BIS Working Papers No 179. An empirical evaluation of structural credit risk models. Monetary and Economic Department. by Nikola A Tarashev*

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1 BIS Working Papers No 179 An empirical evaluaion of srucural credi risk models by Nikola A Tarashev Moneary and Economic Deparmen July 2005

2 BIS Working Papers are wrien by members of he Moneary and Economic Deparmen of he Bank for Inernaional Selemens, and from ime o ime by oher economiss, and are published by he Bank. The views expressed in hem are hose of heir auhors and no necessarily he views of he BIS. Copies of publicaions are available from: Bank for Inernaional Selemens Press & Communicaions CH-4002 Basel, Swizerland publicaions@bis.org Fax: and This publicaion is available on he BIS websie ( Bank for Inernaional Selemens All righs reserved. Brief excerps may be reproduced or ranslaed provided he source is cied. ISSN (prin) ISSN (online)

3 An Empirical Evaluaion of Srucural Credi Risk Models Nikola A Tarashev Bank for Inernaional Selemens Absrac This paper evaluaes empirically he performance of six srucural credi risk models by comparing he probabiliies of defaul (PDs) hey deliver o ex pos defaul raes. In conras o previous sudies pursuing similar objecives, he paper employs firm-level daa and finds ha heory-based PDs end o mach closely he acual level of credi risk and o accoun for is ime pah. A he same ime, nonmodelled macro variables from he financial and real sides of he economy help o subsanially improve he forecass of defaul raes. The finding suggess ha heory-based PDs fail o fully reflec he dependence of credi risk on he business and credi cycles. Mos of he upbea conclusions regarding he performance of he PDs are due o models wih endogenous defaul. For heir par, frameworks ha assume exogenous defaul end o under-predic credi risk. Three borrower characerisics influence maerially he predicions of he models: he leverage raio; he defaul recovery rae; and he risk-free rae of reurn. Key words: probabiliy of defaul, credi risk models, Basel II, macroeconomic facors of credi risk JEL Classificaion Numbers: C52, G1, G3 I am indebed o Kosas Tsasaronis and Claudio Borio for heir encouragemen and for several insighful discussions on he opic. Jeff Amao, Ingo Fender, Michael Gordy, Hiroaka Hideshima, Ricardo Schechman, Richard Vasicek, Haibin Zhu and seminar paricipans a he BIS, Banque de France and he Workshop of he Basel Commiee s Research Task Force provided useful commens on earlier drafs of he paper. I also hank Dimirios Karampaos for valuable help wih he daa and Melanie Sykes for puing he Word documen ogeher. The views expressed in his paper are my own and do no necessarily reflec hose of he BIS. nikola.arashev@bis.org

4 Conens Inroducion The models Models wih an exogenous defaul boundary Models wih an endogenous defaul boundary The MKMV model Daa: sources, filering and descripive saisics Calibraion mehodology The model-implied PDs Firm-level daa and heoreical predicions of defaul raes Time variabiliy of PDs A comparison beween he exogenous defaul and endogenous defaul models A comparison beween he endogenous defaul models The impac of differen calibraions of he risk free rae Theory-based predicions of defaul raes over longer horizons Economic significance of he forecas errors in model-implied PDs Model-implied PDs and urning poins in he oulook of credi risk Models only regressions Macroeconomic variables and credi risk MKMV PDs as predicors of defaul raes...26 Conclusion...27 References...28 Figures and Tables...30 iii

5 Inroducion Predicors of credi (or defaul) risk, ie he risk ha a borrower does no fulfil is deb conrac, are of naural ineres o praciioners in he financial indusry as well as o regulaors. The accuracy of hese predicors is essenial for sound risk managemen and for supervisory evaluaion of he vulnerabiliy of lender insiuions. In an appreciaion of his, he new capial adequacy framework (Basel II) encourages he acive involvemen of banks in measuring he likelihood of defauls. The growing need for reliable measures of credi risk promps he quesion wheher hey can be obained from academic heoreical models. The finance lieraure has produced a variey of models ha aemp o measure defaul risk. In his paper, I consider he family of srucural models, which focus on he sochasic process of a corporae obligor s asses and posulae ha a defaul occurs when hese asses cross a hreshold value. The models can be divided ino an endogenous defaul and an exogenous defaul group. The frameworks in he former group le borrowers choose sraegically he iming of defaul. 1 In conras, he models in he laer group impose an ad hoc defaul rigger bu develop richer sochasic srucures ha capure empirical regulariies of credi markes. I focus on one componen of defaul risk, he probabiliy of defaul (PD), 2 and aemp o answer he quesion: How do srucural credi risk models fare when pu o he es of he daa? In general erms, my conclusions rely on comparisons beween firm-specific model-based PDs of corporae borrowers and he corresponding ex pos defaul raes. The exercise is carried ou in he conex of wo endogenous defaul models, hose developed in Leland and Tof (1996) (henceforh, LT) and Anderson, Sundaresan and Tychon (1996) (AST), and hree exogenous defaul frameworks, developed in Longsaff and Schwarz (1995) (LS), Collin- Dufresne and Goldsein (2001) (CDG) and Huang and Huang (2003) (HH). As benchmarks, I also consider PDs delivered by he model of he commercial service Moody s KMV (henceforh, MKMV). The laer framework is no publicly available bu is known o share key feaures of he academic models, o use proprieary informaion on credi oulooks and o be esimaed on he basis of hisorical defaul raes. One of he main conclusions of he paper is ha, in general, heory-based PDs rack closely ex pos defaul raes and do so for differen forecas horizons. The bes performers are he endogenous defaul models, which feaure virually unbiased forecass. In conras, he exogenous defaul frameworks end o underpredic defaul raes. Considering he enire sample, he bias is small under he exogenous defaul LS and HH models bu is quie pronounced under he CDG model. The 1 2 The erms debor, obligor and borrower are used inerchangeably hroughou he paper. Oher componens of defaul risk, such as loss given defaul and exposure a defaul, are no analysed in his paper. 1

6 MKMV PDs are found o be he highes across models and o generally overpredic defaul risk. Neverheless, owing o he shor available ime series of hese PDs, he resul remains inconclusive. The finding ha academic models closely mach he overall level of defaul raes sands in sharp conras o a conclusion of Leland (2002). Tha paper calibraes he LT and LS models o he represenaive borrower and concludes ha hey subsanially underesimae ex pos defaul raes a shor horizons, such as one year: he horizons ypically used in pracical applicaions. The sark differences beween he findings of his paper and he conclusions of Leland (2002) are due o he srongly non-linear relaionships beween inpus o he models (ie parameer values) and heir implicaions for PDs. The non-lineariies are such ha a high heory-based PD is much more sensiive o parameer changes han a low PD. This gives rise o he so-called Jensen inequaliy effec, whereby he average heoreical PD across borrowers (which is used here and is an unbiased esimaor of defaul raes if is underlying model is correc) is larger han he heoreical PD of he average (or represenaive) borrower (used in Leland (2002)). I also examine he economic significance of he errors in heoreical forecass of defaul raes. To do so, I focus on he foundaion IRB approach of Basel II, which provides a formula for mapping a PD ino minimum capial requiremens. 3 Using he IRB formula, I compare model-implied capial wih he opimal capial, which is based on he rue PDs revealed ex pos. Adoping perfec knowledge of credi risk as a benchmark, he exercise helps appreciae he economic coss of relying on srucural models for regulaory purposes. The resuls of his sylised exercise reveal a mixed message. A one exreme, he regulaory capial implied by he endogenous defaul models racks exremely closely he opimal capial level. In paricular, when filered hrough he IRB approach, he forecas errors of hese models represen a small fracion of he average level and volailiy of he capial requiremens under perfec knowledge of credi risk. A he oher exreme, he exogenous defaul models provide economically significan forecas errors and perform more poorly han even he simples, sandardised, approach of Basel II, which does no consider explicily PD esimaes bu relies exclusively on exernal credi raings. Finally, I analyse he capaciy of he srucural models o explain he evoluion of credi risk over ime. In paricular, I examine he significance of heoreical PDs as regressors of ex pos defaul raes. As conrol variables, I consider macroeconomic indicaors reflecing he real and financial sides of he economy. These variables are moivaed by Esrella and Hardouvelis (1991) and Smes and Tsasaronis (1997), who idenify predicors of economic aciviy, and by Borio and Lowe (2002), who idenify predicors of banking sysem disress. The resuls of he regression analysis indicae ha he srucural models reveal useful informaion regarding he ime paern of defaul raes. The message is sronger when he heoreical implicaions are based on a ime invarian esimae of he risk-free rae because he academic models end o 3 See Basel Commiee on Banking Supervision (2004). IRB sands for inernal raings based. 2

7 associae rises in ha rae wih lower credi risk, which is no suppored by he daa. In addiion, using boh endogenous and exogenous defaul models, as opposed o a single model from eiher group, improves subsanially he forecass of defaul raes. Neverheless, he macroeconomic conrols do add o and, in cerain asse classes even subsiue for, he informaional conen of he models. The finding reveals ha heoreical PDs fail o fully reflec he dependence of credi risk on he business and credi cycles. 4 A parallel analysis of he MKMV PDs conveys a similar message. Dispariies in he performance of differen academic models can be explained by he way hey handle wo key deb characerisics: he defaul-rigger value of asses and he dead-weigh cos incurred a he ime of defaul. Under he endogenous defaul models, hese characerisics are o be in line wih a marke-wide esimae of he defaul recovery rae as well as wih a number of oher borrower-specific feaures, he mos imporan of which urns ou o be firm leverage. 5 This leads o a srong dispersion of he wo deb characerisics across borrowers and, by he Jensen inequaliy effec, raises he heoreical predicions o levels ha closely mach he observed defaul raes. In addiion, he endogenous defaul models imply ha he defaul boundary increases in he defaul cos, which is in accord wih he empirical regulariy ha defaul raes and losses given defaul end o move in he same direcion. By conras, he exogenous defaul models sugges ad hoc values for boh he defaul boundary and he defaul cos. In such a conex, he wo deb characerisics are also calibraed o be consisen wih he defaul recovery rae bu depend lile on borrower-specific feaures. This underplays differences in he credi oulooks across firms, which leads o an underesimaion of defaul raes. Besides he aforemenioned Leland (2002), several oher aricles have also evaluaed he empirical performance of srucural credi-risk models bu from a differen perspecive. Huang and Huang (2003), for example, calibrae he models o observed defaul raes and hen sudy he implied credi risk premiums. In a more recen sudy, Eom, Helwege and Huang (2004) use daa on individual bonds in order o compare heir credi spreads o he predicions of srucural credi risk models. While hese aricles examine heoreical pricing implicaions, which are based on preference-weighed probabiliies ha reflec bond holders risk appeie, he analysis herein focuses on acual (or saisical) PDs. The res of he paper is organised as follows. In Secion 1, I presen he six models and highligh assumpions ha are expeced o have srong implicaions for he implied PDs. Then, in Secion 2, I describe he daa and, in Secion 3, ouline how hey are employed for he calibraion of he models. In Secion 4, I presen he heoreical PDs and explain differences among hem on he basis of he underlying modelling assumpions. In Secion 5, I consider he implicaions of he models for banks minimum capial requiremens. Finally, in Secion 6, I es he saisical significance of he PDs as explanaory variables of defaul raes. 4 5 Duffie and Wang (2004) reach a similar conclusion in heir analysis of non-srucural credi risk models. The defaul recovery rae equals he fracion of deb principal ha is recovered a he ime of defaul. 3

8 1. The models I examine six srucural credi risk models ha exend he analyic framework of Meron (1974). Tha framework focuses on he credi risk of an individual borrower (as opposed o porfolio credi risk), which defauls when is asses fall below a paricular hreshold. The deerminaion of he hreshold differs across he models considered below. Five of he models are developed in academic aricles and, according o he way hey ackle he decision o defaul, can bes be divided ino wo caegories. The models in he firs caegory adop an exogenous defaul-rigger value of asses. In conras, he models in he second caegory derive he decision o defaul endogenously, as par of he borrower s opimisaion problem. Thus, he endogenous defaul rigger is a funcion of borrower characerisics. The sixh model belongs o MKMV and no all of is feaures are publicly available. From wha is known, his model assumes ha he defaul boundary depends on he mauriy srucure of he deb insrumens issued by he obligor. The five academic models assume ha he asse value, V, evolves as follows: dv ( r + λ δ) d + σdw / V = (1) where r denoes he risk-free rae, λ is he asse risk premium, δ is he asse payou raio (reflecing, for example, dividend and coupon paymens), W is a Wiener process and σ is he insananeous asse volailiy. Some of he models allow he ineres rae or he risk premium o be ime varying. Given he process in (1), he PD over a paricular horizon equals he probabiliy ha he firs passage of V below he defaul rigger, academic models. 6 V, occurs wihin ha horizon. V is a consan in all bu one of he five In such a seup and for a given iniial value of asses, V 0, individual parameers affec he heoreical PD as follows: PD V > 0, PD σ > 0, PD r < 0, PD λ < 0, PD δ > 0 (2) The PD increases in he defaul rigger, V, and in he level of risk, capured by he asse volailiy, σ. The implicaions of he remaining hree parameers, which deermine he drif in he value of asses, are bes considered ogeher. Tigh credi condiions, caused by conracionary moneary policy ha raises he risk-free rae, r, and/or by sronger aversion o risk ha raises he risk premium, λ, seem o counerinuiively depress PDs. A he same ime, however, igh credi condiions would end o raise he payou raio, δ, which increases PDs. 4

9 Some models incorporae a second sochasic process: of he risk-free rae, he asse risk premium or he defaul boundary. In such models, he PD depends also on parameers ha define he second process and is co-movemen wih asse values. The discussion of hese parameers is deferred o he nex subsecion, which discusses model-specific issues, and o he secion on he calibraion mehodology. 1.1 Models wih an exogenous defaul boundary In he hree exogenous defaul models, he hreshold level of asses, V, is lef unspecified and is ypically chosen o be in accordance wih aggregae hisorical daa. In paricular, when he fracion of asses los in defaul is α and he face value of deb is P, equal an esimae of he deb recovery rae afer defaul. 7 ( ) V is se so ha he quaniy 1 α V In addiion, all hree models in his caegory assume ha deb is of infinie mauriy. The assumpion delivers analyic racabiliy bu makes i impossible o capure he empirical regulariy ha borrowers are less likely o defaul over a given horizon if hey are o repay he deb principal furher in he fuure. The hree models are disinguished by heir choice of a second sochasic process, which complemens he process of asses in (1). The second process is inroduced in order o allow he seups o capure sylised feaures of he daa ha have poenially imporan implicaions for credi risk. I ouline hese implicaions in he remainder of he subsecion. P The model of Longsaff and Schwarz (1995) In he LS model, he value of asses ineracs wih a sochasic risk-free rae of reurn. The correlaion beween he wo random variables is assumed o be negaive, which inends o capure he cooling effec of higher ineres raes on he macro economy. Specifically, equaion (1) is augmened by: dr = kr r r d + σrdw r (3) where r is he long-run risk-free rae of reurn, k r reflecs he speed of mean reversion, r insananeous volailiy of he risk-free rae of reurn and ( dw, dw ) σ < 0 corr. rv σ r is he Under such a specificaion, a change of r has an ambiguous impac on he PD. By expression (1), a higher ineres rae increases he deerminisic drif in he value of asses and, ceeris paribus, lowers he PD. Neverheless, since σ rv < 0, a higher r ends o be associaed wih a negaive shock o he 6 7 Two of he exogenous boundary models assume ha derive V as a ime-invarian funcion of consan parameers. If he model allows for ime variaion in V is a consan. In conras, he endogenous boundary models V, he above procedure would se he iniial value of he process. 5

10 value of asses, which raises he PD. The relaive imporance of he laer impac of he risk-free rae increases when asse volailiy, σ, is higher and/or he correlaion σ σ r rv σ is closer o The model of Collin-Dufresne and Goldsein (2001) Collin-Dufresne and Goldsein observe ha firms end o issue more (less) deb when heir asse value increases (decreases). This leads o mean reversion in he leverage raio (ie he raio of deb o asses) an empirical regulariy ha he LS model does no accoun for. The CDG model accommodaes he empirical regulariy, which implies ha he defaul rigger, V, moves in sep wih he value of asses, V. Under he mainained assumpion ha V is a consan fracion of deb: d ln V = k ( lnv lnv ν)d, k > 0, ν > 0 (4) l l The parameers ν and k l have direc implicaions for he heoreical PD. In paricular, ν > 0 implies ha, in he absence of (ransiory) shocks o asses, heir value would say above he defaul rigger: ie he firm would be inherenly solven. The closer is ν o zero, however, he sronger is he endency of V and V o converge o a common value. This increases defaul risk, ie he risk of V falling below V. For is par, a higher l k implies ha he raio ln( V ) V is more likely o say close o is long-run value ν. Since he laer value is assumed o be posiive, an increase in k l lowers he PD The model of Huang and Huang (2003) There is empirical evidence ha equiy risk premiums end o move counercyclically and are, hus, negaively correlaed wih reurns on broad equiy indices. On he basis of such evidence, he HH model posulaes negaive correlaion beween he risk premium and unexpeced shocks o he reurn on he asses of he ypical borrower. Specifically, (1) is augmened by: λ λ d λ = k λ λ λ d + σ λdw, corr ( dw, dw ) σ λ v < 0 (5) A higher λ implies a higher long-run drif in he value of asses, which, ceeris paribus, lowers he PD. The impac of λ is sronger he larger is he mean-reversion parameer k λ. In addiion, since σ λv < 0, a negaive value of dw, which pus upward pressure on he PD, ends o be couneraced by an increase of he drif in he value of asses. 1.2 Models wih an endogenous defaul boundary The wo endogenous defaul models le a borrower decide when o defaul. The frameworks differ mainly in he assumpions underlying he defaul decision. The AST model allows obligors o renege on and hen aler he erms of heir deb conrac. In conras, a renegoiaion is no possible in he LT model, in which borrowers service heir deb as long as doing so is jusified by he expeced fuure reurn on equiy. 6

11 The wo models differ also in heir assumpions regarding he ime o mauriy of deb conracs. Namely, he AST framework incorporaes perpeual bonds, whereas he LT model assumes ha he firm coninuously issues deb of a consan bu finie ime o mauriy The model of Anderson, Sundaresan and Tychon (1996) A he ime of defaul, crediors in he AST model can eiher (i) liquidae he borrowing firm and seize is asses ne of a bankrupcy cos or (ii) accep he erms of a new deb conrac. Since a liquidaion of he borrowing firm is he wors possible oucome for is equiy holders, hey propose a pos-defaul conrac ha is accepable o crediors. To rule ou arbirage opporuniies in his seup, i is necessary ha he value of deb increases coninuously in he value of asses. In paricular, he no-arbirage condiion requires a smooh swich beween he pre-defaul and pos-defaul value of deb. On he one hand, given a fixed bankrupcy cos, K, incurred only if he crediors liquidae he borrower, he pos-defaul value of deb is se by equiy holders o equal V K. This renders crediors indifferen beween re-conracing and liquidaing he borrower. On he oher hand, he pre-defaul value of deb is also an increasing funcion of he firm s asses bu is shifed upward by a higher risk-neural drif in heir process (ie, a higher r 8 and / or a lower δ ), a higher deb principal, P, a higher coupon rae, c, a lower asse volailiy, σ, and a lower monioring cos, m. 9 When asses equal he equilibrium defaul rigger, V AST, he pos- and pre-defaul values of deb are he same. A decline in bankrupcy coss K booss he pos-defaul value of deb, decreases debors bargaining power and induces hem o wai longer before renegoiaing, ie o se a lower V AST. In conras, an upward shif in he pre-defaul value of deb promps debors o negoiae a more advanageous conrac earlier: ie se a higher value of leads o he following comparaive saics: V AST. In he ligh of he previous paragraph, his dvast dvast dvast dvast dvast dvast dvast > 0, > 0, < 0, > 0, > 0, < 0, < 0 (6) dk dr dδ dp dc dσ dm 8 9 An increase in r also decreases he presen value of coupon paymens, which lowers, ceeris paribus, he pre-defaul value of he bond. The daa lead, however, o parameerisaions of he model, under which he ne impac of r reflecs he channel specified in he main ex. The original AST model does no incorporae monioring coss. They are inroduced here in order o dampen he sensiiviy of heoreical PDs o changes in coupon paymens, cp. In mahemaical erms, he original formula for V AST is generalised by replacing cp wih ( 1 m)cp. When m = 0 he AST model over-predics ex pos defaul raes of BB-raed firms by 3.6 percenage poins on average. For BBB-raed and B-raed firms he over-predicion is, respecively, by 1.1 and 4.7 percenage poins. 7

12 1.2.2 The model of Leland and Tof (1996) In he LT model, he borrower forfeis is equiy value as soon as i does no fulfil a conraced obligaion. Thus, he willingness o service deb increases (ie he defaul rigger V LT decreases) in he value of equiy, which equals he value of he firm ne of he value of is deb. 10 Ceeris paribus, he value of he firm decreases in he defaul cos, which is assumed o be an exogenous fracion α of asses. In conras, since i is assessed over he infinie horizon, he value of he firm is insensiive o he ime o mauriy, T, of he coninuously issued deb conracs. For is par, he value of he finiely lived deb decreases in α bu by less han he value of he firm. The value of deb decreases also in T, a rise in which heighens he risk of a defaul before he conrac maures. The upsho is ha he value of equiy (he defaul rigger V LT ) decreases (increases) in he defaul cos bu increases (decreases) in he ime o deb mauriy. The implicaions of he oher model parameers are similarly raionalised. The coupon rae, c, he principal, P, and he asse payou rae, δ decrease, while he risk-free rae, r, asse volailiy, σ, and ax benefis, τ, increase he value of equiy. Thus, aking ino accoun he discussion in he previous paragraph, one obains: dv LT dα > 0, dv LT dt < 0, dv LT dc > 0, dv LT dp > 0, dv LT dδ > 0, dv dr LT < 0, dv LT dσ < 0, dv LT dτ < 0 (7) 1.3 The MKMV model The sixh model belongs o he commercial service MKMV. A sep in he MKMV approach esimaes he asse value and asse volailiy of he borrowing firm. The sep is based on: (i) an opion pricing model; (ii) daa including equiy prices and conracual liabiliies; and (iii) informaion abou he borrower s size, indusry, profiabiliy and geographical locaion. Anoher sep of he MKMV approach delivers a defaul-rigger value of asses, which increases in he borrower s book liabiliies. In he deerminaion of he defaul barrier, shor-erm liabiliies are weighed roughly wice as much as longerm liabiliies. In addiion, here is an underlying assumpion ha a defaul occurs as soon as he lender incurs economic loss. Finally, an MKMV proprieary model uses he esimaes of he borrower s asse value, asse volailiy, and defaul boundary o deliver a firm-specific PD. The model is esimaed on he basis of hisorical defaul raes and credi spreads. Those daa are obained from he larges available public-firm defaul daabase, which has been colleced by MKMV. 11 To he exen ha MKMV s rich proprieary daa sources have value added and/or he fuure resembles he pas, he commercial service would produce beer ou-of-sample forecass of defaul In he LT model, he (marke) value of he firm equals he asse value plus he value of ax benefis, less he value of bankrupcy coss, over he infinie horizon. For a more deailed accoun of, and references on, he MKMV approach, see Leland (2002). 8

13 raes han he academic models, he calibraion of which is based exclusively on public daa and is no validaed in sample. This fac moivaes including MKMV PDs in he analysis, even hough, as I explain in he nex secion, hey are available in quie shor ime series. 2. Daa: sources, filering and descripive saisics Daa availabiliy limis he analysis o corporae borrowers domiciled in he Unied Saes. Firm-specific borrower and deb characerisics are provided by Moody s (raing of senior unsecured deb, 12 coupon rae and ime o mauriy of ousanding bond issues), Bloomberg (book value of oal deb 13 and marke capialisaion) and Daasream (price of equiy and dividend rae). In addiion, daa on he face value of defauled deb and is price 30 days afer defaul, provided by Moody s, help esimae defaul recovery raes. Moody s also provides daa on defaul raes over differen ime horizons. 14 The one-year defaul rae, for example, is calculaed as he number of firms ha defauled wihin a year divided by he number of firms ha could have defauled wihin ha year. In mos general erms, Moody s defines defaul as he insance in which he lender incurs an economic loss, which maches an assumpion of all he models considered in his paper. 15 The firms, racked for he calculaion of a paricular defaul rae, are chosen according o he raing of heir senior unsecured deb and, in his paper, he focus is on firms wih a BBB, BB, or B raing. 16 The average number of firms racked for he calculaion of defaul raes is 517 (BBB raing), 389 (BB raing), and 482 (B raing). Firms in higher raing classes only rarely fail on deb obligaions. Thus, he defaul hisory in hose raing classes carries lile informaion wih respec o changes in he crediworhiness of he consiuen firms. Furher, Moody s coverage of firms raed C or below is limied and prevens meaningful analysis. The macroeconomic variables are provided by he IMF, he Congressional Budge Office and he BIS and consis of: an index of US asse prices; 17 he US GDP gap; he US credi-o-gdp raio; he erm spread in he Treasury rae; and he one-year Treasury rae. The firs wo variables are deflaed by he US CPI. The credi-o-gdp raio and he asse-price index reflec he credi cycle and are used as gaps from heir respecive sochasic rends. Following Borio and Lowe (2002), he rends are When a firm does no have senior unsecured deb, Moody s inerpolaes he raing. Toal deb includes all ineres-bearing obligaions. Overall, he paper uses he Defaul Risk and Credi Risk Calculaor daabases of Moody s Invesors Service. The daabases provide informaion abou all bond issues in Moody s raing universe as well as raings and defaul daa. Mood y s definiion of defaul is spelled ou in Moody s Invesors Services (1998). In erms of Moody s raing convenion, BBB corresponds o a raing beween Baa1 and Baa3, BB o Ba1-Ba3, and B o B1- B3. The index is a weighed geomeric mean of equiy prices, residenial and non-residenial propery prices. The weighs change hrough ime and are based on households annual ne wealh. The daa sources are: S&P Corporae 500 (equiy prices); Naional Council of Real Esae Invesmen Fiduciaries (commercial propery prices) and US Office of Federal Housing Enerprise Oversigh (residenial propery prices). 9

14 calculaed on he basis of daa available in real ime. 18 The erm spread and he GDP gap reflec developmens on he real side of he economy. The former variable is se equal o he difference beween he en-year and hree-monh Treasury raes, whereas he laer variable is calculaed as he difference beween log real GDP and log poenial real GDP. Finally, he one-year Treasury rae is used for esimaing he risk-free rae of reurn. The calculaion of firm-specific PDs requires daa from differen sources and i is he inersecion of he Moody s and Bloomberg daases ha resrics he sample size. The upsho is ha he available daa allow for obaining firm-specific PDs a a quarerly frequency: from Q o Q The smalles cross secions of PDs are a he beginning of he sample: 16 BBB-, 15 BB- and 6 B-raed firms. The cross-secions hen expand monoonically hrough ime and aain an average (maximum) size of: 77 (140) for BBB-, 77 (127) BB-, and 59 (172) for B-raed firms. The sample is dominaed by non-financial 19, 20 firms, which consiue 86%, 91% and 92% of, respecively, he BBB, BB and B firms. Besides he PDs implied by srucural models from he academic lieraure, I also examine expeced defaul frequencies (EDFs) calculaed by he commercial service MKMV. The EDFs are MKMV esimaes of corporae borrowers one-year defaul raes and, as such, are he exac counerpars of he one-year PDs implied by he academic models. In comparison o he daa employed for he calculaion of academic PDs, he available MKMV EDFs feaure richer cross secions, consising on average of 319 BBB-, 279 BB- and 277 B-raed firms, bu span wice as shor a ime period, from Q Q Calibraion mehodology In his secion, I ouline he calibraion of he academic models. Excep for rare cases, discussed in Secions , he analyical soluions of he models are obained under he assumpion ha he parameers say consan hrough ime. Since key parameers reflec risk premiums, deb-service paymens, equiy volailiy, ec, he assumpion is unrealisic and should be inerpreed as referring o seady-sae borrower characerisics ha convey a long-erm level of risk. One of he objecives of his paper, however, is o evaluae he ime pah of defaul rae forecass ha have a shor, one-year, horizon. Over such a horizon, he credi risk of a firm depends significanly on ransiory shocks o is characerisics. In he ligh of his, I calibrae he model parameers o heir shor-erm esimaes, Specifically, a dae- poin on he rend is calculaed via a Hodrick-Presco filer, which uses daa only up o ime. The parameer of he HP filer is se o Since financial firms ener Moody s calculaion of defaul raes, he consisency of he analysis requires ha such firms be considered in he derivaion of heoreical PDs. Tha said, excluding financial firms from he sample leaves he resuls virually unchanged. The repored sample sizes are obained afer filering he daa in order o exclude leverage raios, dividend raes and equiy volailiies ha do no belong o he inerval (0,1). (The calibraion of leverage and equiy volailiy is described in Secion 3). Such a filer removes a relaively small number of observaions and is unlikely o influence he analysis. In addiion, I also filer ou firm-quarer observaions ha imply a defaul-rigger value of asses ha is larger han 90% of he asses iniial value. A similar bu more drasic filer is applied by MKMV: whenever he MKMV model delivers a PD greaer han 20%, he PD is repored as equal o 20%. 10

15 obained a he ime when a PD is consruced. Provided ha deb characerisics are expeced o change lile over he PD horizon, he calibraion procedure is largely consisen wih he assumpions of he models and allows hem o capure ime variabiliy in firms credi risk. 21 To he exen ha he daa permi i and subjec o issues of comparabiliy across models (described below), I calibrae he model parameers a he firm level and updae hem in each quarer of he sample. 22 More specifically, I obain yearly values for he coupon rae, c, and ime o mauriy, T, direcly from daa on bond issues. 23 The mainained assumpion is ha c and T are represenaive for all (bank and non-bank deb) of he obligor. Turning o he risk-free rae, r, I esimae i in wo alernaive ways. Firs, I consider heoreical predicions based on a consan value of r, which equals he average one-year Treasury rae over he enire sample. Then, I base anoher se of resuls on a quarerly ime series of r : an enry in ha series equals he average Treasury rae in he corresponding quarer. The iniial value of asses, V 0, eners he parameerisaion of he models only as a fracion of he defaul boundary V. Since, recalling (1), asses are assumed o follow a geomeric Brownian moion, he exac value of V 0 is normalised o 100 wihou loss of generaliy. I se he deb principal P = l V0 afer calculaing he leverage, l, as he raio of he book value of oal deb o he sum of oal deb and marke capialisaion of he firm. In urn, following Huang and Huang (2003), I se he payou raio δ = l c + ( 1 l ) d equiy holders. 24, where d is he dividend rae paid ou o he firm s Boh l and d are calibraed quarerly and a he firm level. I derive firm-specific values of he asse risk premium and volailiy a he quarerly frequency by firs esimaing he corresponding equiy premium, λ e, and equiy volailiy, e σ. Since equaion (1) implies ha he value of equiy follows a geomeric Brownian moion, I esimae a firm-specific σ e as he sandard deviaion of equiy reurns realised over he year ending wih he curren quarer. For is par, e he esimaion of λ proceeds in hree seps. In he firs sep, I make use of Tarashev and Tsasaronis (2005), which employs opions daa o esimae, iner alia, a ime varying risk premium for he S&P 500 sock marke index. On he basis of ha esimae, I obain a quarerly ime series of marke premiums, which peaks in he lae 1990s and averages 8%. In he second sep, I use resuls of Bhandari (1988), For he endogenous defaul models, one also needs o assume ha he parameer variabiliy is small enough o have a negligible imporance on he defaul rigger value of asses. In general, he laer value would depend on boh shor-erm and long-erm borrower characerisics. If a parameer is sochasic in model bu fixed in anoher one, he parameer s iniial value in he former model is se equal o is consan value in he laer model. This procedure follows Huang and Huang (2003). For each firm-year pair in he sample, I se c and T o be equal, respecively, o he average coupon rae and ime o mauriy of he firm s ousanding bond issues. The averages use weighs proporional o he face values of he corresponding bonds. Since, on a firm-by-firm basis, he average ime o mauriy declines ypically hrough ime, he calibraed value of T is roughly wice as large as he average ime o mauriy over he remaining life of he firm s ousanding deb. This is consisen wih he mauriy srucure assumed in he LT model. Ideally, d would also incorporae sales and repurchases of equiy shares. The employed daa sources do no, however, provide a comprehensive coverage of ha variable. 11

16 which derives a firm-level relaionship beween leverage and risk premiums. Employing Bhandari s esimae of ha relaionship and firm-specific leverage raios, I obain quarerly cross-secions of firmspecific equiy premiums. Finally, I ransform each of hese cross secions so ha heir average values mach he corresponding sock marke risk premiums from he firs sep. 25 The conversion of equiy premiums and volailiy ino heir asse counerpars, λ and σ, uses heoryimplied relaionships beween he value of equiy and he value of asses. In order o ulimaely underscore differences in he way he heoreical models process similar informaion, I focus on wo simple (and similar) specificaions of equiy as a funcion of asses. One of hese specificaions is derived wihin he LS model afer seing he volailiy of he risk-free rae o zero; he oher specificaion is implied by he LT model in he limi in which he ime o mauriy of deb shoos o infiniy. 26 The firs specificaion is used for esimaing λ and σ in he hree exogenous defaul models, whereas he second specificaion is employed for he calibraion of he endogenous defaul models. 27 For he calibraion of cerain parameers, which are consan across firms and ime, I rely on he exan lieraure and especially on Leland (2002) and Huang and Huang (2003). Namely, for he LS model, I se: k r = , σ r = , r = , σ r, v = ; for he HH model: k λ = , σ λ = , λ = , σ λ, v = ; and for he CDG model: k l = 0. 2, and ν = In addiion, I adop τ = m = The las wo parameers ha remain o be se are: he fracion of asses los in defaul, α, (or he fixed bankrupcy cos, K, in he AST model) and he defaul-rigger value of asses, V. The deerminaion of hese parameers relies on an esimae of he defaul recovery rae, which is defined as he price of deb 30 days afer defaul divided by he associaed face value. The esimae of he recovery rae, ρ, is allowed o change from year o year bu says consan across firms in each quarer. I base he esimae ρ on informaion available o bond holders and obligors in real ime. Tha informaion is likely o consis of pas recovery raes and addiional news ha is refleced in nex-in-line The ransformaion is necessiaed by he fac ha he esimaes of Bhandari (1988) do no accoun for ime variaion in he marke risk premium. The simplified LS and LT seups allow for analyic expressions of asse premium and volailiy as funcions of equiy premium and volailiy. Those seups are examined by Huang and Huang (2003) where hey are referred o as he base case and endogenous defaul boundary models, respecively. Swiching beween he wo specificaions has no maerial impac on he implied PDs. I also explored he implicaions of alernaive parameerisaions of he HH and CDG models. In hese parameerisaions, λ and ν were calibraed o firm-specific daa: he ime average of a firm s asse risk premium and leverage raio, respecively. Swiching o hese alernaive parameerisaions has virually no effec on he implied PDs. Esimaing he monioring coss, m, is beyond he scope of his paper, whereas he adoped value of τ is as assumed in Leland (2002). In he ligh of he similar implicaions of m and τ for he PDs implied, respecively, by he AST and LT models, I assign he same value o boh parameers. 12

17 defauls. In he ligh of his, I se each year-specific ρ equal o wo-hirds imes he mean recovery rae of all he defauls up o he curren calendar year plus one-hird imes he mean recovery rae in he curren year. 30 For consisency wih he way Moody s calculaes defaul raes, and wih echnical assumpions of he heoreical models under sudy, I calculae recovery raes using only defauls on senior unsecured deb. 31 The procedure for assigning values o he defaul rigger, V, and he defaul cos, α (or K ), is model specific. Namely, he calibraion of V reflecs he fac ha he endogenous defaul and exogenous defaul models rea borrowers decision o defaul differenly. In addiion, he models in he former group allow for deriving he value of α (or K ) on he basis of oher, independenly calibraed, model parameers. By conras, he exogenous defaul models do no provide any guidance regarding he value of α. Neverheless, he wo ypes of models incorporae he defaul coss in a concepually idenical way, via an exogenous consan. This promps aligning he value of α across models. For he wo endogenous defaul models, he values of V and α (or K ) are deermined simulaneously by he requiremen ha, in each quarer-raing class, he average (laen) recovery rae of he riskier 50% of he firms is o equal he curren esimae of he marke-wide recovery rae: 32 ρ = ( 1 α ) V LT i ( α ), i N P i, ρ = i N V AST, i ( K ) P i K (8) where indexes he quarer and i he firm, principal N is he se of he riskier half of he firms and he P i is calculaed as described earlier in he secion. The defaul coss are kep consan across firms bu vary quarerly, whereas he defaul riggers vary boh quarerly and across firms. The firs subscrip of V LT and parameers (recall (6) and (7)). V AST indicaes ha he defaul riggers are also funcions of firm-specific In he case of he exogenous defaul models, I se α = 0. 40, which is beween he value adoped in Leland (2002) (ie, 0.3) and he average α derived in his paper under he LT model (ie, 0.46). Then, I Alernaive calibraion procedures include seing ρ equal o he average recovery rae (i) over he enire sample or (ii) up o he curren year. The former alernaive fixes ρ hrough ime and hus insulaes he analysis from any empirical relaionship beween PDs and losses given defaul. Implemening he laer alernaive, insead of he procedure proposed in he main ex, has a small quaniaive impac on he implicaions of he models. The esimae of he defaul recovery rae in 1990 is based on 33 defauls, whereas he corresponding value of ρ in 2003 is based on 561 defauls. The average number of defauls, which underlie he ime series of ρ, is 161. I consider only he riskier firms because hey are expeced o acually defaul and, hus, deermine he recovery raes in he daa. In he AST model, he value of K increases if, insead, one bases is calibraion on he riskies 25% of he firms. This ranslaes ino higher model-implied PDs: he average PD of B-raed firms increases from 4.5% o 4.7%. For BB-raed firms, he increase is from 1.2% o 1.9%, and for BBB-raed firms from 0.2% o 0.4%. The impac is similar wihin he LT model bu is aained on he back of unrealisically high values of α. In conras, he resuls do no change maerially if one calibraes α and K on he basis of all he firms in he cross secion as opposed o he riskies 50%. 13

18 allow he defaul rigger o vary quarerly and across firms by seing i according o he following version of equaion (8): 33 ( ) 1 α V i ρ = (9) P i Tables 1 3 allow for an appreciaion of he calibraion mehodology and is implicaions. Table 1 recapiulaes all he model parameers and feaures of heir calibraion, whereas Table 2 caalogues how changes in he parameers affec heory implied PDs. For is par, Table 3 repors parameer averages alongside characerisics of he represenaive firm, as adoped by Leland (2002) and/or Huang and Huang (2003). The differences beween he calibraion resuls of his paper and hose of he previous lieraure are due primarily o hree facors. Firs, his paper calibraes separaely hree raing classes whereas he earlier papers base cerain aggregae values on he enire specrum of raings. Second, he daa underlying he fourh column in Table 3 uses daa saring as early as 1970 whereas he daa used herein sar in Third, his paper calculaes he leverage raio on he basis of oal deb, whereas he earlier papers rely on esimaes of leverage ha are provided by Sandard & Poor s (1999) and incorporae oal liabiliies The model-implied PDs Leland (2002) finds ha he LT and (a simplified version of) he LS models imply one-year PDs ha underpredic consisenly and subsanially ex pos defaul raes. He conjecures ha his migh be due o his calibraion of he models, which focuses on he represenaive firm endowed wih he average borrower characerisics. In examining he cross-secional variabiliy of firm-specific PDs, his secion demonsraes iner alia ha Leland s conjecure is indeed borne ou There is no consensus in he lieraure regarding he value of α : a one exreme, Andrade and Kaplan (1998) argue ha i should no exceed 20%; a he oher exreme Leland and Tof (1996) se i o 50%. Reducing he value of α o 30% in he exogenous defaul models does no affec he ime paern of he cross-secional averages of PDs bu shifs hem down subsanially: by percenage poins (on average) for BBB-raed firms, by 0.61 percenage poins for BB-raed firms and by 2.5 percenage poins for B-raed firms. This affecs especially he esimae of he risk-free rae of reurn because he 1990s winessed levels of he ineres rae ha were low by hisorical sandards. The leverage used in previous sudies is larger because oal liabiliies provide a broader measure of financial obligaions. In addiion o deb, oal liabiliies include obligaions ha do no involve ineres paymens (eg promises for physical deliveries). There are hree reasons for choosing o work wih oal deb as opposed o oal liabiliies. Firs, a leverage raio ha is based on oal liabiliies would lead o an overesimaion of coupon paymens and payou raes. In urn, his would lead o oo high values of model-based PDs. Second, oal deb racks more closely he bond insrumens underlying he calculaion of defaul recovery raes. Third, oal deb underlies he calculaion of MKMV PDs, which are used here o benchmark he performance of he academic models. Finally, i is necessary o also acknowledge ha, by ignoring a porion of liabiliies, oal deb is likely o impue oo big of a fracion of he equiy risk premium and volailiy ono he asse risk premium and volailiy. The impac of leverage on he calibraion of he laer wo parameers ranslaes, however, ino a small quaniaive impac on model-implied PDs. 14

19 Irrespecive of which se of parameers one chooses o work wih in Table 3, he implied one-year probabiliies of defaul are orders of magniude smaller han he corresponding defaul rae. This is illusraed by Table 4, which repors average ex pos defaul raes of B, BB and BBB-raed firms ogeher wih wo ses of PDs: one associaed wih he represenaive firm in Leland (2002) and one wih he average firms in his paper. Even hough he heoreical one-year PD of he represenaive firm underesimaes severely defaul risk, he models perform sarkly beer when employed for he calculaion of firm-specific PDs. Focusing on one raing class-quarer a a ime and averaging he PDs in he associaed cross secion produces he ime series of heoreical predicions porrayed in Figure In he figure, an average one-year PD is aligned wih he defaul rae realised over he following year wihin he corresponding raing class. Considered even casually, he figure suggess ha he bias in average firm-specific PDs vis-à-vis ex pos defaul raes is eiher negligible or much smaller han he bias in he PDs of he represenaive or average firms. Table 4 reveals direcly he small bias in heoreical defaul predicions. I is illusraed by he small difference beween he ime averages of he LT forecass porrayed in Figure 1 and he corresponding average defaul raes. 37 Such a small difference is expeced o prevail if a valid credi-risk model is applied o a random selecion of firms in a raing class and he sample period is sufficienly long. Figure 1 includes PDs implied by he endogenous defaul seups, he LT and AST models, and only one of he exogenous defaul seups, he HH model. The reason for no showing he implicaions of he LS model is ha hey are virually idenical o heir counerpars in he HH model. This is due o he wo models differing only in heir choice of a second sochasic process (for he risk-free rae of reurn or he risk premium) whose quaniaive implicaion for PDs urns ou o be negligible. In order o avoid repeiion, I suppress he LS seup from he subsequen analysis. In addiion, he CDG model underpredics consisenly he ex pos defaul raes. The resul is driven by he wo parameers k l and ν, whose values imply ha he defaul boundary ends o say far below he value of asses (recall equaion (4) and he accompanying discussion). This depresses he heoreical PDs. Having idenified a pronounced bias in he implicaions of he CDG model, I do no include i furher in he analysis Firm-level daa and heoreical predicions of defaul raes 39 In his subsecion, I explain he pronounced difference beween defaul-rae predicions based on firmspecific PDs and alernaive predicions based on average borrower characerisics. To illusrae he The series in Figures 1 3 are based on he ime-varying calibraion of he risk free rae, r. The implicaions of fixing he value of r hrough ime are discussed in Secion 4.5. The message of Table 4 changes lile in he conex of he AST, HH and LS models: see Tarashev (2005). As he speed of mean reversion, k l, decreases owards zero, he PDs generaed by he CDG model converge o he ones generaed by he LS and HH models. For exposiional purposes, I focus predominanly on BB-raed firms for he remainder of Secion 4. The predicions of he models as regards he defaul raes of BBB- and B-raed firms can be disseced similarly. 15

20 issues involved, I focus on a cross-secion of one-year PDs ha is derived wihin he LT seup and is associaed wih BB-raed firms in Q Figure 2 provides a hisogram of he cross-secion, which is characerisic of he enire sample and reveals ha only a fracion of he firms represen non-negligible credi risk. These firms are behind he long righ ail (righ skew) of he disribuion. Their relaive number in he cross secion and he magniudes of heir PDs drive he models predicions of he one-year defaul rae. The skew of he disribuion is an illusraion of he so-called Jensen inequaliy effec, which arises because of a non-linear (convex) relaionship beween he parameers of he model and he implied PD. The imporance of he Jensen inequaliy effec is bes appreciaed when he PD is considered as a funcion of only one variable. Figure 3 illusraes such a hypoheical scenario by focusing on he leverage raios of he firms behind he plo in Figure 2. Figure 3 reveals ha he LT model (crysallising in he middle panel) ranslaes a sligh skew in he leverage raios ino a highly skewed disribuion of PDs when all he oher parameers of he model are kep a heir cross-secional averages. The upsho of he Jensen inequaliy effec is ha he average PD in a cross-secion is subsanially larger han he PD of he corresponding represenaive firm, which is endowed wih he average borrower characerisics. In he above example, he average BB-raed firm in Q has a modelimplied PD equal o 0.18%, whereas he average of firm-specific PDs is 3.7%. 4.2 Time variabiliy of PDs Figure 1 reveals subsanial ime variaion in he heory-based PDs, which can be raionalised on he basis of he discussion of he models in Secion 1 and heir calibraion oulined in Secion 3. To illusrae, I consider here he implicaions of he LT model for firms raed BB in Q and Q Beween hese wo quarers, here is a subsanial drop of he average PD: from 3.7% o 1.1%. Zooming on he riskies 20 firms in each cross-secion of firm-specific PDs, ie he firms ha drive he defaul forecass of he model, he average coupon rae and ime o mauriy of ousanding deb are virually idenical in he wo quarers. In conras, he average asse pay-ou rae, δ, decreases from 6% o 5% while he average leverage raio, l, decreases sharply from 62% o 49% and he average asse volailiy, σ, shoos up from 29% o 37%. 40 Expression (7) indicaes ha such changes in δ, l and σ decrease he defaul boundary, V LT and, by expression (2), decrease he probabiliy of defaul. As also indicaed by expression (2), he parameers δ and σ affec he PD via a second channel as well: via heir implicaions for he process of asses. I urns ou, however, ha he firs channel dominaes. 40 The change in parameers may seem oo abrup since i occurs beween wo consecuive quarers. Noe, however, ha i is due parly o exi/enry of firms from/in he BB-raing class and parly o changes in firm characerisics beween he wo quarers. In paricular, roughly half of he riskies 20% of he BB firms in Q are no longer in he BB-raing class in Q1 2002: his conribues o he decrease in leverage raios repored above. In addiion, hose of he riskies BB-raed firms, which say in he same class over he wo quarers, experience on average a 6 percenage poin decline in leverage. 16