interest risk management

8/2/2019 Interest Risk Management

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RiskManagementofFixedIncomePortfolios

N.Gershun


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2

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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4

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.

19


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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)

37


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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

51

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

56

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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58

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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interest risk management

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