HullOFOD8eSolutionsCh04Word版

CHAPTER4InterestRatesPracticeQuestionsProblem4.1.Abankquotesyouaninterestrateof14%perannumwithquarterlycompounding.Whatistheequivalentratewith(a)continuouscompoundingand(b)annualcompounding?(a)Theratewithcontinuouscompoundingisor13.76%perannum.(b)Theratewithannualcompoundingisor14.75%perannum.Problem4.2.WhatismeantbyLIBORandLIBID.Whichishigher?LIBORistheLondonInterBankOfferedRate.ItiscalculateddailybytheBritishBankersAssociationandistherateaAA-ratedbankrequiresondepositsitplaceswithotherbanks.LIBIDistheLondonInterBankBidrate.ItistherateabankispreparedtopayondepositsfromotherAA-ratedbanks.LIBORisgreaterthanLIBID.Problem4.3.Thesix-monthandone-yearzeroratesareboth10%perannum.Forabondthathasalifeof18monthsandpaysacouponof8%perannum(withsemiannualpaymentsandonehavingjustbeenmade),theyieldis10.4%perannum.Whatisthebond’sprice?Whatisthe18-monthzerorate?Allratesarequotedwithsemiannualcompounding.Supposethebondhasafacevalueof$100.Itspriceisobtainedbydiscountingthecashflowsat10.4%.ThepriceisIfthe18-monthzerorateis,wemusthavewhichgives%.Problem4.4.Aninvestorreceives$1,100inoneyearinreturnforaninvestmentof$1,000now.Calculatethepercentagereturnperannumwitha)annualcompounding,b)semiannualcompounding,c)monthlycompoundingandd)continuouscompounding.(a)Withannualcompoundingthereturnisor10%perannum.(b)Withsemi-annualcompoundingthereturniswherei.e.,sothat.Thepercentagereturnistherefore9.76%perannum.(c)Withmonthlycompoundingthereturniswherei.e.sothat.Thepercentagereturnistherefore9.57%perannum.(d)Withcontinuouscompoundingthereturniswhere:i.e.,sothat.Thepercentagereturnistherefore9.53%perannum.Problem4.5.Supposethatzerointerestrateswithcontinuouscompoundingareasfollows:Maturity(months)Rate(%perannum)38.068.298.4128.5158.6188.7Calculateforwardinterestratesforthesecond,third,fourth,fifth,andsixthquarters.TheforwardrateswithcontinuouscompoundingareasfollowstoQtr28.4%Qtr38.8%Qtr48.8%Qtr59.0%Qtr69.2%Problem4.6.AssumingthatzeroratesareasinProblem4.5,whatisthevalueofanFRAthatenablestheholdertoearn9.5%forathree-monthperiodstartinginoneyearonaprincipalof$1,000,000?Theinterestrateisexpressedwithquarterlycompounding.Theforwardrateis9.0%withcontinuouscompoundingor9.102%withquarterlycompounding.Fromequation(4.9),thevalueoftheFRAisthereforeor$893.56.Problem4.7.Thetermstructureofinterestratesisupwardsloping.Putthefollowinginorderofmagnitude:(a)Thefive-yearzerorate(a)Theyieldonafive-yearcoupon-bearingbond(a)Theforwardratecorrespondingtotheperiodbetween4.75andyearsinthefutureWhatistheanswertothisquestionwhenthetermstructureofinterestratesisdownwardsloping?Whenthetermstructureisupwardsloping,.Whenitisdownwardsloping,.Problem4.8.Whatdoesdurationtellyouaboutthesensitivityofabondportfoliotointerestrates?Whatarethelimitationsofthedurationmeasure?Durationprovidesinformationabouttheeffectofasmallparallelshiftintheyieldcurveonthevalueofabondportfolio.Thepercentagedecreaseinthevalueoftheportfolioequalsthedurationoftheportfoliomultipliedbytheamountbywhichinterestratesareincreasedinthesmallparallelshift.Thedurationmeasurehasthefollowinglimitation.Itappliesonlytoparallelshiftsintheyieldcurvethataresmall.Problem4.9.Whatrateofinterestwithcontinuouscompoundingisequivalentto15%perannumwithmonthlycompounding?Therateofinterestiswhere:i.e.,Therateofinterestistherefore14.91%perannum.Problem4.10.Adepositaccountpays12%perannumwithcontinuouscompounding,butinterestisactuallypaidquarterly.Howmuchinterestwillbepaideachquarterona$10,000deposit?TheequivalentrateofinterestwithquarterlycompoundingiswhereorTheamountofinterestpaideachquarteristherefore:or$304.55.Problem4.11.Supposethat6-month,12-month,18-month,24-month,and30-monthzeroratesare4%,4.2%,4.4%,4.6%,and4.8%perannumwithcontinuouscompoundingrespectively.Estimatethecashpriceofabondwithafacevalueof100thatwillmaturein30monthsandpaysacouponof4%perannumsemiannually.Thebondpays$2in6,12,18,and24months,and$102in30months.ThecashpriceisProblem4.12.Athree-yearbondprovidesacouponof8%semiannuallyandhasacashpriceof104.Whatisthebond’syield?Thebondpays$4in6,12,18,24,and30months,and$104in36months.ThebondyieldisthevalueofthatsolvesUsingtheSolverorGoalSeektoolinExcelor6.407%.Problem4.13.Supposethatthe6-month,12-month,18-month,and24-monthzeroratesare5%,6%,6.5%,and7%respectively.Whatisthetwo-yearparyield?Usingthenotationinthetext,,.AlsoTheformulainthetextgivestheparyieldasToverifythatthisiscorrectwecalculatethevalueofabondthatpaysacouponof7.072%peryear(thatis3.5365everysixmonths).Thevalueisverifyingthat7.072%istheparyield.Problem4.14.Supposethatzerointerestrateswithcontinuouscompoundingareasfollows:Maturity(years)Rate(%perannum)12.023.033.744.254.5Calculateforwardinterestratesforthesecond,third,fourth,andfifthyears.Theforwardrateswithcontinuouscompoundingareasfollows:Year2:4.0%Year3:5.1%Year4:5.7%Year5:5.7%Problem4.15.UsetheratesinProblem4.14tovalueanFRAwhereyouwillpay5%forthethirdyearon$1million.Theforwardrateis5.1%withcontinuouscompoundingorwithannualcompounding.The3-yearinterestrateis3.7%withcontinuouscompounding.Fromequation(4.10),thevalueoftheFRAisthereforeor$2,078.85.Problem4.16.A10-year,8%couponbondcurrentlysellsfor$90.A10-year,4%couponbondcurrentlysellsfor$80.Whatisthe10-yearzerorate?(Hint:Considertakingalongpositionintwoofthe4%couponbondsandashortpositioninoneofthe8%couponbonds.)Takingalongpositionintwoofthe4%couponbondsandashortpositioninoneofthe8%couponbondsleadstothefollowingcashflowsbecausethecouponscancelout.$100in10yearstimeisequivalentto$70today.The10-yearrate,,(continuouslycompounded)isthereforegivenbyTherateisor3.57%perannum.Problem4.17.Explaincarefullywhyliquiditypreferencetheoryisconsistentwiththeobservationthatthetermstructureofinterestratestendstobeupwardslopingmoreoftenthanitisdownwardsloping.Iflong-termratesweresimplyareflectionofexpectedfutureshort-termrates,wewouldexpectthetermstructuretobedownwardslopingasoftenasitisupwardsloping.(Thisisbasedontheassumptionthathalfofthetimeinvestorsexpectratestoincreaseandhalfofthetimeinvestorsexpectratestodecrease).Liquiditypreferencetheoryarguesthatlongtermratesarehighrelativetoexpectedfutureshort-termrates.Thismeansthatthetermstructureshouldbeupwardslopingmoreoftenthanitisdownwardsloping.Problem4.18.“Whenthezerocurveisupwardsloping,thezerorateforaparticularmaturityisgreaterthantheparyieldforthatmaturity.Whenthezerocurveisdownwardsloping,thereverseistrue.”Explainwhythisisso.Theparyieldistheyieldonacoupon-bearingbond.Thezerorateistheyieldonazero-couponbond.Whentheyieldcurveisupwardsloping,theyieldonan-yearcoupon-bearingbondislessthantheyieldonan-yearzero-couponbond.Thisisbecausethecouponsarediscountedatalowerratethanthe-yearrateanddragtheyielddownbelowthisrate.Similarly,whentheyieldcurveisdownwardsloping,theyieldonan-yearcouponbearingbondishigherthantheyieldonan-yearzero-couponbond.Problem4.19.WhyareU.S.Treasuryratessignificantlylowerthanotherratesthatareclosetoriskfree?Therearethreereasons(seeBusinessSnapshot4.1).1.TreasurybillsandTreasurybondsmustbepurchasedbyfinancialinstitutionstofulfillavarietyofregulatoryrequirements.ThisincreasesdemandfortheseTreasuryinstrumentsdrivingthepriceupandtheyielddown.2.TheamountofcapitalabankisrequiredtoholdtosupportaninvestmentinTreasurybillsandbondsissubstantiallysmallerthanthecapitalrequiredtosupportasimilarinvestmentinothervery-low-riskinstruments.3.IntheUnitedStates,Treasuryinstrumentsaregivenafavorabletaxtreatmentcomparedwithmostotherfixed-incomeinvestmentsbecausetheyarenottaxedatthestatelevel.Problem4.20.Whydoesaloanintherepomarketinvolveverylittlecreditrisk?Arepoisacontractwhereaninvestmentdealerwhoownssecuritiesagreestosellthemtoanothercompanynowandbuythembacklaterataslightlyhigherprice.Theothercompanyisprovidingaloantotheinvestmentdealer.Thisloaninvolvesverylittlecreditrisk.Iftheborrowerdoesnothonortheagreement,thelendingcompanysimplykeepsthesecurities.Ifthelendingcompanydoesnotkeeptoitssideoftheagreement,theoriginalownerofthesecuritieskeepsthecash.Problem4.21.ExplainwhyanFRAisequivalenttotheexchangeofafloatingrateofinterestforafixedrateofinterest?AFRAisanagreementthatacertainspecifiedinterestrate,,willapplytoacertainprincipal,,foracertainspecifiedfuturetimeperiod.Supposethattherateobservedinthemarketforthefuturetimeperiodatthebeginningofthetimeperiodprovestobe.IftheFRAisanagreementthatwillapplywhentheprincipalisinvested,theholderoftheFRAcanborrowtheprincipalatandtheninvestitat.Thenetcashflowattheendoftheperiodisthenaninflowofandanoutflowof.IftheFRAisanagreementthatwillapplywhentheprincipalisborrowed,theholderoftheFRAcaninvesttheborrowedprincipalat.Thenetcashflowattheendoftheperiodisthenaninflowofandanoutflowof.IneithercaseweseethattheFRAinvolvestheexchangeofafixedrateofinterestontheprincipalofforafloatingrateofinterestontheprincipal.Problem4.22.Afive-yearbondwithayieldof11%(continuouslycompounded)paysan8%couponattheendofeachyear.a)Whatisthebond’sprice?b)Whatisthebond’sduration?c)Usethedurationtocalculatetheeffectonthebond’spriceofa0.2%decreaseinitsyield.d)Recalculatethebond’spriceonthebasisofa10.8%perannumyieldandverifythattheresultisinagreementwithyouranswerto(c).a)Thebond’spriceisb)Thebond’sdurationisc)Since,withthenotationinthechaptertheeffectonthebond’spriceofa0.2%decreaseinitsyieldisThebond’spriceshouldincreasefrom86.80to87.54.d)Witha10.8%yieldthebond’spriceisThisisconsistentwiththeanswerin(c).Problem4.23.Thecashpricesofsix-monthandone-yearTreasurybillsare94.0and89.0.A1.5-yearbondthatwillpaycouponsof$4everysixmonthscurrentlysellsfor$94.84.Atwo-yearbondthatwillpaycouponsof$5everysixmonthscurrentlysellsfor$97.12.Calculatethesix-month,one-year,1.5-year,andtwo-yearzerorates.The6-monthTreasurybillprovidesareturnofinsixmonths.Thisisperannumwithsemiannualcompoundingorperannumwithcontinuouscompounding.The12-monthrateiswithannualcompoundingorwithcontinuouscompounding.Forthe1yearbondwemusthavewhereisthe1yearzerorate.Itfollowsthator11.5%.Forthe2-yearbondwemusthavewhereisthe2-yearzerorate.Itfollowsthator11.3%.Problem4.24.“Aninterestrateswapwheresix-monthLIBORisexchangedforafixedrate5%onaprincipalof$100millionforfiveyearsisaportfolioofnineFRAs.”Explainthisstatement.EachexchangeofpaymentsisanFRAwhereinterestat5%isexchangedforinterestatLIBORonaprincipalof$100million.InterestrateswapsarediscussedfurtherinChapter7.FurtherQuestionsProblem4.25(Excelfile)Afive-yearbondprovidesacouponof5%perannumpayablesemiannually.Itspriceis104.Whatisthebond'syield?YoumayfindExcel'sSolveruseful.Theanswer(withcontinuouscompounding)is4.07%Problem4.26(Excelfile)SupposethatLIBORratesformaturitiesofonemonth,twomonths,threemonths,fourmonths,fivemonthsandsixmonthsare2.6%,2.9%,3.1%,3.2%,3.25%,and3.3%withcontinuouscompounding.Whataretheforwardratesforfutureonemonthperiods?Theforwardratesforthesecond,third,fourth,fifthandsixthmonthsare(seespreadsheet)3.2%,3.5%,3.5%,3.45%,3.55%,respectivelywithcontinuouscompounding.Problem4.27.AbankcanborroworlendatLIBOR.Thetwo-monthLIBORrateis0.28%perannumwithcontinuouscompounding.Assumingthatinterestratescannotbenegative,whatisthearbitrageopportunityifthethree-monthLIBORrateis0.1%peryearwithcontinuouscompounding?Howlowcanthethree-monthLIBORratebecomewithoutanarbitrageopportunitybeingcreated?Theforwardrateforthethirdmonthis0.001×3−0.0028×2=−0.0026or−0.26%.Ifweassumethattherateforthethirdmonthwillnotbenegativewecanborrowforthreemonths,lendfortwomonthsandlendatthemarketrateforthethirdmonth.Thelowestlevelforthethree-monthratethatdoesnotpermitthisarbitrageis0.0028×2/3=0.001867or0.1867%.Problem4.28AbankcanborroworlendatLIBOR.Supposethatthesix-monthrateis5%andthenine-monthrateis6%.TheratethatcanbelockedinfortheperiodbetweensixmonthsandninemonthsusinganFRAis7%.Whatarbitrageopportunitiesareopentothebank?Allratesarecontinuouslycompounded.Theforwardrateisor8%.TheFRArateis7%.Aprofitcanthereforebemadebyborrowingforsixmonthsat5%,enteringintoanFRAtoborrowfortheperiodbetween6and9monthsfor7%andlendingforninemonthsat6%.Problem4.29.Aninterestrateisquotedas5%perannumwithsemiannualcompounding.Whatistheequivalentratewith(a)annualcompounding,(b)monthlycompounding,and(c)continuouscompounding?a)Withannualcompoundingtherateisor5.0625%b)Withmonthlycompoundingtherateisor4.949%.c)Withcontinuouscompoundingtherateisor4.939%.Problem4.30.The6-month,12-month.18-month,and24-monthzeroratesare4%,4.5%,4.75%,and5%withsemiannualcompounding.a)Whataretherateswithcontinuouscompounding?b)Whatistheforwardrateforthesix-monthperiodbeginningin18monthsc)WhatisthevalueofanFRAthatpromisestopayyou6%(compoundedsemiannually)onaprincipalof$1millionforthesix-monthperiodstartingin18months?a)Withcontinuouscompoundingthe6-monthrateisor3.961%.The12-monthrateisor4.4501%.The18-monthrateisor4.6945%.The24-monthrateisor4.9385%.b)Theforwardrate(expressedwithcontinuouscompounding)isfromequation(4.5)or5.6707%.Whenexpressedwithsemiannualcompoundingthisisor5.7518%.c)ThevalueofanFRAthatpromisestopay6%forthesixmonthperiodstartingin18monthsisfromequation(4.9)or$1,124.Problem4.31.Whatisthetwo-yearparyieldwhenthezeroratesareasinProblem4.30?Whatistheyieldonatwo-yearbondthatpaysacouponequaltotheparyield?Thevalue,ofanannuitypayingoff$1everysixmonthsisThepresentvalueof$1receivedintwoyears,,is.FromtheformulainSection4.4theparyieldisor4.983%.Bydefinitionthisisalsotheyieldonatwo-yearbondthatpaysacouponequaltotheparyield.Problem4.32.ThefollowingtablegivesthepricesofbondsBondPrincipal($)TimetoMaturity(yrs)AnnualCoupon($)*BondPrice($)1000.50.0981001.00.0951001.56.21011002.08.0104*Halfthestatedcouponispaideverysixmonthsa)Calculatezeroratesformaturitiesof6months,12months,18months,and24months.b)Whataretheforwardratesfortheperiods:6monthsto12months,12monthsto18months,18monthsto24months?c)Whatarethe6-month,12-month,18-month,and24-monthparyieldsforbondsthatprovidesemiannualcouponpayments?d)Estimatethepriceandyieldofatwo-yearbondprovidingasemiannualcouponof7%perannum.a)Thezerorateforamaturityofsixmonths,expressedwithcontinuouscompoundingis.Thezerorateforamaturityofoneyear,expressedwithcontinuouscompoundingis.The1.5-yearrateiswhereThesolutiontothisequationis.The2.0-yearrateiswhereThesolutiontothisequationis.TheseresultsareshowninthetablebelowMaturity(yrs)ZeroRate(%)ForwardRate(%)ParYield(s.a.%)Paryield(c.c%)0.54.04054.04054.08164.04051.05.12936.21815.18135.11541.55.44296.07005.49865.42442.05.80856.90545.86205.7778b)Thecontinuouslycompoundedforwardratescalculatedusingequation(4.5)areshowninthethirdcolumnofthetablec)Theparyield,expressedwithsemiannualcompounding,canbecalculatedfromtheformulainSection4.4.Itisshowninthefourthcolumnofthetable.Inthefifthcolumnofthetableitisconvertedtocontinuouscompoundingd)ThepriceofthebondisTheyieldonthebond,satisfiesThesolutiontothisequationis.Thebondyieldistherefore5.7723%.Problem4.33.PortfolioAconsistsofaone-yearzero-couponbondwithafacevalueof$2,000anda10-yearzero-couponbondwithafacevalueof$6,000.PortfolioBconsistsofa5.95-yearzero-couponbondwithafacevalueof$5,000.Thecurrentyieldonallbondsis10%perannum.(a)Showthatbothportfolioshavethesameduration.(b)Showthatthepercentagechangesinthevaluesofthetwoportfoliosfora0.1%perannumincreaseinyieldsarethesame.(c)Whatarethepercentagechangesinthevaluesofthetwoportfoliosfora5%perannumincreaseinyields?a)ThedurationofPortfolioAisSincethisisalsothedurationofPortfolioB,thetwoportfoliosdohavethesameduration.b)ThevalueofPortfolioAisWhenyieldsincreaseby10basispointsitsvaluebecomesThepercentagedecreaseinvalueisThevalueofPortfolioBisWhenyieldsincreaseby10basispointsitsvaluebecomesThepercentagedecreaseinvalueisThepercentagechangesinthevaluesofthetwoportfoliosfora10basispointincreaseinyieldsarethereforethesame.c)Whenyieldsincreaseby5%thevalueofPortfolioAbecomesandthevalueofPortfolioBbecomesThepercentagereductionsinthevaluesofthetwoportfoliosare:SincethepercentagedeclineinvalueofPortfolioAislessthanthatofPortfolioB,PortfolioAhasagreaterconvexity.（注：可编辑下载，若有不当之处，请指正，谢谢!）