Download - Interest Risk Management
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RiskManagementofFixedIncomePortfolios
N.Gershun
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Duration:MeasuringtheSensitivityofBondPricestoRateChanges
A.TheRoadmapAhead. Weareinterestedintworisksrelatedtotheownershipofbonds(investorsperspective)andhowtohedge(insureagainst)theserisks.Theyare:
(i)Interestratepricerisk:Thisisthepossibilitythatinterestratesmayrisecausingbondpricestodecline.Allbondsaresusceptibletothisrisk.Itiscloselyinvolvedwithreinvestmentrisk.
a.Ariseininterestratesisalsoarisktoborrowers:theiranticipatedfutureborrowingcostswillrise.Wewillalsodealwithinterestrateriskfromtheborrowersperspective.
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(ii)Defaultrisk: Thisisthepossibilitythatthebondsowner(thelender)maynotreceivehispromised
interestandprincipalpayments(eitheratall,oratthepresettimes).ThisriskappliestoallbondsexceptthoseissuedbytheFederalAuthoritiesoftheestablishedOECDcountries.U.S.Treasuriesaredeemedtohave
zerodefaultrisk,forexample.
Wewillfirstconsiderinterestratepriceriskandits
mitigation.
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Interestrateriskmanagement isnotonlyofconcerntoindividualinvestors,butalsotofinancialinstitutions.
ConsiderahypotheticalPensionFund.Assets Liabilities
20yr,10%couponbond perpetualbenefitsof$1,000peryear
value:$10,000 value:$10,000
(thetermstructureisflatat10%)
Assets Liabilities
20yr,10%couponbond perpetualbenefitsof$1,000peryear
value:$16,231 value:$20,000
ThePensionFundisnowinsolvent.
Nowsupposeinterestratesdeclineto5%
WhyIsInterestRateRiskmanagementImportant?
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2.Asecondexample:ahypotheticalbank.
Howdidthishappen?Thesetwobondpricesreactedverydifferentlytoaratedecline.Bothassetsandliabilitiesroseinvalue,butthe
valueoftheliabilitiesrosemuchfaster.
Assets LiabilitiesandEquity
$10,000,10yr. $9,500DD
loan@7% $500shareholderequity
$10,000 $10,000
(termstructureisflatat7%,semi-annualcompounding).
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Now,supposeinterestratesriseto10%forallmaturities.
Assets LiabilitiesandEquity
$8130-- 7%, $9,500DD
10yr.loan $1,370shareholderequity
$8130 $8,130
Thebankisbankrupt.
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Notethatforthepensionfund,itwasadeclineinratesthatledtobankruptcywhereasforthebankit
wasanincreaseinrates.
Theyareexamplesofadurationmismatch anditisthemostcommonreasonforfinancialdistress
amongfinancialinstitutions.
Thenotionofduration isspecialized(tothebondmarket)languageforbondpricesensitivitytoaratechange.
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Intuitively,bondswithsteeperprice-ratecurveswillbemorepricesensitivetoratechanges,andthustheirslopesshouldbelargerin
absolutevalue.
Astherateschange,sodoestheslopeoftheprice-ratefunction.
Bondwithhighratesensitivity
Bondwithlowratesensitivity
BondPrice
P0
P1
r0 r1 r r
Bond
Price
IntuitionBehindDuration
Highdurationbondsareverysensitivetochangesinrates.
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DurationofDiscountBonds(i) Percentagepricechange:
0
0
P 1T r.
P (1 r)
+
Thetimetomaturity,T,intheaboveexpression
issaidtobethebondsMacaulayduration,orsimplyitsduration;theexpressionD/(1+r)
isdefinedasthemodifiedduration,writtenDM
.
(ii) Absolutepricechanges: P0 -DM P0 r.
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DurationofCouponBondsSinceanycouponbondcanbeexpressedasaportfolioofdiscountbonds,thedurationofacoupon
bondiss theweightedaverageofthepricesensitivitiesofitsconstituentdiscountbonds.t= 0 1 2... T
Tcoupon bond t T
t 1 0 0
PV(C) PV(MV )D t T
P P=
= +
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PortfoliosofBondsSupposewemanagedaportfolioofbondsofvariousdurations.
Bythesamelogic(acouponbondisitselfsimplyaportfolioofdiscountbonds)aswehavejustused:
DP =w1D1+w2D2 +...+wNDN
wi =theproportionoftheportfoliostotalvaluerepresentedbybondsoftypei
Di = thedurationofbondsoftypei
durationoftheportfolio
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TheManagementofBondPortfoli Risk:TheIntuition
1.Sinceinterestratesmaychange,thereispriceriskinherentinholdingaportfolioofbonds.Ifratesrise,inparticular,thevalueoftheportfoliowill
decline.2.Therearemanywaystohedgebuttheyallamounttothesamethingconceptually:append
totheportfolioothersecuritieswhosepricemovements,inresponsetointerestrate
changes,areoppositetothoseofthesecurities
alreadypresent.
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ImmunizationIfwewanttoprotectourselvesperfectlyagainstsmall (!)interestratechanges:
MHedged PortfolioD 0.=
Thismeansthatachangeinrateswillleaveour
hedgedportfoliosvaluelargelyunaltered:lossesononepartwillbeoffsetbygainsintheother.
{Hedged original bond hedgingP .,porfolio securities=
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InsuringAgainstDefaultUsingCreditDefaultSwaps
WhatisaCreditDefaultSwap?
1. Whentheexpressionswap isused,thinkintuitivelyofthepurchaseorsaleof
insurance.a.Inmoststandardinsurancearrangements:(ii) Theinsuredpaysafixed cashflow the
insurancepremium
(ii) Thesellermakesavariable payment:thevalueoftheinsureditemifdisasterstrikes,andzerootherwise.
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b. CreditDefaultSwapsarenodifferentexceptthattheinsurableevent thedisasteragainstwhichyouinsure isthedefaultofsomebond.
2. Aninvestorwhobuysabondmayalsobuyinsuranceagainstaspecificdefaultordefault-likeevent
a. forexample,thedefault-likeeventcouldbearatingscutbytheratingsagenciesfromaAAAratingtoaAArating.Itcouldalso (totaketheextremesituation)representthebankruptcyoftheissuer.
b.- Theownerofthebond(insurancepurchaser)paysafixedstreamofpaymentstotheinsurer;- Theproviderofinsurance(insuranceseller)paysavariablepaymenttotheinsuredonlyifthereissomesortofadefault.
Insurers:AIG,HedgeFunds,Citibank
c. Thetimehorizoncanbeasmuchas5years.
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ASimpleSingleNameCreditDefaultSwapThebuyerpaysanannualfeeorpremiumforparrecoverypayoutin
theeventofadefault/bankruptcy.
1. Themechanism:Ifnodefault:
Ifdefault:
premiumonsomenotionalamount(say$100
MMfacevalueofGMACSeniorDebt)
creditriskisassignedtotheseller
Buyer Seller
Buyer Seller
{BondsofparvalueGMAC$100MM}
$100MMcash
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2. Anextremecasewherethedefaultedbondisworthless.
Thepayofftothebuyeroftheinsurance,contingentondefault
0
Payoff
toBuyer Par
ValuePbond beinginsured
0
Payoff
toSeller
Par
ValuePbond beinginsured
Thebuyerhasessentiallypurchasedaspecialput-style
option.
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FeaturesofCDSsecurities
1. Theyareverydifficulttopriceconceptually.
Whymightthisbeso?
2. Thereisnoorganized,liquidmarketforthese
securitieswherereliablepricesmaybeinferred,althoughtheywereboughtandsoldOTC.Withbondpricesfalling,theyhavebecomeToxicAssets.
3. Itisadifferentsenseofhedging thanwehaveemployedpreviously.
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DynamicImmunizationwithInterestRateFuturesContracts
Introduction:
Adrawbacktoimmunizingviathedurationofbondportfoliosistheneedtorebalanceinresponsetorate
shifts.Thismaycreatelargetransactioncostsasthenumberofbondsboughtorsoldmayendupbeingverylarge.Anotherway,inprinciple,istouse
interestratefuturescontractsofsometype.
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TwoAdvantagesofImmunizingwith
InterestRateFutures
(i) thecompositionofthebondportfolioremainunchangedanddurationadjustedusingthefuturescontracts.
(ii) transactionscostsoftradingfuturesaremuchlessthanbondtradingcosts.
Theseconsiderationsareespeciallyimportantwhenthebondstradeinthin markets.
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FuturesContracts
Afuturescontractisverysimilartoaforwardcontract(thelanguagefuturesprice replacesthelanguageforwardprice evenasthenumberisthesame),butwiththemarkingtomarketfeature.
Nomoneychangeshandsatsigning.
ThesecontractsforTreasurysecuritiesareexchangetraded(verylowtransactionscosts).
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SummaryPointsRegardingFutures
Theyareexchangetraded
Theyaresettleddaily(markingtomarket)
Closingoutafuturespositioniseasilyaccomplishedbyenteringintoanoffsettingtrade.Mostcontractsareclosedoutthiswaypriortoexpiration.
Contractsnotclosedoutbeforematurityaresettledbydelivery(choiceofinstrumentanddateofdelivery).Afewcontractsaresettledincash(e.g.,stockindex
futures).
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DurationofFutures
Thefuturescontractitselfdoesnothaveaduration.
Thefuturesprice,however,anditssensitivitytoratechangesdependsonthedurationandyieldoftheunderlyingsecurityexpectedtoprevailatthecontractmaturitydate.
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HedgingwithFutures:AnExample
Hedge$10MportfolioofBondC
UseaT-BillFuturescontractcallingforthedeliveryof$1MMfacevalueofT-billshaving90
daysremaininguntilmaturity.
=>DurationofT-BillFutures=90days,or.25
year.
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HedgingwithFututures (cont.)InstrumentsUsedintheAnalysis
Coupon Maturity Yield Price Duration
BondC 4% 15yrs 12% 455.13 9.60
T-BillFutures -- 1/4yr. 12% 970,873.00 .25
1,000,000$970,853 Futures Price
.121
4
= =
+
The$10mportfolioofBondCrepresents21,972bonds.
Objective:perfectlyhedgetheportfolioofCbonds.
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Solution:
(i)Vp =PCNC +FPT-Bill NT-Bill
Vp =portfoliovalue Nc =#ofCbonds
PC =bondCprice NT-Bill =#ofT-billfuturesFPT-Bill =T-billfuturesprices
(ii)and:DP VP =DCPCNC +DT-Bill FPT-Bill NT-Bill
VP =$10m NC =21,972
DP =0(desired) DT-Bill =.25yearsDC =9.6years FPT-Bill =$970,873PC =$455.13
Write(sellshort)395.5T-BillFuturescontracts
0=10M(9.6)+.25(970,873)NT-bill
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Now,assumeashiftintermstructurefrom12%to13%.
Therelevantpricesarenow:
PC =$418.39 FPTBill =
$1,000,000
$968,523.131
4
=
+
Lossonportfolio =21,972Pc=21,972x(418.39 455.13)
=- $807,251
GainsonFutures =- 395.5(968,523 970,873)=$929,425;weareoverhedged,as
expected
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OtherInstruments
1. T-billfuturesarenolongertraded.Attheshortendofthe
curve, theactionisnowallinEurodollarfutureswhichhavecashsettlementbasedon3monthLIBOR.
2. Considera10yearT-bondcontract.Thisisfor$100,000
facevalueofdeliverablebonds.Whatwoulddiffervis--vistheabovecalculation?
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TheEurodollarMarketTheBasicsoftheEurodollarMarket
1.Whatisit?
AloanmarketforUS$denominatedborrowingandlending(US$CDdepositsarereceivedandUS$denominatedloansextended)basedoutsidetheUnitedStates.
ThisisalargerinterestratemarketthantheUSTreasury
market.
2.Whereisitbased?
ItisbasedprimarilyinLondon,butalsointheCayman
Islands,Tokyo,andHongKong.
Similaroffshoremarketsexistforothercurrencies,e.g.,thepound,yen,etc.
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HowDoBalancesGetCreated?
SupposeGeneralElectricreceives$1MMfromthesaleofatransformeranddepositsthemoneyinacheckingaccountatJPMorganChaseinNewYork.
Itmightthenpurchasea6-month$1MMEurodollarCDfromHSBCinLondonwhereitremainsasadollardeposit(notexchangedintoanequivalentamountofpounds).
Thismoney,asidefromanyreserverequirementHSBCmaywishtoimposeonitself,canthenbeloanedouttofirmswhowishtotakeout$denominatedloans.
ThisentiresystemoftakingdollardepositsinLondonandlendingthemprimarilyfromLondonisreferredtoastheEurodollarmarket.
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EvolutioninEurodollarMarketSizeinbillionsofUS$ofinternationalbankclaims
0
5000
10000
15000
20000
25000
30000
19641976
19821991
19931995
19971999
20012003
20052007
Gross
Net
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TheNatureoftheLoansMadeintheEurodollarMarket
Exclusivelyfloatingrateloans,withaninterestrateresetperiodofatmost6months.
Loansinthismarketaregenerallyextendedonlytofirst
tierindustrialfirmsandfinancialinstitutions.
Thedurationoftheloans,whicharetheassetsofthebanks,isatmost0.5year.
Theborrowerbearsallassociatedinterestraterisk.
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ThisMarketisBenchmarkedbyLIBOR
(theLondonInterbankOfferRate)1. Liboristherateatwhichlargebanksoperatinginthe
Eurodollarmarketextendloanstooneanother.Itisarate
forwhichtheborrowingbankmaydefault(oneofthesebankscouldconceivablyfailandnotdischargeisloancommitmentstoanotherbank.Thispossibilityisnolongerremote).
2.ManymarketsarebenchmarkedfromtheLiborrate.Thismeansthatmanyotherloans/bondsissuedallovertheworldhavetermsthatallowtheirinterestratetoberesetevery6monthsatarateequaltothe6monthLiborrateforthat6monthperiodplussomepremiumorspread.
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TEDSpread
HowlargeisthedefaultpremiumonLIBOR?Itcanbemeasured:
TEDspread:6-monthLIBORminus6-monthTreasury
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0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
9/30/0512/14/052/27/ 065/13/ 067/27/ 0610/10/0612/ 24/063/9/075/23/ 078/6/0710/20/071/3/083/18/ 086/1/08
TEDspread1/2006- 3/2008
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0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
3-M TED spread in
2008
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FloatingRate:AnExample
AmountofLoan:$100MM Terms:LIBOR+50basispoints(.5%)
6Mo.
LIBOR.02
Rate:
.025 .031 .033
t=0 ...
(6mo.) (1yr.) (1.5yr.) (2yr.)
Loanrates: .025 .030 .036 .038
EndofPeriodPaymentby
Borrower:
2.5M 3M 3.6M 3.8M
(paymentsaremadeatendoftheperiod)
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HowDoBorrowersHedgeTheirInterestRateRisk?
Eurodollarfuturesandforwardcontractsrepresentavailabletoolsforthispurpose.
Whatifaborrowerwishestoremovethisriskona
regularbasis?
Aswapcontractaccomplishesthis.
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HedgingwithEurodollarFutures
TheconceptisidenticaltotheprioruseofT-billfutures:theinvestordesiringtohedgemustwriteEurodollarfuturesintheappropriatenumbers.Thesewrittencontactswillbecomemorevaluableasratesrise(prices
fall).Asidefromafewdetails/conventionsofmeasurement,thecalculationsmimicthoseofT-billfutures.
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HowtheEurodollarFP(FuturesPrice)isto
beInterpreted1. Asof1/27/09,11:27AM,wehavethefollowinginformation:
FuturesPrices2/28/08
April00 last open high low vol.
expirationdate
offutures
contract
98.895 98.915 98.915 178
2. SupposewetransactedforonecontractatthecurrentFP.
t=0 T(April09)
FP=98.895Weinterpret98.895asidentifyingthe(annualized!)forward3monthLIBORraterelativetoApril2009asbeing
100 98.895=1.105%,or.01105(1.105%).
Thecontractamountis$1,000,000for3months.
T+.25April Libor.25f .01105=
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Supposeyousigned(wentlong)inafuturescontractwhenFP=98.895,andduringthelifeofthecontract,theFPrisesto99(forwardLIBORfallsto1%);then:
Longpositionreceives(1,000,000)(.00105)(.25)=+$262.50
Shortpositionreceives: (1,000,000)(.00105)(.25)= $262.50.
(Thereisonlycashsettlement.)
99 98.895100
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Supposeatsomepoint,theFPfallsto97(forwardLIBORrisesto3%,annualized);then,
Longpositionreceives(1,000,000)(.01895)(.25)=$4737.50
97 98.895
100
Shortpositionreceives: (1,000,000)(. 01895)(.25)=$4737.50.
Aswithanyofourhedginginstruments(T-bonds,T-billfutures),shortpositionsinEurodollarfuturesincreaseinvalueasratesrise.
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Example:HedgingaBondPortfoliowith3-MonthLIBORFutures
1. Supposethetermstructureisflat.TheTermstructure
ofLIBORrates andtheforwardLIBORcurve wouldbeessentiallyflataswell.
2. AnExample:
Youown$10MofbondsofD=6whentheinterestrate(LIBOR)environmentisflatatr = 5.14%.Youareconcernedthatratesmayriseby.5%.Hedgeyour
positionwithEurodollarfutures.
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Whatisyourestimatedlossifyoudonothedge?
P PD 6V V r 10 (.005)(1 r) (1.0514)
=
+
$285,334=
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Perfectly hedgingusingEurodollarfutures
meansDP =0.
P P 1 1 1 EDF EDFV D n P D n D $1, 000, 000 = +
EDF0 10MM 6 n (.25) (1, 000, 000= +
EDF
10MM 6n 240 contracts; i.e.,
(.25)(1MM)
= =
write240contracts.
you receive gains
and losses relative
to $1 MM
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Supposeratesdoriseasfeared.
Approximatelossonportfolio= $285,334Gainonfutures:
- (240)(1,000,000)(.25)(-.005)= +$300,000
+$15,000
Weareslightlyoverhedged,astheoryremindsusmustbethe
case.
.94.36 94.86
100
SWAPS d h E d ll M k
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TheNotionofaSWAP
1. ForwardContractsallowborrowerstoremoveinterestrateriskoveraspecificfuturetimeperiod,sayiperiodsaheadfornperiods.
Asaresult,wemustcontinuallysignsuchcontractsifwewishto
removelongtermriskonaregularbasis.AconvenientwaytodothisisviaaSWAP.
SWAPSandtheEurodollarMarket:a
ThirdPerspectiveonInterestRateRiskManagement
a.ThisisimportantbecauseEurodollarloansarefloatingrateloans.
b.Standardforwardcontractsinthismarketare3x3 or3x6meaningthattheforwardcontractlocksinarateforin3monthsfromthe3monthsfollowingorin3monthsforthe6
monthsfollowing.
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InterestRateSWAP:DefinitionAninterestrateSWAPisanenforceableagreementbetweentwopartiestoexchangecashflowsperiodbyperiod.
- Onepartypaysafixedratepaymentandreceivesavariablefloatingratepayment.Thispartyistheinsured.
- Thecounterpartypaysthefloatingrateandreceives
thefixedratepayment.Thispartybearstherisk.
Theseratesareappliedtoanagreed-uponfixednotional amount.Theresultingpaymentsarewhat
isexchanged.
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UsesofSWAPS
1.Theusesofsuchinstrumentsarethreefold:
theygiveaccesstofixedorfloatingratecapitalmarkets
theyallowparticipantstomanagetheirasset/liabilitystructuremoreeffectively
theyprovideatoolforhedginginterestraterisk
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LOANS
8%
BANK
pays7%
SWAPDEALER
receives6mo.LIBOR
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Assets Liabilities
$100MM $100MM
10yearloanat8% 6mo.CDsat5%
Every6monthstheBankhastorefinancetheCDs,whoseratesaretypicallytiedtoLIBOR.
Suppose,forsimplicity,theCDrateistheLIBORrate.ConsideraSWAPwherebytheBankexchangestheirvariableliabilityforafixedrateliabilityat7%.
Nowthebankonlyhastoworryaboutthecreditrisk oftheborrowerandtheSWAPdealer.
DEPOSITORS
Example:ManagingAssetsandLiabilitiesConsideraBankwithaverysimplebalancesheet:
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2. Whowantsfloating?Whowantsfixed?
SWAPScanbethoughtofasinsuranceagainstratechanges.
Wantstoreceivevariableandpayafixedpayment
Wantstoreceivefixedandpayvariable
a.SpeculatorswhobelieveEurodollarrateswillrise
Speculatorswhobelieverateswillfall
b.Bankswhichhavefixedrateloans(mortgages)butvariablerateobligations
(CDs)
Bankswhichhavevariablerateassets(mortgages)andwantto
reduceriskinthispartoftheirportfolio
c.Firmswithvariablerateloans,yetsteadycashflowstreams(e.g.,drug
firms;industrialfirms)
Firmswithvariablestreamsofincomebutwhohavefixed
obligationsandwanttoreducerisk
Th M h i f SWAP
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TheMechanismofaSWAP
1.Whatactuallyisswapped?Itisonlythefixedandfloatingratepaymentsthemselves(coupons) onagivennotionalprincipal thatareexchanged.
2. TimeHorizon:specifiedbytheSWAPcontract.Inprincipalitcanbefor
anytimeperiod:1year,5years,10years,etc.3.ExampleofaSWAPcontract:
XpaysY:10%fixedrateperyearonanotional$50MYpaysX:6monthLIBORrateadjustedevery6months.
Paymentsarethusexchangedevery6months:
XpaysY:$50M=$2.5million(fixedrate).YpaysX:50Mx6monthLIBORrate(whichvaries floating
rate)
ThefixedrateisknownastheSWAPrate.
102
H I th SWAP R t D t i d?
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HowIstheSWAPRateDetermined?
ItisthatratewhichequatesthePVofthetwopaymentstreams. Itiscomputedastheresultofathreestepprocedure:
Step1:Computethefloatingratepaymentsusingtheforward
LIBORrates.Thesearetheno-arbitrageratesintheEurodollarmarket.
Step2:DiscountthefloatingratepaymentsusingtheLIBORtermstructure toobtainthepresentvalueoftheexpectedfloatingrate
payments. Step3:AdjusttheFixedRatetobringaboutequalityinthetwo
presentvalues.
ThisisanNPV=0procedurewithintheuniverseofLIBORsecurities.It
thuscostsnothing, asidefromtransactionsfees,toenterintosuchcontracts.
TheKeyIngredient:theForwardLIBORTermStructure
An Example: Computing the SWAP Rate
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AnExample:ComputingtheSWAPRate
Considera1.5yearSWAPwiththreepaymentsexchanged.NotionalAmount$100M.AlthoughLIBORratesareavailableonlyuptooneyear,forwardLIBORratesareavailablefromBloombergorReuterssinceLIBORforwardcontractsaretraded.
Weneed
1.85% 2.2% 2.72% (annualized)
LIBOR
.5r ,LIBOR
.5 .5f ,LIBOR
1 .5f ,
float.5Cfloat
1Cfloat
1.5C
Thus, float.5C
LIBOR
.5r$100 M $.925 MM2
=
float
1CLIBOR.5 .5f$100 M $1.1 MM2
=
float
1.5CLIBOR
1 .5f$100 M $1.36 MM2
=
=
=
=
Thesearetheexpected noarbitrage-- floatingratepayments.
Problem: LIBOR spot rates are available only up to one year
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t= 0 .5 1 1.5
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Problem:LIBORspotratesareavailableonlyuptooneyear.
However,knowledgeoftheforwardratesallowsus,inaworldof no(LIBOR)arbitrage,toconstructtheconsistentsetofcorrespondingspotrates.
ThisisanothersenseofBootstrapping exceptthatitusesforwardratestoconstructspotrates.
LIBOR
.5 .5f
2
LIBOR
1 .5f
2
LIBOR
.5r
2
LIBOR
1r
2
LIBOR
1.5r
2
LIBOR
.5r .01851 1 1.009252 2
+ = + =
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2LIBOR LIBOR LIBOR
1 .5 .5 .5r r f1 1 12 2 2
+ = + +
= ( )
Libor
1
.022
1.00925 1 1.02035 r .0220492
+ = =
3
LIBOR LIBOR LIBOR LIBOR
1.5 .5 .5 .5 1 .5r r f f 1 1 1 12 2 2 2
+ = + + +
.022 .0272(1.00925) 1 1
2 2
+ +
=
3LIBOR
Libor1.51.5
r1 1.03423 r .02256
2
+ = =
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3Libor1.5r1
2
+
Thus,thevalueofthefloatingpaymentsis:
2Libor
1r12
+
FLOAT .925 MM 1.1MM 1.36 MMPV(1.00925) (1.02035) (1.03423)
= + +
=3.376MM
Lastly we compute the SWAP rate where we discount the fixed
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Lastly,wecomputetheSWAPrate,wherewediscountthefixed
paymentsatthesametermstructureofLIBORrates:LetSdenote theswapcashpayment
PVFIXED =
3 2 3LIBOR LIBOR LIBOR
.5 1 1.5
t 1
S S S
r r r1 1 12 2 2
1.00925 1.02035 1.03423
=
+ +
+ + +
64748 64748 64748
3.376MM=2.9377S
=>S=1.149MM.
Thisisa6month cashflow.Onanannualbasis:
S=2(1.1498)=2.298MMOnanannualizedratebasis,thisisequivalentto
Thisistheswaprate.Itisanoarbitrageratewithinthescopeof
theLIBORfamilyofrates.
2.298 MM2.298%
100 MM=
59
Duration of a SWAP
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1. ConsideranN-yearSWAP
2. DurationofFixedside
- SamedurationasanN-yearbondwithcouponrateS3. DurationofFloatingside
- AlwayshaveaPVof$100.Intuition:wheninterestincreases,youreceivemoreinterest,butalsodiscountmore.Theeffect
offsetseachother.
- Durationoffloatingside:0.
4. Receivefixed/payfloatingSWAPhasapositiveduration.
5. Receivefloating/payfixedSWAPhasanegativeduration.6. SWAPcanhedgeinterestrateriskviadurationadjustment,just
likeanN-yearbond.
DurationofaSWAP
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