On the term structure of default premia in the swap and LIBOR markets 1

Transcription

1 On he erm srucure of defaul premia in he swap and IBOR markes 1 Pierre Collin-Dufresne 2 Bruno Solnik 3 Firs version: Sepember 1997 as revised: May The auhors would like o hank seminar paricipans a he AFA 98 in New York, HEC, ancaser Universiy, ondon Business School, Insead, Inquire, Carnegie Mellon Universiy, Ohio Sae Universiy, Boson Universiy, U.C. Irvine and The Wharon School for very insighful commens. They also graefully acknowledge financial suppor from he Fondaion HEC and of Inquire Europe. René Sulz and wo anonymous referees provided useful commens and suggesions. Of course, he auhors are responsible for any remaining error. 2 Assisan Professor a Carnegie Mellon Universiy, GSIA 315A 5000 Forbes Avenue, Pisburgh PA 15213, el:(412) , dufresne@andrew.cmu.edu 3 Professor a HEC, School of Managemen. Groupe HEC, Jouy-en-Josas cedex, France. el: (33) , Fax: (33) , solnik@hec.fr

2 On he erm srucure of defaul premia in he swap and IBOR markes Absrac Exising heories of he erm srucure of swap raes provide an analysis of he Treasury-swap spread based on eiher a liquidiy convenience yield in he Treasury marke, or defaul risk in he swap marke. While hese models do no focus on he relaion beween corporae yields and swap raes (he IBOR-Swap spread), hey imply ha he erm srucure of corporae yields and swap raes should be idenical. As documened previously (e.g. in Sun, Sundaresan and Wang (1993)) his is counerfacual. Here, we propose a model of he defaul risk imbedded in he swap erm srucure ha is able o explain he IBOR-swap spread. Whereas corporae bonds carry defaul risk, we argue ha swap conracs are free of defaul risk. Because swaps are indexed on refreshed -credi-qualiy IBOR raes, he spread beween corporae yields and swap raes should capure he marke s expecaions of he probabiliy of deerioraion in credi qualiy of a corporae bond issuer. We model his feaure and use our model o esimae he likelihood of fuure deerioraion in credi qualiy from he IBOR-swap spread. The analysis is imporan because i shows ha he erm srucure of swap raes does no reflec he borrowing cos of a sandard IBOR credi qualiy issuer. I also has implicaions for modeling he dynamics of he swap erm srucure.

3 1 Inroducion Exising models of swap raes focus on he spread beween swap raes and Treasury yields. In his aricle, we provide a direc comparison of he erm srucures of swap raes and of corporae bond yields. An ineres rae swap is a conrac by which a fixed paymen sream is exchanged agains a floaing paymen sream. The floaing leg of he swap is usually se a he inerbank ineres rae for he relevan currency (ypically he 6-monh IBOR for dollar swaps). Once he floaing leg is specified, he marke rae for a swap is simply he coupon rae on he fixed leg of he swap. The generic swap rae applies o a op-qualiy clien raed AA or beer. Dealers use his marke rae as a reference when hey quoe an acual swap rae o a clien and adjus for defaul risk and oher characerisics of he clien. In his paper we only consider generic swaps quoed for op-qualiy counerparies. We do no sudy he adjusmen made o he generic swap rae for a more risky counerpary. 1 Swaps are quoed for various mauriies; hence here exiss a erm srucure of swap raes ha can be compared o he erm srucures of Treasury yields and of defaulable corporae bond yields. For illusraive purposes we presen he average erm srucures for he period 10/12/88 o 01/29/97 in figure 1. Figure 1: The average erm srucure of swap raes, corporae and Treasury yields: December 1998 o January All erm srucures are expressed in semi-annual, acual/365 convenion. Daa is aken from Daasream and is described in secion 4. As expeced, he swap curve is well above he Treasury curve. More ineresingly, casual observaion suggess ha he swap curve is below he corporae curve, 2 and ha he IBOR-Swap spread 1 Sudies of he adjusmen in he swap rae done o reflec he credi qualiy of differen counerparies can be found in Sorensen and Bollier (1995), Sun, Sundaresan and Wang (1993), Duffie and Huang (1996). 2 In he empirical work, we use daa on IBOR bonds as measures of he yields on defaulable corporae bonds. IBOR bonds are fixed-coupon bonds negoiaed OTC and issued by op-qualiy corporae bond issuers (usually banks and financial insiuions) raed AA or beer. Hence, we will call IBOR-swap he spread beween yields on IBOR-qualiy bonds and IBOR swap raes for all quoed mauriies; for example, he 5- year IBOR-swap spread is he spread beween he yield on a 5-year IBOR bond and he fixed rae on a 5-year 1

4 increases wih mauriy. The average spread (across mauriies and daes) beween IBOR bond yields and swap raes is around 15 basis poins. I is, by consrucion, zero a 6-monhs o mauriy. Our inen is o develop a model ha explains he spread beween corporae bond yields and swap raes (IBOR-swap spread) and is dynamics. Alhough swap raes are ofen quoed relaive o Treasury yields for pracical reasons (he Treasury erm srucure is widely-available and coninuously-updaed), he imporan comparison for swap raes is wih corporae bond yields of similar credi qualiy. While he IBOR-swap spread only amouns o a few basis poins 3 i can be of significan financial imporance. Corporae issuers measure heir spreads relaive o he swap curve raher han o he Treasury curve which is differen in erms of credi qualiy, and exhibis significan insiuional and regulaory disorions (such as repo specials, axes and perhaps liquidiy). Swaps are ofen used by corporae issuers in complex financing packages involving corporae bonds in order o gain some financing cos reducion compared o issuing plain-vanilla bonds. Bankers use he swap curve, in lieu of he corporae curve, as he basic ool for pricing corporae asses and liabiliies. This pracice originaes from he observaion ha swap raes are coninuously quoed (and raded) for a wide range of mauriies and herefore more readily updaed han corporae yields. Ye i is jusified only if he swap erm srucure ruly reflecs he cos of financing of a op-raed corporae issuer for he various mauriies. In his paper we argue ha he IBOR-swap spread, is no o be dismissed as simply resuling form daa problems (or liquidiy), bu ha i should exis on purely heoreical grounds. The focus of exan models of he swap erm srucure 4 is he analysis of he spread beween Treasury yields and swap raes. ile has been done o explain he spread beween corporae bond yields and swap raes. 5 Recen papers (Grinbla (1995) and Duffie and Singleon (1997)) provide models where he erm srucure of swap raes can be modeled using a radiional wo-facor model of he erm srucure. In boh models he erm srucure of swap raes is equal o a erm srucure of corporae par-bond raes. Grinbla (1995) proposes a model where boh swap conracs and Treasury bonds are free of defaul risk. The swap-treasury spread arises because of a liquidiy convenience yield 6 accruing o he holder of a governmen-issued securiy. As a resul, swap raes are equal o a risk-free par bond raes in his model. In Duffie and Singleon (1997) he swap-treasury spread arises because swap conracs carry defaul risk. In heir model he swap rae is equivalen o a par-bond yield on a credi risky bond, which hey model using a wo-facor model as in Duffie and Singleon (1999). In his paper we provide a model of he spread beween par-bond raes and swap raes. We relax wo assumpions explicily or implicily made in previous lieraure, namely: (i) homogeneous IBORswap credi qualiy and (ii) refreshed credi qualiy of IBOR couner-paries which we discuss below. 7 IBOR swap, wih all paries of op-credi qualiy. 3 The IBOR-swap spread is usually well above he swap bid-ask spread, which only amouns o a couple of basis poins for generic swaps. 4 Earlier work include Solnik (1990), Sundaresan (1991), Cooper and Mello (1991). 5 A noable excepion is Cooper and Mello (1991) who analyze spreads beween risky swap and bond raes in a srucural framework similar o Meron (1974). Their model differs from ours as hey focus on wealh ransfers beween bond, swap and equiy holders of a firm, and hus assume ha swap conracs are credi risky. As a resul, in heir model corporae and swap raes are funcions of he highly sylized liabiliy srucure of he firm analyzed. 6 Grinbla models his as an exogenous facor, similarly o convenience yields in he forward conrac lieraure. 7 See also Duffie and Singleon (1997) for a discussion of hese assumpions. 2

5 The assumpion of homogeneous IBOR-swap credi qualiy implies ha swap conracs and IBOR bonds have he same defaul risk, and hence ha all cash flows peraining o eiher conrac should be discouned under he risk-neural measure using he same risk-adjused rae. However, i is very likely ha swaps be no impaced a all by defaul risk so ha hey should be reaed as defaul risk-free, unlike IBOR bonds which carry AA defaul risk. I is now widely recognized 8 ha corporae bonds bear more credi risk han swaps wrien by he same counerparies. The naure of he swap conrac makes defaul on swaps much less cosly han on bonds. The poenial loss on a swap does no include he principal bu only an ineres rae differenial (e.g. fixed minus floaing), and only in he case where his difference is posiive for he non-defauling pary (i.e. if ineres rae movemens have led o a posiive swap marke value for he non-defauling pary). Furhermore, his poenial loss is ofen reduced or eliminaed by he posing of collaeral or marking-o-marke provisions, as well as oher conracual provisions in case of credi downgrading of a pary. Some furher argue ha a swap beween wo paries of similar credi qualiy should enail no defaul risk premium in eiher direcion because of he symmeric naure of he conrac. 9 So he impac of credi risk on he pricing of a generic swap should a bes be minimal. 10 Hence i seems essenial o use differen risk-adjused raes for corporae bonds and swap conracs issued by he same pary. In his aricle we assume ha he payoffs of a generic swap are basically priced as if free of defaul risk: he discoun facor adjused for defaul risk o be used under he risk-neural measure o price swap conracs for AA paries is he risk-free ineres rae. However, he swap erm srucure will be differen from (and above) he risk-free erm srucure, because he swap rae paymens are indexed on 6-monh IBOR which is a defaul-risky rae. Hence, he swap rae will be higher han he risk-free rae even hough he swap conrac is free of defaul risk. On he oher hand, he fac ha swap conracs are less risky han IBOR bonds, does no necessarily imply ha swap raes be lower han IBOR bond yields. This may a firs sound counerinuiive, bu is, in fac, jus a resul of he swap paymens being indexed on he shor end of he IBOR bond yield curve. For example, he swap rae on a swap wih a 6-monh mauriy is always equal o he 6-monh IBOR rae by design of he conrac, no maer wha he difference in credi risk is beween he swap conrac and he 6-monh IBOR bond. The assumpion of refreshed credi qualiy of IBOR couner-paries presumes ha he counerparies will mainain he same credi qualiy over ime. Our subsequen analysis shows ha his assumpion may be inappropriae o undersand he IBORswap spread. The swap conrac is conracually indexed on he 6-monh IBOR rae, which is a refreshed op-credi-qualiy rae. On he oher hand, long-erm IBOR bonds are priced o reflec he likelihood ha he credi qualiy of a op-raed issuer may deeriorae over he life of he bond. Thus, our analysis implies ha he IBOR-swap spread capures he likelihood ha an issuer s credi qualiy may change over ime. We show ha a model ha accouns for (1) he difference in credi risk beween swap conracs and op-qualiy corporae bonds and (2) he difference in credi qualiy of a consanly updaed, refreshed credi qualiy index and ha of a specific op-raed issuer ha may experience a depreciaion in crediqualiy, can reasonably explain he observed spread and is dynamics. Of course here may be oher facors, which could furher explain he dynamics of he IBORswap spread, such as liquidiy. Alhough we are no aware of documened liquidiy evens in he IBOR-swap marke (e.g. comparable o he repo specialness in he Treasury marke), i is possible 8 See izenberger (1992) and Solnik(1990), 9 See Sorensen and Bollier (1994) and Duffie and Huang (1996). 10 We do no sudy he issue of swap pricing when one of he couner-paries is of lesser credi qualiy. Duffie and Huang (1996) show ha such a difference in credi risk has lile impac on swap raes. 3

6 ha he greaer noional ransacion volume of he swap marke is an indicaor for greaer liquidiy and ha his may affec pricing. A pragmaic answer could be o reinerpre our resuls and consider ha our insananeous credi spread, which eners he adjused rae used o discoun under he riskneural measure, reflecs boh credi risk and swap-ibor liquidiy differenial (in he spiri of Duffie and Singleon (1997)). Bu he wo effecs canno be disenangled. Absen a heory for liquidiy, and in ligh of he widespread use of he swap erm srucure in lieu of a op-qualiy corporae-bond erm srucure, i seems useful o provide an explanaion of he IBOR-swap spread, based solely on a realisic defaul-risk model. The ask of isolaing and quanifying he impac of liquidiy relaive o defaul-risk is lef o furher research. Our paper is srucured as follows. In secion 2, we presen our model for corporae bond yields and swap raes. We examine some of he implicaions of our model for IBOR-swap spreads in secion 3. An empirical validaion is provided in secion 4. We conclude in secion 5. Proofs are provided in an appendix. 2 The model Our inen is o develop a model ha provides some qualiaive and realisic quaniaive implicaions abou he (relaive) pricing of wo securiies: he zero-coupon defaulable IBOR bonds for all mauriies T + wih price P () a ime and he swap conrac (iniiaed a ime ) o exchange he (prese) 6-monh IBOR, Y :5 ( +0:5(i, 1)), agains fixed paymens of Y () every 6-monhs for S years (i.e. a every +0:5i 8i =1; 2;:::; 2). As is usual in his lieraure, we denoe by risk-free, securiies ha are free of defaul risk, bu no necessarily of ineres-rae risk. We denoe by risky securiies wih defaul risk. Hence, wih our previous assumpions, swap conracs are risk-free, bu IBOR bonds are risky. They all clearly carry ineres-rae risk The IBOR bond erm srucure Alhough op qualiy, hese corporae IBOR bonds carry defaul risk. We adop he so-called reduced form o defaul-risk modeling discussed by Duffie and Singleon (1999). In his framework, defaul is an unpredicable sopping ime modeled by he firs occurrence of a poin process wih sochasic inensiy, no necessarily relaed o he value of he corporae bond or he value of he firm s asses. In oher words, we implicily assume ha he bond is small relaive o he overall porfolio of asses of he firm. As shown in Duffie and Singleon (1999) he price of he risky zero-coupon bond is given by: P () =EQ he R i, T R(s)ds (1) This formula saes ha he presen value of risky cash flows may be found by discouning hem a a risk-adjused ineres rae under he equivalen maringale measure. The risk adjused rae R() is equal o he insananeous risk-free rae r() plus an insananeous credi spread, which is he insananeous expeced loss rae under he risk-neural measure. Noice ha formula (1) implies ha risky bonds can be priced as risk-free bonds by expanding he number of facors driving he erm srucure. For example, if we chose o model R() as he sum of 11 Since we focus on pricing securiies in his secion all processes are specified under he risk-neural measure. We ake a risk-neural measure Q as given, and discuss he issue of risk-premia in he empirical secion. 4

7 wo independen facors, he risky erm srucure of ineres raes would become a radiional wo-facor model of he erm srucure. However, i seems unlikely ha such a specificaion of he insananeous credi spread for opraed credi qualiy issuers be appropriae. Indeed, heoreical (Meron (1974), Jarrow, ando and Turnbull (1997)) as well as empirical (Sarig and Warga (1989) and Fons (1994)) evidence shows ha he erm srucure of credi spreads exhibi sysemaic paerns, which are no well-capured by sandard processes used for modeling he risk-free erm srucure. 12 In ligh of his evidence, we pu more srucure on our model of he insananeous credi-spread process o allow for possible deerioraion of credi qualiy of he IBOR bond issuers. We model he risk-adjused discoun rae for an issuer, who is op-raed a ime-, asr(s) = r(s) + (s) 8s, and assume he insananeous credi spread of an issuer ha is op-raed a ime evolves according o (for s ): d (s) = (s) (s), (s) ds + (s)dw (s) + 1 (s)dn (s) (2) d (s) = 2 (s)dn (s) (3) () = (4) w () is a brownian under he risk-neural measure. ; ; 1 ; 2 are deerminisic funcions of ime and is a consan. In words, deerioraion in credi qualiy is riggered by a poin process wih inensiy (s) and associaed couning process N (s). N (s) is equal o he number of jumps in credi qualiy beween and s (N () =0). 13 The inensiy (s) may be sochasic. This model implies ha when he credi qualiy of he issuer deerioraes, his credi spread jumps up by a discree amoun 1. A he same ime, here is an adjusmen in he long erm mean of he credi spread which jumps up by a discree amoun A so-called refreshed op-raed issuer which is guaraneed o remain op-raed forever, has 1 = 2 =0. The dynamics of his insananeous credi spread () () are: d() = (),, () d + ()dw () (5) We make he furher assumpion ha he shor-erm risk-free rae r() follows a gaussian process: dr() = r ()((), r()) d + r ()dw r () (6) where w r () is a Q-brownian moion. r ; r are deerminisic funcions of ime. Furhermore, we assume ha he long-erm mean is iself sochasic and mean-revering: d() = (), (), () d + ()dw () (7) where w () is a Q-brownian moion, and ; ; are deerminisic funcions of ime. All brownian moions are possibly correlaed wih deerminisic correlaion coefficiens given by: dw r dw = 12 For example, he erm srucure of credi spread for op-raed issuers should always be increasing wih mauriy. 13 This poin process is assumed o have no common jumps wih he poin process ha riggers defaul. This is a echnical assumpion which is necessary for expression (1) o be valid. I merely saes ha defaul and deerioraion canno occur a exacly he same insan of ime. Of course, any deerioraion in credi qualiy implies ha he probabiliy of a defaul increases. 14 Since we focus on op-qualiy couner-paries, we consider only deerioraion of credi qualiy. The model can easily be exended o include possible appreciaion in credi qualiy. In he appendix we derive he resuls for a more general model which allows also for appreciaion in credi qualiy. We show ha all our resuls go hrough as long as he expeced depreciaion in credi qualiy is higher han he expeced appreciaion for a op-raed credi qualiy issuer. 5

8 r d dw r dw = r d dw dw = d. The Gaussian processes used o model r and presen some well-known shorcomings (negaive values and homoskedasiciy). We choose he Gaussian framework mainly for racabiliy reasons as our goal is o derive closed-form soluions ha provide inuiion abou he relaive impac of he refreshed credi qualiy and non-homogeneous credi qualiy assumpion on he IBOR-swap spread. 15 One can show (e.g. using sandard echniques developed in, for example, Duffie and Kan (1996), Das and Foresi (1996)) ha he risky zero-coupon bond prices of a a ime op-raed issuer, are given by he following formula: P () =P ()P ()er T, (s;t )ds (8) where: P () = e Ar(;T ),Br(;T )r(),c(;t )() (9) P () = ea (;T ),B (;T )() (s; T ) = (s) 1, e, 2(s)(T,s),(1(s),2(s))B (s;t ) And A r ;B r ;A ;B ;C are he sandard deerminisic funcions appearing in he compuaion of a zerocoupon bond (see appendix). Noice ha P is he price of a risk-free zero-coupon bond paying $1 a ime +, which, when coefficiens are consan, is he special case of angeieg s (1980) model analyzed by Jegadeesh and Pennacchi (1996), and which reduces o he sandard Vasicek-bond price for consan, r, r. oosely speaking, (;T) can be viewed as he marginal increase in he yield on a defaulable zero-coupon bond, issued a by a op-raed firm and mauring a T, due o possible deerioraion in credi qualiy beween and T. As for mos exising models of credi-risk, in our framework, a coupon-paying bond can be priced as a sum of risky -zero-coupon bonds. 16 Hence he coupon, Y () paid semi-annually by a corporae bond issued a par a ime and mauring a ime + is given by: (10) Y () = 1, P P () i=2 P : (11) :5i i=1 () 2.2 The swap erm srucure We consider a plain-vanilla or generic swap indexed on 6-monh IBOR, wih he hree usual characerisics C1 he paymens are indexed on a lagged floaing-index value, C2 he rese lag of he floaing index has he same lengh as he paymen period, and C3 paymen daes correspond exacly o rese daes. e us define Y () as he fixed rae o be paid semi-annually for years in a generic swap enered S a dae agains he six-monh-ibor rae of Y :5 (). 6-monh IBOR is defined by he shor end of he op-qualiy corporae erm srucure according o equaion (11): Y :5 :5 1, P () = () P :5 () (12) 15 Noice also ha negaive credi spreads can be inerpreed as (presumably rare) siuaions in which defaul is expeced o resul in recovery of more han he marke value of he bond jus prior o bankrupcy. For example, when bankrupcy negoiaion is done on he grounds of ousanding principal values, he proporion of ousanding principal reimbursed may be higher han he marke value of he bond. 16 All widely used credi risk models share he feaure ha coupon bonds can be priced from zero-coupon bonds, e.g. Duffie Singleon (1999), Jarrow, ando and Turnbull (1997). 6

9 In a generic swap he floaing leg paymen a dae i +0:5i is Y :5 ( i,1). As discussed above, he swap conrac is considered as risk-free. Consequenly, he discoun rae o use under he risk-neural measure is he risk-free rae r() defined above. By definiion of he swap, Y S is he annuiy ha achieves a zero value for he conrac a iniiaion, such ha: E Q " i=2n X i=1 e, R i r(s)ds Y S () # =E Q " i=2n X i=1 e, R i r(s)ds Y :5 ( i,1) Subsiuing from (12), using he formulas derived above, and afer some calculaions, we find: i=2n X 1+Y S () = i=1 # (13) P :5(i,1) ()! i C(; P :5i () i,1 ; i ) C 0 (; i,1 ; i ) (14) P wih! i = P :5i i=2n ()= P :5i i=1 (), C and C 0 are given in equaions A.7 and A.8 in he appendix. The expression derived above for he fixed rae on a swap looks complicaed. However, i is simple o inerpre. Firs, consider a swap wih only one paymen dae, i.e. wih a 6-monh mauriy (n =0:5). ThefixedraeobepaidY :5 () simplifies o he IBOR rae: S Y :5 S :5 () =Y () (15) The fixed rae paid on longer-erm swaps can be inerpreed as a weighed average of forward IBOR raes correced for defaul risk. P P :5(i,1) ()=P :5i () is he implici (one plus) IBOR forward rae beween i,1 and i (since! i i = 1). There are wo correcion facors C and C 0. The former is essenially a Jensen-inequaliy effec, he laer C 0 (; i,1 ; i ) accouns for he possibiliy of jumps in he insananeous credi spread of he IBOR raes ha serve as a reference for he floaing leg of he swap. The link beween he fixed rae on a swap and a weighed average of forward raes has been noed in previous lieraure. 17 Our formula differs from previous models because i accouns for (1) differences in credi risk beween swap conracs and IBOR bonds, and (2) he difference beween a coninuously upgraded refreshed credi qualiy IBOR rae and he yield on a ypical IBOR couner-pary which reflecs possible fuure jumps in credi qualiy. Before we urn o he discussion of hese issues, we would like o briefly menion he swap-treasury spread. The swap spread is ofen quoed wih respec o he yield on a governmen bond wih equivalen mauriy. Alhough boh conracs are free of defaul risk in our model, he swap rae is differen from he Treasury rae. As we have seen, he 6-monh swap rae is equal o he 6-monh IBOR rae by definiion of he swap conrac (equaion 15). So he swap erm srucure is anchored a he 6-monh IBOR, which is clearly higher han he 6-monh Treasury yield, because he IBOR rae reflecs credi risk. More generally, he swap erm srucure depends on he credi-risk process since he floaing leg of he swap conrac is indexed on he 6-monh IBOR rae. Even hough he swap conrac is free of defaul risk, he swap rae depends on he credi-risk process hrough he floaing leg indexaion (i is a risk-free conrac wrien on a risky underlying rae). As a consequence, he dynamics of he swap raes depend on he dynamics of he credi-risk process and, hence, differ from he dynamics of he Treasury raes. The swap-treasury spread is, ypically, no consan across mauriies in our model. 17 See for example Sundaresan (1991) and Duffie and Singleon (1997). 7

10 3 A beer picure of he IBOR-swap spread? In his secion we provide some inuiion for he respecive impac on he IBOR-swap spread of our wo main assumpions (as defined in he inroducion): (1) homogeneous vs. non homogeneous IBOR-swap credi qualiy and (2) refreshed vs. non refreshed credi qualiy in he IBOR marke. If we were o assume ha here is boh homogeneous IBOR-swap credi qualiy and refreshed credi qualiy of he IBOR couner-paries, hen he swap rae would be given by he following formula: i=2n X 1+Y S () = ()=P i=2n i=1! i P :5(i,1) () ) Y P :5i () S () = 1, P P () 2n P (16) :5i i=1 () wih! i = P :5i P :5i i=1 (). In his case he swap rae is equal o he IBOR-bond yield for all mauriies (see Duffie and Singleon (1997) and Sun, Sundaresan and Wang (1993)). Bu, we observe on average a posiive IBOR-swap spread. Since, as discussed previously, swap conracs carry less defaul risk han corporae bonds, i seems naural o firs invesigae wheher relaxing he assumpion of homogeneous IBOR-swap credi qualiy, can explain he observed IBOR-Swap spread. 3.1 Non-homogeneous credi qualiy beween swap and IBOR markes Firs we consider he case where he swap conrac is risk-free whereas IBOR bonds are risky, bu where here is no possibiliy of jumps in he credi spread. Thus, he corporae bond is assumed o always remain of refreshed credi qualiy. Then our formula (14) for swap raes reduces o: i=2n X 1+Y S () = i=1 P :5(i,1) ()! i C(; P :5i () i,1 ; i ) (17) P wih! i = P :5i i=2n ()= P :5i i=1 () and C is as defined previously. The facor C is in fac jus a Jensen-inequaliy effec which in pracice is very close o Thus, he major effec of inroducing non-homogeneiy beween swap and IBOR bond markes is o change he weighing of forward IBOR raes in compuing he swap rae. Indeed a comparison of equaions (17) wih (16) for he case where C = 1 shows ha he only impac of inroducing non-homogeneous credi qualiy is o change P he weighing from! i = P :5i ()=P i=2n P :5i i=1 () in he homogeneous case o! i = P :5i i=2n ()= P :5i i=1 () in he non-homogeneous case. Some algebra reveals ha he slope of he forward-ibor curve dicaes he relaion beween IBOR bond yields and swap raes. We summarize his relaion in he following proposiion: Proposiion 1 Assume (1) he swap conrac is (defaul-) risk-free, (2) he IBOR bond is defaul risky, (3) IBOR bonds are sure o mainain heir credi qualiy (refreshed credi qualiy), and (4) C is negligible (i.e. close o 1). Then, when he forward-ibor curve is upward-sloping (downwardsloping), he swap rae curve should be above (below) he IBOR bond yield curve. For example, he swap rae curve will be above he IBOR bond yield curve when he forward- IBOR raes are increasing wih mauriy. This resul is purely a consequence of he indexaion mechanism of swap conrac. 18 This saemen is easily checked for reasonable parameer values. For example wih parameer values as esimaed in secion 4, C(0; 5; 5:5) ,6. 8

11 The proposiion above shows ha relaxing he homogeneous swap-ibor credi qualiy alone will no explain he observed IBOR-swap credi spreads. Since on average we observe upwardsloping IBOR curves and increasing forward-ibor curves, he above proposiion implies ha he swap curve should be mosly above he corporae rae curve. Empirically, however, we observe he opposie as documened in Sun, Sundaresan and Wang (1993). 3.2 Relaxing he refreshed-credi-qualiy assumpion We claim ha he IBOR-swap spread reflecs he probabiliy of credi deerioraion in a op-qualiy IBOR couner-pary. Indeed, by conracual definiion, he swap conrac is indexed on a refreshed IBOR rae index, which is coninuously updaed so as o mainain is credi qualiy. On he oher hand a ypical IBOR bond issuer may experience a deerioraion in credi qualiy a anyime which is priced ino he bond yield. Comparing equaions (14) and (17), we see ha his is refleced by he facor C 0, which capures he possible change in credi qualiy over ime and can be viewed as he difference beween wo credi risks. The firs applies o an issuer wih refreshed op-credi qualiy on all rese daes (as implici in he swap rae) and he second applies o an issuer who was of op-credi qualiy a ime of issue, (as implici in he IBOR bond yield) Model-implied spreads beween refreshed-qualiy-ibor yields and IBOR yields In his secion, we use our model o provide some insighs ino he cos paid by a ypical IBOR couner-pary for he likelihood of beeing downgraded over he fuure life of he bond. This cos can be measured wihin our framework as he difference beween he yield paid by a op-raed issuer compued using equaion (8), and ha paid by a refreshed credi qualiy issuer compued using he same formula, bu seing he inensiy of credi-deerioraion o zero ( =0). 20 The non-refreshed bond corresponds o a ypical op-qualiy corporae bond, while he refreshed-qualiy bond is ficiious. The refreshedcredi qualiy bond does no carry any credi-deerioraion risk, bu may be defauled upon anyime. The sandard corporae bond reflecs boh: i may defaul a anyime and i may experience deerioraion in is credi qualiy. Figure 2 shows he spread in bond yields beween op-raed issuers wih consan expeced insananeous downgrading se o 10 basis poins (i.e. in our previous noaions 1 = 2 =, (s) = and =10bp) and op-raed issuers wih refreshed credi qualiy ( =0). Of course, he consan expeced insananeous downgrading can resul from differen combinaions of jump size and inensiy of credi depreciaion. We show wo cases: a high size/low inensiy case ( = 100bp, = 0:1) and a low size/high inensiy case ( = 10bp, = 1). All oher parameer values correspond o hose esimaed in he nex secion, Table 1. The values of he insananeous risk-free rae and of he credi spread are se a heir long-erm means. Figure 2 shows ha he spread beween non-refreshed and refreshed credi qualiy bond yields is economically significan, increasing wih mauriy, reaching 60 bp a a 20-year mauriy. Ineresingly, he figure also reveals ha he spread has a slighly differen sensiiviy o size and inensiy of credi 19 To undersand he inuiion, consider he exposure on a 10-year-mauriy corporae bond versus a 10-year swap. Compare he defaul spread on he cash flow of one paricular mauriy, say in 7 years. Holding a 10-year corporae bond eniles one o receive a coupon in 7 years if here has no been any previous defaul. The value of ha coupon depends on he expeced recovery rae of a cash-flow received in 7 years by a oday op-raed firm. On he oher hand, he cash flow o be received in 7 years in a (defaul-risk-free) swap conrac incorporaes defaul risk only hrough he floaing index, which depends on he expeced recovery rae on a 6-monh defaulable bond issued by a firm ha will be op-raed in 6.5 years. 20 We hank a referee for suggesing his analysis. 9

12 Figure 2: Term srucures of spreads beween yields of non-refreshed and refreshed credi qualiy corporae bonds as implied by he model. All parameer values are aken from he esimaed values in Table 1 below. We use wo differen values for jump size and jump inensiy. Case 1 has = 100bp and = 0:1, case 2 has =10bp and = 1. Noice ha in boh cases we keep he expeced insananeous downgrading consan o =10bp. depreciaion risk. For a consan expeced depreciaion in credi qualiy, he credi spread is acually increasing in jump inensiy bu decreasing in size. In oher words, credi spreads are more sensiive o changes in inensiy han o changes in he size of he jump in credi spreads. 4 Some empirical resuls Using daa on Treasury bond yields, IBOR bond yields and swap raes we now esimae he parameers of our model. This allows us o deermine he significance of he deerioraion in credi qualiy of opraed issuers implici in he IBOR-swap spread. 21 We shorly describe he daa and economeric mehodology used and discuss he empirical resuls. 4.1 Daa and economeric mehodolgy We use weekly daa for Treasury, IBOR par-bond and swap raes from Ocober 12, 1988 o January The daa were obained from Daasream. Daasream repors he mid swap raes 22 quoed by a 21 We use Treasuries as a proxy for he rue risk-free rae even hough hey are ofen claimed o offer advanages over and above he risk-free asse, such as liquidiy and axes. This allows us o isolae he differen componen of he IBOR-swap spread and give some economic inerpreaion o our resuls. Noice ha since we esimae he IBOR-swap spread, we may reasonably hope his will no have a big impac on our esimaion of he insananeous credi-risk process. 22 The bid and ask swap raes quoed depend on he credi qualiy of he cusomer. The bid-mid and mid-ask spreads for a generic swap quoed o a AAA or AA cusomer are generally equal o one basis poin over he period. As menioned in Sun, Sundaresan and Wang (1993) and Cossin and Piroe (1997), he spreads increase by a few basis poins for a lesser-raed cusomer. 10

13 major swap dealer for mauriies of 2, 3, 4, 5, 7 and 10 years. Treasury bond daa covers he mauriies: 1, 2, 3, 4, 5, 7 and 10 years. Finally we use he IBOR yields repored by Daasream for mauriies 0.5, 1, 2, 3, 4, and 5 years. These are quoed yields for fixed-coupon par-bonds negoiaed OTC and issued by corporae issuers (usually banks and financial insiuions) raed AA or beer. 23 In order o subjec our model o empirical scruiny, we make a few simplifying assumpions. We assume ha all parameers are consan. To reduce he number of parameers o be esimaed, we assume ha 1 = 2 = and ha (s) =. In words, we assume ha when he credi qualiy deerioraes, boh he long-erm mean and he level of he credi spreads jump by an equal amoun, and ha he probabiliy of credi deerioraion is consan. Because of a well-known indeerminacy arising in such models (Duffee (1999), Duffie and Singleon (1999)) we canno esimae, he inensiy of he jump, separaely from he size of he jump. We hus esimae he join produc. 24 We also need o make assumpions abou he risk premia associaed wih our hree sochasic facors, because our daa is observed under he hisorical P-measure whereas we have specified he processes under he risk-neural measure. For he empirical implemenaion we assume risk-premia o be consan. 25 We hus have hree addiional parameers o esimae, r ; ; he risk premia associaed wih ineres-rae risk and generic refreshed credi qualiy credi spread. The risk premia capure he shif in disribuion going from he physical measure P o he risk-neural measure Q. 26 To furher reduce he number of parameers o be esimaed, we consrain he auocorrelaion coefficien for all he error erms o be he same. We hus have a oal of 16 parameers o esimae. We use maximum-likelihood esimaion using boh ime-series and cross-secional daa in he spiri of Chen and Sco (1993). The approach consiss of using hree arbirarily chosen yields, e.g. a swap rae and a IBOR bond yield o deermine he sae (r;;) using formulas in (8) and (14) and given a vecor of parameer values. The remaining yields, which, a any poin in ime, are also deerminisic funcions of he sae variables are hen over idenified. Following Chen and Sco (1993), we assume hese oher yields are priced or measured wih error. 27 Given he known ransiion densiy for he sae variables and some assumed disribuion for he error erms, he likelihood can be derived. 4.2 Resuls Esimaed parameers are repored in Table 1. They are reasonable and saisically significan excep for he risk-premia on cenral endency and 23 The marke is prey liquid, see Sun, Sundaresan and Wang (1993) for a discussion of he IBOR bond marke and comparisons of he Daasream-daa wih alernaive daa ses. Furher deails on our daa se can can be found in an appendix., 24 Wih hese assumpions (s) reduces o: (s; T ) = 1, e,s),(t (T, s) for small (empirically i is of he order of 10,4 ). We hus esimae he parameer using he approximaion: (s; T )=(T, s). If here is also a possibiliy for appreciaion in credi qualiy han is equal o he expeced depreciaion in credi qualiy ne of expeced appreciaion (as shown in he appendix). 25 Since we do no observe acual jumps in he jump process, we canno esimae he change of measure (i.e. of inensiy). In oher words, we can only esimae he risk-neural expeced credi-risk depreciaion. 26 In he gaussian framework, risk-premia have a nice inerpreaion. In our noaion, is he amoun which mus be added o he risk-neural long-erm mean o obain he long-erm mean of he shor rae under he hisorical measure, i.e = Q +. Similar inerpreaions apply for he r ; process. Excep of course, ha r denoes he amoun by which he whole pah of has o be shifed. Noice ha our definiion is slighly differen from he radiional risk-premium, because we find he adjusmen in erms of he change in long-erm means more inuiive. Of course, Girsanov s heorem gives he relaion beween he brownian moions and he radiional marke price of risk: dw P = dw Q, d. 27 Duffie and Singleon (1997) use a similar mehod. Alernaively, we could have used a Kalman-filer o avoid making an arbirary assumpion on which yields are priced wihou errors. 11

14 Table 1: Parameer esimaes resuling from he Maximum ikelihood described in secion 4.2. all parameers are presened for sae variables of he form dz = z( z, z)d + zdw P for z = r;;. Mean log-likelihood Parameers Esimaes Sd. err. r r r r r u insananeous credi spread ( and ). The long-run mean of he risk-free rae under he risk-neural measure is 8.78% per year. The risk-premia coefficiens, r ;, are negaive, implying ha erm premia are posiive and increasing wih mauriy. 28 I appears ha he level facor of he risk-free erm srucure has relaively low mean-reversion (10%) and volailiy (1%) compared o he second-facor, he long run endency, which appears o have high mean-reversion (50%) and high volailiy (8%). This is in line wih he resuls of Jegadeesh and Pennacchi (1996) who inerpre he cenral endency as a proxy for a long run ineres rae arge, hus reflecing expecaion abou fuure inflaion raes. On he oher hand, we find a significan negaive correlaion beween he long-run endency and he shor-erm rae. 29 The long-run mean of he insananeous credi spread for op-raed corporae firms is 38 basis poins under he risk-neural measure. The insananeous credi spread is quie volaile (1:3%) and has a srong level of mean-reversion (1:42). The credi-deerioraion parameer is esimaed around 5 bp. I is very significan boh saisically and economically. Recall ha is he expeced depreciaion rae in credi qualiy. 30 Our findings hus imply ha he spread beween IBOR par-bond yields and swap raes is consisen wih op-qualiy IBOR issuers experiencing, on average, a depreciaion in credi qualiy of 5 basis poins per year under he risk-neural measure. This is economically significan and implies an increasing erm srucure of credi spreads for op-raed IBOR issuers. The correlaion beween movemens in he insananeous risk-free rae and credi spread is negaive (,0:27) implying ha he credi spread ends o decrease when he risk-free rae rises. 31 Ineresingly, he correlaion beween he long-run endency of he reasury erm srucure and he credi-risk process 28 Term premia are defined as he expeced reurn on a risk-free bond in excess of he insananeous risk-free rae. They are equal o, r r B r (; T ), C(; T ). 29 Jegadeesh and Pennacchi are unable o precisely esimae ha correlaion, bu hey propose wo possible inerpreaions depending on he sign of he correlaion. We refer he reader o heir discussion, p Unforunaely, as in Duffee (1999), we canno disenangle he probabiliy of downgrading from he jump size in he level of he insananeous credi spread. In principle, if we had ime-series daa on individual credirisky bond prices, our model would allow o esimae boh parameers separaely. Here, since for comparison wih generic swap raes we use only generic IBOR yields a conrac iniiaion, we have no observaion of acual credi-depreciaion evens. I would be ineresing o analyze individual corporae-bond daa, as in Duffee (1999) for example, using our model of corporae bonds. 31 This is also consisen wih he recen resuls in Duffee (1998) and Duffee (1999). 12

15 is posiive. There are also macro-economic explanaions for he correlaion beween ineres raes and he credi spread. For example, he Treasury curve flaens in response o a slow-down in economic aciviy which should ranslae ino higher spreads o compensae for credi risk. Par of he laer effec may acually be capured in our model by he correlaion beween he shor rae and he credi spread. I is ineresing o assess he qualiy of he esimaion by looking a he properies of he error erms for he various swap, IBOR and Treasury raes, wih mauriies ranging from 0.5 o 10 years. Table 2: Mean and sandard deviaion of he condiional errors ( i) in bp resuling from he Maximum ikelihood esimaion described in secion 4.2. Noice ha he 1-year and 5-year Treasury and 1-year IBOR are fied perfecly because hey are chosen for inversion. Mauriy 0.5 years 1 year 2 years 3 years 4 years 5 years 7 years 10 years average mean (Treasury yields) N.A R.M.S.E N.A mean (Swap raes) N.A. N.A R.M.S.E N.A. N.A mean (IBOR raes) N.A. N.A. 0.1 R.M.S.E N.A. N.A. 8.1 The error erms (u i ) are srongly auocorrelaed 32 ( u = 0:92), bu he average and roo mean square errors (R.M.S.E.) of he condiional error erms ( i ) are quie low, as can be seen in Table 2. Depending on he mauriy, he mean condiional error ranges from -0.7 bp (basis poin) o +.5 bp across all mauriies and all raes. The R.M.S.E is less han 9 bp for all raes and mauriies. Noice ha he R.M.S.E is less han 5 bp for swap and Treasury raes and slighly higher for IBOR raes, i.e he model does beer a capuring he dynamics of he swap and Treasury erm srucure. This may also indicae ha he dynamics of he downgrading process chosen for his applicaion is oo simple and could be improved upon Conclusion In his paper we sudy he erm srucure of he spread beween corporae bond yields and swap raes for op-qualiy counerparies. The swap erm srucure is widely used by bankers, invesors and borrowers in lieu of he corporae erm srucure as he basic ool for pricing corporae asses and liabiliies as well as all kinds of financial asses. This pracice originaes because swap raes are coninuously quoed (and raded) for a wide range of mauriies and herefore more readily updaed han corporae yields. Ye, i is jusified only if he swap erm srucure ruly reflecs he cos of financing for a op-raed corporae issuer for he various mauriies. Indeed, we have shown ha realisic modeling of defaul risk leads o a heoreical difference beween he wo curves (we call i he IBOR-swap spread ). 34 Our model is consisen wih he empirical fac ha IBOR-qualiy bond yields are in general higher han swap raes for similar mauriies. Our wo key assumpions are (1) swaps carry less credi risk han corporae bonds, (2) he credi qualiy of op-raed issuers may deeriorae over he life of he conrac and in paricular differ from ha of a coninuously updaed refreshed credi qualiy index. 32 Chen and Sco (1993) and Duffie and Singleon (1997) find similar resuls. 33 In an earlier version we also looked a uncondiional fiing errors and he volailiy of he model implied spread. Resuls show a good fi of he model. 34 Alhough none of he exan heories specifically sudy he spread beween corporae bond yields and swap raes - hey focus on he Treasury-Swap spread - hey imply ha hese wo erm srucures should be idenical. 13

16 Ineresingly, our resuls show ha, he firs assumpion is no sufficien o explain a posiive IBORswap spread. IBOR bond yields should be mosly below (no above) swap raes if swaps are free of defaul risk while IBOR bonds carry defaul risk, and if all counerparies are sure o mainain heir credi qualiy over he life of he conracs. Our second assumpion is hus crucial o explain he observed posiive IBOR-swap spread. Because swap paymens are indexed on he 6-monh IBOR rae, a coninuously updaed, refreshed credi qualiy rae, we argue he IBOR-swap spread capures he expeced credi-qualiy deerioraion of a op-raed credi-qualiy issuer. We provide an explici model of he difference beween a refreshed credi-qualiy erm srucure and an acual op-raed crediqualiy erm srucure ha includes he possible jumps in credi qualiy and derive he swap rae in his framework. Our empirical resuls show he exisence of an economically and saisically significan expeced credi-qualiy deerioraion for op-raed IBOR-bond issuers. There are several ways in which our work could be exended, 35 bu we believe ha our analysis highlighs an imporan dimension in swap pricing ha has been negleced so far in he academic lieraure. 35 For example, we could: use differen processes for he sae variables, inroduce a sochasic inensiy for credi deerioraion, model he fac ha swap conracs carry some defaul risk (alhough less han bonds), add a liquidiy convenience yield or oher facors in he Treasury marke. 14

17 References [1] R. Chen and. Sco. Maximum likelihood esimaion for a mulifacor equilibrium model of he erm srucure of ineres raes. Journal of Fixed Income, december,vol.3 no 3:14 31, [2] I. Cooper and A. Mello. The defaul risk on swaps. Journal of Finance, 46: , [3] D. Cossin and H. Piroe. Swap credi risk: An empirical invesigaion on ransacion daa. HEC- IBFM, Universiy of ausanne, May [4] S. R. Das and S. Foresi. Exac soluions for bond and opion prices wih sysemaic jump risk. The Review of Derivaives Research, v1 n1:7 24, [5] G. R. Duffee. The relaion beween reasury yields and corporae bond yield spreads. Journal of Finance, 53: , [6] G. R. Duffee. Esimaing he price of defaul risk. The Review of Financial Sudies, 12: , [7] D. Duffie and M. Huang. Swap raes and credi qualiy. Journal of Finance, 51: , [8] D. Duffie and R. Kan. A yield-facor model of ineres raes. Mahemaical Finance, 6: , [9] D. Duffie and K. Singleon. An economeric model of he erm srucure of ineres-rae swap yields. Journal of Finance, II.no.4: , [10] D. Duffie and K. Singleon. Modeling erm srucures of defaulable bonds. The Review of Financial Sudies, 12 nb. 4: , [11] J. Fons. Using defaul raes o model he erm srucure of credi risk. Financial Analyss Journal, Sep.-Oc.:25 32, [12] M. Grinbla. An analyical soluion for ineres rae swap spreads. UCA Working Paper, [13] R. A. Jarrow, D. ando, and S. Turnbull. A markov model for he erm srucure of credi spreads. The Review of Financial Sudies, 10: , [14] N. Jegadeesh and G. G. Pennacchi. The behavior of ineres raes implied by he erm srucure of eurodollar fuures. Journal of Money, Credi and Banking, 28: , [15] T. C. angeieg. A mulivariae model of he erm srucure. Journal of Finance, 35:71 97, [16] R. izenberger. Swaps: Plain and fanciful. Journal of Finance, july, [17] R. C. Meron. On he pricing of corporae deb: The risk srucure of ineres raes. Journal of Finance, 29: , [18] O. Sarig and A. Warga. Some empirical esimaes of he risk srucure of ineres raes. Journal of Finance, 44: , [19] B. Solnik. Swap pricing and defaul risk. Journal of Inernaional Financial managemen and Accouning, 2:79 91, [20] E. Sorensen and T. Bollier. Pricing swap defaul risk. Financial Analyss Journal, May/June:23 33, [21] T. Sun, S. Sundaresan, and C. Wang. Ineres rae swaps: An empirical invesigaion. Journal of Financial Economics, 34:77 99,

18 A The formulas This appendix gives he differen formulas used in he ex. All he derivaions, proofs and furher deails abou he empirical analysis can be found in an appendix available a hp:// The risk-free discoun bond price P () = e Ar(;T ),Br(;T )r(),c(;t )() (A.1) A r (; T ) =, B r (; T )= Z T Z T Z T The risky discoun bond price P () =EQ where by sandard argumens: C(v; T ) (v)(v)dv Z T C(u; T )B r (u; T ) r (u) (u) r du r (; s)ds C(; T )= Z T he, R T (r(s)+(s))ds i E Q B r (u; T ) 2 2 r (u)du c(; s)ds c(; s) = he, R T Z s Z T C(v; T ) 2 2 dv + r (u; s) (; u) r (u)du R s i (2+(1,2) (u;s))dn (u)ds (A.2) (A.3) E Q he, R T (r(s)+(s))ds i = P ()e A (;T ),B(;T )() (A.4) A (; T ) =, B (; T ) = Z T 1=2 Z T Z T, B (u; T ) (u), r (u) r (u)b r (u; T ), r (u) (u)c(u; T ) du + B (u; T ) 2 2 (u)du (; s)ds (; s) =e R s, (u)du And as proven in an appendix available a he UR address quoed above, we have: i E Q he, R T R s ( 2 (u)+( 1 (u), 2 (u)) (u;s))dn (u)ds =E Q he R, T (1,e, 2 (u)(t,u),( 1 (u), 2 (u))b (u;t ) ) (u)du Using (A.3),(A.4) and (A.5), we obain he risky discoun bond prices given in 8. The swap rae formula Some calculaions show ha: P i=2n i=1 1+Y () = S P :5(i,1) ()P :5(i,1) ()=P :5i Where in he above, we have defined: ln C(; i,1 ; i ) = R i () e P i=2n i=1 P :5i () i,1 i,1 (s;i)ds C(; i,1 ; i ) Z i,1 B (u; i,1 )(B r (u; i ), B r (u; i,1 )) r r (u) (u)du + Z i,1 Z i,1 B (u; i,1 )(C(u; i ), C(u; i,1 )) (u) (u)du + B (u; i )(B (u; i ), B (u; i,1 )) (u) 2 du Afer some rearranging and algebra, we obain equaion 14 in he ex wih: ln C 0 = Z i i,1 Z i Z i,1 i,1 (s; i )ds, (s; i )ds, (s; i,1 )ds (A.5) (A.6) (A.7) (A.8) i 16