)mustsatisfy:0.96×(1+r–0.005)2>1.0620.96×(1+r–0.005)2>1.1236(1+r–0.005)2>1.17041+r–0.005>1.08191+r>1.0869Therefore:r>0.0869=8.69%b.Ifyouinvestforsixyears,thentheportfolioreturnmustsatisfy:0.96×(1+r–0.005)6>1.066=1.4185(1+r–0.005)6>1.47761+r–0.005>1.06721+r>1.0722r>7.22%Thecutoffrateofreturnislowerforthesix-yearinvestmentbecausethe“fixedcost”(i.e.,theone-timefront-endload)isspreadoutoveragreaternumberofyears.c.Witha12b-1feeinsteadofafront-endload,theportfoliomustearnarateofreturn(r)thatsatisfies:1+r–0.005–0.0075>1.06Inthiscase,rmustexceed7.25%regardlessoftheinvestmenthorizon.2- 课后答案网（http://www.khdaw.com）18.Theturnoverrateis50%.Thismeansthat,onaverage,50%oftheportfolioissoldandreplacedwithothersecuritieseachyear.Tradingcostsonthesellordersare0.4%;andthebuyorderstoreplacethosesecuritiesentailanother0.4%intradingcosts.Totaltradingcostswillreduceportfolioreturnsby:2×0.4%×0.50=0.4%19.Forthebondfund,thefractionofportfolioincomegivenuptofeesis:0.6%=0.150=15.0%4.0%Fortheequityfund,thefractionofinvestmentearningsgivenuptofeesis:0.6%=0.050=5.0%12.0%Feesareamuchhigherfractionofexpectedearningsforthebondfund,andthereforemaybeamoreimportantfactorinselectingthebondfund.Thismayhelptoexplainwhyunmanagedunitinvestmenttrustsareconcentratedinthefixedincomemarket.Theadvantagesofunitinvestmenttrustsarelowturnoverandlowtradingcostsandmanagementfees.Thisisamoreimportantconcerntobond-marketinvestors.20.Supposeyouhave$1,000toinvest.TheinitialinvestmentinClassAsharesis$940netofthefront-endload.Afterfouryears,yourportfoliowillbeworth:$940×(1.10)4=$1,376.25ClassBsharesallowyoutoinvestthefull$1,000,butyourinvestmentperformancenetof12b-1feeswillbeonly9.5%,andyouwillpaya1%back-endloadfeeifyousellafterfouryears.Yourportfoliovalueafterfouryearswillbe:$1,000×(1.095)4=$1,437.66Afterpayingtheback-endloadfee,yourportfoliovaluewillbe:$1,437.66×0.99=$1,423.28ClassBsharesarethebetterchoiceifyourhorizonisfouryears.Withafifteen-yearhorizon,theClassAshareswillbeworth:$940×(1.10)15=$3,926.61FortheClassBshares,thereisnoback-endloadinthiscasesincethehorizonisgreaterthanfiveyears.Therefore,thevalueoftheClassBshareswillbe:$1,000×(1.095)15=$3,901.322- 课后答案网（http://www.khdaw.com）Atthislongerhorizon,ClassBsharesarenolongerthebetterchoice.TheeffectofClassB"s0.5%12b-1feesaccumulatesovertimeandfinallyoverwhelmsthe6%loadchargedtoClassAinvestors.2- 课后答案网（http://www.khdaw.com）21.Supposethatfinishinginthetophalfofallportfoliomanagersispurelyluck,andthattheprobabilityofdoingsoinanyyearisexactly½.Thentheprobabilitythatanyparticularmanagerwouldfinishinthetophalfofthesamplefiveyearsinarowis(½)5=1/32.Wewouldthenexpecttofindthat[350×(1/32)]=11managersfinishinthetophalfforeachofthefiveconsecutiveyears.Thisispreciselywhatwefound.Thus,weshouldnotconcludethattheconsistentperformanceafterfiveyearsisproofofskill.Wewouldexpecttofindelevenmanagersexhibitingpreciselythislevelof"consistency"evenifperformanceisduesolelytoluck.CHAPTER5:LEARNINGABOUTRETURNANDRISKFROMTHEHISTORICALRECORD1.a.The“Inflation-Plus”CDisthesaferinvestmentbecauseitguaranteesthepurchasingpoweroftheinvestment.Usingtheapproximationthattherealrateequalsthenominalrateminustheinflationrate,theCDprovidesarealrateof3.5%regardlessoftheinflationrate.b.Theexpectedreturndependsontheexpectedrateofinflationoverthenextyear.Iftheexpectedrateofinflationislessthan3.5%thentheconventionalCDoffersahigherrealreturnthantheInflation-PlusCD;iftheexpectedrateofinflationisgreaterthan3.5%,thentheoppositeistrue.c.Ifyouexpecttherateofinflationtobe3%overthenextyear,thentheconventionalCDoffersyouanexpectedrealrateofreturnof4%,whichis0.5%higherthantherealrateontheinflation-protectedCD.Butunlessyouknowthatinflationwillbe3%withcertainty,theconventionalCDisalsoriskier.Thequestionofwhichisthebetterinvestmentthendependsonyourattitudetowardsriskversusreturn.Youmightchoosetodiversifyandinvestpartofyourfundsineach.d.No.Wecannotassumethattheentiredifferencebetweentherisk-freenominalrate(onconventionalCDs)of7%andtherealrisk-freerate(oninflation-protectedCDs)of3.5%istheexpectedrateofinflation.PartofthedifferenceisprobablyariskpremiumassociatedwiththeuncertaintysurroundingtherealrateofreturnontheconventionalCDs.Thisimpliesthattheexpectedrateofinflationislessthan3.5%peryear.2.FromTable5.3,theaverageriskpremiumforlarge-capitalizationU.S.stocksfortheperiod1926-2005was:(12.15%−3.75%)=8.40%peryearAdding8.40%tothe6%risk-freeinterestrate,theexpectedannualHPRfortheS&P500stockportfoliois:6.00%+8.40%=14.40%2- 课后答案网（http://www.khdaw.com）3.Probabilitydistributionofpriceandone-yearholdingperiodreturnfora30-yearGermanGovernmentbond(whichwillhave29yearstomaturityatyear’send):CouponEconomyProbabilityYTMPriceCapitalGainHPRInterestBoom0.2011.0%$74.05−$25.95$8.00−17.95%NormalGrowth0.508.0%$100.00$0.00$8.008.00%Recession0.307.0%$112.28$12.28$8.0020.28%2- 课后答案网（http://www.khdaw.com）4.E(r)=[0.35×44.5%]+[0.30×14.0%]+[0.35×(–16.5%)]=14%σ2=[0.35×(44.5–14)2]+[0.30×(14–14)2]+[0.35×(–16.5–14)2]=651.175σ=25.52%Themeanisunchanged,butthestandarddeviationhasincreased,astheprobabilitiesofthehighandlowreturnshaveincreased.5.Forthemoneymarketfund,yourholdingperiodreturnforthenextyeardependsonthelevelof30-dayinterestrateseachmonthwhenthefundrollsovermaturingsecurities.Theone-yearsavingsdepositoffersa7.5%holdingperiodreturnfortheyear.Ifyouforecastthattherateonmoneymarketinstrumentswillincreasesignificantlyabovethecurrent6%yield,thenthemoneymarketfundmightresultinahigherHPRthanthesavingsdeposit.The20-yearU.KGovernmentbondoffersayieldtomaturityof9%peryear,whichis150basispointshigherthantherateontheone-yearsavingsdeposit;however,youcouldearnaone-yearHPRmuchlessthan7.5%onthebondiflong-terminterestratesincreaseduringtheyear.IfU.KGovernmentbondyieldsriseabove9%,thenthepriceofthebondwillfall,andtheresultingcapitallosswillwipeoutsomeorallofthe9%returnyouwouldhaveearnedifbondyieldshadremainedunchangedoverthecourseoftheyear.6.a.Ifbusinessesreducetheircapitalspending,thentheyarelikelytodecreasetheirdemandforfunds.ThiswillshiftthedemandcurveinFigure5.1totheleftandreducetheequilibriumrealrateofinterest.b.Increasedhouseholdsavingwillshiftthesupplyoffundscurvetotherightandcauserealinterestratestofall.c.OpenmarketpurchasesofU.S.TreasurysecuritiesbytheFederalReserveBoardisequivalenttoanincreaseinthesupplyoffunds(ashiftofthesupplycurvetotheright).Theequilibriumrealrateofinterestwillfall.2- 课后答案网（http://www.khdaw.com）7.Theaverageratesofreturnandstandarddeviationsarequitedifferentinthesubperiods:STOCKSStandSakKMreuedwrtaDnoneesivssiastion--120.0.1922.0.306–126620056075%%5319716–3.1--20085.0.0.55646%854%78591926.6–33--19490.0.1.1%300307%2126BONDSSSKtakMuneertdwaoannsiresds2- 课后答案网（http://www.khdaw.com）Dseviation0.1.1925.8.966–609320089015%%341979.6–510.-20070.30.5%370273%2291924.6–44.-0.194230.51%250%03346Themostrelevantstatisticstouseforprojectingintothefuturewouldseemtobethestatisticsestimatedovertheperiod1976-2005,becausethislaterperiodseemstohavebeenadifferenteconomicregime.After1955,theU.S.economyenteredtheKeynesianera,whentheFederalgovernmentactivelyattemptedtostabilizetheeconomyandtopreventextremesinboomandbustcycles.Notethatthestandarddeviationofstockreturnshasdecreasedsubstantiallyinthelaterperiodwhilethestandarddeviationofbondreturnshasincreased.8.Realinterestratesareexpectedtorise.Theinvestmentactivitywillshiftthedemandforfundscurve(inFigure5.1)totheright.Thereforetheequilibriumrealinterestratewillincrease.1+RR−i0.80−0.709.ar=−1===0.0588=5.88%1+i1+i1.70b.r≈R−i=80%−70%=10%Clearly,theapproximationgivesarealHPRthatistoohigh.2- 课后答案网（http://www.khdaw.com）10.FromTable5.2,theaveragerealrateonT-billshasbeen:0.72%a.T-bills:0.72%realrate+3%inflation=3.72%b.Expectedreturnonlargestocks:3.72%T-billrate+8.40%historicalriskpremium=12.12%c.Theriskpremiumonstocksremainsunchanged.Apremium,thedifferencebetweentworates,isarealvalue,unaffectedbyinflation.2- 课后答案网（http://www.khdaw.com）11.E(r)=(0.1×15%)+(0.6×13%)+(0.3×7%)=11.4%12.Theexpecteddollarreturnontheinvestmentinequitiesis$18,000comparedtothe$5,000expectedreturnforT-bills.Therefore,theexpectedriskpremiumis$13,000.13.E(rX)=[0.2×(−20%)]+[0.5×18%]+[0.3×50%]=20%E(rY)=[0.2×(−15%)]+[0.5×20%]+[0.3×10%]=10%14.σ2222X=[0.2×(–20–20)]+[0.5×(18–20)]+[0.3×(50–20)]=592σX=24.33%σ2222Y=[0.2×(–15–10)]+[0.5×(20–10)]+[0.3×(10–10)]=175σX=13.23%15.E(r)=(0.9×20%)+(0.1×10%)=19%16.E(r)=[0.2×(−25%)]+[0.3×10%]+[0.5×24%]=10%17.Theprobabilitythattheeconomywillbeneutralis0.50,or50%.Givenaneutraleconomy,thestockwillexperiencepoorperformance30%ofthetime.Theprobabilityofbothpoorstockperformanceandaneutraleconomyistherefore:0.30×0.50=0.15=15%18.a.ProbabilityDistributionoftheHPRontheStockMarketandPut:STOCKPUTStateoftheEndingPriceProbabilityHPREndingValueHPREconomy+DividendBoom0.30$13434%$0.00−100%NormalGrowth0.50$11414%$0.00−100%Recession0.20$84−16%$29.50146%Rememberthatthecostoftheindexfundis$100pershare,andthecostoftheputoptionis$12.2- 课后答案网（http://www.khdaw.com）b.Thecostofoneshareoftheindexfundplusaputoptionis$112.TheprobabilitydistributionoftheHPRontheportfoliois:EndingPriceStateofthe+ProbabilityHPREconomyPut+DividendBoom0.30$134.0019.6%=(134−112)/112NormalGrowth0.50$114.001.8%=(114−112)/112Recession0.20$113.501.3%=(113.50−112)/112c.BuyingtheputoptionguaranteestheinvestoraminimumHPRof1.3%regardlessofwhathappenstothestock"sprice.Thus,itoffersinsuranceagainstapricedecline.19.TheprobabilitydistributionofthedollarreturnonCDpluscalloptionis:StateoftheEndingValueEndingValueCombinedProbabilityEconomyofCDofCallValueBoom0.30$114.00$19.50$133.50NormalGrowth0.50$114.00$0.00$114.00Recession0.20$114.00$0.00$114.00CHAPTER6:RISKAVERSIONANDCAPITALALLOCATIONTORISKYASSETS1.TheindifferencecurvewithU=.022.PointE3.Utilityforeachinvestment=E(r)–0.5×4×σ2Wechoosetheinvestmentwiththehighestutilityvalue.ExpectedStandardUtilityInvestmentreturndeviationUE(r)σ10.120.30-0.060020.150.50-0.350030.210.160.158840.240.210.1518SelectInvestment3.2- 课后答案网（http://www.khdaw.com）4.Wheninvestorsareriskneutral,thenA=0;theinvestmentwiththehighestutilityisInvestment4becauseithasthehighestexpectedreturn.5.b6.WhenwespecifyutilitybyU=E(r)–0.5Aσ2,theutilitylevelforT-billsis:0.07Theutilitylevelfortheriskyportfoliois:U=0.12–0.5A(0.18)2=0.12–0.0162AInorderfortheriskyportfoliotobepreferredtobills,thefollowinginequalitymusthold:0.12–0.0162A>0.07⇒A<0.05/0.0162=3.09Amustbelessthan3.09fortheriskyportfoliotobepreferredtobills.2- 课后答案网（http://www.khdaw.com）7.PointsonthecurvearederivedbysolvingforE(r)inthefollowingequation:U=0.05=E(r)–0.5Aσ2=E(r)–1.5σ2ThevaluesofE(r),giventhevaluesofσ2,aretherefore:σσ2E(r)0.000.00000.050000.050.00250.053750.100.01000.065000.150.02250.083750.200.04000.110000.250.06250.14375Theboldlineinthefollowinggraph(labeledQ3)depictstheindifferencecurve.E(r)U(Q4,A=4)U(Q3,A=3)5U(Q5,A=0)4σU(Q6,A<0)8.RepeatingtheanalysisinProblem7,utilityisnow:U=E(r)–0.5Aσ2=E(r)–2.0σ2=0.04Theequal-utilitycombinationsofexpectedreturnandstandarddeviationarepresentedinthetablebelow.Theindifferencecurveistheupwardslopinglineinthegraphabove,labeledQ4.2- 课后答案网（http://www.khdaw.com）σσ2E(r)0.000.00000.04000.050.00250.04500.100.01000.06000.150.02250.08500.200.04000.12000.250.06250.1650TheindifferencecurveinProblem8differsfromthatinProblem7inbothslopeandintercept.WhenAincreasesfrom3to4,theincreasedriskaversionresultsinagreaterslopefortheindifferencecurvesincemoreexpectedreturnisneededinordertocompensateforadditionalσ.ThelowerlevelofutilityassumedforProblem8(0.04ratherthan0.05)shiftstheverticalinterceptdownby1%.9.Thecoefficientofriskaversionforariskneutralinvestoriszero.Therefore,thecorrespondingutilityisequaltotheportfolio’sexpectedreturn.Thecorrespondingindifferencecurveintheexpectedreturn-standarddeviationplaneisahorizontalline,labeledQ5inthegraphabove(seeProblem7).10.Arisklover,ratherthanpenalizingportfolioutilitytoaccountforrisk,derivesgreaterutilityasvarianceincreases.Thisamountstoanegativecoefficientofriskaversion.Thecorrespondingindifferencecurveisdownwardslopinginthegraphabove(seeProblem7),andislabeledQ6.11.a.Theexpectedcashflowis:(0.5×$70,000)+(0.5×200,000)=$135,000Withariskpremiumof8%overtherisk-freerateof6%,therequiredrateofreturnis14%.Therefore,thepresentvalueoftheportfoliois:$135,000/1.14=$118,421$118,421×[1+E(r)]=$135,000Therefore,E(r)=14%.Theportfoliopriceissettoequatetheexpectedrateofreturnwiththerequiredrateofreturn.2- 课后答案网（http://www.khdaw.com）6%+12%=18%Thepresentvalueoftheportfolioisnow:$135,000/1.18=$114,407d.Foragivenexpectedcashflow,portfoliosthatcommandgreaterriskpremiamustsellatlowerprices.Theextradiscountfromexpectedvalueisapenaltyforrisk.12.Expectedreturnforequityfund=money-marketrate+riskpremium=6%+10%=16%Expectedreturnofclient’soverallportfolio=(0.6×16%)+(0.4×6%)=12%Standarddeviationofclient’soverallportfolio=0.6×14%=8.4%1013.Rewardtovariabilityratio==0.71148−514.a.E(rC)=8%=5%+y(11%–5%)⇒y==0.511−5b.σC=yσP=0.50×15%=7.5%c.Thefirstclientismoreriskaverse,allowingasmallerstandarddeviation.2- 课后答案网（http://www.khdaw.com）B15.Data:rf=2%,E(rM)=10%,σM=25%,andr=6%fTheCMLandindifferencecurvesareasfollows:E(r)borrowlendCALCMLP1062σ2516.Forytobelessthan1.0(sothattheinvestorisalender),riskaversion(A)mustbelargeenoughsuchthat:E(rM)−rf0.10−0.02y=<1⇒A>=1.2822Aσ0.25MForytobegreaterthan1.0(sothattheinvestorisaborrower),riskaversionmustbesmallenoughsuchthat:E(rM)−rf0.10−0.06y=>1⇒A<=0.6422Aσ0.25MForvaluesofriskaversionwithinthisrange,theclientwillneitherborrownorlend,butinsteadwillholdacompleteportfoliocomprisedonlyoftheoptimalriskyportfolio:y=1for0.64≤Α≤1.282- 课后答案网（http://www.khdaw.com）17.a.ThegraphforProblem15hastoberedrawnhere,with:E(rP)=8%andσP=15%0.08−0.02b.Foralendingposition:A>=2.6720.150.08−0.06Foraborrowingposition:A<=0.8920.15Therefore,y=1for0.89≤A≤2.67E(r)CMLM10FCAL862σ152518.Themaximumfeasiblefee,denotedf,dependsonthereward-to-variabilityratio.Fory<1,thelendingrate,2%,isviewedastherelevantrisk-freerate,andwesolveforfasfollows:8−2−f10−215×8=⇒f=6−=1.2%152525Fory>1,theborrowingrate,6%,istherelevantrisk-freerate.Thenwenoticethat,evenwithoutafee,theactivefundisinferiortothepassivefundbecause:8−610−6=0.13<=0.1615252- 课后答案网（http://www.khdaw.com）Morerisktolerantinvestors(whoaremoreinclinedtoborrow)willnotbeclientsofthefundevenwithoutafee.(Ifyousolvedforthefeethatwouldmakeinvestorswhoborrowindifferentbetweentheactiveandpassiveportfolio,aswedidaboveforlendinginvestors,youwouldfindthatfisnegative:thatis,youwouldneedtopayinvestorstochooseyouractivefund.)Theseinvestorsdesirehigherrisk-higherreturncompleteportfoliosandthusareintheborrowingrangeoftherelevantCAL.Inthisrange,thereward-to-variabilityratiooftheindex(thepassivefund)isbetterthanthatofthemanagedfund.19.Expectedreturn=(0.7×18%)+(0.3×8%)=15%Standarddeviation=0.7×28%=19.6%20.Investmentproportions:30.0%inT-bills0.7×25%=17.5%inStockA0.7×32%=22.4%inStockB0.7×43%=30.1%inStockC18−821.Yourreward-to-variabilityratio:S==0.35712815−8Client"sreward-to-variabilityratio:S==0.357119.622.3025CAL(Slope=0.3571)20E(r)P15%Client1050010203040σ(%)2- 课后答案网（http://www.khdaw.com）23.a.E(rC)=rf+y[E(rP)–rf]=8+y(18−8)Iftheexpectedreturnfortheportfoliois16%,then:16−816=8+10y⇒y==0.810Therefore,inordertohaveaportfoliowithexpectedrateofreturnequalto16%,theclientmustinvest80%oftotalfundsintheriskyportfolioand20%inT-bills.b.Client’sinvestmentproportions:20.0%inT-bills0.8×25%=20.0%inStockA0.8×32%=25.6%inStockB0.8×43%=34.4%inStockCc.σC=0.8×σP=0.8×28%=22.4%24.a.σC=y×28%Ifyourclientprefersastandarddeviationofatmost18%,then:y=18/28=0.6429=64.29%investedintheriskyportfoliob.E(rC)=8+10y=8+(0.6429×10)=8+6.429=14.429%E(rP)−rf0.18−0.080.1025.a.y*====0.364422Aσ3.5×0.280.2744PTherefore,theclient’soptimalproportionsare:36.44%investedintheriskyportfolioand63.56%investedinT-bills.b.E(rC)=8+10y*=8+(0.3644×10)=11.644%σC=0.3644×28=10.203%2- 课后答案网（http://www.khdaw.com）13−826.a.SlopeoftheCML==0.2025Thediagramfollows.b.Myfundallowsaninvestortoachieveahighermeanforanygivenstandarddeviationthanwouldapassivestrategy,i.e.,ahigherexpectedreturnforanygivenlevelofrisk.CMLandCAL1816CAL:Slope=0.357114Expected1210CML:Slope=0.20Retrun864200102030StandardDeviation27.a.With70%ofhismoneyinvestedinmyfund’sportfolio,theclient’sexpectedreturnis15%peryearandstandarddeviationis19.6%peryear.Ifheshiftsthatmoneytothepassiveportfolio(whichhasanexpectedreturnof13%andstandarddeviationof25%),hisoverallexpectedreturnbecomes:E(rC)=rf+0.7[E(rM)−rf]=8+[0.7×(13–8)]=11.5%Thestandarddeviationofthecompleteportfoliousingthepassiveportfoliowouldbe:σC=0.7×σM=0.7×25%=17.5%Therefore,theshiftentailsadecreaseinmeanfrom14%to11.5%andadecreaseinstandarddeviationfrom19.6%to17.5%.Sincebothmeanreturnandstandarddeviationdecrease,itisnotyetclearwhetherthemoveisbeneficial.Thedisadvantageoftheshiftisthat,iftheclientiswillingtoacceptameanreturnonhistotalportfolioof11.5%,hecanachieveitwithalowerstandarddeviationusingmyfundratherthanthepassiveportfolio.2- 课后答案网（http://www.khdaw.com）Toachieveatargetmeanof11.5%,wefirstwritethemeanofthecompleteportfolioasafunctionoftheproportioninvestedinmyfund(y):E(rC)=8+y(18−8)=8+10yOurtargetis:E(rC)=11.5%.Therefore,theproportionthatmustbeinvestedinmyfundisdeterminedasfollows:11.5−811.5=8+10y⇒y==0.3510Thestandarddeviationofthisportfoliowouldbe:σC=y×28%=0.35×28%=9.8%Thus,byusingmyportfolio,thesame11.5%expectedreturncanbeachievedwithastandarddeviationofonly9.8%asopposedtothestandarddeviationof17.5%usingthepassiveportfolio.b.Thefeewouldreducethereward-to-variabilityratio,i.e.,theslopeoftheCAL.Theclientwillbeindifferentbetweenmyfundandthepassiveportfolioiftheslopeoftheafter-feeCALandtheCMLareequal.Letfdenotethefee:18−8−f10−fSlopeofCALwithfee==282813−8SlopeofCML(whichrequiresnofee)==0.2025Settingtheseslopesequalwehave:10−f=0.20⇒10−f=28×0.20=5.6⇒f=10−5.6=4.4%peryear2828.a.Theformulafortheoptimalproportiontoinvestinthepassiveportfoliois:E(r)−rMfy*=2AσMSubstitutethefollowing:E(rM)=13%;rf=8%;σM=25%;A=3.5:0.13−0.08y*==0.228623.5×0.25b.TheanswerhereisthesameastheanswertoProblem27(b).Thefeethatyoucanchargeaclientisthesameregardlessoftheassetallocationmixoftheclient’sportfolio.Youcanchargeafeethatwillequatethereward-to-variabilityratioofyourportfoliotothatofyourcompetition.2- 课后答案网（http://www.khdaw.com）29.a.Iftheperiod1926-2005isassumedtoberepresentativeoffutureexpectedperformance,thenweusethefollowingdatatocomputethefractionallocatedtoequity:A=4,E(rM)−rf=8.39%,σM=20.54%(weusethestandarddeviationoftheriskpremiumfromTable6.8).Theny*isgivenby:E(r)−r0.0839Mfy*===0.497222Aσ4×0.2054MThatis,49.72%oftheportfolioshouldbeallocatedtoequityand50.28%shouldbeallocatedtoT-bills.b.Iftheperiod1986-2005isassumedtoberepresentativeoffutureexpectedperformance,thenweusethefollowingdatatocomputethefractionallocatedtoequity:A=4,E(rM)−rf=8.60%,σM=16.24%andy*isgivenby:E(r)−r0.0860Mfy*===0.815222Aσ4×0.1624MTherefore,81.52%ofthecompleteportfolioshouldbeallocatedtoequityand18.48%shouldbeallocatedtoT-bills.c.Inpart(b),themarketriskpremiumisexpectedtobehigherthaninpart(a)andmarketriskislower.Therefore,thereward-to-variabilityratioisexpectedtobehigherinpart(b),whichexplainsthegreaterproportioninvestedinequity.30.Assumingnochangeinrisktolerance,thatis,anunchangedriskaversioncoefficient(A),thenhigherperceivedvolatilityincreasesthedenominatoroftheequationfortheoptimalinvestmentintheriskyportfolio(Equation6.12).Theproportioninvestedintheriskyportfoliowillthereforedecrease.31.Theportfolioexpectedreturnandvariancearecomputedasfollows:(1)(2)(3)(4)rPortfolioσPortfolioσ2PortfolioWBillsrBillsWIndexrIndex(1)×(2)+(3)×(4)(3)×20%0.05%1.013.5%13.5%=0.13520%=0.200.04000.25%0.813.5%11.8%=0.11816%=0.160.02560.45%0.613.5%10.1%=0.10112%=0.120.01440.65%0.413.5%8.4%=0.0848%=0.080.00640.85%0.213.5%6.7%=0.0674%=0.040.00161.05%0.013.5%5.0%=0.0500%=0.000.00002- 课后答案网（http://www.khdaw.com）32.ComputingutilityfromU=E®–0.5×Aσ2=E®–1.5σ2,wearriveatthevaluesinthecolumnlabeledU(A=3)inthefollowingtable:WBillsWIndexrPortfolioσPortfolioσ2PortfolioU(A=3)U(A=5)0.01.00.1350.200.04000.07500.03500.20.80.1180.160.02560.07960.05400.40.60.1010.120.01440.07940.06500.60.40.0840.080.00640.07440.06800.80.20.0670.040.00160.06460.06301.00.00.0500.000.00000.05000.0500ThecolumnlabeledU(A=3)impliesthatinvestorswithA=3preferaportfoliothatisinvested80%inthemarketindexand20%inT-billstoanyoftheotherportfoliosinthetable.33.ThecolumnlabeledU(A=5)inthetableaboveiscomputedfrom:U=E(r)–0.5Aσ2=E(r)–2.5σ2Themoreriskaverseinvestorsprefertheportfoliothatisinvested40%inthemarketindex,ratherthanthe80%marketweightpreferredbyinvestorswithA=3.34.(0.6×$50,000)+[0.4×(−$30,000)]−$5,000=$13,00035.b2- 课后答案网（http://www.khdaw.com）CHAPTER6:APPENDIX1.Byyearend,the$50,000investmentwillgrowto:$50,000×1.06=$53,000Withoutinsurance,theprobabilitydistributionofend-of-yearwealthis:ProbabilityWealthNofire0.999$253,000Fire0.001$53,000Forthisdistribution,expectedutilityiscomputedasfollows:E[U(W)]=[0.999×ln(253,000)]+[0.001×ln(53,000)]=12.439582Thecertaintyequivalentis:W12.439582CE=e=$252,604.85Withfireinsurance,atacostof$P,theinvestmentintherisk-freeassetis:$(50,000–P)Year-endwealthwillbecertain(sinceyouarefullyinsured)andequalto:[$(50,000–P)×1.06]+$200,000SolveforPinthefollowingequation:[$(50,000–P)×1.06]+$200,000=$252,604.85⇒P=$372.78Thisisthemostyouarewillingtopayforinsurance.Notethattheexpectedlossis“only”$200,soyouarewillingtopayasubstantialriskpremiumovertheexpectedvalueoflosses.Theprimaryreasonisthatthevalueofthehouseisalargeproportionofyourwealth.2.a.Withinsurancecoverageforone-halfthevalueofthehouse,thepremiumis$100,andtheinvestmentinthesafeassetis$49,900.Byyearend,theinvestmentof$49,900willgrowto:$49,900×1.06=$52,894Ifthereisafire,yourinsuranceproceedswillbe$100,000,andtheprobabilitydistributionofend-of-yearwealthis:ProbabilityWealthNofire0.999$252,894Fire0.001$152,894Forthisdistribution,expectedutilityiscomputedasfollows:E[U(W)]=[0.999×ln(252,894)]+[0.001×ln(152,894)]=12.4402225Thecertaintyequivalentis:W12.4402225CE=e=$252,766.772- 课后答案网（http://www.khdaw.com）b.Withinsurancecoverageforthefullvalueofthehouse,costing$200,end-of-yearwealthiscertain,andequalto:[($50,000–$200)×1.06]+$200,000=$252,788Sincewealthiscertain,thisisalsothecertaintyequivalentwealthofthefullyinsuredposition.c.Withinsurancecoveragefor1½timesthevalueofthehouse,thepremiumis$300,andtheinsurancepaysoff$300,000intheeventofafire.Theinvestmentinthesafeassetis$49,700.Byyearend,theinvestmentof$49,700willgrowto:$49,700×1.06=$52,682Theprobabilitydistributionofend-of-yearwealthis:ProbabilityWealthNofire0.999$252,682Fire0.001$352,682Forthisdistribution,expectedutilityiscomputedasfollows:E[U(W)]=[0.999×ln(252,682)]+[0.001×ln(352,682)]=12.4402205Thecertaintyequivalentis:W12.440222CE=e=$252,766.27Therefore,fullinsurancedominatesbothover-andunder-insurance.Over-insuringcreatesagamble(youactuallygainwhenthehouseburnsdown).Riskisminimizedwhenyouinsureexactlythevalueofthehouse.CHAPTER7:OPTIMALRISKYPORTFOLIOS1.Thecorrectchoiceisc.Intuitively,wenotethatsinceallstockshavethesameexpectedrateofreturnandstandarddeviation,wechoosethestockthatwillresultinlowestrisk.ThisisthestockthathasthelowestcorrelationwithStockA.Moreformally,wenotethatwhenallstockshavethesameexpectedrateofreturn,theoptimalportfolioforanyrisk-averseinvestoristheglobalminimumvarianceportfolio(G).WhentheportfolioisrestrictedtoStockAandoneadditionalstock,theobjectiveistofindGforanypairthatincludesStockA,andthenselectthecombinationwiththelowestvariance.Withtwostocks,IandJ,theformulafortheweightsinGis:2σ−Cov(r,r)JIJw(I)=Min22σ+σ−2Cov(r,r)IJIJw(J)=1−w(I)MinMinSinceallstandarddeviationsareequalto20%:2- 课后答案网（http://www.khdaw.com）Cov(rI,rJ)=ρσIσJ=400ρandwMin(I)=wMin(J)=0.5Thisintuitiveresultisanimplicationofapropertyofanyefficientfrontier,namely,thatthecovariancesoftheglobalminimumvarianceportfoliowithallotherassetsonthefrontierareidenticalandequaltoitsownvariance.(Otherwise,additionaldiversificationwouldfurtherreducethevariance.)Inthiscase,thestandarddeviationofG(I,J)reducesto:σ1/2Min(G)=[200(1+ρIJ)]ThisleadstotheintuitiveresultthatthedesiredadditionwouldbethestockwiththelowestcorrelationwithStockA,whichisStockD.TheoptimalportfolioisequallyinvestedinStockAandStockD,andthestandarddeviationis17.03%.2.No,theanswertoProblem1wouldnotchange,atleastaslongasinvestorsarenotrisklovers.Riskneutralinvestorswouldnotcarewhichportfoliotheyheldsinceallportfolioshaveanexpectedreturnof8%.3.No,theanswerstoProblems1and2wouldnotchange.Theefficientfrontierofriskyassetsishorizontalat8%,sotheoptimalCALrunsfromtherisk-freeratethroughG.ThebestPortfolioGis,again,theonewiththelowestvariance.Theoptimalcompleteportfoliodependsonriskaversion.4.b.2- 课后答案网（http://www.khdaw.com）5.False.Iftheborrowingandlendingratesarenotidentical,then,dependingonthetastesoftheindividuals(thatis,theshapeoftheirindifferencecurves),borrowersandlenderscouldhavedifferentoptimalriskyportfolios.6.c.7.σP=30=yσ=40y⇒y=0.75E(rP)=12+0.75(30−12)=25.5%8.d.PortfolioYcannotbeefficientbecauseitisdominatedbyanotherportfolio.Forexample,PortfolioXhasbothhigherexpectedreturnandlowerstandarddeviation.9.Theparametersoftheopportunitysetare:E(rS)=20%,E(rB)=12%,σS=30%,σB=15%,ρ=0.10Fromthestandarddeviationsandthecorrelationcoefficientwegeneratethecovariancematrix[notethatCov(rS,rB)=ρσSσB]:BondsStocksBonds22545Stocks45900Theminimum-varianceportfolioiscomputedasfollows:2σB−Cov(rS,rB)225−45wMin(S)===0.173922σ+σ−2Cov(r,r)900+225−(2×45)SBSBwMin(B)=1−0.1739=0.8261Theminimumvarianceportfoliomeanandstandarddeviationare:E(rMin)=(0.1739×20)+(0.8261×12)=13.39%22221/2σ=[wσ+wσ+2wwCov(r,r)]MinSSBBSBSB=[(0.17392×900)+(0.82612×225)+(2×0.1739×0.8261×45)]1/2=13.92%2- 课后答案网（http://www.khdaw.com）10.ProportionProportionExpectedStandardinstockfundinbondfundreturnDeviation0.00%100.00%12.00%15.00%17.39%82.61%13.39%13.92%minimumvariance20.00%80.00%13.60%13.94%40.00%60.00%15.20%15.70%45.16%54.84%15.61%16.54%tangencyportfolio60.00%40.00%16.80%19.53%80.00%20.00%18.40%24.48%100.00%0.00%20.00%30.00%Graphshownbelow.11.25.00INVESTMENTOPPORTUNITYSET20.00CMLTangencyPortfolioEfficientfrontier15.00ofriskyassets10.00MinimumVariancePortfolio5.000.000.005.0010.0015.0020.0025.0030.00Thegraphindicatesthattheoptimalportfolioisthetangencyportfoliowithexpectedreturnapproximately15.6%andstandarddeviationapproximately16.5%.2- 课后答案网（http://www.khdaw.com）12.Theproportionoftheoptimalriskyportfolioinvestedinthestockfundisgivenby:2[E(r)−r]σ−[E(r)−r]Cov(r,r)SfBBfSBw=S22[E(r)−r]σ+[E(r)−r]σ−[E(r)−r+E(r)−r]Cov(r,r)SfBBfSSfBfSB[(20−8)×225]−[(12−8)×45]==0.4516[(20−8)×225]+[(12−8)×900]−[(20−8+12−8)×45]wB=1−0.4516=0.5484Themeanandstandarddeviationoftheoptimalriskyportfolioare:E(rP)=(0.4516×20)+(0.5484×12)=15.61%σ2×900)+(0.54842×225)+(2×0.4516×0.5484×45)]1/2p=[(0.4516=16.54%13.Thereward-to-variabilityratiooftheoptimalCALis:E(rp)−rf15.61−8==0.4601σ16.54p14.a.Ifyourequirethatyourportfolioyieldanexpectedreturnof14%,thenyoucanfindthecorrespondingstandarddeviationfromtheoptimalCAL.TheequationforthisCALis:E(r)−rpfE(r)=r+σ=8+0.4601σCfCCσPSettingE(rC)equalto14%,wefindthatthestandarddeviationoftheoptimalportfoliois13.04%.b.TofindtheproportioninvestedintheT-billfund,rememberthatthemeanofthecompleteportfolio(i.e.,14%)isanaverageoftheT-billrateandtheoptimalcombinationofstocksandbonds(P).LetybetheproportioninvestedintheportfolioP.ThemeanofanyportfolioalongtheoptimalCALis:E(rC)=(l−y)rf+yE(rP)=rf+y[E(rP)−rf]=8+y(15.61−8)SettingE(rC)=14%wefind:y=0.7884and(1−y)=0.2116(theproportioninvestedintheT-billfund).Tofindtheproportionsinvestedineachofthefunds,multiply0.7884timestherespectiveproportionsofstocksandbondsintheoptimalriskyportfolio:Proportionofstocksincompleteportfolio=0.7884×0.4516=0.35602- 课后答案网（http://www.khdaw.com）Proportionofbondsincompleteportfolio=0.7884×0.5484=0.43242- 课后答案网（http://www.khdaw.com）15.Usingonlythestockandbondfundstoachieveaportfolioexpectedreturnof14%,wemustfindtheappropriateproportioninthestockfund(wS)andtheappropriateproportioninthebondfund(wB=1−wS)asfollows:14=20wS+12(1−wS)=12+8wS⇒wS=0.25Sotheproportionsare25%investedinthestockfundand75%inthebondfund.Thestandarddeviationofthisportfoliowillbe:σ2×900)+(0.752×225)+(2×0.25×0.75×45)]1/2P=[(0.25=14.13%Thisisconsiderablygreaterthanthestandarddeviationof13.04%achievedusingT-billsandtheoptimalportfolio.16.False.Theportfoliostandarddeviationequalstheweightedaverageofthecomponent-assetstandarddeviationsonlyinthespecialcasethatallassetsareperfectlypositivelycorrelated.Otherwise,astheformulaforportfoliostandarddeviationshows,theportfoliostandarddeviationislessthantheweightedaverageofthecomponent-assetstandarddeviations.Theportfoliovarianceisaweightedsumoftheelementsinthecovariancematrix,withtheproductsoftheportfolioproportionsasweights.17.d.18.SinceStockAandStockBareperfectlynegativelycorrelated,arisk-freeportfoliocanbecreatedandtherateofreturnforthisportfolio,inequilibrium,willbetherisk-freerate.Tofindtheproportionsofthisportfolio[withtheproportionwAinvestedinStockAandwB=(1–wA)investedinStockB],setthestandarddeviationequaltozero.Withperfectnegativecorrelation,theportfoliostandarddeviationis:σP=Absolutevalue[wAσA−wBσB]0=5wA−[10×(1–wA)]⇒wA=0.6667Theexpectedrateofreturnforthisrisk-freeportfoliois:E(r)=(0.6667×10)+(0.3333×15)=11.667%Therefore,therisk-freerateis11.667%.2- 课后答案网（http://www.khdaw.com）19.a.25.00OptimalCAL20.00PStocks15.0010.00Gold5.000.00010203040StandardDeviation(%)Eventhoughitseemsthatgoldisdominatedbystocks,goldmightstillbeanattractiveassettoholdasapartofaportfolio.Ifthecorrelationbetweengoldandstocksissufficientlylow,goldwillbeheldasacomponentinaportfolio,specifically,theoptimaltangencyportfolio.b.Ifthecorrelationbetweengoldandstocksequals+1,thennoonewouldholdgold.TheoptimalCALwouldbecomprisedofbillsandstocksonly.Sincethesetofrisk/returncombinationsofstocksandgoldwouldplotasastraightlinewithanegativeslope(seethefollowinggraph),thesecombinationswouldbedominatedbythestockportfolio.Ofcourse,thissituationcouldnotpersist.Ifnoonedesiredgold,itspricewouldfallanditsexpectedrateofreturnwouldincreaseuntilitbecamesufficientlyattractivetoincludeinaportfolio.2- 课后答案网（http://www.khdaw.com）252018Stocks1510Gold5rf00.0010.0020.0030.0040.00StandardDeviation(%)20.a.Restrictingtheportfolioto20stocks,ratherthan40to50stocks,willincreasetheriskoftheportfolio,butitispossiblethattheincreaseinriskwillbeminimal.Supposethat,forinstance,the50stocksinauniversehavethesamestandarddeviation(σ)andthecorrelationsbetweeneachpairareidentical,withcorrelationcoefficientρ.Then,thecovariancebetweeneachpairofstockswouldbeρσ2,andthevarianceofanequallyweightedportfoliowouldbe:212n−12σ=σ+ρσPnnTheeffectofthereductioninnonthesecondtermontheright-handsidewouldberelativelysmall(since49/50iscloseto19/20andρσ2issmallerthanσ2),butthedenominatorofthefirsttermwouldbe20insteadof50.Forexample,ifσ=45%andρ=0.2,thenthestandarddeviationwith50stockswouldbe20.91%,andwouldriseto22.05%whenonly20stocksareheld.Suchanincreasemightbeacceptableiftheexpectedreturnisincreasedsufficiently.b.Hennessycouldcontaintheincreaseinriskbymakingsurethathemaintainsreasonablediversificationamongthe20stocksthatremaininhisportfolio.Thisentailsmaintainingalowcorrelationamongtheremainingstocks.Forexample,inpart(a),withρ=0.2,theincreaseinportfolioriskwasminimal.Asapracticalmatter,thismeansthatHennessywouldhavetospreadhisportfolioamongmanyindustries;concentratingonjustafewindustrieswouldresultinhighercorrelationsamongtheincludedstocks.2- 课后答案网（http://www.khdaw.com）21.Riskreductionbenefitsfromdiversificationarenotalinearfunctionofthenumberofissuesintheportfolio.Rather,theincrementalbenefitsfromadditionaldiversificationaremostimportantwhenyouareleastdiversified.RestrictingHennessyto10insteadof20issueswouldincreasetheriskofhisportfoliobyagreateramountthanwouldareductioninthesizeoftheportfoliofrom30to20stocks.Inourexample,restrictingthenumberofstocksto10willincreasethestandarddeviationto23.81%.The1.76%increaseinstandarddeviationresultingfromgivingup10of20stocksisgreaterthanthe1.14%increasethatresultsfromgivingup30of50stocks.22.Thepointiswelltakenbecausethecommitteeshouldbeconcernedwiththevolatilityoftheentireportfolio.SinceHennessy’sportfolioisonlyoneofsixwell-diversifiedportfoliosandissmallerthantheaverage,theconcentrationinfewerissuesmighthaveaminimaleffectonthediversificationofthetotalfund.Hence,unleashingHennessytodostockpickingmaybeadvantageous.23.a.Systematicriskreferstofluctuationsinassetpricescausedbymacroeconomicfactorsthatarecommontoallriskyassets;hencesystematicriskisoftenreferredtoasmarketrisk.Examplesofsystematicriskfactorsincludethebusinesscycle,inflation,monetarypolicyandtechnologicalchanges.Firm-specificriskreferstofluctuationsinassetpricescausedbyfactorsthatareindependentofthemarketsuchasindustrycharacteristicsorfirmcharacteristics.Examplesoffirm-specificriskfactorsincludelitigation,patents,management,andfinancialleverage.b.Trudyshouldexplaintotheclientthatpickingonlythetopfivebestideaswouldmostlikelyresultintheclientholdingamuchmoreriskyportfolio.Thetotalriskofaportfolio,orportfoliovariance,isthecombinationofsystematicriskandfirm-specificrisk.Thesystematiccomponentdependsonthesensitivityoftheindividualassetstomarketmovementsasmeasuredbybeta.Assumingtheportfolioiswelldiversified,thenumberofassetswillnotaffectthesystematicriskcomponentofportfoliovariance.Theportfoliobetadependsontheindividualsecuritybetasandtheportfolioweightsofthosesecurities.Ontheotherhand,thecomponentsoffirm-specificrisk(sometimescallednonsystematicrisk)arenotperfectlypositivelycorrelatedwitheachotherand,asmoreassetsareaddedtotheportfolio,thoseadditionalassetstendtoreduceportfoliorisk.Hence,increasingthenumberofsecuritiesinaportfolioreducesfirm-specificrisk.Forexample,apatentexpirationforonecompanywouldnotaffecttheothersecuritiesintheportfolio.Anincreaseinoilpricesmighthurtanairlinestockbutaidanenergystock.Asthenumberofrandomlyselectedsecuritiesincreases,thetotalrisk(variance)oftheportfolioapproachesitssystematicvariance.2- 课后答案网（http://www.khdaw.com）24.FundDrepresentsthesinglebestadditiontocomplementStephenson"scurrentportfolio,givenhisselectioncriteria.First,FundD’sexpectedreturn(14.0percent)hasthepotentialtoincreasetheportfolio’sreturnsomewhat.Second,FundD’srelativelylowcorrelationwithhiscurrentportfolio(+0.65)indicatesthatFundDwillprovidegreaterdiversificationbenefitsthananyoftheotheralternativesexceptFundB.TheresultofaddingFundDshouldbeaportfoliowithapproximatelythesameexpectedreturnandsomewhatlowervolatilitycomparedtotheoriginalportfolio.Theotherthreefundshaveshortcomingsintermsofeitherexpectedreturnenhancementorvolatilityreductionthroughdiversificationbenefits.FundAoffersthepotentialforincreasingtheportfolio’sreturn,butistoohighlycorrelatedtoprovidesubstantialvolatilityreductionbenefitsthroughdiversification.FundBprovidessubstantialvolatilityreductionthroughdiversificationbenefits,butisexpectedtogenerateareturnwellbelowthecurrentportfolio’sreturn.FundChasthegreatestpotentialtoincreasetheportfolio’sreturn,butistoohighlycorrelatedtoprovidesubstantialvolatilityreductionbenefitsthroughdiversification.25.a.26.a.SubscriptOPreferstotheoriginalportfolio,Eurotothenewstock,andNPtothenewportfolio.i.E(rNP)=wOPE(rOP)+wEuroE(rEuro)=(0.9×0.67)+(0.1×1.25)=0.728%ii.Cov=r×σOP×σEuro=0.40×2.37×2.95=2.7966≅2.80iii.σNP=[wOP2σOP2+wEuro2σEuro2+2wOPwEuro(CovOP,Euro)]1/2=[(0.92×2.372)+(0.12×2.952)+(2×0.9×0.1×2.80)]1/2=2.2673%≅2.27%b.SubscriptOPreferstotheoriginalportfolio,GStogovernmentsecurities,andNPtothenewportfolio.i.E(rNP)=wOPE(rOP)+wGSE(rGS)=(0.9×0.67)+(0.1×0.042)=0.645%ii.Cov=r×σOP×σGS=0×2.37×0=0iii.σ22221/2NP=[wOPσOP+wGSσGS+2wOPwGS(CovOP,GS)]=[(0.92×2.372)+(0.12×0)+(2×0.9×0.1×0)]1/2=2.133%≅2.13%c.Addingtherisk-freegovernmentsecuritieswouldresultinalowerbetaforthenewportfolio.Thenewportfoliobetawillbeaweightedaverageoftheindividualsecuritybetasintheportfolio;thepresenceoftherisk-free2- 课后答案网（http://www.khdaw.com）securitieswouldlowerthatweightedaverage.2- 课后答案网（http://www.khdaw.com）d.Thecommentisnotcorrect.Althoughtherespectivestandarddeviationsandexpectedreturnsforthetwosecuritiesunderconsiderationareequal,thecovariancesbetweeneachsecurityandtheoriginalportfolioareunknown,makingitimpossibletodrawtheconclusionstated.Forinstance,ifthecovariancesaredifferent,selectingonesecurityovertheothermayresultinalowerstandarddeviationfortheportfolioasawhole.Insuchacase,thatsecuritywouldbethepreferredinvestment,assumingallotherfactorsareequal.e.i.Graceclearlyexpressedthesentimentthattheriskoflosswasmoreimportanttoherthantheopportunityforreturn.Usingvariance(orstandarddeviation)asameasureofriskinhercasehasaseriouslimitationbecausestandarddeviationdoesnotdistinguishbetweenpositiveandnegativepricemovements.ii.Twoalternativeriskmeasuresthatcouldbeusedinsteadofvarianceare:RangeofReturns,whichconsidersthehighestandlowestexpectedreturnsinthefutureperiod,withalargerrangebeingasignofgreatervariabilityandthereforeofgreaterrisk.Semivariance,whichcanbeusedtomeasureexpecteddeviationsofreturnsbelowthemean,orsomeotherbenchmark,suchaszero.EitherofthesemeasureswouldpotentiallybesuperiortovarianceforGrace.Rangeofreturnswouldhelptohighlightthefullspectrumofrisksheisassuming,especiallythedownsideportionoftherangeaboutwhichsheissoconcerned.Semivariancewouldalsobeeffective,becauseitimplicitlyassumesthattheinvestorwantstominimizethelikelihoodofreturnsfallingbelowsometargetrate;inGrace’scase,thetargetratewouldbesetatzero(toprotectagainstnegativereturns).27.Rearrangingthetable(convertingrowstocolumns),andcomputingserialcorrelationresultsinthefollowingtable:NominalRatesSmallLargeLong-termIntermed-termTreasurycompanycompanygovernmentgovernmentInflationbillsstocksstocksbondsbonds1920s-3.7218.363.983.773.56-1.001930s7.28-1.254.603.910.30-2.041940s20.639.113.591.700.375.361950s19.0119.410.251.111.872.221960s13.727.841.143.413.892.521970s8.755.906.636.116.297.361980s12.4617.6011.5012.019.005.101990s13.8418.208.607.745.022.93SerialCorrelation0.46-0.220.600.590.630.232- 课后答案网（http://www.khdaw.com）Forexample:tocomputeserialcorrelationindecadenominalreturnsforlarge-companystocks,wesetupthefollowingtwocolumnsinanExcelspreadsheet.Then,usetheExcelfunction“CORREL”tocalculatethecorrelationforthedata.DecadePrevious1930s-1.25%18.36%1940s9.11%-1.25%1950s19.41%9.11%1960s7.84%19.41%1970s5.90%7.84%1980s17.60%5.90%1990s18.20%17.60%Notethateachcorrelationisbasedononlysevenobservations,sowecannotarriveatanystatisticallysignificantconclusions.Lookingattheresults,however,itappearsthat,withtheexceptionoflarge-companystocks,thereispersistentserialcorrelation.(Thisconclusionchangeswhenweturntorealratesinthenextproblem.)28.Thetableforrealrates(usingtheapproximationofsubtractingadecade’saverageinflationfromthedecade’saveragenominalreturn)is:RealRatesSmallLargeLong-termIntermed-termTreasurycompanycompanygovernmentgovernmentbillsstocksstocksbondsbonds1920s-2.7219.364.984.774.561930s9.320.796.645.952.341940s15.273.75-1.77-3.66-4.991950s16.7917.19-1.97-1.11-0.351960s11.205.32-1.380.891.371970s1.39-1.46-0.73-1.25-1.071980s7.3612.506.406.913.901990s10.9115.275.674.812.09SerialCorrelation0.29-0.270.380.110.00Whiletheserialcorrelationindecadenominalreturnsseemstobepositive,itappearsthatrealratesareseriallyuncorrelated.Thedecadetimeseries(althoughagaintooshortforanydefinitiveconclusions)suggestthatrealratesofreturnareindependentfromdecadetodecade.2- 课后答案网（http://www.khdaw.com）29.Theprobabilitydistributionis:ProbabilityRateofReturn0.7100%0.3−50%Mean=[0.7×100]+[0.3×(−50)]=55%Variance=[0.7×(100−55)2]+[0.3×(−50−55)2]=4725Standarddeviation=47251/2=68.74%30.Sincewedonothaveanyinformationaboutexpectedreturns,wefocusexclusivelyonreducingvariability.StocksAandChaveequalstandarddeviations,butthecorrelationofStockBwithStockC(0.10)islessthanthatofStockAwithStockB(0.90).Therefore,aportfoliocomprisedofStocksBandCwillhavelowertotalriskthanaportfoliocomprisedofStocksAandB.31.c.CHAPTER8:INDEXMODELS1.a.Thetwofiguresdepictthestocks’securitycharacteristiclines(SCL).StockAhashigherfirm-specificriskbecausethedeviationsoftheobservationsfromtheSCLarelargerforStockAthanforStockB.DeviationsaremeasuredbytheverticaldistanceofeachobservationfromtheSCL.b.BetaistheslopeoftheSCL,whichisthemeasureofsystematicrisk.TheSCLforStockBissteeper;henceStockB’ssystematicriskisgreater.c.TheR2(orsquaredcorrelationcoefficient)oftheSCListheratiooftheexplainedvarianceofthestock’sreturntototalvariance,andthetotalvarianceisthesumoftheexplainedvarianceplustheunexplainedvariance(thestock’sresidualvariance):22βσ2iMR=222βσ+σ(e)iMiSincetheexplainedvarianceforStockBisgreaterthanforStockA(the22explainedvarianceisβσ,whichisgreatersinceitsbetaishigher),andBMitsresidualvarianceσ2(e2B)issmaller,itsRishigherthanStockA’s.2- 课后答案网（http://www.khdaw.com）d.AlphaistheinterceptoftheSCLwiththeexpectedreturnaxis.StockAhasasmallpositivealphawhereasStockBhasanegativealpha;hence,StockA’salphaislarger.e.ThecorrelationcoefficientissimplythesquarerootofR2,soStockB’scorrelationwiththemarketishigher.2.TheR2oftheregressionis:0.702=0.49Therefore,51%oftotalvarianceisunexplainedbythemarket;thisisnonsystematicrisk.3.d.4.b.2- 课后答案网（http://www.khdaw.com）5.a.Firm-specificriskismeasuredbytheresidualstandarddeviation.Thus,stockAhasmorefirm-specificrisk:10.3%>9.1%b.Marketriskismeasuredbybeta,theslopecoefficientoftheregression.Ahasalargerbetacoefficient:1.2>0.8c.R2measuresthefractionoftotalvarianceofreturnexplainedbythemarketreturn.A’sR2islargerthanB’s:0.576>0.436d.RewritingtheSCLequationintermsoftotalreturn(r)ratherthanexcessreturn(R):rA–rf=α+β(rM–rf)⇒rA=α+rf(1−β)+βrMTheinterceptisnowequalto:α+rf(1−β)=1+rf(l–1.2)Sincerf=6%,theinterceptwouldbe:1–1.2=–0.2%6.a.Tooptimizethisportfolioonewouldneed:n=60estimatesofmeansn=60estimatesofvariances2n−n=1,770estimatesofcovariances22n+3nTherefore,intotal:=1,890estimates2b.Inasingleindexmodel:ri−rf=αi+βi(rM–rf)+eiEquivalently,usingexcessreturns:Ri=αi+βiRM+eiThevarianceoftherateofreturnoneachstockcanbedecomposedintothecomponents:22(l)Thevarianceduetothecommonmarketfactor:βσiM2(2)Thevarianceduetofirmspecificunanticipatedevents:σ(e)iInthismodel:Cov(r,r)=ββσijij2- 课后答案网（http://www.khdaw.com）Thenumberofparameterestimatesis:n=60estimatesofthemeanE(r)in=60estimatesofthesensitivitycoefficientβin=60estimatesofthefirm-specificvarianceσ2(e)i1estimateofthemarketmeanE(rM)21estimateofthemarketvarianceσMTherefore,intotal,182estimates.Thus,thesingleindexmodelreducesthetotalnumberofrequiredparameterestimatesfrom1,890to182.Ingeneral,thenumberofparameterestimatesisreducedfrom:2⎛n+3n⎞⎜⎟to(3n+2)⎜⎟⎝2⎠7.a.Thestandarddeviationofeachindividualstockisgivenby:2221/2σ=[βσ+σ(e)]iiMiSinceβA=0.8,βB=1.2,σ(eA)=30%,σ(eB)=40%,andσM=22%,weget:σ2221/2A=(0.8×22+30)=34.78%σ2221/2B=(1.2×22+40)=47.93%b.Theexpectedrateofreturnonaportfolioistheweightedaverageoftheexpectedreturnsoftheindividualsecurities:E(rP)=wAE(rA)+wBE(rB)+wfrfwherewA,wB,andwfaretheportfolioweightsforStockA,StockB,andT-bills,respectively.Substitutingintheformulaweget:E(rP)=(0.30×13)+(0.45×18)+(0.25×8)=14%Thebetaofaportfolioissimilarlyaweightedaverageofthebetasoftheindividualsecurities:βP=wAβA+wBβB+wfβfThebetaforT-bills(βf)iszero.Thebetafortheportfolioistherefore:βP=(0.30×0.8)+(0.45×1.2)+0=0.782- 课后答案网（http://www.khdaw.com）Thevarianceofthisportfoliois:2222σ=βσ+σ(e)PPMP222whereβσisthesystematiccomponentandσ(e)isthenonsystematicPMPcomponent.Sincetheresiduals(ei)areuncorrelated,thenon-systematicvarianceis:2222222σ(e)=wσ(e)+wσ(e)+wσ(e)PAABBff=(0.302×302)+(0.452×402)+(0.252×0)=405whereσ2(e2A)andσ(eB)arethefirm-specific(nonsystematic)variancesofStocksAandB,andσ2(ef),thenonsystematicvarianceofT-bills,iszero.Theresidualstandarddeviationoftheportfolioisthus:σ(e1/2P)=(405)=20.12%Thetotalvarianceoftheportfolioisthen:222σ=(0.78×22)+405=699.47PThestandarddeviationis26.45%.8.ForStockA:αA=rA−[rf+βA(rM−rf)]=11−[6+0.8(12−6)]=0.2%ForstockB:αB=14−[6+1.5(12−6)]=−1%StockAwouldbeagoodadditiontoawell-diversifiedportfolio.AshortpositioninStockBmaybedesirable.9.ThestandarddeviationofeachstockcanbederivedfromthefollowingequationforR2:222βiσMExplainedvarianceR==i2σTotalvarianceiTherefore:22222βAσM0.7×20σ===980A2R0.20Aσ=31.30%A2- 课后答案网（http://www.khdaw.com）ForstockB:2221.2×20σ==4,800B0.12σ=69.28%B10.ThesystematicriskforAis:2222βσ=0.70×20=196AMThefirm-specificriskofA(theresidualvariance)isthedifferencebetweenA’stotalriskanditssystematicrisk:980–196=784ThesystematicriskforBis:2222βσ=1.20×20=576BMB’sfirm-specificrisk(residualvariance)is:4800–576=422411.ThecovariancebetweenthereturnsofAandBis(sincetheresidualsareassumedtobeuncorrelated):2Cov(r,r)=ββσ=0.70×1.20×400=336ABABMThecorrelationcoefficientbetweenthereturnsofAandBis:Cov(rA,rB)336ρ===0.155ABσσ31.30×69.28AB2212.NotethatthecorrelationisthesquarerootofR:ρ=RCov(r1/2A,rM)=ρσAσM=0.20×31.30×20=280Cov(r1/2B,rM)=ρσBσM=0.12×69.28×20=4802- 课后答案网（http://www.khdaw.com）13.ForportfolioPwecancompute:σ221/21/2P=[(0.6×980)+(0.4×4800)+(2×0.4×0.6×336]=[1282.08]=35.81%βP=(0.6×0.7)+(0.4×1.2)=0.9022222σ(e)=σ−βσ=1282.08−(0.90×400)=958.08PPPM2Cov(rP,rM)=βPσ=0.90×400=360MThissameresultcanalsobeattainedusingthecovariancesoftheindividualstockswiththemarket:Cov(rP,rM)=Cov(0.6rA+0.4rB,rM)=0.6Cov(rA,rM)+0.4Cov(rB,rM)=(0.6×280)+(0.4×480)=36014.NotethatthevarianceofT-billsiszero,andthecovarianceofT-billswithanyassetiszero.Therefore,forportfolioQ:[2222]1/2σ=wσ+wσ+2×w×w×Cov(r,r)QPPMMPMPM[22]1/2=(0.5×1,282.08)+(0.3×400)+(2×0.5×0.3×360)=21.55%β=wβ+wβ=(0.5×0.90)+(0.3×1)+0=0.75QPPMM22222σ(e)=σ−βσ=464.52−(0.75×400)=239.52QQQM2Cov(r,r)=βσ=0.75×400=300QMQM15.a.Alpha(α)Expectedexcessreturnαi=ri–[rf+βi(rM–rf)]E(ri)–rfα=20%–[8%+1.3(16%–8%)]=1.6%20%–8%=12%Aα=18%–[8%+1.8(16%–8%)]=–4.4%18%–8%=10%Bα=17%–[8%+0.7(16%–8%)]=3.4%17%–8%=9%Cα=12%–[8%+1.0(16%–8%)]=–4.0%12%–8%=4%DStocksAandChavepositivealphas,whereasstocksBandDhavenegativealphas.2- 课后答案网（http://www.khdaw.com）Theresidualvariancesare:σ2(e2A)=58=3,364σ2(e2B)=71=5,041σ2(e2C)=60=3,600σ2(e2D)=55=3,025b.Toconstructtheoptimalriskyportfolio,wefirstdeterminetheoptimalactiveportfolio.UsingtheTreynor-Blacktechnique,weconstructtheactiveportfolio:αα/σ2(e)σ2(e)Σα/σ2(e)A0.000476–0.6142B–0.0008731.1265C0.000944–1.2181D–0.0013221.7058Total–0.0007751.0000Donotbeconcernedthatthepositivealphastockshavenegativeweightsandviceversa.Wewillseethattheentirepositionintheactiveportfoliowillbenegative,returningeverythingtogoodorder.Withtheseweights,theforecastfortheactiveportfoliois:α=[–0.6142×1.6]+[1.1265×(–4.4)]–[1.2181×3.4]+[1.7058×(–4.0)]=–16.90%β=[–0.6142×1.3]+[1.1265×1.8]–[1.2181×0.70]+[1.7058×1]=2.08Thehighbeta(higherthananyindividualbeta)resultsfromtheshortpositionsintherelativelylowbetastocksandthelongpositionsintherelativelyhighbetastocks.σ2(e)=[(–0.6142)2×3364]+[1.12652×5041]+[(–1.2181)2×3600]+[1.70582×3025]=21,809.6σ(e)=147.68%Here,again,theleveredpositioninstockB[withhighσ2(e)]overcomesthediversificationeffect,andresultsinahighresidualstandarddeviation.2- 课后答案网（http://www.khdaw.com）Theoptimalriskyportfoliohasaproportionw*intheactiveportfolio,computedasfollows:2α/σ(e)−16.90/21,809.6w===−0.05124022[E(r)−r]/σ8/23MfMThenegativepositionisjustifiedforthereasonstatedearlier.Theadjustmentforbetais:w0−0.05124w*===−0.04861+(1−β)w1+(1−2.08)(−0.05124)0Sincew*isnegative,theresultisapositivepositioninstockswithpositivealphasandanegativepositioninstockswithnegativealphas.Thepositionintheindexportfoliois:1–(–0.0486)=1.0486c.TocalculateSharpe’smeasurefortheoptimalriskyportfolio,wecomputetheinformationratiofortheactiveportfolioandSharpe’smeasureforthemarketportfolio.Theinformationratiofortheactiveportfolioiscomputedasfollows:A=α/σ(e)=–16.90/147.68=–0.1144A2=0.0131Hence,thesquareofSharpe’smeasure(S)oftheoptimizedriskyportfoliois:2222⎛8⎞S=SM+A=⎜⎟+0.0131=0.1341⎝23⎠S=0.3662Comparethistothemarket’sSharpemeasure:SM=8/23=0.3478Thedifferenceis:0.0184Notethattheonly-moderateimprovementinperformanceresultsfromthefactthatonlyasmallpositionistakenintheactiveportfolioAbecauseofitslargeresidualvariance.2- 课后答案网（http://www.khdaw.com）d.Tocalculatetheexactmakeupofthecompleteportfolio,wefirstcomputethemeanexcessreturnoftheoptimalriskyportfolioanditsvariance.Theriskyportfoliobetaisgivenby:βP=wM+(wA×βA)=1.0486+[(–0.0486)×2.08]=0.95E(RP)=αP+βPE(RM)=[(–0.0486)×(–16.90%)]+(0.95×8%)=8.42%22222(2)σ=βσ+σ(e)=(0.95×23)+(−0.0486)×21,809.6=528.94PPMPσ=23.00%PSinceA=2.8,theoptimalpositioninthisportfoliois:8.42y==0.56850.01×2.8×528.94Incontrast,withapassivestrategy:8y==0.540120.01×2.8×23Thisisadifferenceof:0.0284Thefinalpositionsofthecompleteportfolioare:Bills1–0.5685=43.15%M0.5685×l.0486=59.61%A0.5685×(–0.0486)×(–0.6142)=1.70%B0.5685×(–0.0486)×1.1265=–3.11%C0.5685×(–0.0486)×(–1.2181)=3.37%D0.5685×(–0.0486)×1.7058=–4.71%100.00%[sumissubjecttoroundingerror]NotethatMmayincludepositiveproportionsofstocksAthroughD.16.a.Ifamanagerisnotallowedtosellshorthewillnotincludestockswithnegativealphasinhisportfolio,sohewillconsideronlyAandC:α2ασ2(e)α/σ(e)σ2(e)Σα/σ2(e)A1.63,3640.0004760.3352C3.43,6000.0009440.66480.0014201.00002- 课后答案网（http://www.khdaw.com）Theforecastfortheactiveportfoliois:α=(0.3352×1.6)+(0.6648×3.4)=2.80%β=(0.3352×1.3)+(0.6648×0.7)=0.90σ2(e)=(0.33522×3,364)+(0.66482×3,600)=1,969.03σ(e)=44.37%Theweightintheactiveportfoliois:2α/σ(e)2.80/1,969.03w===0.0940022E(R)/σ8/23MMAdjustingforbeta:w00.094w*===0.09311+(1−β)w1+[(1−0.90)×0.094]0Theinformationratiooftheactiveportfoliois:A=α/σ(e)=2.80/44.37=0.0631Hence,thesquareofSharpe’smeasureis:S2=(8/23)2+0.06312=0.1250Therefore:S=0.3535Themarket’sSharpemeasureis:SM=0.3478Whenshortsalesareallowed(Problem18),themanager’sSharpemeasureishigher(0.3662).ThereductionintheSharpemeasureisthecostoftheshortsalerestriction.Thecharacteristicsoftheoptimalriskyportfolioare:βP=wM+wA×βA=(1–0.0931)+(0.0931×0.9)=0.99E(RP)=αP+βPE(RM)=(0.0931×2.8%)+(0.99×8%)=8.18%222222σ=βσ+σ(e)=(0.99×23)+(0.0931×1,969.03)=535.54PPMPσ=23.14%PWithA=2.8,theoptimalpositioninthisportfoliois:8.18y==0.54550.01×2.8×535.542- 课后答案网（http://www.khdaw.com）Thefinalpositionsineachassetare:Bills1–0.5455=45.45%M0.5455×(1−0.0931)=49.47%A0.5455×0.0931×0.3352=1.70%C0.5455×0.0931×0.6648=3.38%100.00%b.Themeanandvarianceoftheoptimizedcompleteportfoliosintheunconstrainedandshort-salesconstrainedcases,andforthepassivestrategyare:2E(RC)σCUnconstrained0.5685×8.42=4.790.56852×528.94=170.95Constrained0.5455×8.18=4.460.54552×535.54=159.36Passive0.5401×8.00=4.320.54012×529.00=154.31Theutilitylevelsbelowarecomputedusingtheformula:2E(r)−0.005AσCCUnconstrained8+4.79–(0.005×2.8×170.95)=10.40Constrained8+4.46–(0.005×2.8×159.36)=10.23Passive8+4.32–(0.005×2.8×154.31)=10.1617.Allalphasarereducedto0.3timestheirvaluesintheoriginalcase.Therefore,therelativeweightsofeachsecurityintheactiveportfolioareunchanged,butthealphaoftheactiveportfolioisonly0.3timesitspreviousvalue:0.3×−16.90%=−5.07%Theinvestorwilltakeasmallerpositionintheactiveportfolio.Theoptimalriskyportfoliohasaproportionw*intheactiveportfolioasfollows:2α/σ(e)−5.07/21,809.6w===−0.01537022E(r−r)/σ8/23MfMThenegativepositionisjustifiedforthereasongivenearlier.Theadjustmentforbetais:w0−0.01537w*===−0.01511+(1−β)w1+[(1−2.08)×(−0.01537)]0Sincew*isnegative,theresultisapositivepositioninstockswithpositivealphasandanegativepositioninstockswithnegativealphas.Thepositionintheindexportfoliois:1–(–0.0151)=1.01512- 课后答案网（http://www.khdaw.com）TocalculateSharpe’smeasurefortheoptimalriskyportfoliowecomputetheinformationratiofortheactiveportfolioandSharpe’smeasureforthemarketportfolio.Theinformationratiooftheactiveportfoliois0.3timesitspreviousvalue:A=α/σ(e)=–5.07/147.68=–0.0343andA2=0.00118Hence,thesquareofSharpe’smeasureoftheoptimizedriskyportfoliois:S2=S22=(8/23)2+0.00118=0.1222M+AS=0.3495Comparethistothemarket’sSharpemeasure:SM=8/23=0.3478Thedifferenceis:0.0017Notethatthereductionoftheforecastalphasbyafactorof0.3reducedthesquaredinformationratioandtheimprovementinthesquaredSharperatiobyafactorof:0.32=0.0918.Theregressionresultsprovidequantitativemeasuresofreturnandriskbasedonmonthlyreturnsoverthefive-yearperiod.βforABCwas0.60,considerablylessthantheaveragestock’sβof1.0.Thisindicatesthat,whentheS&P500roseorfellby1percentagepoint,ABC’sreturnonaverageroseorfellbyonly0.60percentagepoint.Therefore,ABC’ssystematicrisk(ormarketrisk)waslowrelativetothetypicalvalueforstocks.ABC’salpha(theinterceptoftheregression)was–3.2%,indicatingthatwhenthemarketreturnwas0%,theaveragereturnonABCwas–3.2%.ABC’sunsystematicrisk(orresidualrisk),asmeasuredbyσ(e),was13.02%.ForABC,R2was0.35,indicatingclosenessoffittothelinearregressiongreaterthanthevalueforatypicalstock.βforXYZwassomewhathigher,at0.97,indicatingXYZ’sreturnpatternwasverysimilartotheβforthemarketindex.Therefore,XYZstockhadaveragesystematicriskfortheperiodexamined.AlphaforXYZwaspositiveandquitelarge,indicatingareturnofalmost7.3%,onaverage,forXYZindependentofmarketreturn.Residualriskwas21.45%,halfagainasmuchasABC’s,indicatingawiderscatterofobservationsaroundtheregressionlineforXYZ.Correspondingly,thefitoftheregressionmodelwasconsiderablylessthanthatofABC,consistentwithanR2ofonly0.17.2- 课后答案网（http://www.khdaw.com）Theeffectsofincludingoneortheotherofthesestocksinadiversifiedportfoliomaybequitedifferent.Ifitcanbeassumedthatbothstocks’betaswillremainstableovertime,thenthereisalargedifferenceinsystematicrisklevel.Thebetasobtainedfromthetwobrokeragehousesmayhelptheanalystdrawinferencesforthefuture.ThethreeestimatesofABC’sβaresimilar,regardlessofthesampleperiodoftheunderlyingdata.Therangeoftheseestimatesis0.60to0.71,wellbelowthemarketaverageβof1.0.ThethreeestimatesofXYZ’sβvarysignificantlyamongthethreesources,rangingashighas1.45fortheweeklydataoverthemostrecenttwoyears.OnecouldinferthatXYZ’sβforthefuturemightbewellabove1.0,meaningitmighthavesomewhatgreatersystematicriskthanwasimpliedbythemonthlyregressionforthefive-yearperiod.Thesestocksappeartohavesignificantlydifferentsystematicriskcharacteristics.Ifthesestocksareaddedtoadiversifiedportfolio,XYZwilladdmoretototalvolatility.19.9=3+β(11−3)⇒β=0.7520.a.MerrillLynchadjustsbetabytakingthesampleestimateofbetaandaveragingitwith1.0,usingtheweightsof2/3and1/3,asfollows:adjustedbeta=[(2/3)×1.24]+[(1/3)×1.0]=1.16b.Ifyouuseyourcurrentestimateofbetatobeβt–1=1.24,thenβt=0.3+(0.7×1.24)=1.168CHAPTER9:THECAPITALASSETPRICINGMODEL1.c.2.d.FromCAPM,thefairexpectedreturn=8+1.25(15−8)=16.75%Actuallyexpectedreturn=17%α=17−16.75=0.25%3.Sincethestock’sbetaisequalto1.2,itsexpectedrateofreturnis:6+[1.2×(16–6)]=18%D+P−P110E(r)=P02- 课后答案网（http://www.khdaw.com）6+P−5010.18=⇒P=$531504.Theseriesof$1,000paymentsisaperpetuity.Ifbetais0.5,thecashflowshouldbediscountedattherate:6+[0.5×(16–6)]=11%PV=$1,000/0.11=$9,090.91If,however,betaisequalto1,thentheinvestmentshouldyield16%,andthepricepaidforthefirmshouldbe:PV=$1,000/0.16=$6,250Thedifference,$2,840.91,istheamountyouwilloverpayifyouerroneouslyassumethatbetais0.5ratherthan1.5.UsingtheSML:4=6+β(16–6)⇒β=–2/10=–0.26.a.7.E(rP)=rf+βP[E(rM)–rf]18=6+βP(14–6)⇒βP=12/8=1.52- 课后答案网（http://www.khdaw.com）8.a.False.β=0impliesE(r)=rf,notzero.b.False.Investorsrequireariskpremiumonlyforbearingsystematic(undiversifiableormarket)risk.Totalvolatilityincludesdiversifiablerisk.c.False.Yourportfolioshouldbeinvested75%inthemarketportfolioand25%inT-bills.Then:βP=(0.75×1)+(0.25×0)=0.759.Notpossible.PortfolioAhasahigherbetathanPortfolioB,buttheexpectedreturnforPortfolioAislowerthantheexpectedreturnforPortfolioB.Thus,thesetwoportfolioscannotexistinequilibrium.10.Possible.IftheCAPMisvalid,theexpectedrateofreturncompensatesonlyforsystematic(market)risk,representedbybeta,ratherthanforthestandarddeviation,whichincludesnonsystematicrisk.Thus,PortfolioA’slowerrateofreturncanbepairedwithahigherstandarddeviation,aslongasA’sbetaislessthanB’s.11.Notpossible.Thereward-to-variabilityratioforPortfolioAisbetterthanthatofthemarket.ThisscenarioisimpossibleaccordingtotheCAPMbecausetheCAPMpredictsthatthemarketisthemostefficientportfolio.Usingthenumberssupplied:16−10S==0.5A1218−10S==0.33M24PortfolioAprovidesabetterrisk-rewardtradeoffthanthemarketportfolio.12.Notpossible.PortfolioAclearlydominatesthemarketportfolio.PortfolioAhasbothalowerstandarddeviationandahigherexpectedreturn.13.Notpossible.TheSMLforthisscenariois:E(r)=10+β(18–10)Portfolioswithbetaequalto1.5haveanexpectedreturnequalto:E(r)=10+[1.5×(18–10)]=22%TheexpectedreturnforPortfolioAis16%;thatis,PortfolioAplotsbelowtheSML(αA=–6%),andhence,isanoverpricedportfolio.ThisisinconsistentwiththeCAPM.2- 课后答案网（http://www.khdaw.com）2- 课后答案网（http://www.khdaw.com）14.Notpossible.TheSMListhesameasinProblem13.Here,PortfolioA’srequiredreturnis:10+(0.9×8)=17.2%Thisisgreaterthan16%.PortfolioAisoverpricedwithanegativealpha:αA=–1.2%15.Possible.TheCMListhesameasinProblem11.PortfolioAplotsbelowtheCML,asanyassetisexpectedto.ThisscenarioisnotinconsistentwiththeCAPM.16.Ifthesecurity’scorrelationcoefficientwiththemarketportfoliodoubles(withallothervariablessuchasvariancesunchanged),thenbeta,andthereforetheriskpremium,willalsodouble.Thecurrentriskpremiumis:14–6=8%Thenewriskpremiumwouldbe16%,andthenewdiscountrateforthesecuritywouldbe:16+6=22%Ifthestockpaysaconstantperpetualdividend,thenweknowfromtheoriginaldatathatthedividend(D)mustsatisfytheequationforthepresentvalueofaperpetuity:Price=Dividend/Discountrate50=D/0.14⇒D=50×0.14=$7.00Atthenewdiscountrateof22%,thestockwouldbeworth:$7/0.22=$31.82Theincreaseinstockriskhaslowereditsvalueby36.36%.17.d.18.a.Sincethemarketportfolio,bydefinition,hasabetaof1,itsexpectedrateofreturnis12%.b.β=0meansnosystematicrisk.Hence,thestock’sexpectedrateofreturninmarketequilibriumistherisk-freerate,5%.c.UsingtheSML,thefairexpectedrateofreturnforastockwithβ=–0.5is:E(r)=5+[(–0.5)(12–5)]=1.5%Theactuallyexpectedrateofreturn,usingtheexpectedpriceanddividendfornextyearis:E(r)=[($41+$1)/40]–1=0.10=10%Becausetheactuallyexpectedreturnexceedsthefairreturn,thestockisunderpriced.2- 课后答案网（http://www.khdaw.com）19.a.E(rP)=rf+βP[E(rM)–rf]=5%+0.8(15%−5%)=13%α=14%−13%=1%Youshouldinvestinthisfundbecausealphaispositive.b.Thepassiveportfoliowiththesamebetaasthefundshouldbeinvested80%inthemarket-indexportfolioand20%inthemoneymarketaccount.Forthisportfolio:E(rP)=(0.8×15%)+(0.2×5%)=13%14%−13%=1%=α20.d.[Youneedtoknowtherisk-freerate]21.d.[Youneedtoknowtherisk-freerate]22.a.ExpectedReturnAlphaStockX5%+0.8(14%−5%)=12.2%14.0%−12.2%=1.8%StockY5%+1.5(14%−5%)=18.5%17.0%−18.5%=−1.5%b.i.KayshouldrecommendStockXbecauseofitspositivealpha,comparedtoStockY,whichhasanegativealpha.Ingraphicalterms,theexpectedreturn/riskprofileforStockXplotsabovethesecuritymarketline(SML),whiletheprofileforStockYplotsbelowtheSML.Also,dependingontheindividualriskpreferencesofKay’sclients,thelowerbetaforStockXmayhaveabeneficialeffectonoverallportfoliorisk.ii.KayshouldrecommendStockYbecauseithashigherforecastedreturnandlowerstandarddeviationthanStockX.TherespectiveSharperatiosforStocksXandYandthemarketindexare:StockX:(14%−5%)/36%=0.25StockY:(17%−5%)/25%=0.48Marketindex:(14%−5%)/15%=0.60ThemarketindexhasanevenmoreattractiveSharperatiothaneitheroftheindividualstocks,but,giventhechoicebetweenStockXandStockY,StockYisthesuperioralternative.Whenastockisheldasasinglestockportfolio,standarddeviationistherelevantriskmeasure.Forsuchaportfolio,betaasariskmeasureisirrelevant.Althoughholdingasingleassetisnotatypicallyrecommendedinvestmentstrategy,someinvestorsmayholdwhatisessentiallyasingle-assetportfoliowhentheyholdthestockoftheiremployercompany.Forsuchinvestors,therelevanceofstandarddeviationversusbetaisan2- 课后答案网（http://www.khdaw.com）importantissue.2- 课后答案网（http://www.khdaw.com）23.Theappropriatediscountratefortheprojectis:rf+β[E(rM)–rf]=8+[1.8×(16–8)]=22.4%Usingthisdiscountrate:10150NPV=−400+∑tt=11.224=−400pesos+[150pesos×Annuityfactor(22.4%,10years)=180.92pesosTheinternalrateofreturn(IRR)fortheprojectis35.73%.RecallfromyourintroductoryfinanceclassthatNPVispositiveifIRR>discountrate(or,equivalently,hurdlerate).ThehighestvaluethatbetacantakebeforethehurdlerateexceedstheIRRisdeterminedby:35.73=8+β(16–8)⇒β=27.73/8=3.4724.a.McKayshouldborrowfundsandinvestthosefundsproportionatelyinMurray’sexistingportfolio(i.e.,buymoreriskyassetsonmargin).Inadditiontoincreasedexpectedreturn,thealternativeportfolioonthecapitalmarketlinewillalsohaveincreasedrisk,whichiscausedbythehigherproportionofriskyassetsinthetotalportfolio.b.McKayshouldsubstitutelowbetastocksforhighbetastocksinordertoreducetheoverallbetaofYork’sportfolio.Byreducingtheoverallportfoliobeta,McKaywillreducethesystematicriskoftheportfolio,andthereforereduceitsvolatilityrelativetothemarket.Thesecuritymarketline(SML)suggestssuchaction(i.e.,movingdowntheSML),eventhoughreducingbetamayresultinaslightlossofportfolioefficiencyunlessfulldiversificationismaintained.York’sprimaryobjective,however,isnottomaintainefficiency,buttoreduceriskexposure;reducingportfoliobetameetsthatobjective.BecauseYorkdoesnotwanttoengageinborrowingorlending,McKaycannotreduceriskbysellingequitiesandusingtheproceedstobuyrisk-freeassets(i.e.,lendingpartoftheportfolio).25.d.26.r1=19%;r2=16%;β1=1.5;β2=1a.Todeterminewhichinvestorwasabetterselectorofindividualstockswelookatabnormalreturn,whichistheex-postalpha;thatis,theabnormalreturnisthedifferencebetweentheactualreturnandthatpredictedbytheSML.Withoutinformationabouttheparametersofthisequation(risk-freerateandmarketrateofreturn)wecannotdeterminewhichinvestorwasmoreaccurate.2- 课后答案网（http://www.khdaw.com）b.Ifrf=6%andrM=14%,then(usingthenotationalphafortheabnormalreturn):α1=19–[6+1.5(14–6)]=19–18=1%α2=16–[6+1(14–6)]=16–14=2%Here,thesecondinvestorhasthelargerabnormalreturnandthusappearstobethesuperiorstockselector.Bymakingbetterpredictions,thesecondinvestorappearstohavetiltedhisportfoliotowardunderpricedstocks.c.Ifrf=3%andrM=15%,then:α1=19–[3+1.5(15–3)]=19–21=–2%α2=16–[3+1(15–3)]=16–15=1%Here,notonlydoesthesecondinvestorappeartobethesuperiorstockselector,butthefirstinvestor’spredictionsappearvalueless(orworse).27.Inthezero-betaCAPMthezero-betaportfolioreplacestherisk-freerate,andthus:E(r)=8+0.6(17–8)=13.4%28.a.CalltheaggressivestockAandthedefensivestockD.Betaisthesensitivityofthestock’sreturntothemarketreturn,i.e.,thechangeinthestockreturnperunitchangeinthemarketreturn.Therefore,wecomputeeachstock’sbetabycalculatingthedifferenceinitsreturnacrossthetwoscenariosdividedbythedifferenceinthemarketreturn:−2−38β==2.00A5−256−12β==0.30D5−25b.Withthetwoscenariosequallylikely,theexpectedreturnisanaverageofthetwopossibleoutcomes:E(rA)=0.5×(–2+38)=18%E(rD)=0.5×(6+12)=9%c.TheSMLisdeterminedbythemarketexpectedreturnof[0.5(25+5)]=15%,withabetaof1,andtheT-billreturnof6%withabetaofzero.Seethefollowinggraph.2- 课后答案网（http://www.khdaw.com）ExpectedReturn-BetaRelationship4035SML3025αA2015ADM10ExpectedReturn5000.511.522.53BetaTheequationforthesecuritymarketlineis:E(r)=6+β(15–6)d.Basedonitsrisk,theaggressivestockhasarequiredexpectedreturnof:E(rA)=6+2.0(15–6)=24%Theanalyst’sforecastofexpectedreturnisonly18%.Thusthestock’salphais:αA=actuallyexpectedreturn–requiredreturn(givenrisk)=18%–24%=–6%Similarly,therequiredreturnforthedefensivestockis:E(rD)=6+0.3(15–6)=8.7%Theanalyst’sforecastofexpectedreturnforDis9%,andhence,thestockhasapositivealpha:αD=actuallyexpectedreturn–requiredreturn(givenrisk)=9–8.7=+0.3%Thepointsforeachstockplotonthegraphasindicatedabove.e.Thehurdlerateisdeterminedbytheprojectbeta(0.3),notthefirm’sbeta.Thecorrectdiscountrateis8.7%,thefairrateofreturnforstockD.2- 课后答案网（http://www.khdaw.com）29.a.Agree;Regan’sconclusioniscorrect.Bydefinition,themarketportfolioliesonthecapitalmarketline(CML).Undertheassumptionsofcapitalmarkettheory,allportfoliosontheCMLdominate,inarisk-returnsense,portfoliosthatlieontheMarkowitzefficientfrontierbecause,giventhatleverageisallowed,theCMLcreatesaportfoliopossibilitylinethatishigherthanallpointsontheefficientfrontierexceptforthemarketportfolio,whichisRainbow’sportfolio.BecauseEagle’sportfolioliesontheMarkowitzefficientfrontieratapointotherthanthemarketportfolio,Rainbow’sportfoliodominatesEagle’sportfolio.b.Unsystematicriskistheuniqueriskofindividualstocksinaportfoliothatisdiversifiedawaybyholdingawell-diversifiedportfolio.Totalriskiscomposedofsystematic(market)riskandunsystematic(firm-specific)risk.Disagree;Wilson’sremarkisincorrect.BecausebothportfolioslieontheMarkowitzefficientfrontier,neitherEaglenorRainbowhasanyunsystematicrisk.Therefore,unsystematicriskdoesnotexplainthedifferentexpectedreturns.ThedeterminingfactoristhatRainbowliesonthe(straight)line(theCML)connectingtherisk-freeassetandthemarketportfolio(Rainbow),atthepointoftangencytotheMarkowitzefficientfrontierhavingthehighestreturnperunitofrisk.Wilson’sremarkisalsocounteredbythefactthat,sinceunsystematicriskcanbeeliminatedbydiversification,theexpectedreturnforbearingunsystematiciszero.Thisisaresultofthefactthatwell-diversifiedinvestorsbidupthepriceofeveryassettothepointwhereonlysystematicriskearnsapositivereturn(unsystematicriskearnsnoreturn).30.E(r)=rf+β×[E(rM)−rf]FuhrmanLabs:E(r)=5+1.5×[11.5−5.0]=14.75%GartenTesting:E(r)=5+0.8×[11.5−5.0]=10.20%Iftheforecastrateofreturnislessthan(greaterthan)therequiredrateofreturn,thenthesecurityisovervalued(undervalued).FuhrmanLabs:Forecastreturn–Requiredreturn=13.25%−14.75%=−1.50%GartenTesting:Forecastreturn–Requiredreturn=11.25%−10.20%=1.05%2- 课后答案网（http://www.khdaw.com）Therefore,FuhrmanLabsisovervaluedandGartenTestingisundervalued.31.UndertheCAPM,theonlyriskthatinvestorsarecompensatedforbearingistheriskthatcannotbediversifiedaway(systematicrisk).Becausesystematicrisk(measuredbybeta)isequalto1.0forbothportfolios,aninvestorwouldexpectthesamerateofreturnfrombothportfoliosAandB.Moreover,sincebothportfoliosarewelldiversified,itdoesn’tmatterifthespecificriskoftheindividualsecuritiesishighorlow.Thefirm-specificriskhasbeendiversifiedawayforbothportfolios.CHAPTER10:ARBITRAGEPRICINGTHEORYANDMULTIFACTORMODELSOFRISKANDRETURN22221a.σ=βσ+σ(e)M2222σ=(0.8×20)+25=881A2222σ=(1.0×20)+10=500B2222σ=(1.2×20)+20=976Cb.Ifthereareaninfinitenumberofassetswithidenticalcharacteristics,thenawell-diversifiedportfolioofeachtypewillhaveonlysystematicrisksincethenon-systematicriskwillapproachzerowithlargen.Themeanwillequalthatoftheindividual(identical)stocks.c.Thereisnoarbitrageopportunitybecausethewell-diversifiedportfoliosallplotonthesecuritymarketline(SML).Becausetheyarefairlypriced,thereisnoarbitrage.2.TheexpectedreturnforPortfolioFequalstherisk-freeratesinceitsbetaequals0.ForPortfolioA,theratioofriskpremiumtobetais:(12−6)/1.2=5ForPortfolioE,theratioislowerat:(8–6)/0.6=3.33Thisimpliesthatanarbitrageopportunityexists.Forinstance,youcancreateaPortfolioGwithbetaequalto0.6(thesameasE’s)bycombiningPortfolioAandPortfolioFinequalweights.TheexpectedreturnandbetaforPortfolioGarethen:E(rG)=(0.5×12%)+(0.5×6%)=9%βG=(0.5×1.2)+(0.5×0)=0.6ComparingPortfolioGtoPortfolioE,Ghasthesamebetaandhigherreturn.2- 课后答案网（http://www.khdaw.com）Therefore,anarbitrageopportunityexistsbybuyingPortfolioGandsellinganequalamountofPortfolioE.Theprofitforthisarbitragewillbe:rG–rE=[9%+(0.6×F)]−[8%+(0.6×F)]=1%Thatis,1%ofthefunds(longorshort)ineachportfolio.2- 课后答案网（http://www.khdaw.com）3.Substitutingtheportfolioreturnsandbetasintheexpectedreturn-betarelationship,weobtaintwoequationswithtwounknowns,therisk-freerate(rf)andthefactorriskpremium(RP):12=rf+(1.2×RP)9=rf+(0.8×RP)Solvingtheseequations,weobtain:rf=3%andRP=7.5%4.Equation10.9applieshere:E(rp)=rf+βP1[E(r1)−rf]+βP2[E(r2)–rf]Weneedtofindtheriskpremium(RP)foreachofthetwofactors:RP1=[E(r1)−rf]andRP2=[E(r2)−rf]Inordertodoso,wesolvethefollowingsystemoftwoequationswithtwounknowns:31=6+(1.5×RP1)+(2.0×RP2)27=6+(2.2×RP1)+[(–0.2)×RP2]Thesolutiontothissetofequationsis:RP1=10%andRP2=5%Thus,theexpectedreturn-betarelationshipis:E(rP)=6%+(βP1×10%)+(βP2×5%)5.a.Alongpositioninaportfolio(P)comprisedofPortfoliosAandBwillofferanexpectedreturn-betatradeofflyingonastraightlinebetweenpointsAandB.Therefore,wecanchooseweightssuchthatβP=βCbutwithexpectedreturnhigherthanthatofPortfolioC.Hence,combiningPwithashortpositioninCwillcreateanarbitrageportfoliowithzeroinvestment,zerobeta,andpositiverateofreturn.b.Theargumentinpart(a)leadstothepropositionthatthecoefficientofβ2mustbezeroinordertoprecludearbitrageopportunities.6.Therevisedestimateoftheexpectedrateofreturnonthestockwouldbetheoldestimateplusthesumoftheproductsoftheunexpectedchangeineachfactortimestherespectivesensitivitycoefficient:revisedestimate=12%+[(1×2%)+(0.5×3%)]=15.5%2- 课后答案网（http://www.khdaw.com）7.a.Shortinganequally-weightedportfolioofthetennegative-alphastocksandinvestingtheproceedsinanequally-weightedportfolioofthetenpositive-alphastockseliminatesthemarketexposureandcreatesazero-investmentportfolio.DenotingthesystematicmarketfactorasRM,theexpecteddollarreturnis(notingthattheexpectationofnon-systematicrisk,e,iszero):$1,000,000×[0.02+(1.0×RM)]−$1,000,000×[(–0.02)+(1.0×RM)]=$1,000,000×0.04=$40,000Thesensitivityofthepayoffofthisportfoliotothemarketfactoriszerobecausetheexposuresofthepositivealphaandnegativealphastockscancelout.(NoticethatthetermsinvolvingRMsumtozero.)Thus,thesystematiccomponentoftotalriskisalsozero.Thevarianceoftheanalyst’sprofitisnotzero,however,sincethisportfolioisnotwelldiversified.Forn=20stocks(i.e.,long10stocksandshort10stocks)theinvestorwillhavea$100,000position(eitherlongorshort)ineachstock.Netmarketexposureiszero,butfirm-specificriskhasnotbeenfullydiversified.Thevarianceofdollarreturnsfromthepositionsinthe20stocksis:20×[(100,000×0.30)2]=18,000,000,000Thestandarddeviationofdollarreturnsis$134,164.b.Ifn=50stocks(25stockslongand25stocksshort),theinvestorwillhavea$40,000positionineachstock,andthevarianceofdollarreturnsis:50×[(40,000×0.30)2]=7,200,000,000Thestandarddeviationofdollarreturnsis$84,853.Similarly,ifn=100stocks(50stockslongand50stocksshort),theinvestorwillhavea$20,000positionineachstock,andthevarianceofdollarreturnsis:100×[(20,000×0.30)2]=3,600,000,000Thestandarddeviationofdollarreturnsis$60,000.Noticethat,whenthenumberofstocksincreasesbyafactorof5(i.e.,from20to100),standarddeviationdecreasesbyafactorof5=2.23607(from$134,164to$60,000).8.a.Thisstatementisincorrect.TheCAPMrequiresamean-varianceefficientmarketportfolio,butAPTdoesnot.b.Thisstatementisincorrect.TheCAPMassumesnormallydistributedsecurityreturns,butAPTdoesnot.2- 课后答案网（http://www.khdaw.com）c.Thisstatementiscorrect.2- 课后答案网（http://www.khdaw.com）9.b.SincePortfolioXhasβ=1.0,thenXisthemarketportfolioandE(RM)=16%.UsingE(RM)=16%andrf=8%,theexpectedreturnforportfolioYisnotconsistent.10.a.E(r)=6+(1.2×6)+(0.5×8)+(0.3×3)=18.1%b.Surprisesinthemacroeconomicfactorswillresultinsurprisesinthereturnofthestock:Unexpectedreturnfrommacrofactors=[1.2(4–5)]+[0.5(6–3)]+[0.3(0–2)]=–0.3%E(r)=18.1%−0.3%=17.8%11.d.12.TheAPTfactorsmustcorrelatewithmajorsourcesofuncertainty,i.e.,sourcesofuncertaintythatareofconcerntomanyinvestors.Researchersshouldinvestigatefactorsthatcorrelatewithuncertaintyinconsumptionandinvestmentopportunities.GDP,theinflationrate,andinterestratesareamongthefactorsthatcanbeexpectedtodetermineriskpremiums.Inparticular,industrialproduction(IP)isagoodindicatorofchangesinthebusinesscycle.Thus,IPisacandidateforafactorthatishighlycorrelatedwithuncertaintiesthathavetodowithinvestmentandconsumptionopportunitiesintheeconomy.13.Thefirsttwofactorsseempromisingwithrespecttothelikelyimpactonthefirm’scostofcapital.Botharemacrofactorsthatwouldelicithedgingdemandsacrossbroadsectorsofinvestors.Thethirdfactor,whileimportanttoPorkProducts,isapoorchoiceforamultifactorSMLbecausethepriceofhogsisofminorimportancetomostinvestorsandisthereforehighlyunlikelytobeapricedriskfactor.Betterchoiceswouldfocusonvariablesthatinvestorsinaggregatemightfindmoreimportanttotheirwelfare.Examplesinclude:inflationuncertainty,short-terminterest-raterisk,energypricerisk,orexchangeraterisk.Theimportantpointhereisthat,inspecifyingamultifactorSML,wenotconfuseriskfactorsthatareimportanttoaparticularinvestorwithfactorsthatareimportanttoinvestorsingeneral;onlythelatterarelikelytocommandariskpremiuminthecapitalmarkets.14.c.Investorswilltakeonaslargeapositionaspossibleonlyifthemispricingopportunityisanarbitrage.Otherwise,considerationsofriskanddiversificationwilllimitthepositiontheyattempttotakeinthemispricedsecurity.2- 课后答案网（http://www.khdaw.com）15.d.16.d.17.TheAPTrequired(i.e.,equilibrium)rateofreturnonthestockbasedonrfandthefactorbetasis:requiredE(r)=6+(1×6)+(0.5×2)+(0.75×4)=16%Accordingtotheequationforthereturnonthestock,theactuallyexpectedreturnonthestockis15%(becausetheexpectedsurprisesonallfactorsarezerobydefinition).Becausetheactuallyexpectedreturnbasedonriskislessthantheequilibriumreturn,weconcludethatthestockisoverpriced.18.Anypatternofreturnscanbe“explained”ifwearefreetochooseanindefinitelylargenumberofexplanatoryfactors.Ifatheoryofassetpricingistohavevalue,itmustexplainreturnsusingareasonablylimitednumberofexplanatoryvariables(i.e.,systematicfactors).19.d.20.c.CHAPTER11:THEEFFICIENTMARKETHYPOTHESIS1.b.Thisisthedefinitionofanefficientmarket.2.c.Thisisaclassicfilterrulewhichshouldnotproducesuperiorreturnsinanefficientmarket.3.c.Thisisapredictablepatterninreturnswhichshouldnotoccuriftheweak-formEMHisvalid.4.Thecorrelationcoefficientbetweenstockreturnsintwodifferentcountriesfortwonon-overlappingperiodsshouldbezero.Ifnot,onecouldusereturnsfromoneperiodtopredictreturnsinlaterperiodsandmakeabnormalprofits.5.c.TheP/Eratioispublicinformationandshouldnotbepredictiveofabnormalsecurityreturns.2- 课后答案网（http://www.khdaw.com）6.d.Agradualadjustmenttofundamentalvalueswouldallowfortheuseofstrategiesbasedonpastpricemovementsinordertogenerateabnormalprofits.7.a.8.c.Arandomwalkimpliesthatstockpricechangesareunpredictable,usingpastpricechangesoranyotherdata.9.c.Inanefficientmarket,nosecuritiesareconsistentlyoverpricedorunderpriced.Whilesomesecuritieswillturnoutafteranyinvestmentperiodtohaveprovidedpositivealphas(i.e.,risk-adjustedabnormalreturns)andsomenegativealphas,thesepastreturnsarenotpredictiveoffuturereturns.10.b.Semi-strongformefficiencyimpliesthatmarketpricesreflectallpubliclyavailableinformationconcerningpasttradinghistoryaswellasfundamentalaspectsofthefirm.11.d.IflowP/Estockstendtohavepositiveabnormalreturns,thiswouldrepresentanunexploitedprofitopportunitythatwouldprovideevidencethatinvestorsarenotusingallavailableinformationtomakeprofitableinvestments.2- 课后答案网（http://www.khdaw.com）12.a.Thefullpriceadjustmentshouldoccurjustasthenewsaboutthedividendbecomespubliclyavailable.13.Expectedratesofreturnwilldifferbecauseofdifferentialriskpremiums.14.a.Consistent.Basedonpureluck,halfofallmanagersshouldbeatthemarketinanyyear.b.Inconsistent.Thiswouldbethebasisofan“easymoney”rule:simplyinvestwithlastyear"sbestmanagers.c.Consistent.Incontrasttopredictablereturns,predictablevolatilitydoesnotconveyameanstoearnabnormalreturns.d.Inconsistent.TheabnormalperformanceoughttooccurinJanuarywhenearningsareannounced.e.Inconsistent.Reversalsofferameanstoearneasymoney:justbuylastweek’slosers.15.Whilepositivebetastocksrespondwelltofavorablenewinformationabouttheeconomy’sprogressthroughthebusinesscycle,theyshouldnotshowabnormalreturnsaroundalreadyanticipatedevents.Ifarecovery,forexample,isalreadyanticipated,theactualrecoveryisnotnews.Thestockpricealreadyshouldreflectthecomingrecovery.16.Thequestionregardingmarketefficiencyiswhetherinvestorscanearnabnormalrisk-adjustedprofits.Ifthestockpricerun-upoccurswhenonlyinsidersareawareofthecomingdividendincrease,thenitisaviolationofstrong-form,butnotsemistrong-form,efficiency.Ifthepublicalreadyknowsoftheincrease,thenitisaviolationofsemistrong-formefficiency.17.No.Toyota’scontinuingprofitabilitydoesnotimplythatstockmarketinvestorswhopurchasedToyotasharesafteritssuccesswasalreadyevidentwouldhaveearnedanexceptionallyhighreturnontheirinvestments.18.b.Thecontrariantechnicalanalystwouldnoticethatotherinvestorshavebecomedeeplypessimisticaboutprices,andthereforewouldtakethisasabullishindicator,reasoningthatmarketsentimentswingstoowidelyandthatstockpricesreflectatoo-pessimisticviewoftheeconomy.2- 课后答案网（http://www.khdaw.com）19.Implicitinthedollar-costaveragingstrategyisthenotionthatstockpricesfluctuatearounda“normal”level.Otherwise,thereisnomeaningtostatementssuchas:“whenthepriceishigh.”Howdoweknow,forexample,whetherapriceof$25todaywillturnouttobeviewedashighorlowcomparedtothestockpricesixmonthsfromnow?20.Buy.Inyourview,thefirmisnotasbadaseveryoneelsebelievesittobe.Therefore,youviewthefirmasundervaluedbythemarket.Youarelesspessimisticaboutthefirm’sprospectsthanthebeliefsbuiltintothestockprice.21.No.Thevalueofdividendpredictabilitywouldbealreadyreflectedinthestockprice.22.Overthelonghaul,thereisanexpectedupwarddriftinstockpricesbasedontheirfairexpectedratesofreturn.Thefairexpectedreturnoveranysingledayisverysmall(e.g.,12%peryearisonlyabout0.03%perday),sothatonanydaythepriceisvirtuallyequallylikelytoriseorfall.However,overlongerperiods,thesmallexpecteddailyreturnsaccumulate,andupwardmovesareindeedmorelikelythandownwardones.23.Themarketrespondspositivelytonewnews.Iftheeventualrecoveryisanticipated,thentherecoveryisalreadyreflectedinstockprices.Onlyabetter-than-expectedrecoveryshouldaffectstockprices.24.Thereturnonthemarketis8%.Therefore,theforecastmonthlyreturnforGMis:0.10%+(1.1×8%)=8.9%GM’sactualreturnwas7%,sotheabnormalreturnwas–1.9%.25.a.Basedonbroadmarkettrends,theCAPMindicatesthatAmbChaserstockshouldhaveincreasedby:1.0%+2.0(1.5%–1.0%)=2.0%Itsfirm-specific(nonsystematic)returnduetothelawsuitis$1millionper$100millioninitialequity,or1%.Therefore,thetotalreturnshouldbe3%.(Itisassumedherethattheoutcomeofthelawsuithadazeroexpectedvalue.)b.Ifthesettlementwasexpectedtobe$2million,thentheactualsettlementwasa“$1milliondisappointment,”andsothefirm-specificreturnwouldbe–1%,foratotalreturnof2%–1%=1%.2- 课后答案网（http://www.khdaw.com）2- 课后答案网（http://www.khdaw.com）26.Givenmarketperformance,predictedreturnsonthetwostockswouldbe:Apex:0.2%+(1.4×3%)=4.4%Bpex:–0.1%+(0.6×3%)=1.7%Apexunderperformedthisprediction;Bpexoutperformedtheprediction.WeconcludethatBpexwonthelawsuit.27.a.E(rM)=12%,rf=4%andβ=0.5Therefore,theexpectedrateofreturnis:4%+0.5(12%–4%)=8%Ifthestockisfairlypriced,thenE(r)=8%.b.IfrMfallsshortofyourexpectationby2%(thatis,10%–12%)thenyouwouldexpectthereturnforChangingFortunesIndustriestofallshortofyouroriginalexpectationby:β×2%=1%Therefore,youwouldforecasta“revised”expectationforChangingFortunesof:8%–1%=7%c.Givenamarketreturnof10%,youwouldforecastareturnforChangingFortunesof7%.Theactualreturnis10%.Therefore,thesurpriseduetofirm-specificfactorsis10%–7%=3%whichweattributetothesettlement.Becausethefirmisinitiallyworth$100million,thesurpriseamountofthesettlementis3%of$100million,or$3million,implyingthatthepriorexpectationforthesettlementwasonly$2million.28.Themarketmayhaveanticipatedevengreaterearnings.Comparedtopriorexpectations,theannouncementwasadisappointment.29.a.Theefficientmarkethypothesis(EMH)statesthatamarketisefficientifsecuritypricesimmediatelyandfullyreflectallavailablerelevantinformation.Ifthemarketfullyreflectsinformation,theknowledgeofthatinformationwouldnotallowaninvestortoprofitfromtheinformationbecausestockpricesalreadyincorporatetheinformation.i.TheweakformoftheEMHassertsthatstockpricesreflectalltheinformationthatcanbederivedbyexaminingmarkettradingdatasuchasthehistoryofpastpricesandtradingvolume.Astrongbodyofevidencesupportsweak-formefficiencyinthemajorU.S.securitiesmarkets.Forexample,testresultssuggestthattechnicaltradingrulesdonotproducesuperiorreturnsafteradjustingfortransactioncostsandtaxes.2- 课后答案网（http://www.khdaw.com）ii.Thesemistrongformstatesthatafirm’sstockpricereflectsallpubliclyavailableinformationaboutafirm’sprospects.Examplesofpubliclyavailableinformationarecompanyannualreportsandinvestmentadvisorydata.Evidencestronglysupportsthenotionofsemistrongefficiency,butoccasionalstudies(e.g.,thoseidentifyingmarketanomaliessuchasthesmall-firm-in-Januaryorbook-to-marketeffects)andevents(suchasthestockmarketcrashofOctober19,1987)areinconsistentwiththisformofmarketefficiency.However,thereisaquestionconcerningtheextenttowhichthese“anomalies”resultfromdatamining.iii.ThestrongformoftheEMHholdsthatcurrentmarketpricesreflectallinformation(whetherpubliclyavailableorprivatelyheld)thatcanberelevanttothevaluationofthefirm.Empiricalevidencesuggeststhatstrong-formefficiencydoesnothold.Ifthisformwerecorrect,priceswouldfullyreflectallinformation.Thereforeeveninsiderscouldnotearnexcessreturns.Buttheevidenceisthatcorporateofficersdohaveaccesstopertinentinformationlongenoughbeforepublicreleasetoenablethemtoprofitfromtradingonthisinformation.b.i.Technicalanalysisinvolvesthesearchforrecurrentandpredictablepatternsinstockpricesinordertoenhancereturns.TheEMHimpliesthattechnicalanalysisiswithoutvalue.Ifpastpricescontainnousefulinformationforpredictingfutureprices,thereisnopointinfollowinganytechnicaltradingrule.ii.Fundamentalanalysisusesearningsanddividendprospectsofthefirm,expectationsoffutureinterestrates,andriskevaluationofthefirmtodetermineproperstockprices.TheEMHpredictsthatmostfundamentalanalysisisdoomedtofailure.Accordingtosemistrong-formefficiency,noinvestorcanearnexcessreturnsfromtradingrulesbasedonpubliclyavailableinformation.Onlyanalystswithuniqueinsightachievesuperiorreturns.Insummary,theEMHholdsthatthemarketappearstoadjustsoquicklytoinformationaboutbothindividualstocksandtheeconomyasawholethatnotechniqueofselectingaportfoliousingeithertechnicalorfundamentalanalysiscanconsistentlyoutperformastrategyofsimplybuyingandholdingadiversifiedportfolioofsecurities,suchasthosecomprisingthepopularmarketindexes.c.Portfoliomanagershaveseveralrolesandresponsibilitieseveninperfectlyefficientmarkets.Themostimportantresponsibilityistoidentifytherisk/returnobjectivesforaportfoliogiventheinvestor’sconstraints.Inanefficientmarket,portfoliomanagersareresponsiblefortailoringtheportfoliotomeettheinvestor’sneeds,ratherthantobeatthemarket,whichrequiresidentifyingtheclient’sreturnrequirementsandrisktolerance.Rational2- 课后答案网（http://www.khdaw.com）portfoliomanagementalsorequiresexaminingtheinvestor’sconstraints,includingliquidity,timehorizon,lawsandregulations,taxes,anduniquepreferencesandcircumstancessuchasageandemployment.2- 课后答案网（http://www.khdaw.com）30.Thenegativeabnormalreturns(downwarddriftinCAR)justpriortostockpurchasessuggestthatinsidersdeferredtheirpurchasesuntilafterbadnewswasreleasedtothepublic.Thisisevidenceofvaluableinsideinformation.Thepositiveabnormalreturnsafterpurchasesuggestinsiderpurchasesinanticipationofgoodnews.Theanalysisissymmetricforinsidersales.31.Hereweneedatwo-factormodelrelatingFord’sreturntothoseofboththebroadmarketandtheautoindustry.IfwecallrItheindustryreturn,thenwewouldfirstestimateparametersa,b,cinthefollowingregression:rFORD=a+brM+crI+eGiventheseestimateswewouldcalculateFord’sfirm-specificreturnas:rFORD−[a+brM+crI+e]Thisestimateoffirm-specificnewswouldmeasurethemarket’sassessmentofthepotentialprofitabilityofFord’snewmodel.32.a.SomeempiricalevidencethatsupportstheEMHis:(i)professionalmoneymanagersdonottypicallyearnhigherreturnsthancomparablerisk,passiveindexstrategies;(ii)eventstudiestypicallyshowthatstocksrespondimmediatelytothepublicreleaseofrelevantnews;(iii)mosttestsoftechnicalanalysisfindthatitisdifficulttoidentifypricetrendsthatcanbeexploitedtoearnsuperiorrisk-adjustedinvestmentreturns.b.SomeevidencethatisdifficulttoreconcilewiththeEMHconcernssimpleportfoliostrategiesthatapparentlywouldhaveprovidedhighrisk-adjustedreturnsinthepast.Someexamplesofportfolioswithattractivehistoricalreturns:(i)lowP/Estocks;(ii)highbook-to-marketratiostocks;(iii)smallfirmsinJanuary;(iv)firmswithverypoorstockpriceperformanceinthelastfewmonths.Otherevidenceconcernspost-earnings-announcementstockpricedriftandintermediate-termpricemomentum.c.Aninvestormightchoosenottoindexevenifmarketsareefficientbecauseheorshemaywanttotailoraportfoliotospecifictaxconsiderationsortospecificriskmanagementissues,forexample,theneedtohedge(oratleastnotaddto)exposuretoaparticularsourceofrisk(e.g.,industryexposure).2- 课后答案网（http://www.khdaw.com）33.a.Theearnings(anddividend)growthrateofgrowthstocksmaybeconsistentlyoverestimatedbyinvestors.Investorsmayextrapolaterecentgrowthtoofarintothefutureandtherebydownplaytheinevitableslowdown.Atanygiventime,growthstocksarelikelytorevertto(lower)meanreturnsandvaluestocksarelikelytorevertto(higher)meanreturns,oftenoveranextendedfuturetimehorizon.b.Inefficientmarkets,thecurrentpricesofstocksalreadyreflectallknownrelevantinformation.Inthissituation,growthstocksandvaluestocksprovidethesamerisk-adjustedexpectedreturn.CHAPTER15:THETERMSTRUCTUREOFINTERESTRATES1.Ingeneral,theforwardratecanbeviewedasthesumofthemarket’sexpectationofthefutureshortrateplusapotentialriskor‘liquidity’premium.Accordingtotheexpectationstheoryofthetermstructureofinterestrates,theliquiditypremiumiszerosothattheforwardrateisequaltothemarket’sexpectationofthefutureshortrate.Therefore,themarket’sexpectationoffutureshortrates(i.e.,forwardrates)canbederivedfromtheyieldcurve,andthereisnoriskpremiumforlongermaturities.Theliquiditypreferencetheory,ontheotherhand,specifiesthattheliquiditypremiumispositivesothattheforwardrateislessthanthemarket’sexpectationofthefutureshortrate.Thiscouldresultinanupwardslopingtermstructureevenifthemarketdoesnotanticipateanincreaseininterestrates.Theliquiditypreferencetheoryisbasedontheassumptionthatthefinancialmarketsaredominatedbyshort-terminvestorswhodemandapremiuminordertobeinducedtoinvestinlongmaturitysecurities.2.Expectationshypothesis:Theyieldsonlong-termbondsaregeometricaveragesofpresentandexpectedfutureshortrates.Anupwardslopingcurveisexplainedbyexpectedfutureshort-termratesbeinghigherthanthecurrentshort-termrate.Adownward-slopingyieldcurveimpliesexpectedfutureshortratesarelowerthanthecurrentshortrate.Thusbondsofdifferentmaturitieshavedifferentyieldsifexpectationsoffutureshortratesaredifferentfromthecurrentshortrate.Liquiditypreferencehypothesis:Yieldsonlong-termbondsaregreaterthantheexpectedreturnfromrolling-overshort-termbondsinordertocompensateinvestorsinlong-termbondsforbearinginterestraterisk.Thusbondsofdifferentmaturitiescanhavedifferentyieldsevenifexpectedfutureshort-termratesareallequaltothecurrentshort-termrate.Anupwardslopingyieldcurvecanbeconsistentevenwithexpectationsoffallingshortratesifliquiditypremiumsarehighenough.If,however,theyieldcurveisdownwardsloping2- 课后答案网（http://www.khdaw.com）andliquiditypremiumsareassumedtobepositive,thenwecanconcludethatfutureshortratesareexpectedtobelowerthanthecurrentshortrate.3.True.Undertheexpectationshypothesis,therearenoriskpremiabuiltintobondprices.Theonlyreasonforlong-termyieldstoexceedshort-termyieldsisanexpectationofhighershort-termratesinthefuture.4.d.5.Uncertain.Expectationsoflowerinflationwillusuallyleadtolowernominalinterestrates.Nevertheless,iftheliquiditypremiumissufficientlygreat,long-termyieldsmayexceedshort-termyieldsdespiteexpectationsoffallingshortrates.2- 课后答案网（http://www.khdaw.com）6.MaturityPriceYTMForwardRate1£943.406.00%2£898.475.50%(1.0552/1.06)–1=5.00%3£847.625.67%(1.05673/1.0552)–1=6.00%4£792.166.00%(1.064/1.05673)–1=7.00%7.Theexpectedpricepathofthe4-yearzerocouponbondisshownbelow.(Notethatwediscountthefacevaluebytheappropriatesequenceofforwardratesimpliedbythisyear’syieldcurve.)BeginningExpectedPriceExpectedRateofReturnofYear1£792.16(£839.69/£792.16)–1=6.00%1,0002=£839.69(£881.68/£839.69)–1=5.00%1.05×1.06×1.071,0003=£881.68(£934.58/£881.68)–1=6.00%1.06×1.071,0004=£934.58(£1,000.00/£934.58)–1=7.00%1.078.Thegivenratesareannualrates,buteachperiodisahalf-year.Therefore,theperperiodspotratesare2.5%onone-yearbondsand2%onsix-monthbonds.Thesemiannualforwardrateisobtainedbysolvingforfinthefollowingequation:21.0251+f==1.03001.02Thismeansthattheforwardrateis0.0300=3.00%semiannually,or6.00%annually.9.Thepresentvalueofeachbond’spaymentscanbederivedbydiscountingeachcashflowbytheappropriateratefromthespotinterestrate(i.e.,thepureyield)curve:$10$10$110BondA:PV=++=$98.53231.051.081.11$6$6$106BondB:PV=++=$88.36231.051.081.112- 课后答案网（http://www.khdaw.com）BondAsellsfor$0.13(i.e.,0.13%ofparvalue)lessthanthepresentvalueofitsstrippedpayments.BondBsellsfor$0.02lessthanthepresentvalueofitsstrippedpayments.BondAismoreattractivelypriced.2- 课后答案网（http://www.khdaw.com）10.a.(1+y434)=(1+y3)(1+f4)(1.055)4=(1.05)3(1+f4)1.2388=1.1576(1+f4)⇒f4=0.0701=7.01%b.Theconditionswouldbethosethatunderlietheexpectationstheoryofthetermstructure:riskneutralmarketparticipantswhoarewillingtosubstituteamongmaturitiessolelyonthebasisofyielddifferentials.Thisbehaviorwouldruleoutliquidityortermpremiarelatingtorisk.c.Undertheexpectationshypothesis,lowerimpliedforwardrateswouldindicatelowerexpectedfuturespotratesforthecorrespondingperiod.Sincethelowerexpectedfutureratesembodiedinthetermstructurearenominalrates,eitherlowerexpectedfuturerealratesorlowerexpectedfutureinflationrateswouldbeconsistentwiththespecifiedchangeintheobserved(implied)forwardrate.11.a.A3-yearzerocouponbondwithfacevalue$100willselltodayatayieldof7%andapriceof:$100/1.073=$81.63Nextyear,thebondwillhaveatwo-yearmaturity,andthereforeayieldof7%(fromnextyear’sforecastedyieldcurve).Thepricewillbe$87.34,resultinginaholdingperiodreturnof7%.b.Theforwardratesbasedontoday’syieldcurveareasfollows:YearForwardRate2(1.062/1.05)–1=7.01%3(1.073/1.062)–1=9.03%Usingtheforwardrates,theforecastfortheyieldcurvenextyearis:MaturityYTM17.01%2(1.0701×1.0903)1/2–1=8.02%ThemarketforecastisforahigherYTMon2–yearbondsthanyourforecast.Thus,themarketpredictsalowerpriceandhigherrateofreturn.2- 课后答案网（http://www.khdaw.com）12.a.Weobtainforwardratesfromthefollowingtable:MaturityYTMForwardRatePrice(forpartsc,d)1year10%$1,000/1.10=$909.092years11%(1.112/1.10)–1=12.01%$1,000/1.112=$811.623years12%(1.123/1.112)–1=14.03%$1,000/1.123=$711.78b.Weobtainnextyear’spricesandyieldsbydiscountingeachzero’sfacevalueattheforwardratesfornextyearthatwederivedinpart(a):MaturityPriceYTM1year$1,000/1.1201=$892.7812.01%2years$1,000/(1.1201×1.1403)=$782.9313.02%Notethatthisyear’supwardslopingyieldcurveimplies,accordingtotheexpectationshypothesis,ashiftupwardinnextyear’scurve.c.Nextyear,the2-yearzerowillbea1-yearzero,andwillthereforesellatapriceof:$1,000/1.1201=$892.78Similarly,thecurrent3-yearzerowillbea2-yearzeroandwillsellfor:$782.93Expectedtotalrateofreturn:$892.782-yearbond:−1=1.1000−1=10.00%$811.62$782.933-yearbond:−1=1.1000−1=10.00%$711.78d.Thecurrentpriceofthebondshouldequalthevalueofeachpaymenttimesthepresentvalueof$1tobereceivedatthe“maturity”ofthatpayment.Thepresentvalueschedulecanbetakendirectlyfromthepricesofzero-couponbondscalculatedabove.Currentprice=($120×0.90909)+($120×0.81162)+($1,120×0.71178)=$109.0908+$97.3944+$797.1936=$1,003.68Similarly,theexpectedpricesofzerosoneyearfromnowcanbeusedtocalculatetheexpectedbondvalueatthattime:Expectedprice1yearfromnow=($120×0.89278)+($1,120×0.78293)=$107.1336+$876.8816=$984.02Totalexpectedrateofreturn=$120+($984.02−$1,003.68)=0.1000=10.00%$1,003.682- 课后答案网（http://www.khdaw.com）810813.a.P=+=€101.8821.061.07b.Theyieldtomaturityisthesolutionforyinthefollowingequation:8108+=€101.8821+y(1+y)[Usingafinancialcalculator,entern=2;FV=100;PMT=8;PV=–101.88;Computei]YTM=6.961%c.Theforwardratefornextyear,derivedfromthezero-couponyieldcurve,isthesolutionforf2inthefollowingequation:2(1.07)1+f2==1.0801⇒f2=0.0801=8.01%1.06Therefore,usinganexpectedratefornextyearofr2=8.01%,wefindthattheforecastbondpriceis:108P==€99.991.0801d.Iftheliquiditypremiumis1%thentheforecastinterestrateis:E(r2)=f2–liquiditypremium=8.01%–1.00%=7.01%Theforecastofthebondpriceis:108=€100.931.070114.a.Thecurrentbondpriceis:(€85×0.94340)+(€85×0.87352)+(€1,085×0.81637)=€1,040.20Thispriceimpliesayieldtomaturityof6.97%,asshownbythefollowing:[€85×Annuityfactor(6.97%,3)]+[€1,000×PVfactor(6.97%,3)]=€1,040.17b.Ifoneyearfromnowy=8%,thenthebondpricewillbe:[€85×Annuityfactor(8%,2)]+[€1,000×PVfactor(8%,2)]=€1,008.92Theholdingperiodrateofreturnis:[€85+(€1,008.92–€1,040.20)]/€1,040.20=0.0516=5.16%2- 课后答案网（http://www.khdaw.com）15.ForwardYearPVofSFr1receivedatperiodendRate15%SFr1/1.05=SFr0.952426%SFr1/(1.05×1.06)=SFr0.898538%SFr1/(1.05×1.06×1.08)=SFr0.8319a.Price=(SFr60×0.9524)+(SFr60×0.8985)+(SFr1,060×0.8319)=SFr992.87b.Tofindtheyieldtomaturity,solveforyinthefollowingequation:SFr992.87=[SFr60×Annuityfactor(y,3)]+[SFr1,000×PVfactor(y,3)]Thiscanbesolvedusingafinancialcalculatortoshowthaty=6.27%c.PaymentreceivedWillgrowbyToafuturePeriodatendofperiod:afactorof:valueof:1SFr60.001.06×1.08SFr68.692SFr60.001.08SFr64.803SFr1,060.001.00SFr1,060.00SFr1,193.49SFr992.87×(1+y3realized)=SFr1,193.491/3⎛SFr1,193.49⎞1+yrealized=⎜⎟=1.0633⇒yrealized=6.33%⎝SFr992.87⎠d.Nextyear,thepriceofthebondwillbe:[SFr60×Annuityfactor(7%,2)]+[SFr1,000×PVfactor(7%,2)]=SFr981.92Therefore,therewillbeacapitallossequalto:SFr992.87–SFr981.92=SFr10.95SFr60+(−SFr10.95)Theholdingperiodreturnis:=0.0494=4.94%SFr992.872- 课后答案网（http://www.khdaw.com）16.Thefollowingtableshowstheexpectedshort-terminterestratebasedontheprojectionsofFederalReserveratecuts,thetermpremium(whichincreasesatarateof0.10%per12months),theforwardrate(whichisthesumoftheexpectedrateandtermpremium),andtheYTM,whichisthegeometricaverageoftheforwardrates.ExpectedTermForwardForwardrateYTMTimeshortratepremiumrate(annual)(semi-annual)(semi-annual)05.00%0.00%5.00%2.500%2.500%6months4.500.054.552.2752.38712months4.000.104.102.0502.27518months4.000.154.152.0752.22524months5.000.205.202.6002.30030months5.000.255.252.6252.354Thisanalysisispredicatedontheliquiditypreferencetheoryofthetermstructure,whichassertsthattheforwardrateinanyperiodisthesumoftheexpectedshortrateplustheliquiditypremium.17.a.Basedonthepureexpectationstheory,VanHusen’sconclusionisincorrect.Accordingtothistheory,theexpectedreturnoveranytimehorizonwouldbethesame,regardlessofthematuritystrategyemployed.b.Accordingtotheliquiditypreferencetheory,theshapeoftheyieldcurveimpliesthatshort-terminterestratesareexpectedtoriseinthefuture.Thistheoryassertsthatforwardratesreflectexpectationsaboutfutureinterestratesplusaliquiditypremiumthatincreaseswithmaturity.Giventheshapeoftheyieldcurveandtheliquiditypremiumdataprovided,theyieldcurvewouldstillbepositivelysloped(atleastthroughmaturityofeightyears)aftersubtractingtherespectiveliquiditypremiums:2.90%–0.55%=2.35%3.50%–0.55%=2.95%3.80%–0.65%=3.15%4.00%–0.75%=3.25%4.15%–0.90%=3.25%4.30%–1.10%=3.20%4.45%–1.20%=3.25%4.60%–1.50%=3.10%4.70%–1.60%=3.10%2- 课后答案网（http://www.khdaw.com）18.Thecouponbondscanbeviewedasportfoliosofstrippedzeros:eachcouponcanstandaloneasanindependentzero-couponbond.Therefore,yieldsoncouponbondsreflectyieldsonpaymentswithdatescorrespondingtoeachcoupon.Whentheyieldcurveisupwardsloping,couponbondshaveloweryieldsthanzeroswiththesamematuritybecausetheyieldstomaturityoncouponbondsreflecttheyieldsontheearlierinterimcouponpayments.19.Thepriceofthecouponbond,basedonitsyieldtomaturity,is:[$120×Annuityfactor(5.8%,2)]+[$1,000×PVfactor(5.8%,2)]=$1,113.99Ifthecouponswerestrippedandsoldseparatelyaszeros,then,basedontheyieldtomaturityofzeroswithmaturitiesofoneandtwoyears,respectively,thecouponpaymentscouldbesoldseparatelyfor:$120$1,120+=$1,111.0821.051.06Thearbitragestrategyistobuyzeroswithfacevaluesof$120and$1,120andrespectivematuritiesofoneyearandtwoyears,andsimultaneouslysellthecouponbond.Theprofitequals$2.91oneachbond.20.a.Theone-yearzero-couponbondhasayieldtomaturityof6%,asshownbelow:£100£94.34=⇒y1=0.06000=6.000%1+y1Theyieldonthetwo-yearzerois8.472%,asshownbelow:£100£84.99=⇒y2=0.08472=8.472%2(1+y)2£12£112Thepriceofthecouponbondis:+=£106.5121.06(1.08472)Therefore:yieldtomaturityforthecouponbond=8.333%[Onafinancialcalculator,enter:n=2;PV=–106.51;FV=100;PMT=12]22(1+y2)(1.08472)b.f=−1=−1=0.1100=11.00%21+y1.0612- 课后答案网（http://www.khdaw.com）112c.Expectedprice==€100.901.11(Notethatnextyear,thecouponbondwillhaveonepaymentleft.)Expectedholdingperiodreturn=12+(100.90−106.51)=0.0600=6.00%106.51Thisholdingperiodreturnisthesameasthereturnontheone-yearzero.d.Ifthereisaliquiditypremium,then:E(r2)$100.901+E(r)2E(HPR)>6%21.a.Thereturnontheone-yearzero-couponbondwillbe5.1%.Thepriceofthe4-yearzerotodayis:$1,000/1.0544=$810.28Nextyear,iftheyieldcurveisunchanged,today’s4-yearzerocouponbondwillhavea3-yearmaturity,aYTMof5.3%,andthereforethepricewillbe:$1,000/1.0533=$856.48Theresultingone-yearrateofreturnwillbe5.70%.Therefore,inthiscase,thelonger-termbondisexpectedtoprovidethehigherreturnbecauseitsYTMisexpectedtodeclineduringtheholdingperiod.b.Ifyoubelieveintheexpectationshypothesis,youwouldnotexpectthattheyieldcurvenextyearwillbethesameastoday’scurve.Theupwardslopeintoday"scurvewouldbeevidencethatexpectedshortratesarerisingandthattheyieldcurvewillshiftupward,reducingtheholdingperiodreturnonthefour-yearbond.Undertheexpectationshypothesis,allbondshaveequalexpectedholdingperiodreturns.Therefore,youwouldpredictthattheHPRforthe4-yearbondwouldbe5.1%,thesameasforthe1-yearbond.2- 课后答案网（http://www.khdaw.com）22.a.Five-yearSpotRate:$70$70$70$70$1,070$1,000=++++12345(1+y)(1+y)(1+y)(1+y)(1+y)12345$70$70$70$70$1,070$1,000=++++2345(1.05)(1.0521)(1.0605)(1.0716)(1+y)5$1,070$1,000=$66.67+$63.24+$58.69+$53.08+5(1+y)5$1,070$758.32=5(1+y)55$1,0705(1+y)=⇒y=1.411−1=7.13%55$758.32Five-yearForwardRate:5(1.0713)−1=1.0701−1=7.01%4(1.0716)b.Theyieldtomaturityisthesinglediscountratethatequatesthepresentvalueofaseriesofcashflowstoacurrentprice.Itistheinternalrateofreturn.Thespotrateforagivenperiodistheyieldtomaturityonazero-couponbondthatmaturesattheendoftheperiod.Aspotrateisthediscountrateforeachperiod.Spotratesareusedtodiscounteachcashflowofacouponbondinordertocalculateacurrentprice.Spotratesaretheratesappropriatefordiscountingfuturecashflowsofdifferentmaturities.Aforwardrateistheimplicitratethatlinksanytwospotrates.Forwardratesaredirectlyrelatedtospotrates,andthereforetoyieldtomaturity.Somewouldargue(asintheexpectationshypothesis)thatforwardratesarethemarketexpectationsoffutureinterestrates.Aforwardraterepresentsabreak-evenratethatlinkstwospotrates.Itisimportanttonotethatforwardrateslinkspotrates,notyieldstomaturity.Yieldtomaturityisnotuniqueforanyparticularmaturity.Inotherwords,twobondswiththesamematuritybutdifferentcouponratesmayhavedifferentyieldstomaturity.Incontrast,spotratesandforwardratesforeachdateareunique.2- 课后答案网（http://www.khdaw.com）c.The4-yearspotrateis7.16%.Therefore,7.16%isthetheoreticalyieldtomaturityforthezero-couponU.S.Treasurynote.Thepriceofthezero-couponnotediscountedat7.16%isthepresentvalueof$1,000tobereceivedin4years.Usingannualcompounding:$1,000PV==$758.354(1.0716)23.a.Thetwo-yearimpliedannuallycompoundedforwardrateforadeferredloanbeginningin3yearsiscalculatedasfollows:1/21/255⎡(1+y)⎤⎡1.05⎤5f3(2)=⎢3⎥−1=⎢3⎥−1=0.0716=7.16%⎣(1+y3)⎦⎣1.06⎦b.Assumingaparvalueof€1,000,thebondpriceiscalculatedasfollows:909090901,090P=++++12345(1+y)(1+y)(1+y)(1+y)(1+y)12345909090901,090=++++=1,165.6012345(1.08)(1.07)(1.06)(1.05)(1.05)24.a.MaturityForwardPriceYTM(years)rate1€925.938.00%2€853.398.25%8.50%3€782.928.50%9.00%4€715.008.75%9.50%5€650.009.00%10.00%b.Foreach3-yearzeroissuedtoday,usetheproceedstobuy:€782.92/€715.00=1.095four-yearzerosYourcashflowsarethusasfollows:TimeCashFlow0€03−€1,000The3-yearzeroissuedattime0matures;theissuerpaysout€1,000facevalue4+€1,095The4-yearzerospurchasedattime0mature;receivefacevalueThisisasyntheticone-yearloanoriginatingattime3.Therateonthesyntheticloanis0.095=9.5%,preciselytheforwardrateforyear3.2- 课后答案网（http://www.khdaw.com）2- 课后答案网（http://www.khdaw.com）c.Foreach4-yearzeroissuedtoday,usetheproceedstobuy:€$715.00€/$650.00=1.100five-yearzerosYourcashflowsarethusasfollows:TimeCashFlow0€04−€1,000The4-yearzeroissuedattime0matures;theissuerpaysout€1,000facevalue5+€1,100The5-yearzerospurchasedattime0mature;receivefacevalueThisisasyntheticone-yearloanoriginatingattime4.Therateonthesyntheticloanis0.100=10.0%,preciselytheforwardrateforyear4.25.a.Foreachthree-yearzeroyoubuytoday,issue:€782.92/€650.00=1.2045five-yearzerosThetime-0cashflowequalszero.b.Yourcashflowsarethusasfollows:TimeCashFlow0€03+€1,000.00The3-yearzeropurchasedattime0matures;receive€1,000facevalue5−€1,204.50The5-yearzerosissuedattime0mature;issuerpaysfacevalueThisisasynthetictwo-yearloanoriginatingattime3.c.Theeffectivetwo-yearinterestrateontheforwardloanis:€1,204.50/€1,000−1=0.2045=20.45%d.Theone-yearforwardratesforyears4and5are9.5%and10%,respectively.Noticethat:1.095×1.10=1.2045=1+(two-yearforwardrateonthe3-yearaheadforwardloan)The5-yearYTMis9.0%.The3-yearYTMis8.5%.Therefore,anotherwaytoderivethe2-yearforwardrateforaloanstartingattime3is:55(1+y5)1.09f(2)=−1=−1=0.2046=20.46%333(1+y)1.0853[Note:thereisaslightdiscrepancyhereduetoroundingerrorintheYTMcalculationsabove.]2- 课后答案网（http://www.khdaw.com）CHAPTER16:MANAGINGBONDPORTFOLIOS1.Thepercentagechangeinthebond’spriceis:Duration7.194−×∆y=−×0.005=−0.0327=−3.27%ora3.27%decline1+y1.102.a.YTM=6%(1)(2)(3)(4)(5)TimeuntilPVofCFColumn(1)×PaymentCashFlow(DiscountWeightColumn(4)(years)rate=6%)1$60.00$56.600.05660.05662$60.00$53.400.05340.10683$1,060.00$890.000.89002.6700ColumnSums$1,000.001.00002.8334Duration=2.833yearsb.YTM=10%(1)(2)(3)(4)(5)TimeuntilPVofCFColumn(1)×PaymentCashFlow(DiscountWeightColumn(4)(years)rate=10%)1$60.00$54.550.06060.06062$60.00$49.590.05510.11023$1,060.00$796.390.88442.6532ColumnSums$900.531.00002.8240Duration=2.824years,whichislessthanthedurationattheYTMof6%.2- 课后答案网（http://www.khdaw.com）3.Forasemiannual6%couponbondsellingatpar,weusethefollowingparameters:coupon=3%perhalf-yearperiod,y=3%,T=6semiannualperiods.(1)(2)(3)(4)(5)TimeuntilPVofCFColumn(1)×PaymentCashFlow(DiscountWeightColumn(4)(years)rate=3%)1$3.00$2.9130.029130.029132$3.00$2.8280.028280.056563$3.00$2.7450.027450.082364$3.00$2.6650.026650.106625$3.00$2.5880.025880.129396$103.00$86.2610.862615.17565ColumnSums$100.0001.000005.57971D=5.5797half-yearperiods=2.7899yearsIfthebond’syieldis10%,useasemiannualyieldof5%,andsemiannualcouponof3%:(1)(2)(3)(4)(5)TimeuntilPVofCFColumn(1)×PaymentCashFlow(DiscountWeightColumn(4)(years)rate=5%)1$3.00$2.8570.031800.031802$3.00$2.7210.030290.060573$3.00$2.5920.028840.086534$3.00$2.4680.027470.109885$3.00$2.3510.026160.130816$103.00$76.8600.855445.13265ColumnSums$89.8491.000005.55223D=5.5522half-yearperiods=2.7761years4.a.BondBhasahigheryieldtomaturitythanbondAsinceitscouponpaymentsandmaturityareequaltothoseofA,whileitspriceislower.(Perhapstheyieldishigherbecauseofdifferencesincreditrisk.)Therefore,thedurationofBondBmustbeshorter.b.BondAhasaloweryieldandalowercoupon,bothofwhichcauseBondAtohavealongerdurationthanBondB.Moreover,Acannotbecalled,sothatitsmaturityisatleastaslongasthatofB,whichgenerallyincreasesduration.2- 课后答案网（http://www.khdaw.com）5.a.(1)(2)(3)(4)(5)TimeuntilPVofCFColumn(1)×PaymentCashFlow(Discountrate=WeightColumn(4)(years)2%)1¥100million¥98.04million0.73020.73025¥40million¥36.23million0.26981.3491ColumnSums¥134.27million1.00002.0793D=2.0793years=requiredmaturityofzerocouponbond.b.Themarketvalueofthezeromustbe¥134.27million,thesameasthemarketvalueoftheobligations.Therefore,thefacevaluemustbe:¥134.27million×(1.02)2.0793=¥139.91million6.a.10×0.01×800=80.00b.½×120×(0.015)2=0.0135=1.35%c.9/1.10=8.18d.(i)e.(i)f.(iii)7.Ineachcase,choosethelonger-durationbondinordertobenefitfromaratedecrease.a.TheAaa-ratedbondhastheloweryieldtomaturityandthereforethelongerduration.b.Thelower-couponbondhasthelongerdurationandgreaterdefactocallprotection.c.Thelowercouponbondhasthelongerduration.8.a.Thecallfeatureprovidesavaluableoptiontotheissuer,sinceitcanbuybackthebondataspecifiedcallpriceevenifthepresentvalueofthescheduledremainingpaymentsisgreaterthanthecallprice.Theinvestorwilldemand,andtheissuerwillbewillingtopay,ahigheryieldontheissueascompensationforthisfeature.2- 课后答案网（http://www.khdaw.com）b.Thecallfeaturereducesboththeduration(interestratesensitivity)andtheconvexityofthebond.Ifinterestratesfall,theincreaseinthepriceofthecallablebondwillnotbeaslargeasitwouldbeifthebondwerenoncallable.Moreover,theusualcurvaturethatcharacterizespricechangesforastraightbondisreducedbyacallfeature.Theprice-yieldcurve(seeFigure16.6)flattensoutastheinterestratefallsandtheoptiontocallthebondbecomesmoreattractive.Infact,atverylowinterestrates,thebondexhibitsnegativeconvexity.9.a.PVoftheobligation=$10,000×Annuityfactor(8%,2)=$17,832.65(1)(2)(3)(4)(5)TimeuntilPVofCFColumn(1)×PaymentCashFlow(DiscountWeightColumn(4)(years)rate=8%)1$10,000.00$9,259.2590.519230.519232$10,000.00$8,573.3880.480770.96154ColumnSums$17,832.6471.000001.48077Duration=1.4808yearsb.Azero-couponbondmaturingin1.4808yearswouldimmunizetheobligation.Sincethepresentvalueofthezero-couponbondmustbe$17,832.65,thefacevalue(i.e.,thefutureredemptionvalue)mustbe:$17,832.65×1.081.4808=$19,985.26c.Iftheinterestrateincreasesto9%,thezero-couponbondwoulddecreaseinvalueto:$19,985.26=$17,590.921.48081.09Thepresentvalueofthetuitionobligationwoulddecreaseto$17,591.11.Thenetpositiondecreasesinvalueby$0.19.Iftheinterestratedecreasesto7%,thezero-couponbondwouldincreaseinvalueto:$19,985.26=$18,079.991.48081.07Thepresentvalueofthetuitionobligationwouldincreaseto$18,080.18.Thenetpositiondecreasesinvalueby$0.19.Thereasonthenetpositionchangesatallisthat,astheinterestratechanges,sodoesthedurationofthestreamoftuitionpayments.2- 课后答案网（http://www.khdaw.com）10.a.(i)Currentyield=Coupon/Price=€70/€960=0.0729=7.29%(ii)YTM=3.993%semiannuallyor7.986%annualbondequivalentyield.Onafinancialcalculator,enter:n=10;PV=–960;FV=1000;PMT=35Computetheinterestrate.(iii)Horizonyieldorrealizedcompoundyieldis4.166%(semiannually),or8.332%annualbondequivalentyield.Toobtainthisvalue,firstfindthefuturevalue(FV)ofreinvestedcouponsandprincipal.Therewillbesixpaymentsof€35each,reinvestedsemiannuallyat3%perperiod.Onafinancialcalculator,enter:PV=0;PMT=€35;n=6;i=3%.Compute:FV=€226.39Threeyearsfromnow,thebondwillbesellingattheparvalueof€1,000becausetheyieldtomaturityisforecasttoequalthecouponrate.Therefore,totalproceedsinthreeyearswillbe€1,226.39.Thenfindtherate(yrealized)thatmakestheFVofthepurchasepriceequalto€1,226.39:€960×(1+y6realized)=€1,226.39⇒yrealized=4.166%(semiannual)b.Shortcomingsofeachmeasure:(i)Currentyielddoesnotaccountforcapitalgainsorlossesonbondsboughtatpricesotherthanparvalue.Italsodoesnotaccountforreinvestmentincomeoncouponpayments.(ii)Yieldtomaturityassumesthebondishelduntilmaturityandthatallcouponincomecanbereinvestedatarateequaltotheyieldtomaturity.(iii)Horizonyieldorrealizedcompoundyieldisaffectedbytheforecastofreinvestmentrates,holdingperiod,andyieldofthebondattheendoftheinvestor"sholdingperiod.Note:Thiscriticismofhorizonyieldisabitunfair:whileYTMcanbecalculatedwithoutexplicitassumptionsregardingfutureYTMandreinvestmentrates,youimplicitlyassumethatthesevaluesequalthecurrentYTMifyouuseYTMasameasureofexpectedreturn.11.Thetablebelowshowstheholdingperiodreturnsforeachofthethreebonds:Maturity1year2years3yearsYTMatbeginningofyear7.00%8.00%9.00%Beginningofyearprices$1,009.35$1,000.00$974.69Pricesatyearend(at9%YTM)$1,000.00$990.83$982.41Capitalgain–$9.35–$9.17$7.72Coupon$80.00$80.00$80.001-yeartotal$return$70.65$70.83$87.721-yeartotalrateofreturn7.00%7.08%9.00%Youshouldbuythe3-yearbondbecauseitprovidesa9%holding-periodreturnoverthenextyear,whichisgreaterthanthereturnoneitheroftheotherbonds.2- 课后答案网（http://www.khdaw.com）Macaulayduration1012.a.Modifiedduration===9.26years1+YTM1.08b.Foroption-freecouponbonds,modifieddurationisabettermeasureofthebond’ssensitivitytochangesininterestrates.Maturityconsidersonlythefinalcashflow,whilemodifieddurationincludesotherfactors,suchasthesizeandtimingofcouponpayments,andthelevelofinterestrates(yieldtomaturity).Modifiedduration,unlikematurity,indicatestheapproximatepercentagechangeinthebondpriceforagivenchangeinyieldtomaturity.c.i.Modifieddurationincreasesasthecoupondecreases.ii.Modifieddurationdecreasesasmaturitydecreases.d.Convexitymeasuresthecurvatureofthebond’sprice-yieldcurve.Suchcurvaturemeansthatthedurationruleforbondpricechange(whichisbasedonlyontheslopeofthecurveattheoriginalyield)isonlyanapproximation.Addingatermtoaccountfortheconvexityofthebondincreasestheaccuracyoftheapproximation.Thatconvexityadjustmentisthelastterminthefollowingequation:∆P⎡12⎤=(−D*×∆y)+×Convexity×(∆y)⎢⎥P⎣2⎦13.a.PVofobligation=¥200million/0.16=¥1,250millionDurationofobligation=1.16/0.16=7.25yearsCallwtheweightonthe5-yearmaturitybond(whichhasdurationof4years).Then:(w×4)+[(1–w)×11]=7.25⇒w=0.5357Therefore:0.5357×¥1,250=¥669.6millioninthe5-yearbond,and;0.4643×¥1,250=¥580.4millioninthe20-yearbond.b.Thepriceofthe20-yearbond(per¥100offacevalue)is:[6×Annuityfactor(16%,20)]+[100×PVfactor(16%,20)]=¥40.71Therefore,thebondsellsfor0.4071timesitsparvalue,and:Marketvalue=Parvalue×0.4071¥580.4million=Parvalue×0.4071⇒Parvalue=¥1,425.7million2- 课后答案网（http://www.khdaw.com）14.a.Thedurationoftheperpetuityis:1.04/0.04=26yearsCallwtheweightofthezero-couponbond.Then:(w×5)+[(1–w)×26]=10⇒w=16/21=0.7619Therefore,theportfolioweightswouldbeasfollows:16/21investedinthezeroand5/21intheperpetuity.b.Nextyear,thezero-couponbondwillhaveadurationof4yearsandtheperpetuitywillstillhavea26-yearduration.Toobtainthetargetdurationofnineyears,whichisnowthedurationoftheobligation,weagainsolveforw:(w×4)+[(1–w)×26]=9⇒w=17/22=0.7727So,theproportionoftheportfolioinvestedinthezeroincreasesto17/22andtheproportioninvestedintheperpetuityfallsto5/22.15.a.TheAabondinitiallyhasahigherYTM(yieldspreadof40b.p.versus31b.p.),butitisexpectedtohaveawideningspreadrelativetoTreasuries.Thiswillreducetherateofreturn.TheAaaspreadisexpectedtobestable.Calculatecomparativereturnsasfollows:IncrementalreturnoverTreasuries=Incrementalyieldspread−(Changeinspread×duration)Aaabond:31bp−(0×3.1years)=31bpAabond:40bp−(10bp×3.1years)=9bpTherefore,choosetheAaabond.b.Othervariablestobeconsidered:•Potentialchangesinissue-specificcreditquality.Ifthecreditqualityofthebondschanges,spreadsrelativetoTreasurieswillalsochange.•Changesinrelativeyieldspreadsforagivenbondrating.Ifqualityspreadsinthegeneralbondmarketchangebecauseofchangesinrequiredriskpremiums,theyieldspreadsofthebondswillchangeevenifthereisnochangeintheassessmentofthecreditqualityoftheseparticularbonds.•Maturityeffect.Asbondsneartheirmaturity,theeffectofcreditqualityonspreadscanalsochange.Thiscanaffectbondsofdifferentinitialcreditqualitydifferently.2- 课后答案网（http://www.khdaw.com）16.a.i.Theeffectivedurationofthe4.75%Treasurysecurityis:∆P/P(116.887−86.372)/100−==15.2575∆r0.02ii.Thedurationoftheportfolioistheweightedaverageofthedurationsoftheindividualbondsintheportfolio:PortfolioDuration=w1D1+w2D2+w3D3+…+wkDkwherewi=marketvalueofbondi/marketvalueoftheportfolioDi=durationofbondik=numberofbondsintheportfolioTheeffectivedurationofthebondportfolioiscalculatedasfollows:[($48,667,680/$98,667,680)×2.15]+[($50,000,000/$98,667,680)×15.26]=8.79b.VanHusen’sremarkswouldbecorrectiftherewereasmall,parallelshiftinyields.Durationisafirst(linear)approximationonlyforsmallchangesinyield.Forlargerchangesinyield,theconvexitymeasureisneededinordertoapproximatethechangeinpricethatisnotexplainedbyduration.Additionally,portfoliodurationassumesthatallyieldschangebythesamenumberofbasispoints(parallelshift),soanynon-parallelshiftinyieldswouldresultinadifferenceinthepricesensitivityoftheportfoliocomparedtothepricesensitivityofasinglesecurityhavingthesameduration.17.a.Thedurationoftheannuityifitweretostartinoneyearwouldbe:(1)(2)(3)(4)(5)TimeuntilPVofCFColumn(1)×PaymentCashFlow(DiscountWeightColumn(4)(years)rate=10%)1€18,000€16,363.6360.147950.147952€18,000€14,876.0330.134500.269003€18,000€13,523.6660.122270.366824€18,000€12,294.2420.111160.444635€18,000€11,176.5840.101050.505266€18,000€10,160.5310.091870.551197€18,000€9,236.8460.083510.584608€18,000€8,397.1330.075920.607389€18,000€7,633.7570.069020.6211810€18,000€6,939.7790.062750.62745ColumnSums€110,602.2081.000004.72546D=4.7255yearsBecausethepaymentstreamstartsinfiveyears,insteadofoneyear,weaddfouryearstotheduration,sothedurationis8.7255years.2- 课后答案网（http://www.khdaw.com）b.Thepresentvalueofthedeferredannuityis:18,000×Annuityfactor(10%,10)=€75,54341.10Callwtheweightoftheportfolioinvestedinthe5-yearzero.Then:(w×5)+[(1–w)×20]=8.7255⇒w=0.7516Theinvestmentinthe5-yearzeroisequalto:0.7516×€75,543=€56,778Theinvestmentinthe20-yearzerosisequalto:0.2484×€75,543=€18,765Thesearethepresentormarketvaluesofeachinvestment.Thefacevaluesareequaltotherespectivefuturevaluesoftheinvestments.Thefacevalueofthe5-yearzerosis:€56,778×(1.10)5=€91,442Therefore,between50and51zero-couponbonds,eachofparvalue€1,000,wouldbepurchased.Similarly,thefacevalueofthe20-yearzerosis:€18,765×(1.10)20=€126,24218.Usingafinancialcalculator,wefindthattheactualpriceofthebondasafunctionofyieldtomaturityis:YieldtomaturityPrice7%$1,620.458%$1,450.319%$1,308.21UsingtheDurationRule,assumingyieldtomaturityfallsto7%:⎛Duration⎞Predictedpricechange=⎜−⎟×∆y×P⎜⎟0⎝1+y⎠⎛11.54⎞=⎜−⎟×(−0.01)×$1,450.31=$154.97⎝1.08⎠Therefore:predictednewprice=$1,450.31+$154.97=$1,605.28Theactualpriceata7%yieldtomaturityis$1,620.45.Therefore:$1,605.28−$1,620.45%error==−0.0094=−0.94%(approximationistoolow)$1,620.452- 课后答案网（http://www.khdaw.com）UsingtheDurationRule,assumingyieldtomaturityincreasesto9%:⎛Duration⎞Predictedpricechange=⎜−⎟×∆y×P⎜⎟0⎝1+y⎠⎛11.54⎞=⎜−⎟×0.01×$1,450.31=−$154.97⎝1.08⎠Therefore:predictednewprice=$1,450.31–$154.97=$1,295.34Theactualpriceata9%yieldtomaturityis$1,308.21.Therefore:$1,295.34−$1,308.21%error==−0.0098=−0.98%(approximationistoolow)$1,308.21UsingDuration-with-ConvexityRule,assumingyieldtomaturityfallsto7%⎧⎪⎡⎛Duration⎞⎤2⎫⎪Predictedpricechange=⎨⎢⎜⎜−⎟⎟×∆y⎥+[0.5×Convexity×(∆y)]⎬×P0⎪⎩⎣⎝1+y⎠⎦⎪⎭⎧⎡⎛11.54⎞⎤[2]⎫=⎨⎢⎜−⎟×(−0.01)⎥+0.5×192.4×(−0.01)⎬×$1,450.31=$168.92⎩⎣⎝1.08⎠⎦⎭Therefore:predictednewprice=$1,450.31+$168.92=$1,619.23Theactualpriceata7%yieldtomaturityis$1,620.45.Therefore:$1,619.23−$1,620.45%error==−0.00075=−0.075%(approximationistoolow)$1,620.45UsingDuration-with-ConvexityRule,assumingyieldtomaturityrisesto9%:⎧⎪⎡⎛Duration⎞⎤2⎫⎪Predictedpricechange=⎨⎢⎜⎜−⎟⎟×∆y⎥+[0.5×Convexity×(∆y)]⎬×P0⎪⎩⎣⎝1+y⎠⎦⎪⎭⎧⎡⎛11.54⎞⎤[2]⎫=⎨⎢⎜−⎟×0.01⎥+0.5×192.4×(0.01)⎬×$1,450.31=−$141.02⎩⎣⎝1.08⎠⎦⎭Therefore:predictednewprice=$1,450.31–$141.02=$1,309.29Theactualpriceata9%yieldtomaturityis$1,308.21.Therefore:$1,309.29−$1,308.21%error==0.00083=0.083%(approximationistoohigh)$1,308.21Conclusion:Theduration-with-convexityruleprovidesmoreaccurateapproximationstothetruechangeinprice.Inthisexample,thepercentageerrorusingconvexitywithdurationislessthanone-tenththeerrorusingonlydurationtoestimatethepricechange.2- 课后答案网（http://www.khdaw.com）19.a.Thepriceofthezerocouponbond($1,000facevalue)sellingatayieldtomaturityof8%is$374.84andthepriceofthecouponbondis$774.84.AtaYTMof9%theactualpriceofthezerocouponbondis$333.28andtheactualpriceofthecouponbondis$691.79.Zerocouponbond:$333.28−$374.84Actual%loss==−0.1109=11.09%loss$374.84Thepercentagelosspredictedbytheduration-with-convexityruleis:2Predicted%loss=[(−11.81)×0.01]+[0.5×150.3×0.01]=−0.1106=11.06%lossCouponbond:$691.79−$774.84Actual%loss==−0.1072=10.72%loss$774.84Thepercentagelosspredictedbytheduration-with-convexityruleis:2Predicted%loss=[(−11.79)×0.01]+[0.5×231.2×0.01]=−0.1063=10.63%lossb.Nowassumeyieldtomaturityfallsto7%.Thepriceofthezeroincreasesto$422.04,andthepriceofthecouponbondincreasesto$875.91.Zerocouponbond:$422.04−$374.84Actual%gain==0.1259=12.59%gain$374.84Thepercentagegainpredictedbytheduration-with-convexityruleis:2Predicted%gain=[(−11.81)×(−0.01)]+[0.5×150.3×0.01]=0.1256=12.56%gainCouponbond$875.91−$774.84Actual%gain==0.1304=13.04%gain$774.84Thepercentagegainpredictedbytheduration-with-convexityruleis:2Predicted%gain=[(−11.79)×(−0.01)]+[0.5×231.2×0.01]=0.1295=12.95%gainc.The6%couponbond,whichhashigherconvexity,outperformsthezeroregardlessofwhetherratesriseorfall.Thiscanbeseentobeageneralpropertyusingtheduration-with-convexityformula:thedurationeffectsonthetwobondsduetoanychangeinratesareequal(sincetherespectivedurationsarevirtuallyequal),buttheconvexityeffect,whichisalwayspositive,alwaysfavorsthehigherconvexitybond.Thus,iftheyieldsonthebondschangebyequalamounts,asweassumedinthisexample,thehigherconvexitybondoutperformsalowerconvexitybondwiththesamedurationandinitialyieldtomaturity.2- 课后答案网（http://www.khdaw.com）d.Thissituationcannotpersist.Noonewouldbewillingtobuythelowerconvexitybondifitalwaysunderperformstheotherbond.Thepriceofthelowerconvexitybondwillfallanditsyieldtomaturitywillrise.Thus,thelowerconvexitybondwillsellatahigherinitialyieldtomaturity.Thathigheryieldiscompensationforlowerconvexity.Ifrateschangeonlyslightly,thehigheryield-lowerconvexitybondwillperformbetter;ifrateschangebyasubstantialamount,theloweryield-higherconvexitybondwillperformbetter.20.a.Thefollowingspreadsheetshowsthattheconvexityofthebondis69.740.Thepresentvalueofeachcashflowisobtainedbydiscountingat6%.(Sincethebondhasa6%couponandsellsatpar,itsYTMis6%.)Convexityequals:thesumofthelastcolumn(7,836.031)dividedby:[P×(1+y)2]=100×(1.06)2=112.36TimeCashflow22PV(CF)t+t(t+t)×PV(CF)(t)(CF)165.660211.321265.340632.040365.0381260.453464.7532095.051564.48430134.506664.23042177.650763.99056223.459863.76472271.042963.55190319.6251010659.1901106,510.883Sum:100.0007,836.031Convexity:69.740Thedurationofthebondis:(1)(2)(3)(4)(5)TimeuntilPVofCFColumn(1)×PaymentCashFlow(DiscountWeightColumn(4)(years)rate=6%)1$6$5.6600.056600.056602$6$5.3400.053400.106803$6$5.0380.050380.151134$6$4.7530.047530.190105$6$4.4840.044840.224186$6$4.2300.042300.253797$6$3.9900.039900.279328$6$3.7640.037640.301169$6$3.5510.035510.3196310$106$59.1900.591905.91898ColumnSums$100.0001.000007.80169D=7.802years2- 课后答案网（http://www.khdaw.com）b.Iftheyieldtomaturityincreasesto7%,thebondpricewillfallto92.976%ofparvalue,apercentagedecreaseof7.024%.c.Thedurationrulepredictsapercentagepricechangeof:⎛D⎞⎛7.802⎞⎜−⎟×0.01=⎜−⎟×0.01=−0.07360=−7.360%⎝1.06⎠⎝1.06⎠Thisoverstatestheactualpercentagedecreaseinpriceby0.336%.Thepricepredictedbythedurationruleis7.360%lessthanfacevalue,or92.640%offacevalue.d.Theduration-with-convexityrulepredictsapercentagepricechangeof:⎡⎛7.802⎞⎤[2]⎢⎜−⎟×0.01⎥+0.5×69.740×0.01=−0.07012=−7.012%⎣⎝1.06⎠⎦Thepercentageerroris0.012%,whichissubstantiallylessthantheerrorusingthedurationrule.Thepricepredictedbytheduration-with-convexityruleis7.012%lessthanfacevalue,or92.988%offacevalue.21.Theeconomicclimateisoneofimpendinginterestrateincreases.Hence,wewillseektoshortenportfolioduration.a.Choosetheshortmaturity(2009)bond.b.TheArizonabondlikelyhaslowerduration.TheArizonacouponsareslightlylower,buttheArizonayieldishigher.c.Choosethe123/8%couponbond.Thematuritiesareapproximatelyequal,butthe123/8%couponismuchhigher,resultinginalowerduration.d.ThedurationoftheShellbondisloweriftheeffectofthehigheryieldtomaturityandearlierstartofsinkingfundredemptiondominatesitsslightlylowercouponrate.e.Thefloatingratebondhasadurationthatapproximatestheadjustmentperiod,whichisonly6months.2- 课后答案网（http://www.khdaw.com）22.a.Amanagerwhobelievesthatthelevelofinterestrateswillchangeshouldengageinarateanticipationswap,lengtheningdurationifratesareexpectedtofall,andshorteningdurationifratesareexpectedtorise.b.Achangeinyieldspreadsacrosssectorswouldcallforanintermarketspreadswap,inwhichthemanagerbuysbondsinthesectorforwhichyieldsareexpectedtofallrelativetootherbondsandsellsbondsinthesectorforwhichyieldsareexpectedtoriserelativetootherbonds.c.Abeliefthattheyieldspreadonaparticularinstrumentwillchangecallsforasubstitutionswapinwhichthatsecurityissoldifitsyieldisexpectedtoriserelativetotheyieldofothersimilarbonds,orisboughtifitsyieldisexpectedtofallrelativetotheyieldofothersimilarbonds.23.a.%pricechange=(−Effectiveduration)×ChangeinYTM(%)CIC:(−7.35)×(−0.50%)=3.675%PTR:(−5.40)×(−0.50%)=2.700%b.Sinceweareaskedtocalculatehorizonreturnoveraperiodofonlyonecouponperiod,thereisnoreinvestmentincome.Couponpayment+Year-endprice−InitialPriceHorizonreturn=Initialprice$31.25+$1,055.50−$1,017.50CIC:=0.06806=6.806%$1,017.50$36.75+$1,041.50−$1,017.50PTR:=0.05971=5.971%$1,017.50c.NoticethatCICisnon-callablebutPTRiscallable.Therefore,CIChaspositiveconvexity,whilePTRhasnegativeconvexity.Thus,theconvexitycorrectiontothedurationapproximationwillbepositiveforCICandnegativeforPTR.24.a.Foranoption-freebond,theeffectivedurationandmodifieddurationareapproximatelythesame.Thedurationofthebondis7.1.Usingthedataprovided,thedurationiscalculatedasfollows:∆P/P(100.71−99.29)/100−==7.100∆r0.0022- 课后答案网（http://www.khdaw.com）b.Thetotalpercentagepricechangeforthebondisestimatedasfollows:Percentagepricechangeusingduration=–7.90×–0.02×100=15.80%Convexityadjustment=1.66%Totalestimatedpercentagepricechange=15.80%+1.66%=17.46%c.Theassistant’sargumentisincorrect.Becausemodifiedconvexitydoesnotrecognizethefactthatcashflowsforbondswithanembeddedoptioncanchangeasyieldschange,modifiedconvexityremainspositiveasyieldsmovebelowthecallablebond’sstatedcouponrate,justasitwouldforanoption-freebond.Effectiveconvexity,however,takesintoaccountthefactthatcashflowsforasecuritywithanembeddedoptioncanchangeasinterestrateschange.Whenyieldsmovesignificantlybelowthestatedcouponrate,thelikelihoodthatthebondwillbecalledbytheissuerincreasesandtheeffectiveconvexityturnsnegative.25.a.Theadvantagesofabondindexingstrategyare:•Historically,themajorityofactivemanagersunderperformbenchmarkindexesinmostperiods;indexingreducesthepossibilityofunderperformanceatagivenlevelofrisk.•Indexedportfoliosdonotdependonadvisorexpectationsandsohavelessriskofunderperformingthemarket.•Managementadvisoryfeesforindexedportfoliosaredramaticallylessthanfeesforactivelymanagedportfolios.Feeschargedbyactivemanagersgenerallyrangefrom15to50basispoints,whilefeesforindexedportfoliosrangefrom1to20basispoints(withthehighestofthoserepresentingenhancedindexing).Othernon-advisoryfees(i.e.,custodialfees)arealsolessforindexedportfolios.•Plansponsorshavegreatercontroloverindexedportfoliosbecauseindividualmanagersdonothaveasmuchfreedomtovaryfromtheparametersofthebenchmarkindex.Someplansponsorsevendecidetomanageindexportfolioswithin-houseinvestmentstaff.•Indexingisessentially“buyingthemarket.”Ifmarketsareefficient,anindexingstrategyshouldreduceunsystematicdiversifiablerisk,andshouldgeneratemaximumreturnforagivenlevelofrisk.Thedisadvantagesofabondindexingstrategyare:•Indexedportfolioreturnsmaymatchthebondindex,butdonotnecessarilyreflectoptimalperformance.Insometimeperiods,manyactivemanagersmayoutperformanindexingstrategyatthesamelevelofrisk.•Thechosenbondindexandportfolioreturnsmaynotmeettheclientobjectivesortheliabilitystream.•Bondindexingmayrestrictthefundfromparticipatinginsectorsorotheropportunitiesthatcouldincreasereturns.2- 课后答案网（http://www.khdaw.com）b.Thestratifiedsampling,orcellular,methoddividestheindexintocells,witheachcellrepresentingadifferentcharacteristicoftheindex.Commoncellsusedinthecellularmethodcombine(butarenotlimitedto)duration,coupon,maturity,marketsectors,creditrating,andcallandsinkingfundfeatures.Theindexmanagerthenselectsoneormorebondissuestorepresenttheentirecell.Thetotalmarketweightofissuesheldforeachcellisbasedonthetargetindex’scompositionofthatcharacteristic.c.Trackingerrorisdefinedasthediscrepancybetweentheperformanceofanindexedportfolioandthebenchmarkindex.Whentheamountinvestedisrelativelysmallandthenumberofcellstobereplicatedislarge,asignificantsourceoftrackingerrorwiththecellularmethodoccursbecauseoftheneedtobuyoddlotsofissuesinordertoaccuratelyrepresenttherequiredcells.Oddlotsgenerallymustbepurchasedathigherpricesthanroundlots.Ontheotherhand,reducingthenumberofcellstolimittherequirednumberofoddlotswouldpotentiallyincreasetrackingerrorbecauseofthemismatchwiththetarget.26.Thematurityofthe30-yearbondwillfallto25years,anditsyieldisforecasttobe8%.Therefore,thepriceforecastforthebondis:$893.25[Usingafinancialcalculator,enterthefollowing:n=25;i=8;FV=1000;PMT=70]Ata6%interestrate,thefivecouponpaymentswillaccumulateto$394.60afterfiveyears.Therefore,totalproceedswillbe:$394.60+$893.25=$1,287.85Therefore,the5-yearreturnis:($1,287.85/$867.42)–1=0.4847Thisisa48.47%5-yearreturn,or8.23%annually.Thematurityofthe20-yearbondwillfallto15years,anditsyieldisforecasttobe7.5%.Therefore,thepriceforecastforthebondis:$911.73[Usingafinancialcalculator,enterthefollowing:n=15;i=7.5;FV=1000;PMT=65]Ata6%interestrate,thefivecouponpaymentswillaccumulateto$366.41afterfiveyears.Therefore,totalproceedswillbe:$366.41+$911.73=$1,278.14Therefore,the5-yearreturnis:($1,278.14/$879.50)–1=0.4533Thisisa45.33%5-yearreturn,or7.76%annually.The30-yearbondoffersthehigherexpectedreturn.2- 课后答案网（http://www.khdaw.com）27.Theminimumterminalvaluethatthemanageriswillingtoacceptisdeterminedbytherequirementfora3%annualreturnontheinitialinvestment.Therefore,theflooris:¥500million×(1.03)5=¥579.6millionThreeyearsaftertheinitialinvestment,onlytwoyearsremainuntilthehorizondate,andtheinterestratehasrisento7%.Therefore,atthistime,inordertobeassuredthatthetargetvaluecanbeattained,themanagerneedsaportfolioworth:¥579.6million/(1.07)2=¥506.2millionThisisthetriggerpoint.28.Whileitistruethatshort-termratesaremorevolatilethanlong-termrates,thelongerdurationofthelonger-termbondsmakestheirpricesandtheirratesofreturnmorevolatile.Thehigherdurationmagnifiesthesensitivitytointerest-ratesavings.29.∆P/P=−D*∆yForStrategyI:5-yearmaturity:∆P/P=−4.83×(−0.75%)=3.6225%25-yearmaturity:∆P/P=−23.81×0.50%=−11.9050%StrategyI:∆P/P=(0.5×3.6225%)+[0.5×(−11.9050%)]=−4.1413%ForStrategyII:15-yearmaturity:∆P/P=−14.35×0.25%=−3.5875%30.a.i.Strongeconomicrecoverywithrisinginflationexpectations.Interestratesandbondyieldswillmostlikelyrise,andthepricesofbothbondswillfall.Theprobabilitythatthecallablebondwillbecalledwoulddecrease,andthecallablebondwillbehavemorelikethenon-callablebond.(Notethattheyhavesimilardurationswhenpricedtomaturity).Theslightlylowerdurationofthecallablebondwillresultinsomewhatbetterperformanceinthehighinterestratescenario.ii.Economicrecessionwithreducedinflationexpectations.Interestratesandbondyieldswillmostlikelyfall.Thecallablebondislikelytobecalled.Therelevantdurationcalculationforthecallablebondisnowmodifieddurationtocall.Priceappreciationislimitedasindicatedbythelowerduration.Thenon-callablebond,ontheotherhand,continuestohavethesamemodifieddurationandhencehasgreaterpriceappreciation.2- 课后答案网（http://www.khdaw.com）b.Projectedpricechange=(modifiedduration)×(changeinYTM)=(–6.80)×(–0.75%)=5.1%Therefore,thepricewillincreasetoapproximately$105.10fromitscurrentlevelof$100.c.ForBondA,thecallablebond,bondlifeandthereforebondcashflowsareuncertain.Ifoneignoresthecallfeatureandanalyzesthebondona“tomaturity”basis,allcalculationsforyieldanddurationaredistorted.Durationsaretoolongandyieldsaretoohigh.Ontheotherhand,ifonetreatsthepremiumbondsellingabovethecallpriceona“tocall”basis,thedurationisunrealisticallyshortandyieldstoolow.Themosteffectiveapproachistouseanoptionvaluationapproach.Thecallablebondcanbedecomposedintotwoseparatesecurities:anon-callablebondandanoption:Priceofcallablebond=Priceofnon-callablebond–priceofoptionSincethecalloptionalwayshassomepositivevalue,thepriceofthecallablebondisalwayslessthanthepriceofthenon-callablesecurity.31.a.TimePVofCFuntilCashYears×PeriodDiscountrate=WeightPaymentFlowWeight6%perperiod(Years)A.8%couponbond10.5$40$37.7360.04050.020321.04035.6000.03830.038331.54033.5850.03610.054142.01,040823.7770.88511.7702Sum:$930.6981.00001.8829B.Zero-coupon10.5$0$0.0000.00000.000021.000.0000.00000.000031.500.0000.00000.000042.01,000792.0941.00002.0000Sum:$792.0941.00002.0000Forthecouponbond,theweightonthelastpaymentinthetableaboveislessthanitisinSpreadsheet16.1becausethediscountrateishigher;theweightsforthefirstthreepaymentsarelargerthanthoseinSpreadsheet16.1.Consequently,thedurationofthebondfalls.Thezerocouponbond,bycontrast,hasafixedweightof1.0forthesinglepaymentatmaturity.2- 课后答案网（http://www.khdaw.com）b.TimePVofCFuntilCashYears×PeriodDiscountrate=WeightPaymentFlowWeight5%perperiod(Years)A.8%couponbond10.5$60$57.1430.05520.027621.06054.4220.05260.052631.56051.8300.05010.075142.01,060872.0650.84221.6844Sum:$1,035.4601.00001.8396Sincethecouponpaymentsarelargerintheabovetable,theweightsontheearlierpaymentsarehigherthaninSpreadsheet16.1,sodurationdecreases.32.a.TimeCashPV(CF)t+t2(t+t2)×PV(CF)(t)FlowCoupon=$801$80$72.7272145.455YTM=0.1028066.1166396.694Maturity=538060.10512721.262Price=$924.18448054.641201,092.82251,080670.5953020,117.851Price:$924.184Sum:22,474.0832]=20.097Convexity=Sum/[Price×(1+y)b.TimeCashPV(CF)t2+t(t2+t)×PV(CF)(t)FlowCoupon=$01$0$0.00020.000YTM=0.10200.00060.000Maturity=5300.000120.000Price=$924.184400.000200.00051,000620.9213018,627.640Price:$620.921Sum:18,627.6402]=24.793Convexity=Sum/[Price×(1+y)CHAPTER24:PORTFOLIOPERFORMANCEEVALUATION1.TimeCashflowHoldingperiodreturn03(–$90)=–$2702- 课后答案网（http://www.khdaw.com）1$100(100–90)/90=11.11%2$1000%3$1000%a.Time-weightedgeometricaveragerateofreturn=(1.1111×1.0×1.0)1/3–1=0.0357=3.57%b.Time-weightedarithmeticaveragerateofreturn=(11.11%+0+0)/3=3.70%Thearithmeticaverageisalwaysgreaterthanorequaltothegeometricaverage;thegreaterthedispersion,thegreaterthedifference.c.Dollar-weightedaveragerateofreturn=IRR=5.46%[Usingafinancialcalculator,enter:n=3,PV=–270,FV=0,PMT=100.Thencomputetheinterestrate.]TheIRRexceedstheotheraveragesbecausetheinvestmentfundwasthelargestwhenthehighestreturnoccurred.2.a.StockAStockB(i)Alpha=regressionintercept1.0%2.0%(ii)Informationratio=αP/σ(eP)0.09710.1047(iii)*Sharpemeasure=(rP–rf)/σP0.49070.3373(iv)**Treynormeasure=(rP–rf)/βP8.83310.500*TocomputetheSharpemeasure,notethatforeachstock,(rP–rf)canbecomputedfromtheright-handsideoftheregressionequation,usingtheassumedparametersrM=14%andrf=6%.Thestandarddeviationofeachstock’sreturnsisgivenintheproblem.**ThebetatousefortheTreynormeasureistheslopecoefficientoftheregressionequationpresentedintheproblem.b.(i)Ifthisistheonlyriskyassetheldbytheinvestor,thenSharpe’smeasureistheappropriatemeasure.SincetheSharpemeasureishigherforStockA,thenAisthebestchoice.(ii)Ifthestockismixedwiththemarketindexfund,thenthecontributiontotheoverallSharpemeasureisdeterminedbytheappraisalratio;therefore,StockBispreferred.(iii)Ifthestockisoneofmanystocks,thenTreynor’smeasureistheappropriatemeasure,andStockBispreferred.2- 课后答案网（http://www.khdaw.com）3.a.Arithmeticaverage:rABC=10%;rXYZ=10%b.Dispersion:σABC=7.07%;σXYZ=13.91%StockXYZhasgreaterdispersion.(Note:Weused5degreesoffreedomincalculatingstandarddeviations.)c.Geometricaverage:r1/5ABC=(1.20×1.12×1.14×1.03×1.01)–1=0.0977=9.77%r1/5XYZ=(1.30×1.12×1.18×1.00×0.90)–1=0.0911=9.11%Despitethefactthatthetwostockshavethesamearithmeticaverage,thegeometricaverageforXYZislessthanthegeometricaverageforABC.ThereasonforthisresultisthefactthatthegreatervarianceofXYZdrivesthegeometricaveragefurtherbelowthearithmeticaverage.d.Intermsof“forwardlooking”statistics,thearithmeticaverageisthebetterestimateofexpectedrateofreturn.Therefore,ifthedatareflecttheprobabilitiesoffuturereturns,10%istheexpectedrateofreturnforbothstocks.4.a.Thealphasforthetwoportfoliosare:αA=12%–[5%+0.7(13%–5%)]=1.4%αB=16%–[5%+1.4(13%–5%)]=–0.2%Ideally,youwouldwanttotakealongpositioninPortfolioAandashortpositioninPortfolioB.b.Ifyouwillholdonlyoneofthetwoportfolios,thentheSharpemeasureistheappropriatecriterion:12−5S==0.583A1216−5S==0.355B31UsingtheSharpecriterion,PortfolioAisthepreferredportfolio.2- 课后答案网（http://www.khdaw.com）5.a.Time-weightedaveragereturnsarebasedonyear-by-yearratesofreturn:YearReturn=(capitalgains+dividend)/price2002−2003[($120–$100)+$4]/$100=24.00%2003−2004[($90–$120)+$4]/$120=–21.67%2004−2005[($100–$90)+$4]/$90=15.56%Arithmeticmean:(24%–21.67%+15.56%)/3=5.96%Geometricmean:(1.24×0.7833×1.1556)1/3–1=0.0392=3.92%b.CashDateFlowExplanation1/1/02–$300Purchaseofthreesharesat$100each1/1/03–$228Purchaseoftwosharesat$120lessdividendincomeonthreesharesheld1/1/04$110Dividendsonfivesharesplussaleofoneshareat$901/1/05$416Dividendsonfoursharesplussaleoffoursharesat$100each416110Date:1/1/021/1/031/1/041/1/05−228−300Dollar-weightedreturn=Internalrateofreturn=–0.1607%2- 课后答案网（http://www.khdaw.com）6.a.Themanager’salphais:10%–[6%+0.5(14%–6%)]=0b.FromBlack-Jensen-Scholesandothers,weknowthat,onaverage,portfolioswithlowbetahavehistoricallyhadpositivealphas.(TheslopeoftheempiricalsecuritymarketlineisshallowerthanpredictedbytheCAPM.)Therefore,giventhemanager’slowbeta,performancemightactuallybesub-pardespitetheestimatedalphaofzero.7.a.Managerreturn:(0.30×20)+(0.10×15)+(0.40×10)+(0.20×5)=12.50%Benchmark(bogey):(0.15×12)+(0.30×15)+(0.45×14)+(0.10×12)=13.80%Addedvalue:–1.30%b.Addedvaluefromcountryallocation:(1)(2)(3)=(1)×(2)ExcessweightIndexReturnContributiontoCountry(Manager–benchmark)minusbogeyperformanceU.K.0.15%−1.8%−0.27%Japan–0.20%1.2%–0.24%U.S.−0.05%0.2%−0.01%Germany0.10%−1.8%−0.18%Contributionofcountryallocation:−0.70%c.Addedvaluefromstockselection:(1)(2)(3)=(1)×(2)DifferentialreturnManager’sContributiontoCountrywithincountrycountryweightperformance(Manager–Index)U.K.8%0.30%2.4%Japan0%0.10%0.0%U.S.−4%0.40%−1.6%Germany−7%0.20%−1.4%Contributionofstockselection:−0.6%Summary:Countryallocation–0.70%Stockselection−0.60%Excessperformance–1.30%8.Theuseofuniversesofmanagerstoevaluaterelativeinvestmentperformancedoes,tosomeextent,overcomestatisticalproblems,aslongasthosemanagergroupscanbemadesufficientlyhomogeneouswithrespecttostyle.2- 课后答案网（http://www.khdaw.com）9.Support:Amanagercouldbeabetterperformerinonetypeofcircumstancethaninanother.Forexample,amanagerwhodoesnotiming,butsimplymaintainsahighbeta,willdobetterinupmarketsandworseindownmarkets.Therefore,weshouldobserveperformanceoveranentirecycle.Also,totheextentthatobservingamanageroveranentirecycleincreasesthenumberofobservations,itwouldimprovethereliabilityofthemeasurement.Contradict:Ifweadequatelycontrolforexposuretothemarket(i.e.,adjustforbeta),thenmarketperformanceshouldnotaffecttherelativeperformanceofindividualmanagers.Itisthereforenotnecessarytowaitforanentiremarketcycletopassbeforeevaluatingamanager.10.a.Bogey:(0.60×2.5%)+(0.30×1.2%)+(0.10×0.5%)=1.91%Actual:(0.70×2.0%)+(0.20×1.0%)+(0.10×0.5%)=1.65%Underperformance:0.26%b.SecuritySelection:(1)(2)(3)=(1)×(2)DifferentialreturnManager"sContributiontoMarketwithinmarketportfolioweightperformance(Manager–index)Equity–0.5%0.70−0.35%Bonds–0.2%0.20–0.04%Cash0.0%0.100.00%Contributionofsecurityselection:−0.39%c.AssetAllocation:(1)(2)(3)=(1)×(2)ExcessweightIndexContributiontoMarket(Manager–benchmark)ReturnperformanceEquity0.10%2.5%0.25%Bonds–0.10%1.2%–0.12%Cash0.00%0.5%0.00%Contributionofassetallocation:0.13%Summary:Securityselection–0.39%Assetallocation0.13%Excessperformance–0.26%2- 课后答案网（http://www.khdaw.com）11.Weneedtodistinguishbetweenmarkettimingandsecurityselectionabilities.Theinterceptofthescatterdiagramisameasureofstockselectionability.Ifthemanagertendstohaveapositiveexcessreturnevenwhenthemarket’sperformanceismerely“neutral”(i.e.,haszeroexcessreturn),thenweconcludethatthemanagerhasonaveragemadegoodstockpicks.Stockselectionmustbethesourceofthepositiveexcessreturns.Timingabilityisindicatedbythecurvatureoftheplottedline.Linesthatbecomesteeperasyoumovetotherightalongthehorizontalaxisshowgoodtimingability.Thesteeperslopeshowsthatthemanagermaintainedhigherportfoliosensitivitytomarketswings(i.e.,ahigherbeta)inperiodswhenthemarketperformedwell.Thisabilitytochoosemoremarket-sensitivesecuritiesinanticipationofmarketupturnsistheessenceofgoodtiming.Incontrast,adecliningslopeasyoumovetotherightmeansthattheportfoliowasmoresensitivetothemarketwhenthemarketdidpoorlyandlesssensitivewhenthemarketdidwell.Thisindicatespoortiming.Wecanthereforeclassifyperformanceforthefourmanagersasfollows:SelectionTimingAbilityAbilityA.BadGoodB.GoodGoodC.GoodBadD.BadBad12.a.13.d.14.Geometricaverage=(1.15×0.90)1/2–1=0.0173=1.73%15.a.Eachofthesebenchmarkshasseveraldeficiencies,asdescribedbelow.Marketindex:•Amarketindexmayexhibitsurvivorshipbias.Firmsthathavegoneoutofbusinessareremovedfromtheindex,resultinginaperformancemeasurethatoverstatesactualperformancehadthefailedfirmsbeenincluded.•Amarketindexmayexhibitdoublecountingthatarisesbecauseofcompaniesowningothercompaniesandbothbeingrepresentedintheindex.•Itisoftendifficulttoexactlyandcontinuallyreplicatetheholdingsinthemarketindexwithoutincurringsubstantialtradingcosts.•Thechosenindexmaynotbeanappropriateproxyforthemanagementstyleofthemanagers.2- 课后答案网（http://www.khdaw.com）•Thechosenindexmaynotrepresenttheentireuniverseofsecurities.Forexample,theS&P500Indexrepresents65%to70%ofU.S.equitymarketcapitalization.•Thechosenindex(e.g.,theS&P500)mayhavealargecapitalizationbias.•Thechosenindexmaynotbeinvestable.Theremaybesecuritiesintheindexthatcannotbeheldintheportfolio.Benchmarknormalportfolio:•Thisisthemostdifficultperformancemeasurementmethodtodevelopandcalculate.•Thenormalportfoliomustbecontinuallyupdated,requiringsubstantialresources.•Consultantsandclientsareconcernedthatmanagerswhoareinvolvedindevelopingandcalculatingtheirbenchmarkportfoliomayproduceaneasily-beatennormalportfolio,makingtheirperformanceappearbetterthanitactuallyis.Medianofthemanageruniverse:•Itcanbedifficulttoidentifyauniverseofmanagersappropriatefortheinvestmentstyleoftheplan’smanagers.•Selectionofamanageruniverseforcomparisoninvolvessome,perhapsmuch,subjectivejudgement.•Comparisonwithamanageruniversedoesnottakeintoaccounttherisktakenintheportfolio.•Themedianofamanageruniversedoesnotrepresentan“investable”portfolio;thatis,aportfoliomanagermaynotbeabletoinvestinthemedianmanagerportfolio.•Suchabenchmarkmaybeambiguous.Thenamesandweightsofthesecuritiesconstitutingthebenchmarkarenotclearlydelineated.•Thebenchmarkisnotconstructedpriortothestartofanevaluationperiod;itisnotspecifiedinadvance.•Amanageruniversemayexhibitsurvivorshipbias;managerswhohavegoneoutofbusinessareremovedfromtheuniverse,resultinginaperformancemeasurethatoverstatestheactualperformancehadthosemanagersbeenincluded.b.i.TheSharperatioiscalculatedbydividingtheportfolioriskpremium(i.e.,actualportfolioreturnminustherisk-freereturn)bytheportfoliostandarddeviation:Sharperatio=(rP–rf)/σPTheTreynormeasureiscalculatedbydividingtheportfolioriskpremium(i.e.,actualportfolioreturnminustherisk-freereturn)bytheportfoliobeta:Treynormeasure=(rP–rf)/βP2- 课后答案网（http://www.khdaw.com）Jensen’salphaiscalculatedbysubtractingthemarketriskpremium,adjustedforriskbytheportfolio’sbeta,fromtheactualportfolioexcessreturn(riskpremium).ItcanbedescribedasthedifferenceinreturnearnedbytheportfoliocomparedtothereturnimpliedbytheCapitalAssetPricingModelorSecurityMarketLine:αP=rP–[rf+βP(rM−rf)]ii.TheSharperatioassumesthattherelevantriskistotalrisk,anditmeasuresexcessreturnperunitoftotalrisk.TheTreynormeasureassumesthattherelevantriskissystematicrisk,anditmeasuresexcessreturnperunitofsystematicrisk.Jensen’salphaassumesthattherelevantriskissystematicrisk,anditmeasuresexcessreturnatagivenlevelofsystematicrisk.16.b.17.Internalrateofreturn=7.5%18.Geometricaverage=(0.91×1.23×1.17)1/3–1=0.0941=9.41%19.Time-weightedaveragereturn=(15%+10%)/2=12.5%Tocomputedollar-weightedrateofreturn,cashflowsare:CF0=−$500,000CF1=−$500,000CF2=($500,000×1.15×1.10)+($500,000×1.10)=$1,182,500Dollar-weightedrateofreturn=11.71%20.Treynormeasure=(17–8)/1.1=8.18221.Sharpemeasure=(24–8)/18=0.88822.i.Thestatementisincorrect.Validbenchmarksareunbiased.Medianmanagerbenchmarks,however,aresubjecttosignificantsurvivorshipbias,whichresultsinseveraldrawbacks,includingthefollowing:•Theperformanceofmedianmanagerbenchmarksisbiasedupwards.•Theupwardbiasincreaseswithtime.•Survivorbiasintroducesuncertaintywithregardtomanagerrankings.•Survivorbiasskewstheshapeofthedistributioncurve.2- 课后答案网（http://www.khdaw.com）ii.Thestatementisincorrect.Validbenchmarksareunambiguousandabletobereplicated.Themedianmanagerbenchmark,however,isambiguousbecausetheweightsoftheindividualsecuritiesinthebenchmarkarenotknown.Theportfolio’scompositioncannotbeknownbeforetheconclusionofameasurementperiodbecauseidentificationasamedianmanagercanoccuronlyafterperformanceismeasured.Validbenchmarksarealsoinvestable.Themedianmanagerbenchmarkisnotinvestable.Thatis,amanagerusingamedianmanagerbenchmarkcannotforegoactivemanagementand,takingapassive/indexedapproach,simplyholdthebenchmark.Thisisaresultofthefactthattheweightsofindividualsecuritiesinthebenchmarkarenotknown.iii.Thestatementiscorrect.Themedianmanagerbenchmarkmaybeinappropriatebecausethemedianmanageruniverseencompassesmanyinvestmentstylesand,therefore,maynotbeconsistentwithagivenmanager’sstyle.23.a.Sharperatio=(rP–rf)/σPWilliamsonCapital:Sharperatio=(22.1%−5.0%)/16.8%=1.02JoynerAssetManagement:Sharperatio=(24.2%−5.0%)/20.2%=0.95Treynormeasure=(rP–rf)/βPWilliamsonCapital:Treynormeasure=(22.1%−5.0%)/1.2=14.25JoynerAssetManagement:Treynormeasure=(24.2%−5.0%)/0.8=24.00b.ThedifferenceintherankingsofWilliamsonandJoynerresultsdirectlyfromthedifferenceindiversificationoftheportfolios.JoynerhasahigherTreynormeasure(24.00)andalowerSharperatio(0.95)thandoesWilliamson(14.25and1.202,respectively),soJoynermustbelessdiversifiedthanWilliamson.TheTreynormeasureindicatesthatJoynerhasahigherreturnperunitofsystematicriskthandoesWilliamson,whiletheSharperatioindicatesthatJoynerhasalowerreturnperunitoftotalriskthandoesWilliamson.24.a.ManagerAStrength.AlthoughManagerA’sone-yeartotalreturnwassomewhatbelowtheinternationalindexreturn(–6.0percentversus–5.0percent),thismanagerapparentlyhassomecountry/securityreturnexpertise.Thislargelocalmarketreturnadvantageof2.0percentexceedsthe0.2percentreturnfortheinternationalindex.Weakness.ManagerAhasanobviousweaknessinthecurrencymanagementarea.Thismanagerexperiencedamarkedcurrencyreturnshortfall,withareturnof–8.0percentversus–5.2percentfortheindex.2- 课后答案网（http://www.khdaw.com）ManagerBStrength.ManagerB’stotalreturnexceededthatoftheindex,withamarkedpositiveincrementapparentinthecurrencyreturn.ManagerBhada–1.0percentcurrencyreturncomparedtoa–5.2percentcurrencyreturnontheinternationalindex.Basedonthisoutcome,ManagerB’sstrengthappearstobeexpertiseinthecurrencyselectionarea.Weakness.ManagerBhadamarkedshortfallinlocalmarketreturn.Therefore,ManagerBappearstobeweakinsecurity/marketselectionability.b.Thefollowingstrategieswouldenablethefundtotakeadvantageofthestrengthsofeachofthetwomanagerswhileminimizingtheirweaknesses.1.Recommendation:OnestrategywouldbetodirectManagerAtomakenocurrencybetsrelativetotheinternationalindexandtodirectManagerBtomakeonlycurrencydecisions,andnoactivecountryorsecurityselectionbets.Justification:ThisstrategywouldmitigateManagerA’sweaknessbyhedgingallcurrencyexposuresintoindex-likeweights.ThiswouldallowcaptureofManagerA’scountryandstockselectionskillswhileavoidinglossesfrompoorcurrencymanagement.ThisstrategywouldalsomitigateManagerB’sweakness,leavinganindex-likeportfolioconstructandcapitalizingontheapparentskillincurrencymanagement.2.Recommendation:AnotherstrategywouldbetocombinetheportfoliosofManagerAandManagerB,withManagerAmakingcountryexposureandsecurityselectiondecisionsandManagerBmanagingthecurrencyexposurescreatedbyManagerA’sdecisions(providinga“currencyoverlay”).Justification:ThisrecommendationwouldcapturethestrengthsofbothManagerAandManagerBandwouldminimizetheircollectiveweaknesses.25.a.TreynormeasuresPortfolioX:(10–6)/0.6=6.67S&P500:(12–6)/1.0=6.00SharpemeasuresPortfolioX:(10–6)/18=0.222S&P500:(12–6)/13=0.462PortfolioXoutperformsthemarketbasedontheTreynormeasure,butunderperformsbasedontheSharpemeasure.2- 课后答案网（http://www.khdaw.com）b.Thetwomeasuresofperformanceareinconflictbecausetheyusedifferentmeasuresofrisk.PortfolioXhaslesssystematicriskthanthemarket,asmeasuredbyitslowerbeta,butmoretotalrisk(volatility),asmeasuredbyitshigherstandarddeviation.Therefore,theportfoliooutperformsthemarketbasedontheTreynormeasurebutunderperformsbasedontheSharpemeasure.26.a.MethodIdoesnothingtoseparatelyidentifytheeffectsofmarkettimingandsecurityselectiondecisions.Italsousesaquestionable“neutralposition,”thecompositionoftheportfolioatthebeginningoftheyear.b.MethodIIisnotperfect,butisthebestofthethreetechniques.Itatleastattemptstofocusonmarkettimingbyexaminingthereturnsforportfoliosconstructedfrombondmarketindexesusingactualweightsinvariousindexesversusyear-averageweights.Theproblemwiththismethodisthattheyear-averageweightsneednotcorrespondtoaclient’s“neutral”weights.Forexample,whatifthemanagerwereoptimisticovertheentireyearregardinglong-termbonds?Heraverageweightingcouldreflectheroptimism,andnotaneutralposition.c.MethodIIIusesnetpurchasesofbondsasasignalofbondmanageroptimism.Butsuchnetpurchasescanbemotivatedbywithdrawalsfromorcontributionstothefundratherthanthemanager’sdecisions.(Notethatthisisanopen-endedmutualfund.)Therefore,itisinappropriatetoevaluatethemanagerbasedonwhethernetpurchasesturnouttobereliablebullishorbearishsignals.27.a.Indeed,theoneyearresultswereterrible,butoneyearisapoorstatisticalbasefromwhichtodrawinferences.Moreover,theboardoftrusteeshaddirectedKarltoadoptalong-termhorizon.TheBoardspecificallyinstructedtheinvestmentmanagertogiveprioritytolongtermresults.b.ThesampleofpensionfundshadamuchlargershareinvestedinequitiesthandidAlpine.Equitiesperformedmuchbetterthanbonds.YetthetrusteestoldAlpinetoholddownrisk,investingnotmorethan25%oftheplan’sassetsincommonstocks.(Alpine’sbetawasalsosomewhatdefensive.)Alpineshouldnotbeheldresponsibleforanassetallocationpolicydictatedbytheclient.c.Alpine’salphameasuresitsrisk-adjustedperformancecomparedtothemarket:α=13.3%–[7.5%+0.90(13.8%–7.5%)]=0.13%(actuallyabovezero)2- 课后答案网（http://www.khdaw.com）d.Notethatthelast5years,andparticularlythemostrecentyear,havebeenbadforbonds,theassetclassthatAlpinehadbeenencouragedtohold.Withinthisassetclass,however,Alpinedidmuchbetterthantheindexfund.Moreover,despitethefactthatthebondindexunderperformedboththeactuarialreturnandT-bills,Alpineoutperformedboth.Alpine’sperformancewithineachassetclasshasbeensuperioronarisk-adjustedbasis.Itsoveralldisappointingreturnswereduetoaheavyassetallocationweightingtowardsbonds,whichwastheBoard’s,notAlpine’s,choice.e.Atrusteemaynotcareaboutthetime-weightedreturn,butthatreturnismoreindicativeofthemanager’sperformance.Afterall,themanagerhasnocontroloverthecashinflowsandoutflowsofthefund.2-'