The intersection of market and credit risk q

Transcription

1 Journal of Banking & Finance 24 (2000) 271±299 The inersecion of marke and credi risk q Rober A. Jarrow a,1, Suar M. Turnbull b, * a Johnson Graduae School of Managemen, Cornell Universiy, Ihaca, New York, USA b Canadian Imperial Banck of Commerce, Global Analyics, Marke Risk Managemen Division, BCE Place, Level 11, 161 Bay Sree, Torono, On., Canada M5J 2S8 Absrac Economic heory ells us ha marke and credi risks are inrinsically relaed o each oher and no separable. We describe he wo main approaches o pricing credi risky insrumens: he srucural approach and he reduced form approach. I is argued ha he sandard approaches o credi risk managemen ± CrediMerics, CrediRisk+ and KMV ± are of limied value when applied o porfolios of ineres rae sensiive insrumens and in measuring marke and credi risk. Empirically reurns on high yield bonds have a higher correlaion wih equiy index reurns and a lower correlaion wih Treasury bond index reurns han do low yield bonds. Also, macro economic variables appear o in uence he aggregae rae of business failures. The CrediMerics, CrediRisk+ and KMV mehodologies canno reproduce hese empirical observaions given heir consan ineres rae assumpion. However, we can incorporae hese empirical observaions ino he reduced form of Jarrow and Turnbull (1995b). Drawing he analogy. Risk 5, 63±70 model. Here defaul probabiliies are correlaed due o heir dependence on common economic facors. Defaul risk and recovery rae uncerainy may no be he sole deerminans of he credi spread. We show how o incorporae a convenience yield as one of he deerminans of he credi spread. For credi risk managemen, he ime horizon is ypically one year or longer. This has wo imporan implicaions, since he sandard approximaions do no apply over a one q The views expressed in his paper are hose of he auhors and do no necessarily re ec he posiion of he Canadian Imperial Bank of Commerce. * Corresponding auhor. Tel.: ; fax: address: urnbus@cibc.ca (S.M. Turnbull). 1 Tel.: /00/$ - see fron maer Ó 2000 Elsevier Science B.V. All righs reserved. PII: S (99)

2 272 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271±299 year horizon. Firs, we mus use pricing models for risk managemen. Some praciioners have aken a di eren approach han academics in he pricing of credi risky bonds. In he even of defaul, a bond holder is legally eniled o accrued ineres plus principal. We discuss he implicaions of his fac for pricing. Second, i is necessary o keep rack of wo probabiliy measures: he maringale probabiliy for pricing and he naural probabiliy for value-a-risk. We discuss he bene s of keeping rack of hese wo measures. Ó 2000 Elsevier Science B.V. All righs reserved. JEL classi caion: G28; G33; G2 Keywords: Credi risk modeling; Pricing; Defaul probabiliies 1. Inroducion In he curren regulaory environmen, he BIS (1996) requiremens for speci c risk specify ha ``concenraion risk'', ``spread risk'', ``downgrade risk'' and ``defaul risk'' mus be ``appropriaely'' capured. The principle focus of he recen Federal Reserve Sysems Task Force Repor (1998) on Inernal Credi Risk Models is he allocaion of economic capial for credi risk, which is assumed o be separable from oher risks such as marke risk. Economic heory ells us ha marke and credi risk are inrinsically relaed o each oher and, more imporanly, hey are no separable. If he marke value of he rmõs asses unexpecedly changes ± generaing marke risk ± his a ecs he probabiliy of defaul ± generaing credi risk. Conversely, if he probabiliy of defaul unexpecedly changes ± generaing credi risk ± his a ecs he marke value of he rm ± generaing marke risk. The lack of separabiliy beween marke and credi risk a ecs he deerminaion of economic capial, which is of cenral imporance o regulaors. I also a ecs he risk adjused reurn on capial used in measuring he performance of di eren groups wihin a bank. 2 Is omission is a serious limiaion of he exising approaches o quanifying credi risk. The modern approach o defaul risk and he valuaion of coningen claims, such as deb, sars wih he work of Meron (1974). Since hen, MeronÕs model, ermed he srucural approach, has been exended in many di eren ways. Unforunaely, implemening he srucural approach faces signi can pracical di culies due o he lack of observable marke daa on he rmõs value. To circumven hese di culies, Jarrow and Turnbull (1995a, b) infer he condiional maringale probabiliies of defaul from he erm srucure of credi spreads. In he Jarrow±Turnbull approach, ermed he reduced form approach, 2 For an inroducion o risk adjused reurn on capial, see Crouhy e al. (1999).

3 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271± marke and credi risk are inherenly iner-relaed. These wo approaches are described in Secion 2. CrediMerics, CrediRisk+ and KMV have become he sandard mehodologies for credi risk managemen. The CrediMerics and KMV mehodologies are based on he srucural approach, and he CrediRisk+ mehodology originaes from an acuarial approach o moraliy. The KMV mehodology has many advanages. Firs, by relying on he marke value of equiy o esimae he rmõs volailiy, i incorporaes marke informaion on defaul probabiliies. Second, he graph relaing he disance o defaul o he observed defaul frequency implies ha he esimaes are less dependen on he underlying disribuional assumpions. There are also a number of disadvanages. Many of he basic inpus o he KMV model ± he value of he rm, he volailiy and he expeced value of he rae of reurn on he rmõs asses ± canno be direcly observed. Implici esimaion echniques mus be used and here is no way o check he accuracy of he esimaes. Second, ineres raes are assumed o be deerminisic. While his assumpion probably has lile e ec on he esimaed defaul probabiliy over a one year horizon, i limis he usefulness of he KMV mehodology when applied o loans and oher ineres rae sensiive insrumens. Third, an implicaion of he KMV opion model is ha as he mauriy of a credi risky bond ends o zero, he credi spread also ends o zero. Empirically, we do no observe his implicaion. Fourh, hisorical daa are used o deermine he expeced defaul frequency and consequenly here is he implici assumpion of saionariy. This assumpion is probably no valid. For example, in a recession, he rue curve may shif upwards implying ha for a given disance o defaul, he expeced defaul frequency has increased. Consequenly, he KMV mehodology underesimaes he rue probabiliy of defaul. The reverse occurs if he economy is experiencing srong economic growh. Finally, an ad hoc and quesionable liabiliy srucure for a rm is used in order o apply he opion heory. CrediMerics represens one of he rs publicly available aemps using probabiliy ransiion marices o develop a porfolio credi risk managemen framework ha measures he marginal impac of individual bonds on he risk and reurn of he porfolio. The CrediMerics mehodology has a number of limiaions. Firs, i considers only credi evens because he erm srucure of defaul free ineres raes is assumed o be xed. CrediMerics assumes no marke risk over a speci ed period. Alhough his is reasonable for oaing rae and shor daed noes, i is less reasonable for zero-coupon bonds, and inaccurae for CLOs, CMOs, and derivaive ransacions. Second, he Credi- Merics defaul probabiliies do no depend upon he sae of he economy. This is inconsisen wih he empirical evidence and wih curren credi pracices. Third, he correlaion beween asse reurns is assumed o equal he correlaion beween equiy reurns. This is a crude approximaion given

4 274 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271±299 uncerain bond reurns. The CrediMerics oupus are sensiive o his assumpion. A key di culy in he srucural-based approaches of KMV and Credi- Merics is ha hey mus esimae he correlaion beween he raes of reurn on asses using equiy reurns, as asse reurns are unobservable. Iniial resuls sugges ha he credi VARs produced by hese mehodologies are sensiive o he correlaion coe ciens on asse reurns and ha small errors are imporan. 3 Unforunaely, because asse reurns canno be observed, here is no direc way o check he accuracy of hese mehodologies. The CrediRisk+ mehodology has some advanages. Firs, CrediRisk+ has closed form expressions for he probabiliy disribuion of porfolio loan losses. Thus, he mehodology does no require simulaion and compuaion is relaively quick. Second, he mehodology requires minimal daa inpus of each loan: he probabiliy of defaul and he loss given defaul. No informaion is required abou he erm srucure of ineres raes or probabiliy ransiion marices. However, here are a number of disadvanages. Firs, CrediRisk+ ignores he sochasic erm srucure of ineres raes ha a ec credi exposure over ime. Exposures in CrediRisk+ are predeermined consans. The problems wih ignoring ineres rae risk made in he previous secion on CrediMerics are also perinen here. Second, even in is mos general form where he probabiliy of defaul depends upon several sochasic facors, no aemp is made o relae hese facors o how exposure changes. Third, he CrediRisk+ mehodology ignores non-linear producs such as opions, or even foreign currency swaps. Praciioners and regulaors ofen calculae VAR measures for credi and marke risk separaely and hen add he wo numbers ogeher. This is jusi ed by arguing ha i is di cul o esimae he correlaion beween marke and credi risk. Therefore, o be conservaive assume perfec correlaion, compue he separae VARs and hen add. This argumen is simple and unsaisfacory. I is no clear wha is mean by he saemen ha marke risk and credi risk are perfecly correlaed. There is no one bu many facors ha a ec marke risk exposure, he probabiliy of defaul and he recovery rae. These facors have di eren correlaions, which may be posiive or negaive. If he addiive mehodology suggesed by regulaors is conservaive, how conservaive? Risk capial under he BIS 1988 Accord was iself viewed as conservaive. Excessive capial may be inappropriaely required. By no having a model ha explicily incorporaes he e ecs of credi risk upon price, i is no clear ha marke risk iself is being correcly esimaed. For example, if he even of defaul is modeled by a jump process and defauls are correlaed, hen i is well known 3 See Crouhy and Mark (1998).

5 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271± ha he sandard form of he capial asse pricing model used for risk managemen is mis-speci ed. 4 Anoher criicism voiced by regulaors is ha we do no have enough daa o es credi models. ``A credi even (read defaul) is a rare even. Therefore we need daa exending over many years. These daa do no exis and herefore we should no allow credi models o be used for risk managemen.'' 5 This is a narrow perspecive. For markes where here is su cien daa o consruc erm srucures of credi spreads, we can es credi models such as he reduced form model described in Secion 4, using he same crieria as for esing marke risk models. Since he esing procedures for marke risk are well acceped, his nulli es his criicism raised by regulaors. We brie y review he empirical research examining he deerminans of credi spreads in Secion 3. I is empirically observed ha reurns on high yield bonds have a higher correlaion wih equiy index reurns and a lower correlaion wih Treasury bond index reurns han do low yield bonds. The KMV and CrediMerics mehodologies are inconsisen wih hese empirical observaions due o heir assumpion of consan ineres raes. Alman (1983/1990) and Wilson (1997a, b) show ha macro-economic variables a ec he aggregae number of business failure. In Secion 4 we show how o incorporae hese empirical ndings ino he reduced form model of Jarrow and Turnbull. This is done by modeling he defaul process as a muli-facor Cox process; ha is, he inensiy funcion is assumed o depend upon di eren sae variables. This srucure faciliaes using he volailiy of credi spreads o deermine he facor inpus. In a Cox process, defaul probabiliies are correlaed due o heir dependence upon he same economic facors. Because defaul risk and an uncerain recovery rae may no be he sole deerminans of he credi spread, we show how o incorporae a convenience yield as an addiional deerminan. This incorporaes a ype of liquidiy risk ino he esimaion procedure. Anoher issue relaing o credi risk in VAR compuaions is he selecion of he ime horizon. For marke risk managemen in he BIS 1988 Accord and he 1996 Amendmen, ime horizons are ypically quie shor ± 10 days ± allowing he use of dela±gamma±hea-approximaions. For credi risk managemen ime horizons are ypically much longer han 10 days. A liquidaion horizon of one year is quie common. This has wo imporan implicaions. Firs, i implies ha he pricing approximaions used for marke risk managemen are inadequae. I is necessary o employ 4 See Jarrow and Rosenfeld (1984). 5 This view is repeaed in he recen Basle repor: ``Credi Risk Modelling'' (1998).

6 276 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271±299 exac valuaion models because second order Taylor series expansions leave oo much error. In he academic lieraure i is ofen assumed ha he recovery value of a bond holderõs claim is proporional o he value of he bond jus prior o defaul. This is a convenien mahemaical assumpion. Cours, a leas in he Unied Saes, recognize ha bond holders can claim accrued ineres plus he face value of he bond in he even of defaul. This is a di eren recovery rae srucure. The legal approach is ofen preferred by indusry paricipans. In Secion 4 we show how o exend he exising credi risk models o incorporae hese di eren recovery rae assumpions. The second issue in credi risk model implemenaion is ha i is necessary o keep rack of wo disinc probabiliy measures. One is he naural or empirical measure. For pricing derivaive securiies, his naural probabiliy measure is changed o he maringale measure ( he so-called ``risk-neural'' disribuion). For risk managemen i is necessary o use boh disribuions. The maringale disribuion is necessary o value he insrumens in he porfolio. The naural probabiliy disribuion is necessary o calculae value-a-risk. We clarify his disincion in he ex. We also show ha we can infer he markeõs assessmen of he probabiliy of defaul under he naural measure. This provides a check on he esimaes generaed by MoodyÕs, Sandard and PoorÕs and KMV. A summary is provided in Secion Pricing credi risky insrumens This secion describes he wo approaches o credi risk modeling ± he srucural and reduced form approaches. The rs approach ± see Meron (1974) ± relaes defaul o he underlying asses of he rm. This approach is ermed he srucural approach. The second approach ± see Jarrow and Turnbull (1995a,b) ± prices credi derivaives o he observable erm srucures of ineres raes for he di eren credi classes. This approach is ermed he reduced form approach Srucural approach The srucural approach is bes exempli ed by Meron (1974, 1977), who considers a rm wih a simple capial srucure. The rm issues one ype of deb ± a zero-coupon bond wih a face value F and mauriy T. A mauriy, if he value of he rmõs asses is greaer han he amoun owed o he deb holders ± he face amoun F ± hen he equiy holders pay o he deb holders and reain he rm. If he value of he rmõs asses is less han he face value, he equiy holders defaul on heir obligaions. There are no coss associaed wih defaul

7 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271± and he absolue prioriy rule is obeyed. In his case, deb holders ake over he rm and he value of equiy is zero, assuming limied liabiliy. 6 In his simple framework, Meron shows ha he value of risky deb, m 1 ; T, is given by m 1 ; T ˆ F B ; T p V Š; 2:1 where B ; T is he ime value of a zero-coupon bond ha pays one dollar for sure a ime T ; V is he ime value of he rmõs asses, and p V Š is he value of a European pu opion 7 on he asses of he rm ha maures a ime T wih a srike price of F. To derive an explici valuaion formula, Meron imposed a number of addiional assumpions. Firs, he erm srucure of ineres raes is deerminisic and a. Second, he probabiliy disribuion of he rmõs asses is described by a lognormal probabiliy disribuion. Third, he rm is assumed o pay no dividends over he life of he deb. In addiion, he sandard assumpions abou perfec capial markes apply. 8 The Meron model has a leas ve implicaions. Firs, when he pu opion is deep ou-of-he-money V F, he probabiliy of defaul is low and corporae deb rades as if i is defaul free. Second, if he pu opion rades inhe-money, he volailiy of he corporae deb is sensiive o he volailiy of he underlying asse. 9 Third, if he defaul free ineres rae increases, he spread associaed wih corporae deb decreases. 10 Inuiively, if he defaul free spo ineres rae increases, keeping he value of he rm consan, he mean of he asseõs probabiliy disribuion increases and he probabiliy of defaul declines. As he marke value of he corporae deb increases, he yieldo-mauriy decreases, and he spread declines. The magniude of his change is larger he higher he yield on he deb. Fourh, marke and credi risk are no separable. To see his, suppose ha he value of he rmõs asses unexpecedly decreases, giving rise o marke risk. The decrease in he asseõs value increases he probabiliy of defaul, giving rise o credi risk. The converse is also rue. This ineracion of marke and credi risk is discussed in Crouhy e al. (1998). Fifh, as he mauriy of he zero-coupon bond ends o zero, he credi spread also ends o zero. 6 See Halpern e al. (1980). 7 For an inroducion o he pricing of opions, see Jarrow and Turnbull (1996b). 8 These assumpions are described in deail in Jarrow and Turnbull (1996b, p. 34) 9 Using pu±call pariy, expression (2.1) can be wrien m 1 ; T ˆ V c V Š; where c V Š is he value of a European call opion wih srike price F and mauring a ime T. If V F hen c V Š is `small' and m 1 ; T is rading like unlevered equiy. 10 Le m 1 0; T FB 0; T exp S p T ; where S p denoes he spread. p Then os p =or ˆ V 0 =v 1 0; T N d 1 6 0; where d 1 fln V 0 =FB 0; T r 2 T =2g=r T ; N is he cumulaive normal disribuion funcion, and r is he free ineres rae.

8 278 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271±299 There are a leas four pracical limiaions o implemening he Meron model. Firs, o use he pricing formulae, i is necessary o know he marke value of he rmõs asses. This is rarely possible as he ypical rm has numerous complex deb conracs ousanding raded on an infrequen basis. Second, i is also necessary o esimae he reurn volailiy of he rmõs asses. Given ha marke prices canno be observed for he rmõs asses, he rae of reurn canno be measured and volailiies canno be compued. Third, mos corporaions have complex liabiliy srucures. In he Meron framework, i is necessary o simulaneously price all he di eren ypes of liabiliies senior o he corporae deb under consideraion. This generaes signi can compuaional di culies. 11 Fourh, defaul can only occur a he ime of a coupon and/or principal paymen. Bu in pracice, paymens o oher liabiliies oher han hose explicily modeled may rigger defaul. Nielson e al. (1993) and Longsa and Schwarz (1995a, b) ake an alernaive roue in an aemp o avoid some of hese pracical limiaions. In heir approach, capial srucure is assumed o be irrelevan. Bankrupcy can occur a any ime and i occurs when an idenical bu unlevered rmõs value his some exogenous boundary. In defaul he rmõs deb pays o some xed fracional amoun. Again he issue of measuring he reurn volailiy of he rmõs asses mus be addressed. 12 In order o faciliae he derivaion of ÔclosedÕ form soluions, ineres raes are assumed o follow an Ornsein±Uhlenbeck process. Unforunaely, Cahcar and El-Jahel (1998) demonsrae ha for long-erm bonds he assumpion of normally disribued ineres raes, implici in an Ornsein±Uhlenbeck process, can cause problems. Cahcar and El-Jahel assume a square roo process wih parameers suiably chosen o rule ou negaive raes. 13 However, hey impose an addiional assumpion which implies ha spreads are independen of changes in he underlying defaul free erm srucure, conrary o empirical observaion Reduced form approach One of he earlies examples of he reduced form approach is Jarrow and Turnbull (1995b). Jarrow and Turnbull (1995b) allocae rms o credi risk classes. 15 Defaul is modeled as a poin process. Over he inerval ; DŠ he 11 See Jones e al. (1984). 12 See Wei and Guo (1997) for an empirical comparison of he Meron and Longsa and Schwarz models. 13 Cahcar and El-Jahel formulae he model in erms of a Ôsignaling variable.õ They never idenify his variable and o er no hin of how o apply heir model in pracice. 14 Kim e al. (1993) assume a square roo process for he spo ineres rae ha is correlaed wih he reurn on asses. 15 See Lierman and Iben (1991).

9 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271± defaul probabiliy condiional upon no defaul prior o ime is approximaely k D where k is he inensiy (hazard) funcion. Using he erm srucure of credi spreads for each credi class, hey infer he expeced loss over ; DŠ, ha is he produc of he condiional probabiliy of defaul and he recovery rae under he equivalen maringale (he Ôrisk neuralõ) measure. In essence, hey use observable marke daa ± credi spreads ± o infer he markeõs assessmen of he bankrupcy process and hen price credi risk derivaives. In he simple numerical examples conained in Jarrow and Turnbull (1995a, b, 1996a,b), sochasic changes in he credi spread only occur if defaul occurs. To model he volailiy of credi spreads, a more deailed speci caion is required for he inensiy funcion and/or he recovery funcion. Das and Tufano (1996) keep he inensiy funcion deerminisic and assume ha he recovery rae is correlaed wih he defaul free spo rae. Das and Tufano assume ha he recovery rae depends upon sae variables in he economy and is subjec o idiosyncraic variaion. The ineres rae proxies he sae variable. Monkkonen (1997) generalizes he Das and Tufano model by allowing he probabiliy of defaul o depend upon he defaul free rae of ineres. He develops an e cien algorihm for inferring he maringale probabiliies of defaul. The formulaion in Jarrow and Turnbull (1995b) is quie general and allows for he inensiy (hazard) funcion o be an arbirary sochasic process. Lando (1994/1997) assumes ha he inensiy funcion depends upon di eren sae variables. This is referred o as a Cox process. Roughly speaking, a Cox process when condiioned on he sae variables acs like a Poisson process. Lando (1994/1997) derives a simple represenaion for he valuaion of credi risk derivaives. Lando derives hree resuls. Firs, consider a coningen claim ha pays some random amoun X a ime T provided defaul has no occurred, zero oherwise. The ime value of he coningen claim is E Q exp r s ds X 1 C > T ˆ 1 C > E Q exp r s k s ds X ; 2:2 where r is he insananeous spo defaul free rae of ineres, C denoes he random ime when defaul occurs and 1 C > is an indicaor funcion ha equals 1 if defaul has no occurred by ime, zero oherwise. The superscrip Q is used o denoe he equivalen maringale measure. Expression (2.2) represens he expeced discouned payo where he discoun rae r s k s is adjused for he defaul probabiliy. Similar expressions can be obained for alernaive payo srucures.

10 280 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271±299 Second, consider a securiy ha pays a cash ow Y s per uni ime a ime s provided defaul has no occurred, zero oherwise. The ime value of he securiy is Z s Y s 1 C > s exp r u du ds E Q ˆ 1 C > E Q Y s exp Z s r u k u du ds : 2:3 Third, consider a securiy ha pays Z C if defaul occurs a ime C, zero oherwise. The ime value of he securiy is Z C exp r s ds Z C E Q ˆ 1 C > E Q Z s k s exp Z s r u k u du ds : 2:4 The speci caion of he recovery rae process is an imporan componen in he reduced form approach. In he Jarrow and Turnbull (1995a, b) model, i is assumed ha if defaul occurs on, say, a zero-coupon bond, he bond holder will receive a known fracion of he bondõs face value a he mauriy dae. To deermine he presen value of he bond in he even of defaul, he defaul free erm srucure is used. Alernaively, Du e and Singleon (1998) assume ha in defaul he value of he bond is equal o some fracion of he bondõs value jus prior o defaul. This assumpion allows Du e and Singleon o derive an inuiively simple represenaion for he value of a risky bond. For example, he value of a zero-coupon risky bond paying a promised dollar a ime T is m ; T ˆ 1 C > E Q exp r s k s L s ds ; 2:5 where he loss funcion L 1 d and d is he recovery rae funcion. Hughson (1997) shows ha he same resul can be derived in he J±T framework. 16 Modeling he inensiy funcion as a Cox process allows us o model he empirical observaions of Du ee (1998), Das and Tufano (1996) and Shane (1994) ha he credi spread depends on boh he defaul free erm srucure and an equiy index. The work of Jarrow and Turnbull (1995a, b), Du e and Singleon (1998), Hughson (1997) and Lando (1994/1997) implies ha for many credi derivaives we need only model he expeced loss, ha is he produc of he inensiy funcion and he loss funcion. 16 This also implies ha we can inerpre he work of Ramaswamy and Sundaresan (1986) as an applicaion of his heory.

11 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271± For valuing credi derivaives whose payo s depend on credi raing changes, Jarrow e al. (1997) describe a simple model ha explicily incorporaes a rmõs credi raing as an indicaor of defaul. This model can also be used for risk managemen purposes as i is possible o price porfolios of corporae bonds and credi derivaives in a consisen fashion. Ineresingly, he CrediMerics mehodology described in Secion 4 of his paper can be viewed as a special case of he JLT model, where here is no ineres rae risk. 3. Empirical evidence There is considerable empirical evidence consisen wih changes in credi spreads and changes in defaul free ineres raes being negaively correlaed. Du ee (1998) s a regression of he form DSpread ˆ b 0 b 1 DY b 2 DTerm e using monhly corporae bond daa from he period January 1985 o March 1995, where Spread is he spread a ime for a bond mauring a ime T, DSpread he change in he spread from o 1 keeping mauriy T xed, DY denoes he change in he hree monh Treasury yield, Term denoes he difference beween he 30 year consan Treasury bond yield and he hree monh Treasury bill yield, DTerm denoes he change in Term over he period, o 1 and e denoes a zero mean uni variance random erm. The esimaed coe ciens, b 1 and b 2, are negaive and increase in absolue magniude as he credi qualiy decreases irrespecive of mauriy. Similar resuls are also repored by Das and Tufano (1996). 17 Longsa and Schwarz (1995a,b) using annual daa from 1977 o 1992 a regression of he form DSpread ˆ b 0 b 1 DYield b 2 I e ; where DYield denoes he change in he 30 year Treasury, I denoes he reurn on he appropriae equiy index and e denoes a zero mean uni variance random erm. For credi classes Aaa, Aa, A, and Baa indusrials, he esimaed coe ciens are negaive. 18 Irrespecive of mauriy, he coe ciens b 1 and b 2 increase in absolue magniude as he credi qualiy decreases. However, he 17 Das and Tufano used monhly daa for he period 1971±1991. I is no clear if hey lered heir daa o eliminae bonds wih opionaliy. 18 The esimaed negaive coe ciens are no surprising, given he work of Meron (1974). An increase in he Treasury bill rae increases he expeced rae of reurn on a rmõs asses, and hence lowers he probabiliy of defaul. This increases he price of he risky deb and lowers is yield. An increase in he index proxies for an increase in he values of he rmõs asses. This lowers he probabiliy of defaul and hence he yield on he risky deb.

12 282 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271±299 Longsa and Schwarz resuls mus be reaed cauiously as heir daa include bonds wih embedded opions. This cauion is jusi ed by he work of Du ee (1998) who shows ha his can have a major impac on regression resuls. Shane (1994) using monhly daa over he period 1982±1992 found ha reurns on high yield bonds have a higher correlaion wih he reurn on an equiy index han low yield bonds and a lower correlaion wih he reurn on a Treasury bond index han low yield bonds. I is no repored wheher Shane lered her daa o eliminae bonds wih embedded opions. Wilson (1997a, b) examined he e ecs of macro-economic variables ± GDP growh rae, unemploymen rae, long-erm ineres raes, foreign exchange raes and aggregae saving raes ± in esimaing defaul raes. While he R- squares are impressive, he explanaory imporance of he macro-economics variables is debaable. If an economic variable has explanaory power, hen a change in he variable should cause a change in he defaul rae, provided he explanaory variables are no co-inegraed. To examine his, an esimaion based on changes in variables is needed. Unforunaely, Wilson does he esimaion using only levels. Alman (1983/1990) uses rs order di erences, he explanaory variables being he percenage change in real GNP, percenage change in he money supply, percenage change in he Sandard & Poor index and he percenage change in new business formaion. Alman nds a negaive relaion beween changes in hese variables and changes in he aggregae number of business failures. No surprisingly, he repored R-squares are subsanially lower han hose repored in Wilson. All of hese sudies sugges ha credi spreads are a eced by common economic underlying in uences 19. We show in he nex secion how o incorporae hese empirical ndings using he reduced form model of Jarrow and Turnbull. 4. The reduced form model of Jarrow and Turnbull The CrediMerics, CrediRisk+ and KMV mehodologies do no consider boh marke and credi risk. These mehodologies assume ineres raes are consan and consequenly hey canno value derivaive producs ha are sensiive o ineres rae changes, such as bonds and swaps. In his secion we show how o incorporae boh marke and credi risk ino he reduced form model of Jarrow and Turnbull (1995a, b) in a fashion consisen wih he empirical ndings discussed in he las secion. Following Lando (1994/1997), we model he inensiy funcion as a muli-facor Cox process. One can use he 19 See Pedrosa and Roll (1998) for furher evidence.

13 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271± volailiy of credi spreads o esimae he sensiiviy of he inensiy funcion o hese di eren facors. We will also discuss he quesion of correlaion and is role in he Jarrow±Turnbull model. The ypical ime horizon used for credi risk models is one year. This is jusi ed on he basis of he ime necessary o liquidae a porfolio of credi risky insrumens. The relaively long ime horizon implies ha we canno use he approximaions employed in marke risk managemen where he ime horizon is ypically of he order of 10 days. Consequenly we need o use for risk managemen he same models ha are used for pricing. Here praciioners have gone a slighly di eren roue han academics. Du e and Singleon (1998) assume ha in he even of defaul an insrumenõs value is proporional o is value jus prior o defaul. In acualiy, cours in he Unied Saes recognize ha in he even of defaul, bond holders can claim accrued ineres plus he face value of he bond. This di eren recovery rae srucure is ofen used by praciioners in he pricing of credi sensiive insrumens. We examine is implicaions for he valuaion of coupon bonds. Defaul risk and recovery rae uncerainy may no be he sole deerminans of he credi spread. Liquidiy risk may also be an imporan componen. Praciioners, when applying reduced form models such as he Jarrow±Turnbull, ofen use LIBOR insead of he Treasury curve in an aemp o miigae such di culies. We show how o incorporae a convenience yield in he deerminaion of he credi spread. A second consequence of he longer ime horizon employed in credi risk managemen is he need o keep rack of wo probabiliy measures: he naural and maringale. For pricing derivaives, he maringale measure is used (he socalled risk-neural disribuion). For risk managemen i is necessary o use boh disribuions. The naural measure is used in he deerminaion of VAR. A he end of he speci ed ime horizon, i is necessary o value he insrumens in he porfolio and his again requires he use of he maringale disribuion Two facor model We know from he work of Alman (1983/1990) and Wilson (1997a, b) ha macro-economic facors have explanaory power in predicing he number of defauls. We also know ha high yield bonds have a higher correlaion wih he reurn on an equiy index and a lower correlaion wih he reurn on a Treasury bond index han do low yield bonds. One can incorporae hese correlaions ino he probabiliy of defaul k D over he inerval ; DŠ. To describe he dependence of he probabiliy of defaul on he sae of he economy, we use wo proxy variables: he spo ineres rae and he unexpeced change in he marke index. Changes in he defaul free spo ineres rae and he marke index are readily observable on a daily basis, unlike many macro-economic variables ha are only repored quarerly.

14 284 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271±299 Le I denoe a marke index such as he Sandard and Poor 500 sock index. Under he equivalen maringale measure Q i is assumed ha changes in he index are described by a geomeric Brownian moion di ˆ r d r I dw I ŠI ; 4:1 where r I is he reurn volailiy on he index and W I is a Brownian moion. Le y ˆ ln I Š so ha and dy ˆ r r 2 I =2 d r IdW I y y 0 ˆ Z 0 Z r s r 2 I =2 ds r I dw I s : For racabiliy, we assume ha he inensiy funcion is of he form k ˆ a 0 a 1 r br I W I ; 0 4:2 4:3 where a 1 and b are consans, and a 0 is a deerminisic funcion ha can be used o calibrae he model o he observed erm srucure. The coe cien a 1 measures he sensiiviy of he inensiy funcion o he level of ineres raes, and b measures he sensiiviy o he cumulaive unanicipaed changes in he marke index. The assumpion of normaliy allows he derivaion of closed form soluions, such as expression (4.5) below. One of he disadvanages of his assumpion is ha he inensiy funcion can be negaive. In laice-based models, his di culy can be avoided via he use of non-linear ransformaions ± see Jarrow and Turnbull (1997a). 20 We assume ha he insananeous defaul free forward raes are normally disribued: df ; T ˆ r 2 exp # T Šb ; T d rexp # T Š dw 1 4:4 under he equivalen maringale measure Q ± see Heah e al. (1992) ± where b ; T f1 exp # T Šg=#. If # ˆ 0; b ; T ˆ T. The parameer # is ofen referred o as a mean reversion or a volailiy reducion facor (see Jarrow and Turnbull (1996a, ch. 16) for a more deailed discussion). This assumpion implies ha he spo ineres rae is normally disribued. Under his assumpion, he value of a credi risky zero-coupon bond is given by 20 I is someimes argued ha when considering a long daed bond, one should replace he spo rae wih a long daed yield. To he exen ha he spo ineres rae measures he sae of he economy over he life of he bond, expression (4.3) is appropriae. In a muli-facor model of he erm srucure, as described in Heah e al. (1992), hen he spo ineres rae is no su cien. For many applicaions, however, a one facor model will su ce.

15 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271± m ; T ˆ 1 C > B ; T expf a 3 b ; T r f 0; Š b 1 T W I c 2 ; T g 4:5 where a 3 a 1 L; b 2 r 1 L, and L is he consan loss rae, c 2 ; T c 1 ; T 1 2 b2 1 c 1 ; T L T u 2 du a 2 b 1 q b u; T T u du; a 0 u du a 3 r 2 =2 b 0; 2 b ; T b 0; u 2 du a 3 f 0; u du r 2 =2 2a 3 a 2 3 b s; T 2 ds; and q is he correlaion coe cien beween changes in he index and he erm srucure. A proof is given in Appendix A. Expression (4.5) has an inuiive reformulaion. Le v ; T denoe he credi spread de ned implicily by he expression m ; T ˆ 1 C > B ; T exp v ; T T Š: Using expression (4.5), his implies ha v ; T T ˆ a 3 b ; T r f 0; Š b 1 T W I c 2 ; T : 4:6 We see ha changes in he level of ineres raes and unanicipaed changes in he marke index a ec he credi spread. The volailiy of he spread, ignoring he even of defaul, is given by n 1=2: r v ; T T ˆ a 2 3 b ; T 2 r 2 b 2 1 T 2 2a 3 b ; T b 1 T rqo 4:7 The credi spread can be used o esimae he parameers a 3 and b 1 in expression (4.6). Given hese parameers, he funcion fa 0 g can be used o calibrae he iniial erm srucure of credi spreads. Expression (4.6) can alernaively be wrien in he form v ; T T ˆ a 1 Ly T T b 1 T dw I u 0 a 0 u L du b 3 ; T ; where L is he consan loss rae, and B ; T exp y T T and

16 286 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271±299 b 3 ; T ˆ a 3 1 a 3 b 2 ; T 1=2 b 2 1 T 3 =3 a 3 qb 1 b u; T T u du: Leing Dv v 1; T v ; T ; we obain Dv ˆ a 1 L y T 1 1 y T Š bl DI I r D c 3 ; T ; where c 3 ; T ˆ b 3 1; T = T 1 b 3 ; T = T Š: This expression is similar in form o an expression used by Longsa and Schwarz (1995a, b). I can be used o faciliae esimaion of he modelõs parameers or esing he validiy of he model. This addresses one of he concerns raised in he recen Basle Commiee on Banking Supervision (1999) repor Correlaion The issue of correlaion is of cenral imporance in all he credi risk mehodologies. Two ypes of correlaion are ofen ideni ed: defaul correlaion and even correlaion. Defaul correlaion refers o rm defaul probabiliies being correlaed due o common facors in he economy. For example, defaul raes increase if he economy goes ino a recession (see Alman, 1983/ 1990; Wilson, 1997a,b). Even correlaion refers o how a rmõs defaul probabiliy is a eced by defaul of oher rms. This has been modeled by he use of indicaor funcions and copula funcions. 21 The di culy wih modeling even correlaions is ha hey are, in general, sae dependen. For example, consider an indusry where one of he major players defauls. Wheher his has a posiive or negaive e ec upon he remaining rms depends upon wheher defaul is caused by he economy being in recession or poor managemen. In he rs case, he even correlaion may be minimal. In he second case, he even correlaion may be signi can. The probabiliies of defaul may decrease for he remaining rms because he demise of a compeior allows hem o sell more producs. Alernaively, if a rm sells he majoriy of is oupu o he defauling rm hen his will have a derimenal e ec upon he surviving rm. 21 Copula funcions are described in Bowers e al. (1997). For a di eren approach see Du e and Singleon (1998).

17 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271± In he wo facor model described by expression (4.3), defaul probabiliies across obligors are correlaed due o heir dependence upon common facors. If he coe ciens a and b in expression (4.3) are idenically zero, hen he correlaion among defaul probabiliies is zero. This does no imply, however, ha he change in he values of credi risky bonds are independen. Their values will be relaed due o heir common dependence upon he underlying erm srucure of defaul free ineres raes. The e ecs of correlaion mus also be considered when esimaing he dollar cos of counerpary risk. 22 This cos is ignored by mos sandard pricing models Claims of bond holders The modeling of he recovery rae process is a crucial componen in any credi risk model. A common assumpion in he academic lieraure for he recovery rae, following Du e and Singleon (1997), is ha he value of, say, a zero-coupon bond in defaul is proporional o is value jus prior o defaul. An alernaive assumpion ofen used in indusry is based upon he legal claims of bond holders in defaul. Under his assumpion, he value of a zero-coupon bond in defaul is proporional o he implici accrued ineres. For coupon bonds, he bond holders in defaul is accrued ineres plus face value. We consider he implicaion of hese wo di eren assumpions for pricing risky zero-coupon and coupon bonds Risky zero-coupon bonds This secion values risky zero-coupon bonds under he wo di eren recovery rae assumpions. Proporional loss. Du e and Singleon (1997) assume ha if defaul occurs, he value of he zero-coupon bond is m ; T ˆ dm ; T ; 4:8 where m ; T denoes he value of he bond an insan before defaul, d is he recovery rae, and m ; T is he value of he bond given defaul. Following Lando (1994/1997), Du e and Singleon (1998) and Hughson (1997), he value of a risky zero-coupon bond is given by m 1 ; T ˆ 1 C > E Q exp r u k u L u du ; 4:9 where L u 1 d u denoes he proporional loss in he even of defaul. 22 Jarrow and Turnbull (1996a, pp. 577±579), show how o esimae he cos arising from counerpary risk.

18 288 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271±299 Legal claim approach 23. An alernaive approach consisen wih he legal claims of he bond holders assumes ha if defaul occurs, he bond holderõs claim is limied o he implici accrued ineres. Le he bond be issued a ime 0 and is value a he ime of issue be denoed by m 0. The implici ineres rae, r i, is de ned by m 0 ˆ 1 1 r i T 0 4:10a or r i ˆ : 4:10b T 0 m 0 In he even of defaul a ime C, he bond holderõs claim is m 0 1 r i C 0 Š. The payo o he zero-coupon bond considering defaul is 1 if C > T ; 4:11 dm 0 1 r i C 0 Š if C 6 T : The ime value of he zero-coupon bond is m 2 ; T ˆ E Q exp r s ds 1 C > T Z C m 0 E Q exp r s ds d 1 r 0 C 0 Š : 4:12 Using he resuls of Lando (1994/1997), as described in Secion 2.2, we can wrie he above expression as m 2 ; T ˆ 1 C > E Q exp r u k u du Z s m 1 C > m 0 E Q d 1 r i s 0 Šk s exp r u k u du ds : 4:13 The recovery rae process deermines he form of he zero-coupon bond price. This is imporan for boh pricing and esimaion. If he recovery rae is given by expression (4.8), hen expression (4.9) describes he bond price. If he recovery rae is given by expression (4.11), hen expression (4.13) describes he bond price. 23 The legal claims approach is used by a number of praciioners. This secion simply collecs ogeher wha seems o be common ÔsreeÕ knowledge.

19 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271± Credi risky coupon bonds This secion values a coupon bond under he wo di eren assumpions abou he recovery rae process in he even of defaul. Consider a risky bond ha promises o make coupon paymens {c j } a ime { j }, j ˆ 1;... ; n where n is he number of promised paymens. The principal, F, is paid a ime n. Le m c denoe he ime value of he bond, condiional upon no defaul. Proporional loss. Using expression (3.2) gives m 1c ˆ E Q ( X n jˆ1 c j exp Z j r u k u L u du Z n ) F exp r u k u L u du 4:14 ˆ Xn jˆ1 c j m 1 ; j F m 1 ; n : The usual value addiiviy resul holds. Legal claim approach. In he even of defaul, he bond holdersõ claim is limied o accrued ineres plus principal. The implici legal assumpion is ha bonds are rading a par value. If defaul occurs over he rs period, he payo is d Fr i C 0 F Š for < C 6 1 ; where 0 is he ime of he las coupon paymen, and r i he coupon rae. The rs erm inside he square brackes is he accrued ineres and he second erm is he principal. Condiional upon no defaul prior o ime j 1, if defaul occurs over he period j 1 ; j, he paymen o bond holders is f j C d Fr i C j 1 F Š for j 1 < C 6 j where jˆ1;... ; n: 4:15 The value of his claim a ime j 1 is m j j 1 ˆ E Q j 1 exp ˆ 1 C > j 1 E Q j 1 Z C " j 1 r u du f j C Z! # s f j s k s exp r u k u du ds : j 1 j 1 Z j 4:16

20 290 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271±299 The value of his claim a ime is Z j 1 v j ˆ E Q exp r x dx m j j 1 1 C > j 1 Z j 1 ˆ 1 C > E Q exp r x k x dx m j j 1 ( Z j 1 ˆ 1 C > E Q exp r x k x dx Z j Z! ) s f j s k s exp r u k u du ds j 1 j 1 " Z j Z s # ˆ 1 C > E Q f j s k s exp r u k u du ds : 4:17 j 1 Using he above resul, he value of he coupon bond is given by (" X n Z j m 2c ˆ E Q c j exp r u du jˆ1 ˆ 1 C > E Q F exp " 1 C > E Q X n jˆ1 Z n c j exp F exp " X n jˆ1 Z j r u du # 1 C > n Z j Z n ) Xn! r u k u du j 1 # r u k u du jˆ1 m j df 1 r i s j 1 k s j 1 Z s # exp r u k u du ds : 4:18 This resul is addiive, bu no he form of expression (4.14). This implies ha he sandard sripping procedures used o deermine he implied zero-coupon bonds do no apply Convenience yields on reasury securiies In he Jarrow±Turnbull model he credi spread is used o infer he defaul probabiliy under he equivalen maringale measure. Many facors, such as resricions on shor selling, illiquidiies, regulaory requiremens and axaion,

21 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271± may a ec he spread. Babbs (1991) and Grinbla (1994) argue ha a convenience yield parly explains he spread beween he Euro-dollar and Treasury erm srucure. This convenience yield is an implicaion of shor sale consrains on Treasury securiies ha occasionally exis ± see Cornell and Shapiro (1986), Du e (1996), and Chaerjea and Jarrow (1998). Following Jarrow and Turnbull (1997b), we show how o augmen he Jarrow±Turnbull model o include a convenience yield. Le b ; T denoe he ime price of a non-shorable zero-coupon Treasury bond ha maures a ime T. 24 The no-arbirage relaionship beween b ; T and a zero-coupon Treasury bond no subjec o shor sell resricions is b ; T P B ; T : 4:19 A sric inequaliy is possible if he shor selling consrain is binding. Le y ; T denoe he forward convenience yield. Using he forward convenience yield, he above expression can be wrien as b ; T exp y ; s ds ˆ B ; T ; 4:20 where y ; s P 0 for all s 6 T. Recall ha no arbirage beween B ; T and m ; T implies ha B ; T =A and m ; T =A are Q-maringales. Using expression (4.20), his implies ha b ; T exp y ; s ds A 4:21 is a Q maringale. Given exogenous speci caions for B ; T and y ; T under he measure Q, expression (4.21) deermines he arbirage free sochasic process for b ; T. B ; T can be modeled using sandard echniques (see Heah e al., 1992). To model y ; T, we rewrie expression (4.21). The spread beween a credi risky zero-coupon bond and a zero-coupon non-shorable Treasury bond is m ; T =b ; T ˆ m ; T =B ; T Šexp y ; s ds : 4:22 De ne Y ; T exp y ; s ds : 4:23 Y ; T has he properies of a zero-coupon bond and y ; T he properies of a non-negaive forward rae. Consequenly, fy ; T g can also be modeled along he lines described in Heah e al. (1992). 24 The erm ``non-shorable'' refers o a case where here are resricions on he amoun of securiies ha can be shored.

22 292 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271± Change of probabiliy measure In credi risk managemen, i is necessary o keep rack of wo probabiliy measures: he naural measure and he equivalen maringale measure. While his adds an exra layer of complexiy, i also generaes some ineresing bene s. From he use of bond spreads, we can infer he probabiliy of no defaul over a speci ed horizon T under he probabiliy measure Q: Pr Q C > T Š ˆ E Q exp k u du : 4:24 0 If we can esimae he price of risk for he underlying facors, we can change he probabiliy measure and esimae he probabiliy of no defaul under he naural measure Pr P C > T Š ˆ E P exp k u du : 4:25 0 This has an imporan pracical implicaion. I provides a mehod o check he esimaes of defaul probabiliies generaed eiher inernally, by credi raing agencies, or by oher commercial packages such as KMV. Some forms of credi derivaives are coningen upon credi evens, such as credi raing downgrades. To price such insrumens requires a model ha explicily incorporaes credi raing changes. The pricing of such derivaives is usually done using he equivalen maringale or risk-neural probabiliies. Jarrow e al. (1997) show how o incorporae credi raings ino he arbirage free pricing of credi risky derivaives. They show how o infer he maringale ransiion probabiliies given he ransiion probabiliies under he naural measure. The Jarrow±Lando±Turnbull model has been exended by Das and Tufano (1996) and Monkkonen (1997). Das and Tufano assume ha in he even of defaul he recovery rae is a random variable correlaed wih he defaul free rae of ineres. The independence assumpion beween he ransiion probabiliies and he defaul free rae of ineres is mainained. Monkkonen generalizes Das and Tufano by allowing he probabiliies of defaul o depend upon he defaul free rae of ineres. The work of Monkkonen can be generalized furher by modeling he ransiion probabiliies as Cox processes (see Lando, 1994/1997). The only di culy is ha of esimaing he ransiion marix coe ciens. 5. Summary Economic heory ells us ha marke and credi risk are relaed o each oher and no separable. This lack of separabiliy a ecs he deerminaion of

23 R.A. Jarrow, S.M. Turnbull / Journal of Banking & Finance 24 (2000) 271± economic capial. I a ecs he risk adjused reurn on capial used in measuring he performance of di eren groups wihin a bank, and i a ecs he calculaion of he value-a-risk, all of which are imporan o regulaors. Wih accrual accouning he only risk associaed wih a loan is defaul. Curren mehodologies such as CrediMerics, CrediRisk+, and KMV emphasize he accrual accouning perspecive and focus on only defaul risk. Ineres raes are assumed o be consan, implying ha hese mehodologies canno assess he risk associaed wih ineres rae derivaives. In conras, reduced form models, such as he Jarrow±Turnbull model, consider marke and credi risk. They can be calibraed using observable daa and consequenly incorporae marke informaion. They can be used for pricing and for risk managemen. 6. For furher reading The following references are also of ineres o he reader: Alman (1968, 1987, 1989, 1993, 1996); Alman and Kao (1992a,b); Wei (1995); Weiss (1990); Alman and Nammacher (1985); Amin and Jarrow (1991, 1992); Anderson and Sundaresan (1996); Asquih e al. (1994, 1989); Barclay and Smih (1995); Basle Commiee (1996); Bensoussan e al. (1994); Black and Cox (1976); Black e al. (1990); Blume e al. (1991); Cahcar and El-Jahel (1998); Chance (1990); Cooper and Mello (1990a,b); Cornell and Shapiro (1989); Cornell and Mello (1991) Credi Marics (1997); Crouhy and Galai (1997); Delienedis and Geske (1998); Duan (1994); Du ee (1997, 1999); Du e and Huang (1996); Du e and Singleon (1994a,b); Eberhar e al. (1990); Flesaker e al. (1994); Harrison and Pliska (1981); Helwege and Kleiman (1997); Ho and Lee (1986); Ho and Singer (1982, 1984); Hull and Whie (1996); Jacod and Shiryaev (1987); Jarrow and Madan (1995); Jarrow and Turnbull (1994, 1998); Alman and Bencivenga (1995); Johnson and Sulz (1987); Kijima (1998); Kim e al. (1993); Lando (1997, 1998); Leibowiz e al. (1995); Li (1998); Alman and Eberhar (1994); Madan and Unal (1994); Meron (1976); Musiela e al. (1993); Schonbucher (1998); Schwarz (1993, 1997, 1998); Shimko e al. (1993); Singleon (1997); Skinner (1994); Timan and Torous (1989); Wakeman (1996). Acknowledgemens We hank he Edior and wo anonymous referees and seminar paricipans a he ``CrediRisk Modelling and he Regulaory Implicaions'' conference organized by he Bank of England and he Financial Services Auhoriy, he Bank of Japan, he Federal Reserve Board of he Unied Saes, and he