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1、Lecture#5:SwapsAswapisanagreementbetweentwoormorepartiestoexchangesetsofcashflowsoveraperiodinthefuture.Thepartiesthatagreetotheswapareknownascounter-parties.Thecashflowsthatthecounter-partiesmakearegenerallytiredtothevalueofdebtinstrumentsortothevalueofforeigncurrencies.Therefore,thetwobasickindsof
2、swapsareinterestrateswapsandcurrencyswaps.TheSwapsMarket-Swapsarecustomtailoredtotheneedsofthecounter-parties.-Theswapsmarkethasvirtuallynogovernmentregulation.-Defaultrisk-ValueofOutstandingSwaps$BillionofPrincipalYearTotal Interest Rate SwapTotalCurrency Swap19878889909192939495682.91,010.21,539.3
3、2,311.53,065.13,850.86,177.88,815.610,617.4182.8316.8434.8577.5807.2860.4899.6914.8993.6-PlainVanillaSwaps1.Interestrateswaps2.CurrencySwaps-Motivationsforswaps1.mercialneeds:Asanexampleofprimecandidateforaninterestrateswaps,consideratypicalsavingsandloanassociation.Savingsandloanassociationsacceptd
4、epositsandlendthosefundsforlong-termmortgages.Becausedepositorscanwithdrawtheirfundsonshotnoticedepositratesmustadjusttochanginginterestrateconditions.MostmortgagorswishtoborrowatafixedrateforalongtimeinUS.Isthereanyinterestrisk?Canswapscontracthelp?2.parativeadvantage:Inmanysituations,onefirmmayhav
5、ebetteraccesstothecapitalmarketthananotherfirm.Forexample,aU.S.firmmaybeabletoborroweasilyintheU.S.,butitmightnothavesuchfavorableaccesstothecapitalmarketinGermany.Similarly,aGermanfirmmayhavegoodborrowingopportunitiesdomesticallybutpooropportunitiesintheStates.FirmUSD rateGEM rateGermanfirm10%7%USf
6、irm9%8%InterestRateSwaps-TwoPartiesexchangeperiodicinterestpaymentsoveraperiod.Typically,onepartyspaymentsarebasedonafixedratewhereasitscounterpartyspaymentsarebasedonafloatingrate.Interestpaymentsareputedusinganotionalprincipal.-Example:BothAandBneedtoborrow$100millionfor3years.Thefinancingratesfac
7、ingthemaresummarizedasfollows:FixedFloatingA7.5%6-monthLIBOR+0.85%B6.3%6-monthLIBOR+0.25%-ItisparativelycheaperforAtousethefloatingratedebt.ForB,fixedrateborrowingwillbecheaper.Why?1.IfAdesiresthefloatingratedebtandBprefersthefixedratedebt,thereisnoneedforthemtoengageinaswap.2.IfAdesiresthefixedrate
8、debtandBprefersthefloatingratedebt,AshouldstillborrowfloatingrateandBborrowfixedrate.Theycanthenenteraswaptobetterbothparties.6.3%panypany|LIBOR+AB6.3%0.85%LIBORa.panyA:Borrowsfloatingrateandenterstheaboveswap.b.panyB:Borrowsfixedrateandenterstheaboveswap-Theresultsa.panyA:Onasemiannualbasis,receive
9、sLIBOR-6.3%*50mfromtheswap,andpaysthefloatingratedebtserviceLIBOR+0.85%*50m.Thenetpaymentis7.15%*50m,whichislessthan7.5%*50m.b.panyB:Onasemiannualbasis,receives6.3%-LIBOR*50mfromtheswap,andpaysthefixedratedebtservice6.3%*50m.ThenetpaymentisLIBOR*50m,whichislessthanLIBOR+0.25%*50m.-Note:Swapraterefer
10、stofixedrateswap.-Swapsthroughanintermediary6.4%6.25%panySwappany|LIBOR+ADealerB6.3%0.85%LIBORLIBOR-Theresultsc.panyA:Onasemiannualbasis,receivesLIBOR-6.4%*50mfromtheswap,andpaysthefloatingratedebtserviceLIBOR+0.85%*50m.Thenetpaymentis7.25%*50m,whichislessthan7.5%*50m.d.panyB:Onasemiannualbasis,rece
11、ives6.25%-LIBOR*50mfromtheswap,andpaysthefixedratedebtservice6.3%*50m.ThenetpaymentisLIBOR+0.05%*50m,whichislessthanLIBOR+0.25%*50m.Swapdealer:Makes6.4%-6.25%*$50m=$75,000-PricingSchedulesThefixedrateintheswapisquotedasacertainnumberofbasispointsabovetheT-noteyield.Table:Indicationpricingforinterest
12、rateswapsat1:30pm,NewYorkTimeonMay11,1995MaturityyearsBankPaysFixedRateBankreceivesFixedRateCurrentTNRate%23457102-yrTN+17bps3-yrTN+19bps4-yrTN+21bps5-yrTN+23bps7-yrTN+27bps10-yrTN+31bps2-yrTN+20bps3-yrTN+22bps4-yrTN+24bps5-yrTN+26bps7-yrTN+30bps10-yrTN+34bps6.236.356.426.496.586.72-Netting:interest
13、paymentsaremadebyonecounter-partytotheotherafternettingoutthefixedandfloatinginterestpayments.Assume:Notionalamount=Q;fixedratepayment=k;Floatingrateusedintimet=Rt-1LIBORattimet-1.NETpaymentattimet:Fixedrateattimet:Fixed-ratepayerreceivesRt-1Q-kandfloating-ratepayerreceivesk-Rt-1Q.Thefollowingisapos
14、siblescenarioofcashflowsforthefixed-ratepayerundera$100million,5-yearswapat5.6%withsemiannualcashflowexchanges.#TimeyearsLIBOR FloatingPaymentFixedPaymentNet0123456789100.00.51.01.52.02.53.03.54.04.55.05.25.76.15.85.55.65.35.75.95.85.52.602.853.052.902.752.802.652.852.952.90-2.80-2.80-2.80-2.80-2.80
15、-2.80-2.80-2.80-2.80-2.80-0.20+0.05+0.25+0.10-0.05+0.00-0.15+0.05+0.15+0.10-Whatistheimplicationofnettingaboutcreditdefaultrisk?-Pricinginterestrateswaps:a.Setthefixedrateofswapsothattheswaphasazerovalueatthetimeofinitiation.Thisiscalledparswap.b.Supposethatpaymentdatesaret1,t2,tn.Thevalueofaswapatt
16、imet,Vt,fromtheperspectiveofthefloating-ratepayer:Vt=B1t-B2tc.B1t:valueoffixed-ratebondunderlyingtheswapwhentitti+1,B1t=nj=i+1ke-rt,tjtj-t+Qe-rt,tntn-t.d.B2t:valueoffloating-ratebondunderlyingtheswap.Atthefloatingrateresettingday,i.e.,t=t1,t2,tn,immediatelyafterthepaymentismade,B2t=Q.Why?Inbetween,i
17、.e.,titti+1,B2t=Q+k*exp-rt,ti+1ti+1-t,wherek*isthefloatingratepaymentattimeti+1alreadyknownattimet.Determiningtheswaprateattime0:V0k=ni=1kexp-r0,tjtj+Qexp-r0,tntn-Q=0Q=ni=1kexp-r0,tjtj+Qexp-r0,tntn.Thatis,setanappropriatecouponratesothatthebondispricedatpar.e.Example:Counter-partyAinathree-yearswapp
18、ays6-monthLIBORandreceivesafixedrateonanotionalprincipalof$100million.Theswaphas1.25yearstomaturity.Theswapratewasdeterminedoneyearandnine-monthago.Atthetimeofinitiation,3-year8%bondwaspricedatpar.TheLIBORatthelastpaymentdatewas10.2%semiannualpounding.Discountratesfor3-month,9-monthand15-monthmaturi
19、tiesare10%,10.5%,and11%,respectively.Thefixedrate=8%perannum.B1=4e-0.25*0.1+4e-0.75*0.105+104e1.25*0.11=98.24,B2=100+5.1e-0.25*0.1=102.51.V=98.24-102.51=-4.27milliontoAand4.27millionB.f.Portfolioofforwards:Aswapsemiannualinterestexchangescanbeviewedasasequenceofforwardswithmaturities:t1,t2,tnwithamo
20、nforwardprice.DefinePtasthetime-tvalueofzero-couponbondmaturingattimefor$1facevalue.Fortitti+1,1.Atti+1:k-k*,evaluatedatt,k-k*Ptti+12.Atti+2:k-0.5Rti+1Q,evaluatedatt,PVt,ti+2k-0.5Rti+1Q=k-0.5Rti+1,ti+2Qexp-rt,ti+2ti+2-t,whereRti+1,ti+2istheforwardratesemiannualpoundingattimetoverti+1,ti+2.Why?3.Simi
21、larlyforti+3,ti+4,4.Thetotalvalueoftheswapattimet:k-k*exp-rt,ti+1ti+1-t+nj=i+1k-0.5Rtj,tj+1Qexp-rt,tj+1tj+1-t-Example:ContinuethepreviousexampleR3m,9m=2exp0.5*0.75*0.105-0.25*0.1/0.75-0.25-1=11.04%R9m,15m=2exp0.5*1.25*0.11-0.75*0.105/1.25-0.75-1=12.10%V=4-5.1e-0.1*0.25+4-0.5*0.1104*100e-0.105*0.75+4
22、-0.5*0.121*100e-0.11*1.25=-4.27Variationofinterestrateswaps-Indexamortizedswaps:thenotionalprincipalisreducedoverthelifeoftheswap.-Constantyieldswaps:bothpartsarefloating.Forexample,onepartmaybelinkedtotheyieldonthe30-yearT-bondandtheothermaybelinkedonthe10-yearT-note.-Rate-cappedswaps:floatingratei
23、scapped.-PutableandCallableswaps:oneorbothcounter-partieshavetherighttocanceltheswapatcertaintimeswithoutadditionalcosts.-Forwardswaps:theswaprateissetbuttheswapdoesnotmenceuntilalaterdate.Currencyswaps-Twopartiesexchangeperiodicinterestpaymentsandprincipalsintwocurrencies.-Example:BothAandBneedtobo
24、rrowUSD50millionorDEMequivalentof84millionbasedon1.68DEM/USDforthree-year.Thefinancingratesfacingthemaresummarizedasfollows:ItisparativelycheaperforAtousetheDEMdebt.ForB,USDborrowingwillbecheaper.Why?1.IfAdesirestheDEMdebtandBpreferstheUSDdebt,thereisnoneedforthemtoengageinaswap.2.IfAdesirestheUSDde
25、btandBpreferstheDEMdebt,AshouldstillborrowDEMandBborrowUSD.Theycanenteracurrencyswaptobetterbothparties.USDDEMAB7.5%6.9%4.2%4.0%a.Interestpaymentflows6.9%USDpanypany|4.2%DEMAB6.9%USD3.9%DEMb.Initialprincipalflow84mDEMpanypany|84DEMAB50mUSD50mUSDb.Terminalprincipalflow84mDEMpanypany|84DEMAB50mUSD50mU
26、SDb.panyA:BorrowsDEMdebtandenterstheaboveswap.c.panyB:BorrowsUSDdebtandenterstheaboveswap.d.Theresults:1.panyA:Beginning:ExchangeDEM84millionforUSD50million,afairtransactionatthecurrentexchangerateDEM168/USD1.In-between:Onasemiannualbasis,receivesDEM4.2m*3.9%andpaysUSD25m*6.9%duetotheswap,andpaysDEM
27、42m*4.2%duetoitsDEMdebt.ThenetpaymentisUSD25m*6.9%+DEM42m*0.3%,paringtoUSD25m*7.5%.End:ExchangeUSD50mforDEM84m,notafairexchangeattheprevailingexchangerate.2.panyB:Beginning:ExchangeUSD50millionforDEM84million,afairtransactionatthecurrentexchangerateDEM168/USD1.In-between:Onasemiannualbasis,receivesU
28、SD25m*6.9%andpaysDEM4.2m*3.9%duetotheswap,andpaysUSD25m*6.9%duetoitsUSDdebt.ThenetpaymentisDEM42m*3.9%,whichislessthanDEM42m*4.0%,End:ExchangeDEM84mforUSD50m,notafairexchangeattheprevailingexchangerate.-Swapthroughanintermediary7.4%$6.9%$panySwappany|4.2%DMADealerB9%USD4.2%DM3.9%DM-Theresults1.panyA
29、:Beginning:ExchangeDEM84millionforUSD50million,afairtransactionatthecurrentexchangerateDEM168/USD1.In-between:Onasemiannualbasis,receivesDEM4.2m*4.2%andpaysUSD25m*7.4%duetotheswap,andpaysDEM42m*4.2%duetoitsDEMdebt.ThenetpaymentisUSD25m*7.4%,whichislessthanUSD25m*7.5%.End:ExchangeUSD50mforDEM84m,nota
30、fairexchangeattheprevailingexchangerate.2.panyB:Beginning:ExchangeUSD50millionforDEM84million,afairtransactionatthecurrentexchangerateDEM168/USD1.In-between:Onasemiannualbasis,receivesUSD25m*6.9%andpaysDEM4.2m*3.9%duetotheswap,andpaysUSD25m*6.9%duetoitsUSDdebt.ThenetpaymentisDEM42m*3.9%,whichislesst
31、hanDEM42m*4.0%End:ExchangeDEM84mforUSD50m,notafairexchangeattheprevailingexchangerate3.Swapdealer:Onasemiannualbasis,earnsUSD7.4%-6.9%*25mandlossDEM4.2%-3.9%*$2m.-Pricing currency swapsa.Setthetwofixedratesofaswapsothattheswaphasazerovalueatthetimeofinitiation.b.Supposethatpaymentdatesaret1,t2,tn.Th
32、evalueofaswapattimet,Vt,basedonthedomesticcurrency:Vt=StBFt-BDtc.St:exchangeratedomesticpriceofoneunitforeigncurrencyattimet.d.BDt:valueofdomesticfixed-ratebondunderlyingtheswapwhentitti+1,BDt=nj=i+1kDe-rdt,tjtj-t+QDe-rdt,tntn-t,wherekDisthepaymentinthedomesticcurrency,QDistheprincipalamountinthedom
33、esticcurrency.e.BFt:valueofforeignfixed-ratebondunderlyingtheswapmeasuredintheforeigncurrencywhentitti+1,BFt=nj=i+1kFe-rft,tjtj-t+QFe-rft,tntn-t,wherekFisthepaymentintheforeigncurrency,QFistheprincipalamountintheforeigncurrency.f.Determiningthefixedrateattime0SetkDandkFsuchthatQD=nj=1kDe-rd0,tjtj+QD
34、e-rd0,tntnQF=nj=i+1kFe-rf0,tjtj+QFe-rf0,tntnThisimpliesV0=S0BF0-BD0=S0QF-QD=0Thatis,settwoappropriatecouponratessothatbothbondsarepricedatpar.g.Example:Counter-partyAinathree-yearswappaysafixedrateonaprincipalofUSD100mandreceivesafixedrateonaprincipalofDEM168m.Thepaymentsaremadeonasemiannualbasis.Th
35、eprincipalsweresetaccordingtotheexchangerateatthetimeofinitiation.Thecurrentexchangerateis1.52DEM/USD.Theswaphas1.25yearstomaturity.Theswapratewasdeterminedoneyearandnine-monthago.Atthetimeofinitiation,3-year7.2%USDbondwaspricedatpar,and3-year4.2%DEMbondwasalsopricedatpar.ThecurrenttermstructureforU
36、SDandDEMarebothflatat8%and4%respectively.h.BD=3.6e-0.25*0.08+3.6e-0.75*0.08+103.6e-1.25*0.08=100.66mi.BF=1.68*2.1e-0.25*0.04+2.1e-0.75*0.04+102.1e-1.25*0.04=170.08mj.ToA:V=1708/1.52-100.66=USD11.23mandtoB:V=-USD11.23mk.Portfolioofforwards:Acurrencyswapcanbeviewedasasequenceofforwardswithmaturities:t
37、1,t2,tnwithamonforwardprice.Fortitti+1,1.Atti+1:Sti+1kF-kD,evaluatedatt,ithasavalueequaltoFtti+1kF-kDexp-rDtti+1ti+1-t2.Atti+2:Sti+2kF-kD,evaluatedatt,ithasavalueequaltoFtti+2kF-kDexp-rDtti+2ti+2-t3.Similarlyforti+3,ti+4,ti+n-14.Atti+n:Sti+nkF+QF-kD+QD,evaluatedatt,ithasavalueequaltoFtti+nkF+QF-kD+Q
38、Dexp-rDtti+nti+n-t5.Thetotalvalueoftheswapattimetisthesumofalltheterms.-Example:Continuethepreviousexample.F0.25=1/1.52exp0.08-0.040.25=0.6645;F0.75=1/1.52exp0.08-0.040.75=0.6679;F1.25=1/1.52exp0.08-0.041.25=0.6916V=0.6645*2.1*1.68-3.6*e-0.08*0.25+0.6779*2.1*1.68-3.6e0.08*0.75+0.6916*102.1*1.68-103.
39、6e-0.08*1.25=-1.2308-1.1380+13.5986=USD11.2298mEquityswaps-Twopartiesexchangeperiodicpaymentsoverafixedduration.Typically,onepartyspaymentsarebasedonastockindexreturnwhereasitscounter-partyspaymentsarebasedonabenchmark-floatingrate.Paymentsareputedusinganotionalprincipal.-Example:Notionalprincipal$1
40、00m.Counter-partyAreceives3-monthLIBORandpaysS&P500indexreturnplusaswapspreadof-0.1%.S&P500return-0.1%panypanyABLIBORDateDaysLIBOR%SP500SP500returnLIBORPaymentS&PPaymentNetPaymentJan29.00469.75Apr2909.15479.152.00%225,000190,106-34,894Jul2919.35507.425.90231,292580,003348,711Oct2928.65491.70-3.10238
41、,944-319,803-558,747Jan292499.101.50221,056140,498-80,558-Thevalueofequityswapa.ThevalueofthisequityswapwaszeroonJan2,thetimeofinitiation.ThesameistrueforApril2,July2,Oct2andJan2immediatelyafterthepaymentismade.Why?b.Thevalueofthisequityswapon,sayMarch1,willnotbezero,however.Assumethatthefuturespric
42、eofS&P500indexfuturescontractmaturinginAprilcontractfinishedat460.1onthatday.ThediscountrateonMarch1forthematurityofApril2is9.1%.1.WhatisthevalueofswaptotheLIBORpayer?TheLIBORpaymentonApril2isknowntobe225,000.Itspresentvalueis225,000*exp-0.091*32/365=223,212.ThereceiptonApril2subjecttotheS&P500index
43、performanceisIA2-IJ2/IJ2-0.1%*100m.Itspresentvalueis460.1-469.75-0.001*469.75*100m/469.75*exp-0.091*32/365=-2,137,166.Thetotalvalue=-223,212-2,137,166=-2,360,378.modityswaps-Inatypicalmodityswap,onecounter-partymakesperiodicpaymentstothesecondcounter-partyatafixedpriceperunitforagivennotionalquantit
44、yofsomemodity.Thesecondcounter-partypaysthefirstcounter-partyafloatingpriceforagivennotionalquantityofsomemodity.Themoditiesareusuallythesame.Thefloatingpriceisusuallycalculatedasanaverageprice.CreditDefaultSwapsWillbediscussedinthesectionofcreditrisk.Procter&GambleBankersTrustLeveragedSwap1Thestory
45、OnNovember2,1993,P&GandBTenteredafiveyear,semiannualsettlement,$200millionnotionalprincipalinterestrateswapcontractknownasthe“5/30swap.BTpaysafixedrateof5.30%andP&Gpaysafloatingratedependsonthirty-daymercialpaperCPdailyaverageratelessthen75basispoints,plussomespread.Thekeyfactorsintheagreementarethe
46、spreadandthe75basispointsaplainvanillaswapwouldhavebeen5.3%versustheCPdailyaveragerateflat.TheswapwasscheduledtolockinonMay4,1994.Becausethespreadonthelock-in-datewas2,750basispoints,P&Gexperiencedsignificantlossesandfiledalawsuit.Anout-of-courtsettlementwasreachedinMay1996.BTagreedtoabsorb$157milli
47、on.2TheP&G-BTleveragedswapTerm:5yearFrequency:SemiannualpaymentsFixedratepayer:BankersTrustat5.3%Floatingpayer:P&Gat30-daymercialpaperdailyaverageratesless75Basispointsplusaspread.3ThespreadThespreadiszeroforthefirst6-monthsettlementperiod,andthenwouldbefixedfortheremainingninesemiannualperiods,depe
48、ndingonTreasuryyieldsandpricesonthefirstsettlementdate,May4,1994,accordingtotheformula.Spread=max0,98.55-yearCMT%/5.78%-30-yearTYSPrice/1005-yearCMT%istheyieldonthe5-yearconstant-maturityTreasurynote.The30-yearTreasuryTSYbondpriceisthemidpointofthebidandofferpricesonthe6.25%T-bondmaturinginAugust202
49、3,notincludingaccruedinterest.ThespreadonNovember2,1993waszerobecause98.5*5.02%/5.78%-102.57811/100=-0.1703ThespreadonMay4,1994wasMax0,98.5*6.71%/5.78%-86.84375/100=0.2750Thusinreturnforreceivingafixedrateof5.3%,theP&Gwouldhavebeenobligatedtopaythe30-dayCPdailyaveragerateplus26.75%27.50%-0.75%forthe
50、nextfourandonehalfyearsonthe$200millionswapiftheformulahadnotbeenamendedpriortoMay1994.4TheamendmentTheswapwasamendedinJanuary1994tomovethedeterminationdateofthespreadfromMay4,1994toMay19,1994inexchangefor13basispointsimprovementinthefloatingratesideoftheswap,i.e.,75basispointshasbeenchangedto88basi