An Introductory Note on Two Curve Discounting 1

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1 An Inroducory Noe on Two Curve Discouning LCH.Clearne Ld (LCH.Clearne), which operaes he world s leading ineres rae swap (IRS) clearing service, SwapClear, is o begin using he overnigh index swap (OIS) rae curves o discoun is $8 rillion IRS porfolio. Previously, in line wih marke pracice, he porfolio was discouned using LIBOR.... Afer exensive consulaion wih marke paricipans, LCH.Clearne has decided o move o OIS o ensure he mos accurae valuaion of is porfolio for risk managemen purposes. LCH.Clearne, June 7, Ten years ago if you had suggesed ha a sophisicaed invesmen bank did no know how o value a plain vanilla ineres rae swap, people would have laughed a you. Bu ha isn oo far from he case oday. A wha rae should we discoun (or wha is he risk free rae)? Deus Ex Machiao 3, June 3,. The old old model: The risk free rae is he rae a which he governmen borrows. This model assumes ha governmens do no defaul and ha governmen bond yields are no conaminaed by liquidiy or ax premiums. The old model: The risk free rae is he (Libor) rae a which he big inernaional (Libor raed) banks borrow a shor mauriies (say hree monhs). This model assumes ha he credi spread over a shor horizon (say hree monhs) of a highly raed (say AA) bank is pracically zero. This is jusified in a Meron ype model because shor horizon defaul raes are zero for highly raed eniies. In he Meron model, firms slide ino defaul afer repeaed downgrades, and over shor horizons, highly raed firms can experience downgrades and no defauls. In his model, he (TED 4 ) spread beween he governmen bond yield and Libor is a liquidiy or ax premium and is no a credi spread. This model ignores jump o defaul risk. Prof. Jayanh R. Varma, Indian Insiue of Managemen, Ahmedabad, jrvarma@iimahd.erne.in. This inroducory noe is my inerpreaion of he resuls presened in various papers lised in he bibliography and does no claim o conain any original conen. I would like o acknowledge valuable commens received from Prof. Sidharh Sinha on earlier drafs of his noe. Commens and suggesions are mos welcome. hp:// 3 hp://blog.rivas.com/?p=366 4 TED sands for Treasury-EuroDollar and denoes he spread beween he Treasury Bill yield and he Libor rae for he same mauriy (usually hree monhs).

2 3. The new model: The risk free rae is he OIS/Eonia rae obained by compounding he overnigh ineres rae a which Libor raed banks borrow. One way o jusify his is o argue ha he Meron syle argumen for zero credi risk a shor horizons is valid for one day bu no for longer periods. Anoher jusificaion is ha OIS/Eonia is he rae ha is paid on cash collaeral and is herefore he correc discouning rae for collaeralized forward conracs and swaps. The laer argumen does no require ha Libor raed banks have zero credi risk a even overnigh horizons. Why did he global financial crisis change he idea of he risk free rae? The Meron ype argumen ha highly raed eniies have zero credi risk a shor horizons requires perfec observabiliy of he balance shee (asse values and deb). The Meron model saes ha defaul akes place when he marke value of asses drops below he value of he deb (or more generally a defaul boundary ha depends on he level of deb). If he asse value follows a diffusion process (wihou jumps), and he curren value of asses is far above he defaul boundary, hen defaul is pracically impossible over a shor ime horizon because a diffusion akes ime o move he asse value ha far. However, he crisis of 8 showed ha bank balance shees are erribly opaque. Banks valued illiquid asses no by marking o marke bu by marking o model or someimes marking o myh. Similarly, he rue liabiliies of he bank were hidden using various off balance shee vehicles (for example, he ABCP conduis and oher SPV 3 s). This mean ha he rue disance o defaul 4 could be much lower han he esimaed disance o defaul based on observable parameers. The unreliabiliy of accouning informaion creaes a jump o defaul risk and creaes non negligible credi spreads a shor ime horizons 5. Jumps in asse values are anoher explanaion for jump o defaul risk bu his does no appear o have been relevan for he banks during he global financial crisis 6. The opaqueness of bank balance shees and he speed wih which some banks became disressed means ha even one-monh Libor can no longer be regarded as risk free. Wheher overnigh Libor can be regarded as risk free is an open quesion. OIS (Overnigh Index Swap) is he erm used in he US while Eonia (European OverNigh Index Average) is he erm used in Europe. Asse backed commercial paper 3 Special Purpose Vehicles 4 The gap beween he curren asse value and he defaul boundary measured in sandard deviaions of he asse price diffusion. 5 See Duffie and Lando () 6 Jumps could also happen in he liabiliies of a company as seen in he case of he seep rise in credi spreads of BP afer he oil spill in he Gulf of Mexico. Bu his oo does no appear o be relevan o he banks during he global financial crisis.

3 The Tenor Swap Spread and CCS Spread Pos crisis, we assume ha Libor is a risky rae ha includes a credi spread (excep possibly a he overnigh horizon). Recall also ha fuure Libor raes are no he rae a which a specific Libor raed eniy can borrow in fuure. For example, if Bank X is Libor raed oday, i can borrow a Libor oday, bu here is no assurance ha i can borrow a Libor nex quarer because i may no longer be Libor raed nex quarer. The Libor rae nex quarer will be he rae a which a bank can borrow if i is Libor raed on ha dae. There is herefore a big difference beween a six monh Libor deposi and a hree monh Libor raed deposi ha is rolled over afer hree monhs ino anoher hree monh Libor raed deposi. The laer is less risky because he deposi is known o be wih a Libor raed bank a he beginning of he second quarer while here is no such assurance in he second case. The six monh Libor deposi should herefore yield more han he rae obained by compounding he curren hree monh Libor rae wih he expeced hree monh Libor rae nex quarer. This implies ha an ineres rae swap whose floaing leg is hree monh Libor is no he same as a swap whose floaing leg is six monh Libor. The floaing leg paymens on he firs swap are expeced o be lower and herefore he fixed leg should also be lower. A enor swap in which boh legs are floaing say hree monh Libor on one leg and six monh Libor on he oher leg should include a enor spread on he hree monh leg o ensure ha he swap is fair a incepion. We are assuming here ha he ineres rae swaps are free of credi (counerpary) risk as discussed laer when we urn o collaeralizaion. Similar explanaions can be given for he Cross Currency Swap (CCS) spread. In a CCS, boh legs are floaing bu in differen currencies. For example, US dollar Libor on a dollar noional amoun may be exchanged for Japanese yen Libor on an equivalen yen noional wih an exchange of principals a he end. If Libor is regarded as risk free, hen he wo legs are floaing rae bonds ha mus be worh par and he swap should rade fla. In realiy however, for several years, here has been a premium on he yen leg of his swap. Credi qualiy differences could be one explanaion for his, hough liquidiy issues migh also play an imporan role. Old Syle Forward Raes Pre-crisis, when Libor is regarded as risk free, forward raes was compued along any curve (say he six monh ineres rae swap curve) by boosrapping as follows: A he peak of he crisis, even his was perhaps no rue leading o he joke ha Libor is he rae a which banks do no lend o each oher. Bu ouside hose ruly dark monhs, i is reasonable o assume ha op raed banks can borrow a Libor. A a fundamenal level, i is very difficul if no impossible o disinguish beween funding liquidiy risk and credi risk. 3

4 The curren six monh Libor, L L, can be used o compue he price oday of a six monh zero coupon bond as P P = L The one year ineres rae swap rae based on six monh Libor, S y, is he yield of a one year par bond wih semi-annual coupons equal o S y. This par bond has a year one cash flow of S y and he presen value of his cash y S flow is P (subracing he presen value of he firs coupon from he par value of he bond. The price of a one year zero coupon bond is herefore y P S P y P y = S y. The forward rae oday for six monh Libor six monhs from now is hen given by F F =[ P P y ] The above analysis can also be jusified in erms of risk neural valuaion because he forward rae compued a ime using he above equaion F =[ P P y ] = P P y () P y is he price a ime of he porfolio P P y relaive o he numeraire P y and is herefore a maringale using he risk neural probabiliies associaed wih his numeraire. A ime =six monhs, he righ hand side becomes simply he six monh Libor prevailing a ha dae given by [ P y ]. The forward rae is herefore he risk neural expecaion of Libor. F =E [ # P y ] () where E # denoes he risk neural expecaion wih he one year Libor bond as he numeraire. Alernaively, we can also hink of his equaion in erms of a replicaion sraegy o make a forward six monh deposi beginning six monhs from now, we borrow for six monhs and make a one year deposi righ now. This formula is he same as Eq 3 of Mercurio (9). 4

5 Under he pos crisis assumpions, his mehodology is no valid. The replicaion approach does no work because if we make a one year deposi now wih a Libor raed bank, here is no assurance ha a he end of six monhs, ha bank would remain Libor raed. Hence we do no replicae he forward Libor deposi which by definiion is wih a Libor raed bank a incepion (six monhs from now). y The boosrapping approach is no valid because S a par bond. is no longer he yield of The maringale approach does no work because P y is no longer he price of a raded asse and canno herefore be used as a numeraire. New Syle FRA Raes The new mehod does no use he boosrapped forward rae, bu relies insead on he Forward Rae Agreemen (FRA) rae he rae quoed in he marke for a forward Libor deposi. Under risk neural valuaion, he FRA rae can also be compued as he expecaion of he Libor rae using he OIS rae as he risk free discoun rae. When we ake OIS as he risk free rae, we are implicily reaing he excess of he Libor rae (or he swap rae) over he OIS rae for he same mauriy as he expeced defaul loss. For example, assume for simpliciy ha he recovery from a defauling bank is zero and ha Q * is he risk neural probabiliy ha a bank ha is Libor raed bank six monhs from oday will no defaul during he six monhs hereafer. The FRA rae FRA for a six monh Libor deposi beginning six monhs from oday is fair if he expeced presen value of he deposi six monhs is equal o he he expeced presen value of he repaymen afer one year. Leing P OIS denoe he prices of zero coupon bonds based on he OIS curve, we ge P OIS =P y OIS FRA Q* or FRA =[ OIS P P y OIS Q * ] (3) I is seen ha he FRA rae equals he boosrapped forward rae if here is no defaul in which case Q * is equal o uniy and he OIS curves and Libor curves are he same. The difference beween he FRA rae and he forward raes can be illusraed by he following example. Suppose he price of he and m zero coupon OIS bonds are.9839 and corresponding o simple ineres raes of 4.% and 4.5%. The OIS forward rae is seen o be =4.9 %. Suppose ha during any six monh period, hree hings can happen o a Libor raed bank: This formula is he same as ha in Table of Mercurio (9). 5

6 I defauls wih risk neural probabiliy.5% wih zero recovery. I ges downgraded wih risk neural probabiliy 5.%. Afer downgrade, i defauls wih probabiliy risk neural 4% during he subsequen six monhs wih zero recovery. I remains a Libor raed bank wih risk neural probabiliy 94.5%. This implies ha he six monh risk neural survival probabiliy of a Libor raed bank is.5 %=99.5% and he welve monh neural survival probabiliy is 94.5% x 99.5% 5.% x 96.%=94.3% 4.8%=98.83%. These imply raher high defaul probabiliies and I am using hese admiedly unrealisic assumpions only o highligh he phenomenon more clearly By risk neural valuaion, six monh Libor should be 5.5% and welve monh Libor should be % as shown below: 5.5% x.9839 x 99.5%= % x x %=. The Libor forward rae is hen seen o be 6.3% as shown below: % 5.5 %/ =6.3 % However, he FRA rae is 5.93% as compued below using Eq (3): x 99.5% =5.93% The FRA rae is differen from boh he OIS forward rae and he Libor forward rae. The former difference is simply a reflecion of he higher defaul risk of he Libor. The laer difference is more suble i reflecs he possibiliy of he Libor raed bank being downgraded in he firs six monhs and hen defauling a much higher raes hereafer. The welve monh Libor includes a.% probabiliy of defaul in he second six monhs following a downgrade in he firs six monhs ( 5% x 4%=.% ). The forward Libor compued from six monh and welve monh Libor includes he effec of his defaul probabiliy. The FRA rae does no ake his ino accoun because he FRA does no conemplae lending o a downgraded bank. This basis poins of exra defaul probabiliy during six monhs (4 basis poins annualized) is wha causes he forward Libor rae o be 37 basis poins more han he FRA rae. Needless o say, his is an exreme example wih unrealisically high defaul and downgrade probabiliies. In normal marke siuaions, we would expec raher smaller differences. 6

7 Evidenly, i is possible o run his exercise in reverse o exrac defaul probabiliies from OIS, Libor and FRA raes. An alernaive and perhaps more convenien approach is o compue he FRA rae direcly using risk neural valuaion : FRA = E * [ P y ] (4) where E * denoes he risk neural expecaion using he OIS as he risk free rae. Comparing he above equaion for he FRA rae wih Eq () for he (old model) Libor forward rae, we see ha we are aking he expecaion of he same quaniy bu using a differen se of risk neural probabiliies. Essenially, we are moving from regarding he Libor/swap rae as risk free o regarding he OIS as risk free. This implies ha he difference beween he Libor forward rae and he Libor FRA rae is he familiar quano adjusmen ha arises in any change of measure: FRA =QA x F where QA=exp F X F, X and he numeraire raio X = P Y P OIS Y (5) If we ake he correlaion o be equal o for simpliciy and assume ha he volailiy of forward Libor is 3% and he volailiy of he numeraire raio is 5%, hen F X F, X =3% x 5% x.=4.5% and inegraing his for six monhs and exponeniaing gives QA=exp.5 % = Under hese assumpions, a forward rae of 6.3% would imply an FRA rae of.9778 x 6.3 %=6.%. This difference of basis poins is much less han he 37 basis poins ha we obained under exreme assumpions regarding defaul raes and recovery. Two curve mechanics The wo curve discouning process can be summarized as follows: Boosrap 3 he OIS curve o ge zero raes for various mauriies. Value all insrumens by using risk neural valuaion wih he OIS raes as he discoun raes. This formula of course requires he assumpion of zero recovery. In case of non zero recovery, he formula would need o be modified. This formula is he same as he unnumbered equaion jus above Eq 5 of Mercurio (9). 3 In pracice, i would be necessary o use splines or some oher inerpolaion scheme o perform he boosrap. 7

8 When cash flows ha depend on fuure Libor, he risk neural expecaions are based on he OIS rae as he numeraire. Insead of he old model assumpion ha forward Libor raes are realized, we assume ha FRA raes are realized. To value caps and swapions, he following modificaion of he sandard Libor Marke Model (LMM) procedure can be used: For each Libor curve ( monh, 3 monh, 6 monh and so on), compue he FRA raes as he risk neural expecaion of he forward raes wih he OIS raes as he risk free rae. Assume ha he FRAs are lognormally disribued. Value Libor caps and swapions using he sandard LMM wih only one change use he FRA raes insead of he forward raes obained by boosrapping he Libor curves. Collaeralizaion I only remains o jusify he use of he OIS rae as he risk free rae. Afer all he opaqueness of bank balance shees implies ha he jump o defaul risk canno be ignored a even he one day horizon. The risk has o be furher aenuaed o jusify ignoring he risk. In he pos crisis environmen, mos derivaive ransacions beween even highly raed banks end o be collaeralized on a daily basis. The risk in hese swaps is exremely low for wo reasons:. A an overnigh horizon, he risk of a Libor raed bank defauling is quie low even wih all he opaqueness of bank balance shees. Even during he wors phase of he crisis in 8, i ook several days or even weeks for a bank o move from apparen healh o serious disress.. In a swap wih daily collaeral flows, he exposure of one bank o anoher is no he noional value of he swap bu only he poenial change in is marke value over he course of one day. This exposure is only a minuscule fracion of he noional value of he swap. The effecive risk is he produc of wo small risks and can be regarded as negligible even when neiher of hese wo small risks is negligible by iself. For modelling purposes, we look a a formulaion in which he collaeralizaion happens on a coninuous basis and he risk is heoreically zero. The daily collaeralized swap is regarded as a very close approximaion o his ideal siuaion and he risk hough non zero may be regarded as negligible. In his formulaion, i can be shown ha he discoun rae should be he rae paid on he collaeral ha is exchanged beween he banks. To undersand his resul, i is useful o consider he difference beween forward conracs, fuures conracs, and collaeralized forward conracs. 8

9 Forward Conracs To analyse forward conracs, we use he zero coupon bond as he numeraire, le S denoe he price of he underlying and le F be he value of a forward conrac on S a he forward price f prevailing a ime. By risk neural valuaion, we have F =P,T E * [ F T ] =P,T E * [S T f ] (6) The discoun facor is he price of he zero coupon bond mauring a expiry of he forward conrac and his bond is also he numeraire ha deermines he risk neural probabiliies. Wih his numeraire and he associaed risk neural probabiliies, he forward price a incepion (ime ) is he expeced price of he underlying: =F =P,T E * [S T f ] E * [S T ]= f (7) Fuures Conracs To analyse he fuures conrac, we change he numeraire o he money marke accoun and consider a rading sraegy ha sars a ime, ends a ime T, and holds e r i fuures conracs a ime ( T ) and invess all mark o marke gains and losses ino a money marke accoun. This is he inverse of he ailing of he hedge sraegy where we sar wih less han one fuures conracs and build he posiion up gradually o end wih one fuures conrac. Here we sar wih one fuures conrac and le i grow gradually o a random number of conracs a mauriy. Since he fuures conracs are always worh zero, he value of his sraegy is he value of he money marke accoun. A ime T, his conains all mark o marke cash flows plus accumulaed ineres on each of hese cash flows. We derive he expression for he value M i of he money marke accoun as follows. Consider he mark o marke cash flow a ime : [ e fuure accumulaed ineres up o ime T, his becomes [e r i ][e T r i ] F F =[e T r i ] r i ] F F F F =B[,T ] F F. Togeher wih where B[,T ] is he accumulaed value a ime T of a money marke accoun sars wih uni value a ime. Adding all hese accumulaed values in his elescoping sum, we ge he Here =/365. For simpliciy, we are assuming ha he fuures conrac is marked o marke daily and he money marke accoun is reinvesed every day a he hen prevailing overnigh rae. The same analysis could be carried ou using coninuous mark o marke and he coninuous money marke accoun. 9

10 erminal value of he money marke accoun M T as B,T [F T F ] or B,T [S T F ] since F T =S T. Noing ha M = and applying risk neural valuaion o M gives us: *[ =M =E F =E * [S T ] B,T B,T [S T F ] ] =E * [ S T F ] So he fuure price is he expeced price of he underlying under he maringale measure if he money marke accoun is he numeraire. The difference beween he forward and fuures price comes from he difference beween he wo maringale measures. Collaerized Forward Raes Le r i be he risk free rae on day i and c i be he collaeral rae he rae ha he receiving bank pays on he cash collaeral posed wih i. Consider a rading sraegy ha sars a ime, ends a ime T, and holds e y i (8) collaeralized forward conracs a ime ( T ) where y i =r i c i. Le he value of he collaeralized forward conrac a ime i be given by h i. Consider he mark o marke cash flow a ime : [ e y i ] h h. The receiving bank can inves his cash flow a he risk free rae, bu mus pay he collaeral rae on i. Is ne reurn on his cash flow is herefore y i. Togeher wih fuure accumulaed ne reurn up o ime T, his becomes [e y i ][e T y i ] h h =[e T y i ] where C [, T ] is he h h =C [,T ] h h accumulaed value a ime T of a collaeral accoun sars wih uni value a ime. consider a collaeral accoun ha sars wih h a ime and ino which all he mark o marke cash flows are added along wih he ne reurn Adding all hese accumulaed values in his elescoping sum, we ge he erminal value of he collaeral accoun V T as V T =C,T [h h T h ]=C, T h T. Discouning his a he overnigh risk free rae, and aking risk neural expecaions (wih he money marke accoun as he numeraire), we ge: h =E *[e r i *[e C,T ht]=e r i y i *[e e ht]=e c i ht] (9) The discussion in his secion closely follows secion 3. of Masaaki Fujii, Yasufumi Shimada, and Akihiko Takahashi (a).

11 The risk free rae compleely drops ou of he valuaion formula and we obain he resul ha discouning mus be done a he collaeral rae even if i is no he rue risk free rae. The collaeralizaion does no however change he numeraire which remains he money marke accoun which earns he risk free rae. For example, we may believe ha he overnigh US Treasury repo rae in he US is he rue risk free rae and he OIS rae may be above he repo rae because of he credi risk of even overnigh lending. Tha does no invalidae discouning a he OIS rae if he swap is collaeralized. However, he numeraire is no he collaeral accoun; i is he money marke accoun. In some sense, we could regard his as a hree curve model he repo rae providing he numeraire, he OIS rae providing he discoun rae, and he cash flows are deermined by Libor (if he underlying of he collaeralized forward conrac is Libor).

12 Seleced Bibliography Marco Bianchei (9) Two Curves, One Price: Pricing & Hedging Ineres Rae Derivaives Using Diff eren Yield Curves for Discouning and Forwarding, available a hp:// Fixed_Income/-A-Bianchei-DoubleCurvePricing-v.4.pdf Duffie and Lando (), Term Srucures of Credi Spreads wih Incomplee Accouning Informaion, Economerica, 69(3), Chrisian P. Fries () Discouning Revisied. Valuaions under Funding Coss, Counerpary Risk and Collaeralizaion, available a hp://ssrn.com/absrac=69587 Masaaki Fujii, Yasufumi Shimada and Akihiko Takahashi (9) A Marke Model of Ineres Raes wih Dynamic Basis Spreads in he presence of Collaeral and Muliple Currencies, available a hp://ssrn.com/absrac=568 Masaaki Fujii, Yasufumi Shimada, and Akihiko Takahashi (a), A Noe on Consrucion of Muliple Swap Curves wih and wihou Collaeral, available a hp://ssrn.com/absrac=44633 Masaaki Fujii, Yasufumi Shimada, and Akihiko Takahashi (b), Collaeral Posing and Choice of Collaeral Currency Implicaions for Derivaive Pricing and Risk Managemen, available a hp:// Fabio Mercurio (9) Ineres Raes and The Credi Crunch: New Formulas and Marke Models, available a hp://ssrn.com/absrac=335 Massimo Morini (9) Solving he puzzle in he ineres rae marke (Par & ), available a hp://ssrn.com/absrac=5646