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HullOFOD8eSolutionsCh
CHAPTER4
InterestRates
PracticeQuestions
Problem4.1.
Abankquotesyouaninterestrateof14%perannumwithquarterlycompounding.Whatistheequivalentratewith（a）continuouscompoundingand（b）annualcompounding?
（a）Theratewithcontinuouscompoundingis
or13.76%perannum.
（b）Theratewithannualcompoundingis
or14.75%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%.Thepriceis
Ifthe18-monthzerorateis
wemusthave
whichgives
%.
Problem4.4.
Aninvestorreceives$1,100inoneyearinreturnforaninvestmentof$1,000now.Calculatethepercentagereturnperannumwitha）annualcompounding,b）semiannualcompounding,c）monthlycompoundingandd）continuouscompounding.
（a）Withannualcompoundingthereturnis
or10%perannum.
（b）Withsemi-annualcompoundingthereturnis
where
i.e.,
sothat
.Thepercentagereturnistherefore9.76%perannum.
（c）Withmonthlycompoundingthereturnis
where
i.e.
sothat
.Thepercentagereturnistherefore9.57%perannum.
（d）Withcontinuouscompoundingthereturnis
where:
i.e.,
sothat
.Thepercentagereturnistherefore9.53%perannum.
Problem4.5.
Supposethatzerointerestrateswithcontinuouscompoundingareasfollows:
Maturity（months）
Rate（%perannum）
3
8.0
6
8.2
9
8.4
12
8.5
15
8.6
18
8.7
Calculateforwardinterestratesforthesecond,third,fourth,fifth,andsixthquarters.
Theforwardrateswithcontinuouscompoundingareasfollowsto
Qtr2
8.4%
Qtr3
8.8%
Qtr4
8.8%
Qtr5
9.0%
Qtr6
9.2%
Problem4.6.
AssumingthatzeroratesareasinProblem4.5,whatisthevalueofanFRAthatenablestheholdertoearn9.5%forathree-monthperiodstartinginoneyearonaprincipalof$1,000,000?
Theinterestrateisexpressedwithquarterlycompounding.
Theforwardrateis9.0%withcontinuouscompoundingor9.102%withquarterlycompounding.Fromequation（4.9）,thevalueoftheFRAistherefore
or$893.56.
Problem4.7.
Thetermstructureofinterestratesisupwardsloping.Putthefollowinginorderofmagnitude:
（a）Thefive-yearzerorate
（b）Theyieldonafive-yearcoupon-bearingbond
（c）Theforwardratecorrespondingtotheperiodbetween4.75and
yearsinthefuture
Whatistheanswertothisquestionwhenthetermstructureofinterestratesisdownwardsloping?
Whenthetermstructureisupwardsloping,
.Whenitisdownwardsloping,
.
Problem4.8.
Whatdoesdurationtellyouaboutthesensitivityofabondportfoliotointerestrates?
Whatarethelimitationsofthedurationmeasure?
Durationprovidesinformationabouttheeffectofasmallparallelshiftintheyieldcurveonthevalueofabondportfolio.Thepercentagedecreaseinthevalueoftheportfolioequalsthedurationoftheportfoliomultipliedbytheamountbywhichinterestratesareincreasedinthesmallparallelshift.Thedurationmeasurehasthefollowinglimitation.Itappliesonlytoparallelshiftsintheyieldcurvethataresmall.
Problem4.9.
Whatrateofinterestwithcontinuouscompoundingisequivalentto15%perannumwithmonthlycompounding?
Therateofinterestis
where:
i.e.,
Therateofinterestistherefore14.91%perannum.
Problem4.10.
Adepositaccountpays12%perannumwithcontinuouscompounding,butinterestisactuallypaidquarterly.Howmuchinterestwillbepaideachquarterona$10,000deposit?
Theequivalentrateofinterestwithquarterlycompoundingis
where
or
Theamountofinterestpaideachquarteristherefore:
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.Thecashpriceis
Problem4.12.
Athree-yearbondprovidesacouponof8%semiannuallyandhasacashpriceof104.Whatisthebond’syield?
Thebondpays$4in6,12,18,24,and30months,and$104in36months.Thebondyieldisthevalueof
thatsolves
UsingtheSolverorGoalSeektoolinExcel
or6.407%.
Problem4.13.
Supposethatthe6-month,12-month,18-month,and24-monthzeroratesare5%,6%,6.5%,and7%respectively.Whatisthetwo-yearparyield?
Usingthenotationinthetext,
.Also
Theformulainthetextgivestheparyieldas
Toverifythatthisiscorrectwecalculatethevalueofabondthatpaysacouponof7.072%peryear（thatis3.5365everysixmonths）.Thevalueis
verifyingthat7.072%istheparyield.
Problem4.14.
Supposethatzerointerestrateswithcontinuouscompoundingareasfollows:
Maturity（years）
Rate（%perannum）
1
2.0
2
3.0
3
3.7
4
4.2
5
4.5
Calculateforwardinterestratesforthesecond,third,fourth,andfifthyears.
Theforwardrateswithcontinuouscompoundingareasfollows:
Year2:
4.0%
Year3:
5.1%
Year4:
5.7%
Year5:
5.7%
Problem4.15.
UsetheratesinProblem4.14tovalueanFRAwhereyouwillpay5%forthethirdyearon$1million.
Theforwardrateis5.1%withcontinuouscompoundingor
withannualcompounding.The3-yearinterestrateis3.7%withcontinuouscompounding.Fromequation（4.10）,thevalueoftheFRAistherefore
or$2,078.85.
Problem4.16.
A10-year,8%couponbondcurrentlysellsfor$90.A10-year,4%couponbondcurrentlysellsfor$80.Whatisthe10-yearzerorate?
（Hint:
Considertakingalongpositionintwoofthe4%couponbondsandashortpositioninoneofthe8%couponbonds.）
Takingalongpositionintwoofthe4%couponbondsandashortpositioninoneofthe8%couponbondsleadstothefollowingcashflows
becausethecouponscancelout.$100in10yearstimeisequivalentto$70today.The10-yearrate,
（continuouslycompounded）isthereforegivenby
Therateis
or3.57%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.
ExplainwhyanFRAisequivalenttotheexchangeofa
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