(完整版)HullOFOD10eSolutionsCh04

1、CHAPTER4InterestRatesPracticeQuestionsProblem4.1.Abankquotesyouaninterestrateof7%perannumwithquarterlycompounding.Whatistheequivalentratewith(a)continuouscompoundingand(b)annualcompounding?(a) Theratewithcontinuouscompoundingis(0074In1+二0.0694I4丿or6.94%perannum.(b) Theratewithannualcompoundingisor7.

2、19%perannum.Problem4.2.ExplainhowLIBORisdetermined(0.07¥1+亍丿-1=0.0719LIBORistheLondonInterBankOfferedRate.ItiscalculateddailyfromborrowingratesestimatedbyapanelofbanksProblem4.3.Thesix-monthandone-yearzeroratesareboth5%perannum.Forabondthathasalifeof18monthsandpaysacouponof4%perannum(withsemian

3、nualpaymentsandonehavingjustbeenmade),theyieldis5.2%perannum.Whatisthebond'sprice?Whatisthe18-monthzerorate?Allratesarequotedwithsemiannualcompounding.Supposethebondhasafacevalueof$100.Itspriceisobtainedbydiscountingthecashflowsat5.2%.Thepriceis22102+1.0261.02621.0263=98.29Ifthe18-monthzeroratei

4、sR,wemusthave22+1.0251.0252102(1+R/2)3二98.29whichgivesR=5.204%.Problem4.4.Aninvestorreceives$1,100inoneyearinreturnforaninvestmentof$1,000now.Calculatethepercentagereturnperannumwitha)annualcompounding,b)semiannualcompounding,c)monthlycompoundingandd)continuouscompounding.(a) Withannualcompoundingth

5、ereturnis1100-1二0.11000or10%perannum.(b) Withsemi-annualcompoundingthereturnisRwhere(R)210001+_=1100I2丿i.e.,R1+£11二1.04882sothatR=0.0976.Thepercentagereturnistherefore9.76%perannum.(c) WithmonthlycompoundingthereturnisRwhere(R)1210001+=1100I12丿i.e.1+A=血皿797sothatR=0.0957.Thepercentagereturnisth

6、erefore9.57%perannum.(d) WithcontinuouscompoundingthereturnisRwhere:1000eR=1100i.e.,eR=1.1sothatR=ln1.1=0.0953.Thepercentagereturnistherefore9.53%perannum.Problem4.5.Supposethatzerointerestrateswithcontinuouscompoundingareasfollows:Maturity(months)Rate(%perannum)33.063.293.4123.5153.6183.7Calculatef

7、orwardinterestratesforthesecond,third,fourth,fifth,andsixthquarters.TheforwardrateswithcontinuouscompoundingareasfollowstoQtr23.4%Qtr33.8%Qtr43.8%Qtr54.0%Qtr64.2%Problem4.6.Assumingthatrisk-freeratesareasinProblem4.5,whatisthevalueofanFRAwheretheholderwillpayLIBORandreceive4.5%(quarterlycompounded)f

8、orathree-monthperiodstartinginoneyearonaprincipalof$1,000,000.TheforwardLIBORrateforthethreemonthperiodis5%quarterlycompounded.Fromequation(4.9),thevalueoftheFRAistherefore1,000,000X0.25X(0.045-0.050)e-0.036x1.25二-1,195or$1,195Problem4.7.Thetermstructureofinterestratesisupwardsloping.Putthefollowing

9、inorderofmagnitude:(a) Thefive-yearzerorate(b) Theyieldonafive-yearcoupon-bearingbond(c) Theforwardratecorrespondingtotheperiodbetween4.75and5yearsinthefutureWhatistheanswertothisquestionwhenthetermstructureofinterestratesisdownwardsloping?Whenthetermstructureisupwardsloping,c>a>b.Whenitisdown

10、wardsloping,b>a>c.Problem4.8.Whatdoesdurationtellyouaboutthesensitivityofabondportfoliotointerestrates?Whatarethelimitationsofthedurationmeasure?Durationprovidesinformationabouttheeffectofasmallparallelshiftintheyieldcurveonthevalueofabondportfolio.Thepercentagedecreaseinthevalueoftheportfolio

11、equalsthedurationoftheportfoliomultipliedbytheamountbywhichinterestratesareincreasedinthesmallparallelshift.Thedurationmeasurehasthefollowinglimitation.Itappliesonlytoparallelshiftsintheyieldcurvethataresmall.Problem4.9.Whatrateofinterestwithcontinuouscompoundingisequivalentto8%perannumwithmonthlyco

12、mpounding?TherateofinterestisRwhere:i.e.,(0.08)12eR=1+I12丿R二12lnfi+008I12丿=0.0797Therateofinterestistherefore7.97%perannum.Problem4.10.Adepositaccountpays4%perannumwithcontinuouscompounding,butinterestisactuallypaidquarterly.Howmuchinterestwillbepaideachquarterona$10,000deposit?Theequivalentrateofin

13、terestwithquarterlycompoundingisRwhereeo.04=orR二4(eo.o1-1)二0.0402Theamountofinterestpaideachquarteristherefore:0.040210,ooox-：二100.504or$100.50.Problem4.11.Supposethat6-month,12-month,18-month,24-month,and30-monthzeroratesare,respectively,4%,4.2%,4.4%,4.6%,and4.8%perannumwithcontinuouscompounding.Es

14、timatethecashpriceofabondwithafacevalueof100thatwillmaturein30monthsandpayacouponof4%perannumsemiannually.Thebondpays$2in6,12,18,and24months,and$102in30months.Thecashpriceis2e-0.04x0.5+2e-0.042x1.0+2e-0.044x1.5+2e-0.046x2+102e-0.048x2.5=98.04Problem4.12.Athree-yearbondprovidesacouponof8%semiannually

15、andhasacashpriceof104.Whatisthebond'syield?Thebondpays$4in6,12,18,24,and30months,and$104in36months.Thebondyieldisthevalueofythatsolves4e-0.5y+4e-1.0y+4e-1.5y+4e-2.0y+4e-2.5y+104e-3.0y=104UsingtheSolverorGoalSeektoolinExcel,y=0.06407or6.407%.Problem4.13.Supposethatthe6-month,12-month,18-month,and

16、24-monthzeroratesare5%,6%,6.5%,and7%respectively.Whatisthetwo-yearparyield?Usingthenotationinthetext,m=2,d=e-0.07x2=0.8694.AlsoA=e-0.05x0.5+e-0.06x1.0+e-0.065x1.5+e-0.07x2.0=3.6935Theformulainthetextgivestheparyieldas(100-100x0.8694)x2=7创3.6935_*Toverifythatthisiscorrectwecalculatethevalueofabondtha

17、tpaysacouponof7.0741%peryear(thatis3.5370everysixmonths).Thevalueis3.537e-0.05x0.5+3.537e-0.06x1.0+3.537e-0.065x1.5+103.537e-0.07x2.0=100verifyingthat7.0741%istheparyield.Problem4.14.Supposethatrisk-freezerointerestrateswithcontinuouscompoundingareasfollows:Maturity(years)Rate(%perannum)12.023.033.7

18、44.254.5Calculateforwardinterestratesforthesecond,third,fourth,andfifthyears.Theforwardrateswithcontinuouscompoundingareasfollows:Year2:4.0%Year3:5.1%Year4:5.7%Year5:5.7%Problem4.15.Usetherisk-freeratesinProblem4.14tovalueanFRAwhereyouwillpay5%(compoundedannually)andreceiveLIBORforthethirdyearon$1mi

19、llion.TheforwardLIBORrate(annuallycompounded)forthethirdyearis5.5%.WevaluetheFRAbyassumingthattheforwardLIBORwillberealized.ThevalueoftheFRAis1,000,000x(0.0550.050).637x3=$4,474.69Problem4.16.A10-year,8%couponbondcurrentlysellsfor$90.A10-year,4%couponbondcurrentlysellsfor$80.Whatisthe10-yearzerorate

20、?(Hint:Considertakingalongpositionintwoofthe4%couponbondsandashortpositioninoneofthe8%couponbonds.)Takingalongpositionintwoofthe4%couponbondsandashortpositioninoneofthe8%couponbondsleadstothefollowingcashflowsYear0:90一2x80=-70Year10:200-100=100becausethecouponscancelout.$100in10yearstimeisequivalent

21、to$70today.The10-yearrate,R,(continuouslycompounded)isthereforegivenby100=70e10RTherateis丄in100二0.03571070or3.57%perannumProblem4.17.Explaincarefullywhyliquiditypreferencetheoryisconsistentwiththeobservationthatthetermstructureofinterestratestendstobeupwardslopingmoreoftenthanitisdownwardsloping.Ifl

22、ong-termratesweresimplyareflectionofexpectedfutureshort-termrates,wewouldexpectthetermstructuretobedownwardslopingasoftenasitisupwardsloping.(Thisisbasedontheassumptionthathalfofthetimeinvestorsexpectratestoincreaseandhalfofthetimeinvestorsexpectratestodecrease).Liquiditypreferencetheoryarguesthatlo

23、ngtermratesarehighrelativetoexpectedfutureshort-termrates.Thismeansthatthetermstructureshouldbeupwardslopingmoreoftenthanitisdownwardsloping.Problem4.18.“Whenthezerocurveisupwardsloping,thezerorateforaparticularmaturityisgreaterthantheparyieldforthatmaturity.Whenthezerocurveisdownwardsloping,thereve

24、rseistrue.”Explainwhythisisso.Theparyieldistheyieldonacoupon-bearingbond.Thezerorateistheyieldonazero-couponbond.Whentheyieldcurveisupwardsloping,theyieldonanN-yearcoupon-bearingbondislessthantheyieldonanN-yearzero-couponbond.ThisisbecausethecouponsarediscountedatalowerratethantheN-yearrateanddragth

25、eyielddownbelowthisrate.Similarly,whentheyieldcurveisdownwardsloping,theyieldonanN-yearcouponbearingbondishigherthantheyieldonanN-yearzero-couponbond.Problem4.19.WhyareU.S.Treasuryratessignificantlylowerthanotherratesthatareclosetoriskfree?Tworeasonsgiveninthechapterare1.Theamountofcapitalabankisreq

26、uiredtoholdtosupportaninvestmentinTreasurybillsandbondsissubstantiallysmallerthanthecapitalrequiredtosupportasimilarinvestmentinothervery-low-riskinstruments.2.IntheUnitedStates,Treasuryinstrumentsaregivenafavorabletaxtreatmentcomparedwithmostotherfixed-incomeinvestmentsbecausetheyarenottaxedatthest

27、atelevel.Problem4.20.Whydoesaloanintherepomarketinvolveverylittlecreditrisk?Arepoisacontractwhereaninvestmentdealerwhoownssecuritiesagreestosellthemtoanothercompanynowandbuythembacklaterataslightlyhigherprice.Theothercompanyisprovidingaloantotheinvestmentdealer.Thisloaninvolvesverylittlecreditrisk.I

28、ftheborrowerdoesnothonortheagreement,thelendingcompanysimplykeepsthesecurities.Ifthelendingcompanydoesnotkeeptoitssideoftheagreement,theoriginalownerofthesecuritieskeepsthecash.Problem4.21.ExplainwhyanFRAisequivalenttotheexchangeofafloatingrateofinterestforafixedrateofinterest?AFRAisanagreementthata

29、certainspecifiedinterestrate,R,willapplytoacertainKprincipal,L,foracertainspecifiedfuturetimeperiod.SupposethattherateobservedinthemarketforthefuturetimeperiodatthebeginningofthetimeperiodprovestobeR.IftheMFRAisanagreementthatRwillapplywhentheprincipalisinvested,theholderoftheKFRAcanborrowtheprincip

30、alatRandtheninvestitatR.ThenetcashflowattheendofMKtheperiodisthenaninflowofRLandanoutflowofRL.IftheFRAisanagreementthatKMRwillapplywhentheprincipalisborrowed,theholderoftheFRAcaninvesttheKborrowedprincipalatR.ThenetcashflowattheendoftheperiodisthenaninflowofMRLandanoutflowofRL.Ineithercaseweseethatt

31、heFRAinvolvestheexchangeofaMKfixedrateofinterestontheprincipalofLforafloatingrateofinterestontheprincipal.Problem4.22.Afive-yearbondwithayieldof7%(continuouslycompounded)paysan8%couponattheendofeachyear.a) Whatisthebond'sprice?b) Whatisthebond'sduration?c) Usethedurationtocalculatetheeffecto

32、nthebond'spriceofa0.2%decreaseinitsyield.d)Recalculatethebond'spriceonthebasisofa6.8%perannumyieldandverifythattheresultisinagreementwithyouranswerto(c).a) Thebond'spriceis8e-0.07+8e-0.07x2+8e-omx3+8e-0.07x4+108e-0.07x5二103.05b) Thebond'sdurationis18e-0.07+2x8e-0.07x2+3x8e-0.07x3+4x8

33、e-0.07x4+5x108e-0.07x5103.05J=4.3235yearsc) Since,withthenotationinthechapterAB二-BDAytheeffectonthebond'spriceofa0.2%decreaseinitsyieldis103.05x4.3235x0.002=0.89Thebond'spriceshouldincreasefrom103.05to103.94d) Witha6.8%yieldthebond'spriceis8e-0.068+8e-0.068x2+8e-0.068x3+8e-0.068x4+108e-0

34、.068x5二103.95Thisisclosetotheanswerin(c).Problem4.23.Thecashpricesofsix-monthandone-yearTreasurybillsare94.0and89.0.A1.5-yearTreasurybondthatwillpaycouponsof$4everysixmonthscurrentlysellsfor$94.84.Atwo-yearTreasurybondthatwillpaycouponsof$5everysixmonthscurrentlysellsfor$97.12.Calculatethesix-month,

35、one-year,1.5-year,andtwo-yearTreasuryzerorates.The6-monthTreasurybillprovidesareturnof6/94=6.383%insixmonths.Thisis2x6.383=12.766%perannumwithsemiannualcompoundingor2ln(1.06383)=12.38%perannumwithcontinuouscompounding.The12-monthrateis11/89=12.360%withannualcompoundingorln(1.1236)=11.65%withcontinuo

36、uscompounding.Forthe1+yearbondwemusthave24e-0.1238x0.5+4e-0.1165x1+104e-1.5r=94.84whereRisthe1十yearzerorate.Itfollowsthat23.76+3.56+104e-1.5r=94.84e-1.5R=0.8415R=0.115or11.5%.Forthe2-yearbondwemusthave5e-0.1238x0.5+5e-0.1165x1+5e-0.115x1.5+105e-2r=97.12whereRisthe2-yearzerorate.Itfollowsthate-2R=0.7

37、977R=0.113or11.3%.Problem4.24.“Aninterestrateswapwheresix-monthLIBORisexchangedforafixedrate5%onaprincipalof$100millionforfiveyearsisaportfolioofnineFRAs.”Explainthisstatement.Thefirstexchangeofpaymentsisknown.EachsubsequentexchangeofpaymentsisanFRAwhereinterestat5%isexchangedforinterestatLIBORonapr

38、incipalof$100million.InterestrateswapsarediscussedfurtherinChapter7.FurtherQuestionsProblem4.25.Whencompoundedannuallyaninterestrateis11%.Whatistheratewhenexpressedwith(a)semiannualcompounding,(b)quarterlycompounding,(c)monthlycompounding,(d)weeklycompounding,and(e)dailycompounding.Wemustsolve1.11=(

39、1+R/n)nwhereRistherequiredrateandthenumberoftimesperyeartherateiscompounded.Theanswersarea)10.71%,b)10.57%,c)10.48%,d)10.45%,e)10.44%Problem4.26.ThefollowingtablegivesTreasuryzeroratesandcashflowsonaTreasurybond:Maturity(yearsZerorateCouponpaymentPrincipal0.52.0%$201.02.3%$201.527%$202.03.2%$20$1000

40、Zeroratesarecontinuouslycompounded(a) Whatisthebond'stheoreticalprice?(b)Whatisthebond'syieldassumingthebondsellsforitstheoreticalprice?Thebond'stheoreticalpriceis2Oxe-o.o2xo.5+2Oxe-o.o23xi+2Oxe-o.o27xi.5+lO2Oxe-o.o32x2=1015.32Thebond'syieldassumingthatitsellsforitstheoreticalpriceis

41、obtainedbysolving20xe-yx0.5+20xe-yxi+20xe-yxi.5+1020xe-yx2=1015.32Itis3.18%.Problem4.27.(Excelfile)Afive-yearbondprovidesacouponof5%perannumpayablesemiannually.Itspriceis104.Whatisthebond'syield?YoumayfindExcel'sSolveruseful.Theanswer(withcontinuouscompounding)is4.07%Problem4.28(Excelfile)Su

42、pposethat3-month,6-month,12-month,2-year,and3-yearOISratesare2.0%,2.5%,3.2%,4.5%,and5%,respectively.The3-month,6-monthand12-monthOISsinvolveasingleexchangeatmaturity;the2-yearand3-yearOISsinvolvequarterlyexchanges.Thecompoundingfrequenciesusedforexpressingtheratescorrespondtothefrequencyofexchanges.

43、CalculatetheOISzeroratesusingcontinuouscompounding.Interpolatelinearlybetweencontinuouslycompoundedratestodetermineratesbetween6monthsand12months,between12monthsand2years,andbetween2yearsand3years.YoumayfindExcel'sSolveruseful.Thetwo-yearandthreeyearOISratesareparyields.Thecalculationsareindicat

44、edontheExcelfile.The3-month,6-month,12-month,2-yearand3-yearzeroratesare1.9950%,2.4845%,3.1499%,4.5153%,and5.0264%,respectively.ThecalculationscanalsobecarriedoutwithDerivaGem.Problem4.29.Aninterestrateisquotedas5%perannumwithsemiannualcompounding.Whatistheequivalentratewith(a)annualcompounding,(b)m

45、onthlycompounding,and(c)continuouscompounding?2.5%ispaideverysixmonths.a) Withannualcompoundingtherateis1.0252-1=0.050625or5.0625%b) Withmonthlycompoundingtherateis12x(1.0251/6-1)=0.04949or4.949%.c) Withcontinuouscompoundingtherateis2xln1.025=0.04939or4.939%.Problem4.30.The6-month,12-month.18-month,

46、and24-monthzeroratesare4%,4.5%,4.75%,and5%withsemiannualcompounding.a) Whataretherateswithcontinuouscompounding?b) Whatistheforwardrateforthesix-monthperiodbeginningin18months?c) Whatisthetwo-yearparyield?a) Withcontinuouscompoundingthe6-monthrateis2ln1.02=0.039605or3.961%.The12-monthrateis2ln1.0225

47、=0.044501or4.4501%.The18-monthrateis2ln1.02375=0.046945or4.6945%.The24-monthrateis2ln1.025=0.049385or4.9385%.b) Theforwardrate(expressedwithcontinuouscompounding)isfromequation(4.5)4.9385x2-4.6945x1.505or5.6707%.Whenexpressedwithsemiannualcompoundingthisis2(e0.056707x0.5-1)=0.057518or5.7518%.c) Thef

48、ormulafortheparyieldis(100-100d)mc=AInthiscased=(1/1.025)4=0.90595,m=2andA=丄+-+-+-=3.77481.021.022521.0237531.0254sothatc=(10°一9°.595)x2=4.9833.7748or4.983%.Bydefinitionthisisalsotheyieldonatwo-yearbondthatpaysacouponequaltotheparyield.Problem4.31.Supposethatrisk-freeratesareasinProblem4.3

49、0.WhatisthevalueofanFRAwheretheholderpaysLIBORandreceives7%(semiannuallycompounded)forasix-monthperiodbeginningin18months.Thecurrentforwardrateforthisperiodis6%(semiannuallycompounded).Theprincipalis$10millionThevalueoftheFRAis10,000,000x(0.070.06)x0.5x1=45,2981.0254or$45,298.Problem4.32.Thefollowin

50、gtablegivesthepricesofbondsBondPrincipal($)TimetoMaturity(yrs)AnnualCoupon($)*BondPrice($)1000.50.0981001.00.0951001.56.21011002.0&0104*Halfthestatedcouponispaideverysixmonthsa) Calculatezeroratesformaturitiesof6months,12months,18months,and24months.b) Whataretheforwardratesfortheperiods:6monthst

51、o12months,12monthsto18months,18monthsto24months?c) Whatarethe6-month,12-month,18-month,and24-monthparyieldsforbondsthatprovidesemiannualcouponpayments?d) Estimatethepriceandyieldofatwo-yearbondprovidingasemiannualcouponof7%perannum.a) Thezerorateforamaturityofsixmonths,expressedwithcontinuouscompoun

52、dingis21n(l+2/98)=4.0405%.Thezerorateforamaturityofoneyear,expressedwithcontinuouscompoundingisln(1+5/95)=5.1293.The1.5-yearrateisRwhere3.1e-0.040405x0.5+3.1e-0.051293x1+103.1e-Rx1.5-101ThesolutiontothisequationisR=0.054429.The2.0-yearrateisRwhere4e-0.040405x0.5+4e-0.051293x1+4e-0.054429x1.5+104e-Rx

53、2=104ThesolutiontothisequationisR=0.058085.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) Thecontinuouslycompoundedforwardratescalcula

54、tedusingequation(4.5)areshowninthethirdcolumnofthetablec) Theparyield,expressedwithsemiannualcompounding,canbecalculatedfromtheformulainSection4.6.Itisshowninthefourthcolumnofthetable.Inthefifthcolumnofthetableitisconvertedtocontinuouscompoundingd) Thepriceofthebondis3.5e-0.040405x0.5+3.5e-0.051293x

55、1+3.5e-0.054429x1.5+103.5e-0.058085x2=102.13Theyieldonthebond,ysatisfies3.5e-yx0.5+3.5e-yx1.0+3.5e-yx1.5+103.5e-yx2.0=102.13Thesolutiontothisequationisy=0.057723.Thebondyieldistherefore5.7723%.Problem4.33.PortfolioAconsistsofaone-yearzero-couponbondwithafacevalueof$2,000anda10-yearzero-couponbondwit

56、hafacevalueof$6,000.PortfolioBconsistsofa5.95-yearzero-couponbondwithafacevalueof$5,000.Thecurrentyieldonallbondsis10%perannum.(a) Showthatbothportfolioshavethesameduration.(b) Showthatthepercentagechangesinthevaluesofthetwoportfoliosfora0.1%perannumincreaseinyieldsarethesame.(c) Whatarethepercentag

57、echangesinthevaluesofthetwoportfoliosfora5%perannumincreaseinyields?a)ThedurationofPortfolioAis1x2000e-0.1x1+10x6000e-o.ixio=5.952OOOe-o.ixi+6OOOe-o.ixioSincethisisalsothedurationofPortfolioB,thetwoportfoliosdohavethesameduration.b)ThevalueofPortfolioAis2OOOe-o.i+6OOOe-o.ixio=4Oi6.95WhenyieldsincreasebyiObasispointsitsvaluebecomes2OOOe-o.ioi+6OOOe-o.ioixio二3993.i8Thepercentagedecreaseinvalueis23.77xiOO4Oi6.95二O.59%ThevalueofPortfolioBis5OOOe-o.ix5.95二2757.8iWhenyieldsincrease