HullOFOD8eSolutionsCh07文檔良心出品

CHAPTER7

Swaps

PracticeQuestions

Problem7.1.

CompaniesAandBhavebeenofferedthefollowingratesperannumona$20millionfive-yearloan:

FixedRate

FloatingRate

CompanyA

5.0%

LIBOR+0.1%

CompanyB

6.4%

LIBOR+0.6%

CompanyArequiresafloating-rateloan;companyBrequiresafixed-rateloan.Designaswapthatwillnetabank,actingasintermediary,0.1%perannumandthatwillappearequallyattractivetobothcompanies.

Ahasanapparentcomparativeadvantageinfixed-ratemarketsbutwantstoborrowfloating.Bhasanapparentcomparativeadvantageinfloating-ratemarketsbutwantstoborrowfixed.Thisprovidesthebasisfortheswap.Thereisa1.4%perannumdifferentialbetweenthefixedratesofferedtothetwocompaniesanda0.5%perannumdifferentialbetweenthefloatingratesofferedtothetwocompanies.Thetotalgaintoallpartiesfromtheswapistherefore140509%perannum.Becausethebankgets0.1%perannumofthisgain,theswapshouldmakeeachofAandB0.4%perannumbetteroff.ThismeansthatitshouldleadtoAborrowingatLIBOR03%andtoBborrowingat6.0%.TheappropriatearrangementisthereforeasshowninFigureS7.1.

FigureS7.1SwapforProblem7.1

Problem7.2.

CompanyXwishestoborrowU.S.dollarsatafixedrateofinterest.CompanyYwishestoborrowJapaneseyenatafixedrateofinterest.Theamountsrequiredbythetwocompaniesareroughlythesameatthecurrentexchangerate.Thecompanieshavebeenquotedthefollowinginterestrates,whichhavebeenadjustedfortheimpactoftaxes:

Yen

Dollars

CompanyX

5.0%

9.6%

CompanyY

6.5%

10.0%

Designaswapthatwillnetabank,actingasintermediary,50basispointsperannum.Maketheswapequallyattractivetothetwocompaniesandensurethatallforeignexchangeriskisassumedbythebank.

Xhasacomparativeadvantageinyenmarketsbutwantstoborrowdollars.Yhasacomparativeadvantageindollarmarketsbutwantstoborrowyen.Thisprovidesthebasisfortheswap.Thereisa1.5%perannumdifferentialbetweentheyenratesanda0.4%perannumdifferentialbetweenthedollarrates.Thetotalgaintoallpartiesfromtheswapistherefore150411%perannum.Thebankrequires0.5%perannum,leaving0.3%perannumforeachofXandY.TheswapshouldleadtoXborrowingdollarsat960393%perannumandtoYborrowingyenat650362%perannum.TheappropriatearrangementisthereforeasshowninFigureS7.2.Allforeignexchangeriskisbornebythebank.

FigureS7.2SwapforProblem7.2

Problem7.3.

A$100millioninterestrateswaphasaremaininglifeof10months.Underthetermsoftheswap,six-monthLIBORisexchangedfor7%perannum(compoundedsemiannually).Theaverageofthebid-offerratebeingexchangedforsix-monthLIBORinswapsofallmaturitiesiscurrently5%perannumwithcontinuouscompounding.Thesix-monthLIBORratewas4.6%perannumtwomonthsago.Whatisthecurrentvalueoftheswaptothepartypayingfloating?Whatisitsvaluetothepartypayingfixed?

Infourmonths$3.5million(05007$100million)willbereceivedand$2.3million(050046$100million)willbepaid.(Weignoredaycountissues.)In10months$3.5millionwillbereceived,andtheLIBORrateprevailinginfourmonths 'timewillbep

Thevalueofthefixed-ratebondunderlyingtheswapis

005412 0051012亠

5e 1035e $102718million

Thevalueofthefloating-ratebondunderlyingtheswapis

005412

(10023)e $100609million

Thevalueoftheswaptothepartypayingfloatingis$102718$100609$2109million.

Thevalueoftheswaptothepartypayingfixedis$2109million.

Theseresultscanalsobederivedbydecomposingtheswapintoforwardcontracts.Considerthepartypayingfloating.ThefirstforwardcontractinvoIvespaying$2.3millionandreceiving$3.5millioninfourmonths.Ithasavalueof12e005412 $1180million.To

valuethesecondforwardcontract,wenotethattheforwardinterestrateis5%perannumwithcontinuouscompounding,or5.063%perannumwithsemiannualcompounding.The

valueoftheforwardcontractis

100(0070500506305)e0051012 $0929million

Thetotalvalueoftheforwardcontractsistherefore$1180$0929$2109million.

Problem7.4.

Explainwhataswaprateis.Whatistherelationshipbetweenswapratesandparyields?

AswaprateforaparticularmaturityistheaverageofthebidandofferfixedratesthatamarketmakerispreparedtoexchangeforLIBORinastandardplainvaniliaswapwiththatmaturity.TheswaprateforaparticularmaturityistheLIBOR/swapparyieldforthatmaturity.

Problem7.5.

Acurrencyswaphasaremaininglifeof15months.ItinvoIvesexchanginginterestat10%on盤0millionforinterestat6%on$30milliononceayear.ThetermstructureofinterestratesinboththeUnitedKingdomandtheUnitedStatesiscurrentlyflat,andiftheswapwerenegotiatedtodaytheinterestratesexchangedwouldbe4%indollarsand7%insterling.Allinterestratesarequotedwithannualcompounding.Thecurrentexchangerate(dollarsperpoundsterling)is1.8500.Whatisthevalueoftheswaptothepartypayingsterling?Whatisthevalueoftheswaptothepartypayingdollars?

TheswapinvoIvesexchangingthesterlinginterestof20010or^millionforthedollarinterestof30006$18million.Theprincipalamountsarealsoexchangedattheendofthelifeoftheswap.Thevalueofthesterlingbondunderlyingtheswapis

2

14

(107)

22

54 22182millionpounds

(107)54

Thevalueofthedollarbondunderlyingtheswapis

18 318

14 54

(104) (104)

$32061million

Thevalueoftheswaptothepartypayingsterlingistherefore

32061 (22182185) $8976million

Thevalueoftheswaptothepartypayingdollarsis$8976million.Theresultscanalsobeobtainedbyviewingtheswapasaportfolioofforwardcontracts.Thecontinuouslycompoundedinterestratesinsterlinganddollarsare6.766%perannumand3.922%perannum.The3-monthand15-monthforwardexchangeratesare

(003922006766)025 (003922006766)125

85e 18369and185e 17854.Thevaluesofthetwoforwardcontractscorrespondingtotheexchangeofinterestforthepartypayingsterlingaretherefore

(18218369)e003922025 $1855million

and

003922125亠

(18217854)e $1686million

Thevalueoftheforwardcontractcorrespondingtotheexchangeofprincipalsis

003922125

(302017854)e $5435million

Thetotalvalueoftheswapis$1855$1686$5435 $8976million.

Problem7.6.

Explainthedifferencebetweenthecreditriskandthemarketriskinafinancialcontract.

Creditriskarisesfromthepossibilityofadefaultbythecounterparty.Marketriskarisesfrommovementsinmarketvariablessuchasinterestratesandexchangerates.Acomplicationisthatthecreditriskinaswapiscontingentonthevaluesofmarketvariables.Acompanypositioninaswaphascreditriskonlywhenthevalueoftheswaptothecompanyispositive.

Problem7.7.

Acorporatetreasurertellsyouthathehasjustnegotiatedafive-yearloanatacompetitivefixedrateofinterestof5.2%.Thetreasurerexplainsthatheachievedthe5.2%ratebyborrowingatsix-monthLIBORplus150basispointsandswappingLIBORfor3.7%.Hegoesontosaythatthiswaspossiblebecausehiscompanyhasacomparativeadvantageinthefloating-ratemarket.Whathasthetreasureroverlooked?

Therateisnottrulyfixedbecause,ifthecompany atingdecisieBeditwillnotbeable

torolloveritsfloatingrateborrowingsatLIBORplus150basispoints.Theeffectivefixedborrowingratethenincreases.Supposeforexamplethatthetreasurer 'sspreadoverI

increasesfrom150basispointsto200basispoints.Theborrowingrateincreasesfrom5.2%to5.7%.

Problem7.8.

Explainwhyabankissubjecttocreditriskwhenitentersintotwooffsettingswapcontracts.

Atthestartoftheswap,bothcontractshaveavalueofapproximatelyzero.Astimepasses,itislikelythattheswapvalueswillchange,sothatoneswaphasapositivevaluetothebankandtheotherhasanegativevaluetothebank.Ifthecounterpartyontheothersideofthepositive-valueswapdefaults,thebankstillhastohonoritscontractwiththeothercounterparty.Itisliabletoloseanamountequaltothepositivevalueoftheswap.

Problem7.9.

CompaniesXandYhavebeenofferedthefollowingratesperannumona$5million10-yearinvestment:

FixedRate

FloatingRate

CompanyX

8.0%

LIBOR

CompanyY

8.8%

LIBOR

CompanyXrequiresafixed-rateinvestment;companyYrequiresafloating-rateinvestment.Designaswapthatwillnetabank,actingasintermediary,0.2%perannumandwillappearequallyattractivetoXandY.

ThespreadbetweentheinterestratesofferedtoXandYis0.8%perannumonfixedrateinvestmentsand0.0%perannumonfloatingrateinvestments.Thismeansthatthetotalapparentbenefittoallpartiesfromtheswapis08%perannumOfthis0.2%perannumwillgotothebank.Thisleaves0.3%perannumforeachofXandY.Inotherwords,companyXshouldbeabletogetafixed-ratereturnof8.3%perannumwhilecompanyYshouldbeabletogetafloating-ratereturnLIBOR+0.3%perannum.TherequiredswapisshowninFigureS7.3.Thebankearns0.2%,companyXearns8.3%,andcompanyYearnsLIBOR+0.3%.

FigureS7.3 SwapforProblem7.9

Problem7.10.

AfinancialinstitutionhasenteredintoaninterestrateswapwithcompanyX.Underthetermsoftheswap,itreceives10%perannumandpayssix-monthLIBORonaprincipalof$10millionforfiveyears.Paymentsaremadeeverysixmonths.SupposethatcompanyXdefaultsonthesixthpaymentdate(endofyear3)whentheinterestrate(withsemiannualcompounding)is8%perannumforallmaturities.Whatisthelosstothefinancialinstitution?Assumethatsix-monthLIBORwas9%perannumhalfwaythroughyear3.

Attheendofyear3thefinancialinstitutionwasduetoreceive$500,000(0510%of$10million)andpay$450,000(059%of$10million).Theimmediatelossistherefore$50,000.Tovaluetheremainingswapweassumethanforwardratesarerealized.Allforwardratesare8%perannum.Theremainingcashflowsarethereforevaluedontheassumptionthatthefloatingpaymentis0500810000000 $400000andthenetpaymentthat

wouldbereceivedis500000400000 $100000.Thetotalcostofdefaultisthereforethe

costofforegoingthefollowingcashflows:

year: $50,000

year: $100,000

year: $100,000

year: $100,000

year: $100,000

Discountingthesecashflowstoyear3at4%persixmonthsweobtainthecostofthedefaultas$413,000.

Problem7.11.

CompaniesAandBfacethefollowinginterestrates(adjustedforthedifferentialimpactoftaxes):

A

B

USdollars(floatingrate)

LIBOR+0.5%

LIBOR+1.0%

Canadiandollars(fixedrate)

5.0%

6.5%

AssumethatAwantstoborrowU.S.dollarsatafloatingrateofinterestandBwantstoborrowCanadiandollarsatafixedrateofinterest.Afinancialinstitutionisplanningtoarrangeaswapandrequiresa50-basis-pointspread.IftheswapisequallyattractivetoAandB,whatratesofinterestwillAandBenduppaying?

CompanyAhasacomparativeadvantageintheCanadiandollarfixed-ratemarket.CompanyBhasacomparativeadvantageintheU.S.dollarfloating-ratemarket.(Thismaybebecauseoftheirtaxpositions.)However,companyAwantstoborrowintheU.S.dollarfloating-ratemarketandcompanyBwantstoborrowintheCanadiandollarfixed-ratemarket.Thisgivesrisetotheswapopportunity.

ThedifferentialbetweentheU.S.dollarfloatingratesis0.5%perannum,andthedifferentialbetweentheCanadiandollarfixedratesis1.5%perannum.Thedifferencebetweenthedifferentialsis1%perannum.Thetotalpotentialgaintoallpartiesfromtheswapistherefore1%perannum,or100basispoints.Ifthefinancialintermediaryrequires50basispoints,eachofAandBcanbemade25basispointsbetteroff.ThusaswapcanbedesignedsothatitprovidesAwithU.S.dollarsatLIBOR0.25%perannum,andBwithCanadiandollarsat6.25%perannum.TheswapisshowninFigureS7.4.

FigureS7.4 SwapforProblem7.11

Principalpaymentsflowintheoppositedirectiontothearrowsatthestartofthelifeoftheswapandinthesamedirectionasthearrowsattheendofthelifeoftheswap.Thefinancialinstitutionwouldbeexposedtosomeforeignexchangeriskwhichcouldbehedgedusingforwardcontracts.

Problem7.12.

Afinancialinstitutionhasenteredintoa10-yearcurrencyswapwithcompanyY.Underthetermsoftheswap,thefinancialinstitutionreceivesinterestat3%perannuminSwissfrancsandpaysinterestat8%perannuminU.S.dollars.Interestpaymentsareexchangedonceayear.Theprincipalamountsare7milliondollarsand10millionfrancs.SupposethatcompanyYdeclaresbankruptcyattheendofyear6,whentheexchangerateis$0.80perfranc.Whatisthecosttothefinancialinstitution?Assumethat,attheendofyear6,theinterestrateis3%perannuminSwissfrancsand8%perannuminU.S.dollarsforallmaturities.Allinterestratesarequotedwithannualcompounding.

Wheninterestratesarecompoundedannually

T

F°&—

1rf

whereF0istheT-yearforwardrate,S0isthespotrate,risthedomesticrisk-freerate,andrfistheforeignrisk-freerate.Asr008andrf003,thespotandforwardexchangeratesattheendofyear6are

Spot: 0.8000

yearforward: 0.8388

yearforward: 0.8796

yearforward: 0.9223

yearforward: 0.9670

Thevalueoftheswapatthetimeofthedefaultcanbecalculatedontheassumptionthatforwardratesarerealized.Thecashflowslostasaresultofthedefaultarethereforeas

follows:

Year

DollarPaid

CHFReceived

ForwardRate

DollarEquivofCHFReceived

CashFlowLost

6

560,000

300,000

0.8000

240,000

-320,000

7

560,000

300,000

0.8388

251,600

-308,400

8

560,000

300,000

0.8796

263,900

-296,100

9

560,000

300,000

0.9223

276,700

-283,300

10

7,560,000

10,300,000

0.9670

9,960,100

2,400,100

Discountingthenumbersinthefinalcolumntotheendofyear6at8%perannum,thecostofthedefaultis$679,800.

Notethat,ifthisweretheonlycontractenteredintobycompanyY,itwouldmakenosenseforthecompanytodefaultattheendofyearsixastheexchangeofpaymentsatthattimehasapositivevaluetocompanyY.Inpractice,companyYislikelytobedefaultinganddeclaringbankruptcyforreasonsunrelatedtothisparticularcontractandpaymentsonthecontractarelikelytostopwhenbankruptcyisdeclared.

Problem7.13.

Afterithedgesitsforeignexchangeriskusingforwardcontracts,isthefinancialinstitutionsaveragespreadinFigure7.11likelytobegreaterthanorlessthan20basispoints?Explainyouranswer.

Thefinancialinstitutionwillhavetobuy1.1%oftheAUDprincipalintheforwardmarketforeachyearofthelifeoftheswap.SinceAUDinterestratesarehigherthandollarinterestrates,AUDisatadiscountinforwardmarkets.ThismeansthattheAUDpurchasedforyear2islessexpensivethanthatpurchasedforyear1;theAUDpurchasedforyear3islessexpensivethanthatpurchasedforyear2;andsoon.Thisworksinfavorofthefinancialinstitutionandmeansthatitsspreadincreaseswithtime.Thespreadisalwaysabove20basispoints.

Problem7.14.

“Companieswithhighcreditriskaretheonesthatcannotaccessfixed-ratemarketsdirectly.Theyarethecompaniesthataremostlikelytobepayingfixedandreceivingfloatinginaninterestrateswap.”Assumethatthisstatementistrue.Doyouthinkitincreasesordecreasestheriskofafinancialinstitution 'sswapportfolio?Assumethatcompaniesaremostlikelyto

defaultwheninterestratesarehigh.

Consideraplain-vanillainterestrateswapinvolvingtwocompaniesXandY.WesupposethatXispayingfixedandreceivingfloatingwhileYispayingfloatingandreceivingfixed.

ThequotesuggeststhatcompanyXwillusuallybelesscreditworthythancompanyY.(CompanyXmightbeaBBB-ratedcompanythathasdifficultyinaccessingfixed-ratemarketsdirectly;companyYmightbeaAAA-ratedcompanythathasnodifficultyaccessingfixedorfloatingratemarkets.)PresumablycompanyXwantsfixed-ratefundsandcompanyYwantsfloating-ratefunds.

ThefinancialinstitutionwillrealizealossifcompanyYdefaultswhenratesarehighorifcompanyXdefaultswhenratesarelow.Theseeventsarerelativelyunlikelysince(a)Yisunlikelytodefaultinanycircumstancesand(b)defaultsarelesslikelytohappenwhenratesarelow.Forthepurposesofillustration,supposethattheprobabilitiesofvariouseventsareasfollows:

TOC\o"1-5"\h\z

DefaultbyY: 0.001

DefaultbyX: 0.010

Rateshighwhendefaultoccurs: 0.7

Rateslowwhendefault occurs: 0.3

Theprobabilityofalossis

00010700100300037

IftherolesofXandYintheswaphadbeenreversedtheprobabilityofalosswouldbe

00010300100700073

Assumingcompaniesaremorelikelytodefaultwheninterestratesarehigh,theaboveargumentshowsthattheobservationinquoteshastheeffectofdecreasingtheriskofafinancialinstitution 'sswapportfolio.Itisworthnotingthattheassumptionthatdefaultsare

morelikelywheninterestratesarehighisopentoquestion.Theassumptionismotivatedbythethoughtthathighinterestratesoftenleadtofinancialdifficultiesforcorporations.

However,thereisoftenatimelagbetweeninterestratesbeinghighandtheresultantdefault.

Whenthedefaultactuallyhappensinterestratesmayberelativelylow.

Problem7.15.

Whyistheexpectedlossfromadefaultonaswaplessthantheexpectedlossfromthedefaultonaloanwiththesameprincipal?

Inaninterest-rateswapafinancialinstitution 'sexposuredependsonthediffereneebetween

fixed-rateofinterestandafloating-rateofinterest.Ithasnoexposuretothenotionalprincipal.Inaloanthewholeprincipalcanbelost.

Problem7.16.

Abankfindsthatitsassetsarenotmatchedwithitsliabilities.Itistakingfloating-ratedepositsandmakingfixed-rateloans.Howcanswapsbeusedtooffsettherisk?

Thebankispayingafloating-rateonthedepositsandreceivingafixed-rateontheloans.Itcanoffsetitsriskbyenteringintointerestrateswaps(withotherfinancialinstitutionsorcorporations)inwhichitcontractstopayfixedandreceivefloating.

Problem7.17.

Explainhowyouwouldvalueaswapthatistheexchangeofafloatingrateinonecurrencyforafixedrateinanothercurrency.

ThefloatingpaymentscanbevaluedincurrencyAby(i)assumingthattheforwardratesarerealized,and(ii)discountingtheresultingcashflowsatappropriatecurrencyAdiscountrates.

SupposethatthevalueisVA.ThefixedpaymentscanbevaluedincurrencyBby

discountingthemattheappropriatecurrencyBdiscountrates.SupposethatthevalueVB.

IfQisthecurrentexchangerate(numberofunitsofcurrencyAperunitofcurrencyB),thevalueoftheswapincurrencyAisVAQVB.Alternatively,itisVAQVBincurrencyB.

Problem7.18.

TheLIBORzerocurveisflatat5%(continuouslycompounded)outto1.5years.Swapratesfor2-and3-yearsemiannualpayswapsare5.4%and5.6%,respectively.EstimatetheLIBORzeroratesformaturitiesof2.0,2.5,and3.0years.(Assumethatthe2.5-yearswap

rateistheaverageofthe2-and3-yearswaprates.)

Thetwo-yearswaprateis5.4%.Thismeansthatatwo-yearLIBORbondpayinga

semiannualcouponattherateof5.4%perannumsellsforpar.IfR2isthetwo-yearLIBORzerorate

27e0050527e0051027e005151027eR220100

SolvingthisgivesR2005342.The2.5-yearswaprateisassumedtobe5.5%.Thismeansthata2.5-yearLIBORbondpayingasemiannualcouponattherateof5.5%perannumsellsforpar.IfR25isthe2.5-yearLIBORzerorate

00505005100051500534220 R2525

275e00505275e00510275e00515275e0053422010275eR2525100

SolvingthisgivesR25005442.The3-yearswaprateis5.6%.Thismeansthata3-year

LIBORbondpayingasemiannualcouponattherateof5.6%perannumsellsforpar.IfR3

isthethree-yearLIBORzerorate

28e

00505

28e

00510

28e

00515

28e

00534220

28e

00544225

R330

1028eR330100

SolvingthisgivesR3005544.Thezeroratesformaturities2.0,2.5,and3.0yearsaretherefore5.342%,5.442%,and5.544%,respectively.

FurtherQuestions

Problem7.19

(a)CompanyAhasbeenofferedtheratesshowninTable7.3.Itcanborrowforthreeyearsat6.45%.Whatfloatingratecanitswapthisfixedrateinto?

(b)CompanyBhasbeenofferedtheratesshowninTable7.3.Itcanborrowfor5yearsatLIBORplus75basispoints.Whatfixedratecanitswapthisfloatingrateinto?

(a)CompanyAcanpayLIBORandreceive6.21%forthreeyears.Itcanthereforeexchangealoanat6.45%intoaloanatLIBORplus0.24%orLIBORplus24basispoints

(b)CompanyBcanreceiveLIBORandpay6.51%forfiveyears.ItcanthereforeexchangealoanatLIBORplus0.75%foraloanat7.26%.

Problem7.20

(a)CompanyXhasbeenofferedtheratesshowninTable7.3.Itcaninvestforfouryearsat5.5%.Whatfloatingratecanitswapthisfixedrateinto?

(b)CompanyYhasbeenofferedtheratesshowninTable7.3.Itcaninvestfor10yearsatLIBORminus50basispoints.Whatfixedratecanitswapthisfloatingrateinto?

(a)CompanyXcanpay6.39%forfouryearsandreceiveLIBOR.Itcanthereforeexchangetheinvestmentat5.5%foraninvestmentatLIBORminus0.89%orLIBORminus89basispoints.

(b)CompanyYcanreceive6.83%andpayLIBORfor10years.ItcanthereforeexchangeaninvestmentatLIBORminus0.5%foraninvestmentat6.33%.

Problem7.21.

Theone-yearLIBORrateis10%withannualcompounding.Abanktradesswapswhereafixedrateofinterestisexchangedfor12-monthLIBORwithpaymentsbeingexchangedannually.Two-andthree-yearswaprates(expressedwithannualcompounding)are11%and12%perannum.Estimatethetwo-andthree-yearLIBORzerorates.

Thetwo-yearswaprateimpliesthatatwo-yearLIBORbondwithacouponof11%sellsforpar.IfR2isthetwo-yearzerorate

2

11/1.10 111/(1R)100

sothatR2 01105.Thethree-yearswaprateimpliesthatathree-yearLIBORbondwitha

couponof12%sellsforpar.IfR3isthethree-yearzerorate

2 3

12/1.1012/1.1105 112/(1R3) 100

sothatR3 01217.Thetwo-andthree-yearratesaretherefore11.05%and12.17%with

annualcompounding.

Problem7.22.

CompanyA,aBritishmanufacturer,wishestoborrowU.S.dollarsatafixedrateofinterest.CompanyB,aUSmultinational,wishestoborrowsterlingatafixedrateofinterest.Theyhavebeenquotedthefollowingratesperannum(adjustedfordifferentialtaxeffects):

Sterling

USDollars

CompanyA

11.0%

7.0%

CompanyB

10.6%

6.2%

Designaswapthatwillnetabank,actingasintermediary,10basispointsperannumandthatwillproduceagainof15basispointsperannumforeachofthetwocompanies.

ThespreadbetweentheinterestratesofferedtoAandBis0.4%(or40basispoints)onsterlingloansand0.8%(or80basispoints)onU.S.dollarloans.Thetotalbenefittoallpartiesfromtheswapistherefore

8040 40basispoints

Itisthereforepossibletodesignaswapwhichwillearn10basispointsforthebankwhilemakingeachofAandB15basispointsbetteroffthantheywouldbebygoingdirectlytofinancialmarkets.OnepossibleswapisshowninFigureS7.5.CompanyAborrowsataneffectiverateof6.85%perannuminU.S.dollars.

CompanyBborrowsataneffectiverateof10.45%perannuminsterling.Thebankearnsa10-basis-pointspread.Thewayinwhichcurrencyswapssuchasthisoperateisasfollows.Principalamountsindollarsandsterlingthatareroughlyequivalentarechosen.Theseprincipalamountsflowintheoppositedirectiontothearrowsatthetimetheswapisinitiated.Interestpaymentsthenflowinthesamedirectionasthearrowsduringthelifeoftheswapandtheprincipalamountsflowinthesamedirectionasthearrowsattheendofthelifeoftheswap.

Notethatthebankisexposedtosomeexchangerateriskintheswap.Itearns65basispointsinU.S.dollarsandpays55basispointsinsterling.Thisexchangerateriskcouldbehedgedusingforwardcontracts.

fihno一

童」1A1

￡11

t,J.1Lru

Compjny

Fimncud

■

CcitLpdli\

-?

■

A J

f

B

FigureS7.5OnePossibleSwapforProblem7.22

Problem7.23.

Underthetermsofaninterestrateswap,afinancialinstitutionhasagreedtopay10%perannumandreceivethree-monthLIBORinreturnonanotionalprincipalof$100millionwithpaymentsbeingexchangedeverythreemonths.Theswaphasaremaininglifeof14months.Theaverageofthebidandofferfixedratescurrentlybeingswappedforthree-monthLIBORis12%perannumforallmaturities.Thethree-monthLIBORrateonemonthagowas11.8%perannum.Allratesarecompoundedquarterly.Whatisthevalueoftheswap?

Theswapcanberegardedasalongpositioninafloating-ratebondcombinedwithashortpositioninafixed-ratebond.Thecorrectdiscountrateis12%perannumwithquarterlycompoundingor11.82%perannumwithcontinuouscompounding.

Immediatelyafterthenextpaymentthefloating-ratebondwillbeworth$100million.Thenextfloatingpayment($million)is

0118100025295

Thevalueofthefloating-ratebondistherefore

10295e01182212100941

Thevalueofthefixed-ratebondis

01182212 01182512 01182812

25e25e25e

25e0118211121025e01182141298678

Thevalueoftheswapistherefore

10094198678$2263million

Asanalternativeapproachwecanvaluetheswapasaseriesofforwardrateagreements.Thecalculatedvalueis

(295

01182212

25)e

(3025)e01182512

(30

01182812

25)e

(3025)e011821112

(30

011821412

25)e

$2263million

whichisinagreementwiththeanswerobtainedusingthefirstapproach.

Problem7.24.

SupposethatthetermstructureofinterestratesisflatintheUnitedStatesand