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1、CHAPTER4InterestRatesPracticeQuestionsProblem4.1.Abankquotesyouaninterestrateof7%perannumwithquarterlycompounding.Whatistheequivalentratewith(a)continuouscompoundingand(b)annualcompounding?(a) Theratewithcontinuouscompoundingis(0074Inl+二0.06944Jor6.94%perannum.(b) Theratewithannualcompoundingis(0.07
2、iec+亍j-1=0.0719or7.19%erannum.Problem4.2.ExpIainhowLIBORisdetermined1.lBoRistheLondonlnterBankofferedRatejtiscalculateddailyfromborrowingratesestimatedbyapanelofbanksProblem4.3.Thesix-monthandone-yearzeroratesareboth5%perannum.Forabondthathasalifeof18monthsandpaysacouponof4%perannum(withsemiannualpa
3、ymentsandonehavingjustbeenmade),theyieldis5.2%pra7HMw.H7zrz6/wd7ice?Wza”s泌e18-monthzerorate?AlIrateSareqUotedWithSemiannUalComPOUnding.SupposethebondhasafacevalueofS100.Itspriceisobtainedbydiscountingthecashflowsat5.2%,Thepriceis221021.0261.0262263=98.29Ifthe18-monthzerorateisR,wemusthave102(l+R2)3二
4、 98.29221.0251+0252whichgivesR=5.204%.Problen4.4.AninvestorreceivesSI,1OoinoneyeannreturnforaninvestmentofSI,OOOnow.Calculatethepercentagereturnperannumwitha)annualcompounding,b)semiannualcompounding,c)monthlycompoundingandd)continuouscompounding.(a) WithannualcompoundingthereturnisIl(X)-I二0.1100Oor
5、lO%perannum.(b) Withsemi-annualcompoundingthereturnisRwhere(R)210001+=HOO2)i.e.,R1+11二1.04882sothat/?=0.0976.Thepercentagereturnistherefore9.76%perannum.(c) WithmonthlycompoundingthereturnisRwhere(R)1210001+-=1100112J1 .e.二A=血皿沏sothatR=0.0957.Thepercentagereturnistherefore9.57%perannum.(d) Withconti
6、nuouscompoundingtheretumisRwhere:100OeR=IlOO1 .e.,eR=l.1SothatR=In1.1=0.0953.Thepercentagereturnistherefore9.53%perannum.Problem4.5.Supposethatzeroimerestrateswhhcontinuouscompoiindingareasfollows:Maturity(months)Raie(%perannum)33.063.293.4123.5153.6183.7Calculateforwardinterestratesforthesecond,thi
7、rd,fourth,fifth,andsixthquarters.TheforwardrateswithcontinuouscompoundingareasfbllowstoQtr23.4%Qtr33.8%Qtr43.8%Qtr54.0%Qtr64.2%Problen4.6.Assumingthatrisk-freeratesareasinProblem4.5,whatisthevalueofanFRAwheretheholderwillpayLIBORandreceive4.5%(quarterlycompounded)forathree-monthperiodstartinginoneye
8、aronaprincipalof$1,000,Ooo-TheforwardLIBORrateforthethreemonthperiodis5%quarterlycompounded.Fromcquation(4.9),thevalueoftheF7RAistherefore1,000,000X0.25X(0.045-0.050)eo.o36xi,25Zl-1J95or$l,195Problem4.7.Thetermstructureofinterestratesisupwardsloping-Putthefollowinginorderofmagnitude:(a) Thefive-year
9、zerorate(b) Theyieldonafive-yearcoupo-bearingbond(c) TheforwardratecorrespondingtotheperiodbetweenAJSandSyearsinthefutureWhatistheanswertothisquestionwhenthetermstructureofinterestratesisdownwardsloping?Whenthetermstructureisupwardsloping,cab.Whenitisdownwardsloping,Z?6rc.Problem4.8.Whatdoesduration
10、tellyouaboutthesensitivityofabondportfoliotointerestrates?WhatarethelimitationsOfthedurationmeasure?Durationprovidesinformationabouttheeffectofasmallparallelshiftintheyieldcurveonthevalueofabondportfblio.ThepercentagedecreaseinthevalueoftheportfolioequalsthedurationoftheportfoliomuItipliedbytheamoun
11、tbyWhichinterestratesareincreasedinthesmallparallelshift-Thedurationmeasurehasthefollowinglimitation.Itappliesonlytoparallelshiftsintheyieldcurvethataresmall.Problem4.9.Whatrateofinterestwithcontinuouscompoundingisequivalentto8%perannumwithmonthlycompounding?TherateofinterestisRwhere:(0.08)12r=1+Ii.
12、e.,1.mnfi+008A-nI%j=0.0797Therateofinterestistherefre7.97%perannum.Problem4.10.Adepositaccountpays4%perannumwithcontinuouscompounding,butinterestisactuallypaidquarterly.HoWmUChimereStW川bepaideachquarterona$10,00OdePOSit?TheequivalentrateofinterestwithquarterlycompoundingisRwhereo.04=orR二4(eoe-l)二0.0
13、402Theamountofinterestpaideachquarteristherefnre:0.040210,ooox-:二100.504or$100.50.Problem4.il.Supposethat6-month,12-month,18-month,24-month,and30-monthzeroratesare,respectively,4%2%,44%,4.6%,and4.8%perannumwithcontinuouscompounding.Estimatethecashpriceofabondwithafacevalueof100thatw川maturein30months
14、andpayacojponof4%perannumsemiannually.Thebondpays$2in6,12,18,and24months,and$102in30months.Thecashpriceis2e-0.04x0.5+2e-0.042xl.(H-2e-0.044xl.5+2e.(M6x2102-0.(M8x2.5=98.04Problem4.12.Athree-yearbondprovidesacouponof8%semiannuallyandhasacashpriceof104.Whatisthebondsyield?Thebondpays$4in6,12,18,24,and
15、30months,and$104in36months.Thebondyieldisthevalueofythatsolves4e-o.5y+4e-1.oy+4e-1.5y+4e-2.oy+4e-25y+104e-3.oy=104UsingtheSo/izen)rGoa/SeektoolinExcel,y=0.06407or6.407%.Problem4.13.Supposethatthe6-month,12-month,18-month,and24-monthzeroratesare5%,6%,6.5%,and7%respectively.Whatisthetwo-yearparyield?U
16、singthenotationinthetext,m=2,d=e-oo7x2=O.8694.AlsoA=-0.05x0.5+-0.06x1.0+-0.065x1.5+0.07x2.0=3.6935Thefbrmulainthetextgivestheparyieldas(Ioo-100XO.8694)-2-7创3.6935_*ToVerifythatIhiSiSCOrreeIWeCaICUlatetheVaIUeOfabondlhatPaySaCoUPOnOf7.0741%Peryear(IhatiS3.5370everysixmonths).Thevalueis3.537-o.o5xo.5+
17、3.537-o.o6x.o+3.537e-o.o65xi.5+103.537-o.o7x2.o=1OOverifyingthat7.0741%istheparyield.Probiem4.14.Supposethatrisk-freezerointerestrateswithcontinuouscompoundingareasfollows:Maturity(years)Rate(%perannun)12.023.033.744.254.5Calculateforwardinterestratesforthesecond,third,fourth,andfifthyears.Theforwar
18、drateswithcontinuouscoiTipoundingareasfbIlows:Year2:4.0%Year3r5.1%Year4:5.7%Year55.7%Problem4.15.Usetherisk-freeratesinProblem4.14tovalueanFRAwhereyouwillpay5%(compoundedannually)andreceiveLIBORforthethirdyearon$1million.TheforwardLIBORrate(annuallycompounded)forthethirdyearis5.5%.WevaluetheFRAbyass
19、umingthattheforwardLlBoRwillberealized.ThevalueoftheFRAis1,000,000x(0.0550.050).637x3=M,474.69Problem4.16.A10-year,8%couponbondcurrentlysellsfor$90.A10-year,4%couponbondcurrentlysellsfor$80.Whatisthe10-yearzerorate?(Hint:COnSidertakingaIOngPOSitiOnintWOOfthe4%CoUPOnbondSandaShortPoSitiOninOneofthe8%
20、couponbonds.)Takingalongpositionintwoofthe4%couponbondsandashortpositioninoneofthe8%couponbondsleadstothefbllowingcashflowsYear0:90-2x80=-70Yearlo:200-10(MoObecausethecouponscancelout.$1OOin1OyearstimeisequivalenttoSVOloday-The10-yearrate,R,(continuouslycompoundcd)isthcreforegivenby!OO=7OeoRTheratei
21、s1.nH)O二0.03571070or3.57%perannumProblen4.17.ExplaincarefullywhyliquiditypreferencetheoryisconsistentwiththeobservationthatthetermstructureofiIiterestratestendstobeupwardslopingmoreoftenthanitisdownwardsloping.Iflong-termratesweresimplyareflectionofexpectedfutureshort-termrates,Wewouldexpccttheterms
22、tructuretobcdownwardslopingasoftcnasitisupwardslopingXThisisbaSedontheassumptionthathalfbfthctimeinvestorsexpectratestoincreaseandhalfofthetimeinvestorsexpectratestodecrease).Liquiditypreferencetheoarguesthatlongtermratesarehighrelativetoexpectedfutureshort-termrates.Thismcansthatthetenstructureshou
23、ldbeupwardslopingmorcoftenthanitisdownwardsloping.Problem4.18.llIvhenthezerocurveisupwardsloping,thezeroratefraparticulannatutyisgreateharheparyeorhatmatuty.Whenthezerocurveisdwnwardsloping,thereverseisZM.Explainwhythisisso.Theparyieldistheyicldonacoupon-bearingbond.Thezerorateistheyieldonazero-Coup
24、onbond-Whentheyieldcurveisupwardsloping,theyieldonanN-yearcoupon-bearingbondislessthantheyieldonanN-yearzero-Couponbond.ThisisbecausethecouponsarediscountedatalowerratethantheN-yearrateanddragtheyielddownbelowthisrate.Similarly,WhentheyieldcurveisdownwardsIopingJheyieIdonanN-yearcouponbearingbondish
25、igherthantheyieldonanN-yearzero-couponbond.Problem4.19.Whyareu-S.Treasuryratessignificantlylowerthanotherratesthatareclosetoriskfree?TWoreasonsgiveninthechapterare!.TheamountofcapitalabankisrequircdtoholdtosupportaninvestmentinTreasurybillsandbondsissubstantiallysmallerthanthecapitalrequiredtosuppor
26、tasimilarinvestmentinothervery-low-riskinstruments.2.IntheUnitedStates,Treasuryinstrumentsaregivenafavorabletaxtreatmentcomparedwithmostotherfixed-incomeinvestmentsbecausetheyarenottaxedatthestatelevel.Problem4.20.Whydoesaloanintherepomarketinvolveverylittlecreditrisk?Arepoisacontractwhereaninvestme
27、ntdealerwhoownssecuritiesagreestosellthemtoanothercompanynowandbuythembacklaterataslightlyhigherprice.Theothercompanyisprovidingaloantotheinvestmentdealer.Thisloaninvolvesverylittlecreditriskjftheborrowerdoesnothonortheagreement,thelendingcompanysimPlykeepsthesecurities.Ifthelendingcompanydoesnotkee
28、ptoitssideoftheagreement,theoriginalownerofthesecuritieskeepsthecash.Problem4.21.ExplainwhyanFRAisequivalenttotheexchangeofafloatingrateofinterestforafixedrateofinterest?AFRAisanagreementthatacertainspecifiedinterestrate,R,Willapplytoacertainprincipal,L,fracertainspecifiedfuturetimeperiod.Supposetha
29、ttherateobservedinthemarketforthefuturetimeperiodatthebeginningofthetimeperiodprovestobeR.IftheMFRAisanagreementthatRwillapplyWhentheprincipalisinvested,theholderoftheFRACanborrOWthCPrinCiPalatRandthCninVeStitatR.ThenetcashflowatthccndofMKthcpcriodisthcnaninflowofRLandanoutflowofKL.IfthcFRAisanagree
30、mcntthatMRwillapplyWhentheprincipalisborrowed,theholderoftheFRAcaninvesttheborrowcdprincpalatR.ThenetcashflowattheendofthcpcriodisthcnaninflowofMRLandanoutflowofRL-IneithercaseweseethattheFRAinvolvestheexchangeofaMKfixedrateofinterestontheprincipalofLforaoatingrateofinterestontheprincpal.Problem4.22
31、.Afive-yearbondwithayieldof7%(continuouslycompounded)paysan8%couponattheendofeachyear.a) Whatisthebond,sprice?b) Whatisthehond1Sduration?c) Usethedurationtocalculatetheeffectonthebond,spriceofa0.2%decreaseinitsye6.d)ccwJrc7Aamofa6.8%perannumyieldandverifythattheresultisinagreementwithyouranswerto(c)
32、.a) Thebond1Spriceis8eo.o7+8eo,o7x2+8e-om3+8eo.o7x4l08eoo7x5二103.05b) Thebond1Sdurationis18-0.07+2x8-0.07x2+3x8-0.07x3+4x8-0.07x4+5x108-O.O7x5103.05=4.3235yearsc) Since5WiththenotationinthechapterAB二-BDAytheeffectonthebond,spriceofa0.2%decreaseinitsyieldis103.05x4.3235x0.002=0.89ThebondlSpriceshould
33、increasefrom103.05to103.94d) Witha6.8%yieldthebondspriceis8e-0.068+8e-0.068x2+8e-0.068x3+8e-0.06Sx4+108e-0.06Sx5二103.95ThisiscIosetotheanswerin(C).Problem4.23.Thecashpricesofsix-monthandone-yearTreasurybillsare94.0and89.0.A1.5-yearTreasurybondthatwillpaycouponsof$4everysixmonthscurrentlysellsfor$94.
34、84.Atwo-yearTreasurybondthatwillpaycouponsof$5everysixmonthscurrentlysellsfor$97.12.Calculatethesix-month,one-year,1.5-year,andtwo-yearTreasuryzerorates.The6-monthTreasurj,billprovidesareturnof694=6.383%insixmonths.Thisis2x6.383=12.766%perannumwithsemiannualcompoundingor21n(l.06383)=12.38%perannumwi
35、thcontinuouscompounding.The12-Inonthrateisl1/89=12.360%withannualcompoundingorln(l.1236)=11.65%withcontinuouscompoundingFortheRyearbondwemusthave24e-o.1238x5+4e-o.i65xi+104e,5r=94.84whereisthel十yearzerorate.Itfollowsthat23.76+3.56+104e-,5R=94.84e-.5R=0.8415R=O.115or11.5%.Forthe2-yearbondwemuslhave5e
36、-(),1238o,5+5e-x2=1015.32Itis3.18%.Problen4.27.(ExcelfiIe)Afive-yearbondprovidesacouponof5%perannumpayablesemiannually.ltspriceis104.Whatisthebondsyield7YoumayfindExcesSolveruseful.Theanswer(withcontinuouscompounding)is4.07%Problem4.28(Excelfile)Supposethat3-month,6-month112-month,2-yearsand3-yearOI
37、Sratesare2.0%,2.5%,3.2%s45%,and5%,respectively.The3-month16-monthand12-monthOISsinvolveasingleexchangeatmaturity;the2-yearand3-yearOISsinvolvequarterlyexchanges.ThecompoundingfrequenciesusedforexpressingtheratescorreSpondtothefrequencyofexchanges.CalculatetheoiszeroratesusingcontinuouscompoundingJnt
38、erpoIatelinearlybetweencontinuouslycompoundedratestodetermineratesbetweenemonthsandl2months,between12monthsand2years,andbetween2yearsand3years.YoumayfindExcesSoveruseu.Thetwo-yearandthreeyearOISratesareparyields.ThecalculationsareindicatedontheExcelfile.The3-month,6-month,l2-month,2-yearand3-yearzer
39、oratesare1.9950%,2.4845%,3.1499%,4.5153%,and5.0264%,respectively.ThecalculationscanalSobecarriedoutwithDerivaGem.Problem4.29.Aninterestrateisquotedas5%perannumwithsemiannualcompounding.Whatistheequivalentratewith(a)annualcompounding,(b)monthlycompounding,and(c)continuouscompounding?2.5%ispaideverysi
40、xmonths.a) Withannualcompoundingtherateis1.0252-1=0.050625or5.0625%b) Withmonthlycompoundingtherateis12x(1.0256-1)=0.04949or4.949%.c) WithcontinuouscompoundingtherateisZxln1.025=0.04939or4.939%.Problem4.30.The6-month,12-month.18-month,and24-monthzeroratesare4%,4.5%,475%,and5%withsemiannualcompoundin
41、g.a) Whataretherateswithcontinuouscompounding?b) Whatistheforwardrateforthesix-monthperiodbeginningin18months?c) Whatisthetwo-yearparyield?a) Withcontinuouscompoundingthe6-monthrateis21n1.02=0.039605or3.961%.The12-monthrateis2ln1.0225=0.04450lor4.4501%.The18-monthrateis21n1.02375=0.046945or4.6945%.T
42、he24-monthrateis21n1.025=0.049385or4.9385%.b) Theforwardrate(expressedwithcontinuouscompounding)isfromequation(4.5)4.9385x2-4.6945xl.505or5.6707%.Whenexpressedwithsemiannualcompoundingthisis2(eo.o567o7xo.5-1)=0.057518or5.7518%.c) Thefbrmulafortheparyieldis(IoO-IOod)mC=AInthiscased=(1/1.025)4=0.90595,m=2andA=+-+-+-=3.77481.021.022521.0237531.0254sothatc=(lo-9o.595)r2=4.9833.7748or4.983%.Bydefinitionthisisalsotheyieldonatwo-yearbondthatpaysacouponequaltotheparyield.Problem4.31.Supposethatrisk-freeratesareasinProblem4.30.WhatisthevalueofanFRAwheretheholderpaysLIBORandreceives7%(semiann
