How Well Does the Vasicek-Basel AIRB Model Fit the Data? Evidence from a Long Time Series of Corporate Credit Ratings Data

Transcription

1 How Well Does he Vasicek-Basel AIRB Model Fi he Daa? Evidence from a Long ime Series of Corporae Credi Raings Daa by Paul H. Kupiec Preliminary Sepember 2009 EXENDED ABSRAC he Basel II AIRB framework uses Vasicek s asympoic single facor model o se minimum regulaory capial requiremens for bank credi risk. I develop an esimaion approach ha produces consisen esimaes of he parameers in he Vasicek model as well as consisen esimaes of he common macro facor realizaions ha drive credi defauls. he model is esimaed using Moody s daa on he defaul raes of raed corporae credis over he period Model fi is assessed using robus saisics and small samples esimaion issues are examined. he reamen of observaions wih recorded defaul raes of zero is idenified as an imporan esimaion issue ha requires furher sudy. he Vasicek-Basel II AIRB defaul model is no capable of reproducing he observed variaion in Moody s corporae defaul rae daa. he analysis shows ha he rue correlaion among defaul paerns for differen credi grades is inconsisen wih he Vasicek single common facor model specificaion. Over he period , corporae raed credis eiher do no defaul or defaul a much higher frequencies han are prediced by he model. In conras o he model s assumpions, he macro facor ha drives corporae defauls exhibis srong posiive auocorrelaion. Observed credi cycles las many years on average. As a consequence, long ime series are required o produce reliable model parameer esimaes. I is impossible, for example, o produce reliable esimaes of uncondiional probabiliy of defaul from sample as shor as 5- or 10-years. In conras o he Basel AIRB framework, highly-raed corporae credis exhibi much higher defaul rae correlaions compared o lower-raed credis. I use he economeric model o develop procedures o correc for business cycle effecs when esimaing uncondiional defaul raes when a credi class has only a limied sample of defaul daa. hese procedures are useful no only for esimaion, bu also for back esing or aemping o validae a bank s Basel AIRB parameer assignmens. Federal Deposi Insurance Corporaion. he views expressed are hose of he auhor and do no reflec he views of he FDIC. I am graeful o Ed Kane and Ma Prisker for commens on an early draf of hissudy. pkupiec@fdic.gov 1

2 HOW WELL DOES HE VASICEK-BASEL AIRB MODEL FI HE DAA? EVIDENCE FROM A LONG IME SERIES OF CORPORAE CREDI RAINGS DAA I. INRODUCION he Basel II Advanced Inernal Raings-Based (AIRB) framework is used o se minimum regulaory capial requiremens of he larges, mos sophisicaed inernaionally-acive banks. For example, he Financial Sabiliy Insiue (2006) repors ha 95 counries plan o implemen Basel II by 2015, and more han 60 percen of he Basel II adopers plan on including he AIRB opion for credi risk capial requiremens. 1 he AIRB regulaory framework uses an asympoic version of Vasicek s (1987) porfolio credi loss model o approximae he annual defaul rae disribuions on porfolios of credis ha are differeniaed by a bank-assigned credi raing. he AIRB framework uses he Vasicek defaul rae disribuion and esimaes of loss given defaul (LGD) and exposure a defaul (EAD) o approximae he credi loss disribuion for each credi grade porfolio. 2 Regulaory capial requiremens are hen se equal o he 99.9 percen upper-ail criical value of a credi grade s poenial porfolio loss disribuion. Much has been wrien abou Basel II, bu few if any sudies have formally analyzed how well he Basel II Advanced Raing Based (AIRB) model fis credi defaul daa produced by porfolios of credis caegorized under a consisen credi raing sysem. his paper develops a new approach for esimaing he parameers of he Basel AIRB 1 he 60 percen figure includes boh Basel Commiee members and nonmember counries ha plan on adoping he AIRB approach as an opion for credi risk regulaory capial calculaions. 2 he LGD and EAD esimaes are no par of he Vasicek model. hey are esimaed independenly of he parameers of he Vasicek model. 2

3 model and new echniques for assessing he model fi relaive o hisorical defaul rae daa on corporae credis raed by Moody s Invesors Services. he approach I propose uses panel regression mehods o esimae he Vasicek- Basel AIRB model parameers using ime series daa on a cross secion of failure raes from a consisen credi raing sysem. he mehodology produces consisen esimaes of he uncondiional defaul raes associaed wih each credi grade by correcing for business cycle effecs ha are modeled using a single common facor. A consisen esimae of he AIRB correlaion parameer is derived direcly from he defaul rae daa. he mehodology also produces esimaes of he common macro facor ha is assumed o drive defaul correlaions. 3 his common facor can be used o conrol for macroeconomic effecs when circumsances require an esimae of he uncondiional defaul raes for addiional credi grades for which only brief ime series hisories are available. he abiliy o idenify he common marke facor and o conrol for is impac on observed defaul rae realizaions is paricularly useful for back esing and model validaion analysis when rue defaul raes (and hus he common marke facor realizaions) are auocorrelaed. he mehodology allows he researcher o esimae or es he uncondiional defaul rae associaed wih a raing grade while correcing for he sample dependen common facor realizaion. his approach offers a significan 3 he mehodology can also be used o esimae muliple correlaion parameers if he daa includes a sufficienly large number of credi grades. 3

4 improvemen over ess ha ignore ime dependence in defaul rae realizaions and rea sample defaul rae esimaes as uncondiional mean defaul rae values. 4 he proposed esimaion mehodology is implemened using defaul rae daa from Moody s Invesors Services on raed corporae bond issues from he daa are used o derive consisen esimaes for he uncondiional defaul raes associaed wih he Aa, A, Baa, Ba, B and CaaC raing caegories as well as for he Vasicek defaul correlaion parameer. he esimaion exercise highlighs he imporance of zero defaul rae observaions in hisorical daa. Under he asympoic Vasicek-Basel AIRB model assumpions, zero defaul raes should almos never occur, and ye one-hird of he Moody s sample repors a zero annual defaul rae. A repored defaul rae of zero can arise because a porfolio is no ruly asympoic and so he repored defaul rae is downward biased as a consequence of measuremen error. Esimaes can be adjused o accoun for a reasonable upper bound on he magniude of he measuremen error associaed wih zero defaul rae observaions, bu he resuling model parameer esimaes are very sensiive o he reamen accorded zero defaul rae observaions. An imporan issue relaed o model parameer esimaion is he behavior of uncondiional defaul parameer esimaes from small samples. Because of he assumed imporance of he macro facor, defaul realizaions are driven by marke condiions which mus be conrolled for when esimaing he uncondiional defaul raes associaed wih a raing grade. While he Vasicek model specificaion assumes ha he macro facor 4 Uncondiional defaul rae ess include hose proposed by Canor and Falkensein (undaed copy), Pluo and asche (undaed copy), Canor Hamilon and ennan (2007) and Schuermann and Hanson (2004). 4

5 realizaions are independen across ime bu in realiy hey have srong posiive auocorrelaion. his posiive auocorrelaion makes i impossible o esimae uncondiional defaul raes from small samples unless he common facor effecs are properly conrolled for in he esimaion process using esimaes of he common facor realizaions derived from exernal daa. he economeric model specificaion provides an inuiive process for correcing for he aforemenioned small sample bias in uncondiional defaul raes. If consisen esimaes of he Vasicek common facor realizaions and correlaion parameer can be recovered from a long ime series panel daa se of defaul raes, hese esimaes can be used o consruc consisen esimaes of he uncondiional defaul raes for auxiliary raing grades when he new credi class has only a limied sample realized defaul raes. I derive he algorihm o make hese adjusmens and demonsrae he echnique for defaul daa on Moody s alpha-numeric raing scale over he period he nex secion reviews he Vasicek-Basel AIRB porfolio defaul rae model. Secion III discusses he new echnique for esimaing he Vasicek-Basel model parameers. Secion IV discusses imporan esimaion issues ha arise because observed defaul raes exhibi srong posiive correlaion. Secion V discusses he Moody s corporae defaul rae daa. Secion VI discusses specific esimaion issues ha arise because of he prevalence of zero defaul raes in he daa. Secion VII discusses he model parameer esimaes and parameer es saisics. Secion VIII analyzes he small sample esimaion bias problem. Secion IX discusses he small sample correcion algorihm ha conrols for he common facor realizaions and produces consisen esimaes. A final secion summarizes he resuls and concludes. 5

6 II. HE VASICEK PORFOLIO CREDI LOSS DISRIBUION MODEL he Gaussian single facor model of porfolio credi losses (a.k.a. he Vasicek model), developed by Vasicek (1987), Finger (1999), Schönbucher (2000), Gordy (2003) and ohers, provides an approximaion for he disribuion of he defaul rae on a welldiversified credi porfolio. he Vasicek model is a defaul-mode model meaning ha all credis are assumed o eiher perform or defaul wihin he models risk measuremen horizon. he asympoic version of Vasicek model focuses on a large diversified porfolio in which idiosyncraic risk is fully diversified and he only source of porfolio loss uncerainy is he defaul rae ha is driven by he common laen Gaussian facor. 5 he model measures he aggregae value of he losses generaed by defauling credis and he income earned on non-defauling credis is no recognized. 6 he Vasicek model assumes ha uncerainy on credi i is driven by a laen unobserved facor, V ~ i, wih he following properies: ~ Vi V e~ ~ ( e M e~ ) eid ~ ( e ), ( ~ i E e e~ ) E( e ~ i j M M M 1 e~ ) 0, j V e~ i i, j. (1) where () represens he sandard normal densiy funcion. V ~ i is disribued sandard ~ normal, E, and V EV EV 1. V i ~ ~ ~ 2 i i i i V ~ is ofen inerpreed as a proxy for 5 he model assumes he uncondiional probabiliy of defaul, exposure a defaul, and loss raes in defaul (LGD) are known non-sochasic quaniies for all obligors. 6 Kupiec (2006, 2007) develops a more general version of he Vasicek model in which ineres income on defauling credis is recognized and offses losses on defauling credis. 6

7 he marke value of he firm ha issued credi i. he common facor, e ~ M, induces ~ ~ CovV i, V j correlaion beween individual credi laen facor realizaions, V ~ ~ V V. Credi i is assumed o defaul when is laen facor akes on a value less han a ~ credi-specific hreshold, V D. he uncondiional probabiliy ha credi i defauls is i i PD D i, where represens he cumulaive sandard normal densiy funcion. ime is no an independen facor in his model, bu is implicily recognized hrough he calibraion of inpu values for PD. he loss rae on a porfolio of credis ha have idenical correlaions,, and defaul hresholds, D i D, is deermined as follows. Define a defaul indicaor funcion i j for each credi, ~ I i 1 0 if ~ V i D oherwise (2) I ~ has a binomial disribuion wih an expeced value of D. Define X ~ o be he i proporion of credis in he porfolio ha defaul, n ~ I i ~ i1 X. n In an asympoic porfolio, he number of individual credis is assumed o increase wihou bound, n. In he limi, idiosyncraic risks are compleely diversified wihin he porfolio and porfolio defaul rae uncerainy is driven by he common facor alone. he uncondiional disribuion funcion of X ~, he asympoic porfolio s defaul rae, is given by, Pr 1 1 ~ 1 x PD X x, x 0,1 (3) Condiional on a specific draw of he common facor, e ~ e, he condiional M M value of he defaul indicaor for a single credi is, 7

8 ~ I e~ i M e M 1 0 if D e 1 oherwise M e~ i (4) When he correlaion and defaul hresholds are idenical for all credis in he porfolio, he condiional indicor funcions assigned for each credi are independen and idenically disribued. As he number of credis in he porfolio increase wihou bound, condiional on a value of e M, he porfolio defaul rae disribuion converges (almos surely) o a non-random value ha depends on e, M lim n lim n n ~ I e~ e i M M ~ ~ i1 ~ X e e EI e~ e M M n D e 1 M a. s i 1 M PD M 1 e M. (5) Equaion (5) implies, condiional on a specific realizaion of he common facor, an asympoic porfolio s defaul rae is a non-random value fully deermined by wo parameers, he credis uncondiional defaul rae, PD, and he Vasicek correlaion parameer,. Inuiion can be gained by examining a simulaed ime series of defaul raes on a se of asympoic porfolios generaed by equaion (5). Consider a simulaed imes series of defaul raes generaed by four porfolios, each represening a differen credi grade wihin a raing sysem. hese porfolios are disinguished by heir credis uncondiional probabiliies of defaul which are assumed o be: ( P , P2.70, P3 1.00, and P ). All of porfolios are assumed o have an idenical correlaion parameer,

9 Figure 1: Simulaed ime Series of Defaul Raes on Four Asympoic Porfolios Porfolio Defaul Rae 8.00% 7.00% 6.00% 5.00% 4.00% 3.00% 2.00% 1.00% P1 P2 P3 P4 0.00% ime Figure 1 plos simulaed imes series of defaul raes on hese four hypoheical asympoic porfolios. he plos show ha he defaul raes on hese porfolios are very highly correlaed because he idiosyncraic risk of defaul is compleely diversified and he defaul rae is driven by only a single common facor. able 1 repors he sample defaul rae correlaions. he Vasicek model correlaion parameer is 20 percen for all credis and ye he porfolio defaul rae realizaions are nearly perfecly correlaed. 7 Figure 1 and able 1 show ha, under he asympoic Vasicek model, realized porfolio defaul raes will be nearly perfecly correlaed regardless of he magniude of he correlaion parameer. 8 7 he defaul raes would be exacly perfecly correlaed excep equaion (5) applies differen non-linear ransformaions o he common Gaussian erm e M. 8 Somewha paradoxically, able A1 in he Appendix shows ha he correlaions among porfolio defaul raes acually decline as he Vasicek correlaion parameer,, increases. 9

10 able 1: Simulaed Asympoic Porfolio Sample Defaul Rae Correlaions P1 P2 P3 P4 P P P P III. ESIMAION OF HE ASYMPOIC VASICEK MODEL PARAMEERS he parameers of he asympoic Vasicek porfolio model can be consisenly esimaed using panel daa regression echniques. I adop he common pracice of idenifying a credi raing or grade wih is uncondiional probabiliy of defaul. Equaion (5) implies ha he defaul rae realizaion on an asympoic porfolio of credis from an idenical grade is a nonlinear funcion of he uncondiional defaul rae and correlaion parameers of is consiuen credis. Le P j represen he realized defaul rae on porfolio Pj in year. Equaion (5) implies, 1 ( P j ) 1 PD 1 j e 1 M (6) Where PD j is he uncondiional probabiliy of defaul for a credi in raing caegory j ; e is he realized value of he unobserved common Gaussian facor; and j 1,, M, M represen M individual credi raing caegories. Equaion (6) is consisen wih he heoreical predicions of he asympoic Vasick defaul rae model, bu in realiy, observed defaul raes my deviae from heir heoreically prediced value by some mean-zero error erm, ~ i. o complee he 10

11 empirical model, I assume ha error erms are independen and idenically disribued across ime, and uncorrelaed cross-secionally, E i 0 i, E( ~ ~ i j ) 0 j i E~ ~ i jk 0, for k 1, 2,3,, and j, (7) E( ~ ~ ) 0 j i E i ~ 2 2 i j i he final condiion allows each credi grade o be characerized by a differen error variance. Recognizing he possibiliy of he mean-zero model error, he empirical specificaion of he Vasicek model is, 1 ~ ( P ) j 1 PD 1 A e 1 M ~ i (8) Consisen Esimaion of he ransformed Model Parameers Define y 1 j ( P j ) ; y j is he observed porfolio defaul rae ransformed hrough he inverse normal disribuion funcion. Unlike he observed defaul rae, no bounded beween 0 and 1, bu is insead a coninuous variable in he range y j is. Le 1 PD j a j ; a j is a ime-invarian consan deermined by he characerisic credi 1 class j' s uncondiional probabiliy of defaul and he Vasicek correlaion parameer. Define b e 1 M ; b is a scalar muliple of he common Gaussian facor realizaion. Noice ha he scalar muliple for e M is no dependen of he asympoic porfolio s credi raing bu only depends of he credis correlaion parameer. Using hese definiions, equaion (8) can be wrien in more simplified noaion as, 11

12 ~ y a b ~ (9) j j j Under he model assumpions, he parameer a j can be consisenly esimaed as he sample average from a ime series of defaul rae realizaions on he credis in raing caegory j. he sample average esimae is, 1 ~ y j a j 1 b 1 ~ j aˆ j (10) b 1 as 0, a. s ~ j 1 and 0 a. s under he model assumpions. A consisen esimae for b can be derived from he average of he residuals from a cross secion of credi grades afer each of he observed defaul raes are correced for a ime series esimae of is consan erm, M j1 y j M aˆ j b M j1 M j bˆ. (11) j1 he model assumpions require 0 a. s. M M j as he number of independen credi grades becomes large. Ignoring he precision of he consan parameer esimaes ( ˆ ' s), he precision of he scaled macro facor esimaes will depend on he number of independen credi grades included in a cross secion as well as he magniude of he model error variance associaed each credi grade. Smaller model error variances ( i, i 1, 2,3,, M ) and a larger number of cross secions will improve he precision on he scaled macro facor 2 a j 12

13 esimaes. If he model fis he daa from each credi caegory poorly (large residual variances) and here are few independen credi grades, he esimaes canno be expeced o provide a very accurae represenaion of he acual macro facor realizaions. I will provide a more deailed analysis of he small sample properies of he common facor esimaes from he model in a subsequen secion. he empirical model can be wrien in a panel regression forma. o simplify he noaion and mainain clariy, I wrie he model in erms of hree credi grades bu he noaion generalizes o include addiional credi grades. Le X D1 D2 D3 i be a vecor of selecion covariaes ha indicae membership in a credi grade (raing caegory); for example, if y i is he ransformed defaul rae associae wih credi grade 1, and X yi y1 1. Similarly, if y i is he ransformed defaul rae associaed i i i wih credi grade 2, and yi y2 X. Define 1 2 ) o 2 i ( i i2 i be a selecion marix ha idenifies he year associaed wih observaion y i. For example, when yi y i 1, a defaul rae observaion from year 1, i1 ( ) ; when he observaion is from year 3, ( ). Using his noaion, an i3 empirical model for a generic porfolio defaul raes observaion is, y i a a a X i b b b b i i ~ (12) where ~ i is he residual erm. Normally, in an analysis of covariance seing, he parameers in equaion (12) would be esimaed using leas squares (OLS) or weighed leas squares afer dropping a 13

14 dummy variable for one year. 9 In he presen case, we can make use of an addiional model resricion o idenify a complee se of individual year effecs. he Vasicek model assumes ha he common Gaussian facor is a sandard normal variable. his assumpion imposes a resricion ha he average ime effec is 0. his addiional assumpion allows for he consisen esimaion of all of he model parameers by esimaing equaion (12) under he resricion, b 0. he efficiency of model parameer esimaes can be 1 improved by correcing heeroskedasiciy using generalized leas squares, as he residual variances differ among raing classes. Esimaion of he Correlaion and Common Facor Realizaions Resriced OLS provides consisen esimaes of he models parameers, including he raing-grade dependen inerceps and consisen esimaes of he individual ime effecs, ˆ 1, 2,3,,. Because he Gaussian common facor, ~ e, in he Vasicek b model has a sandard deviaion of 1 by assumpion and he bˆ esimaors are consisen, M 1 b ˆ 2 1 is a consisen esimaor for,. I follows ha he consisen esimaors 1 for he e M series are given by, is a consisen esimaor for. bˆ eˆ M ( 1, 2,3,, ) and 1 bˆ 1 Finally, ˆ 1 ˆ i 1 ˆ 1 1 b ˆ a is a consisen esimaor of he uncondiional probabiliy of defaul for raing grade i, i 1, 2,3, and so all he bˆ 2 9 Weighed leas squares could correc for heeroskedasiciy if he variance of he error erm was differen for each raing grade. 14

15 underlying parameers of he Vasicek model are idenified using his panel regression approach o model esimaion. IV. AUOCORRELAION ISSUES I is inuiively reasonable o hink ha, should he model be esimaed using a long ime series of porfolio defaul rae daa, he economic effec of he common facor on he defaul rae should average ou over he sample since he Vasicek model assumes ha he common facor s average effec is 0. he longer he ime series, he beer he sample average of he unobserved common facor realizaions should approximae 0 because, under model assumpions, e~ M are independen idenically disribued random variables wih mean 0. In realiy, observed defaul raes are posiively correlaed across ime which may be indicaive of posiive auocorrelaion in he common facor realizaions. he Vasicek model srucure does no include he possibiliy of business cycles in he daa as here is no auocorrelaion or ime-series srucure in he common Gaussian facor specificaion. If credi raings are updaed annually o projec a consan condiional defaul rae for each credi grade, and hese implici performance forecass are efficien, here is no reason o expec auocorrelaion in he deviaions from he credi grade s uncondiional defaul rae. hese argumens nowihsanding, observed defaul rae daa on raed corporae bonds exhibi posiive auocorrelaion; corporae defaul raes exhibi clear evidence of credi cycles. able 2 repors firs-order auocorrelaion esimaes for he annual defaul raes repored on seleced corporae credis raed by Moody s Invesors Services and for he ransformed defaul raes using he inverse 15

16 cumulaive normal disribuion. hese esimaes show evidence of srong posiive auocorrelaion for he realized defaul raes and ransformed defaul raes for all credis excep hose raed Aa by Moody s. able 2: Credi Cycles in he Realized Defaul Raes on Raed Corporae Credis lagged dependen dependen variable inercep p-value variable p-value R 2 Aa defaul rae A defaul rae < Baa defaul rae < Ba defaul rae < B defaul rae < CaaC defaul rae < Φ ¹(Aa defaul rae) < Φ ¹(A defaul rae) < < Φ ¹(Baa defaul rae) < < Φ ¹(Ba defaul rae) < < Φ ¹(B defaul rae) < < Φ ¹(CaaC defaul rae) < Esimaes are based on Moody's Corporae Defaul Rae Daa, Defaul raes are he number of defauls in he year following a Moody's raing designaion divided by he number of raed credis in a credi grade. Defaul raes are measured as perc here is lile published lieraure on he properies of credi cycles, e.g., heir average lengh, symmery, ampliude of boom and bus phases, or oher feaures. he AR 1 defaul rae model esimaes repored in able 2 sugges ha shocks o defaul raes have lingering measurable effecs for abou hree years, bu he simple specificaion may be oo simplisic o fully capure he full dynamics of defaul rae behavior. While he Vasicek model does no recognize he possibiliy of auocorrelaion in realized defaul raes, if he model is esimaed over a long ime series, even if defaul raes are auocorrelaed, he effec of credi cycles will average ou o 0, and he idenifying resricion b 0 will be appropriae. Consider a case in which a sample 1 includes muliple complee cycles and a parial cycle. Esimaing equaion (12) while 16

17 imposing he resricion b 0 will resul in biased esimaes of he ime effecs as he 1 acual sample will no be balanced beween posiive and negaive deviaions from he uncondiional mean so he resricion is inappropriae. he effec of parial cycles in a sample will diminish as he sample ime series lenghens and includes more and more complee cycles. In he limi, even if here are credi cycles in he defaul daa, he resricion b 0 will hold exacly as. 1 he imporance of using a long ime series o esimae equaion (12) can be illusraed in he conex of a simple auoregressive model. Le generaed by a saionary auoregressive process, R ~ represen a ime series ~ R R ~, ~ u u ~ (0, ), 1. (13) he uncondiional sample mean of he process is, 0. he esimaor for he 1 1 uncondiional sample mean is, 1 R 1 R u~ ~ u 1 As he sample size becomes large, 0, a. s and 1 1 R a. s 1 R. hus, as as, 1 R 0. a. s. 1 1 In equaion (12), ransformed defaul raes are equal o an uncondiional mean value plus a mean zero independen idenically disribued innovaion ha includes he effec of a macroeconomic facor. Wha happens if rue ransformed defaul raes follow 17

18 a saionary auoregressive process similar o equaion (13) bu we esimae equaion (12) wihou accouning for auocorrelaion? Le R represen he ransformed auocorrelaed defaul rae series and define vˆ o be he realized deviaions from he sample mean, vˆ R 1 R. In any sample, OLS esimaion will ensurevˆ 0 by consrucion. 1 he prior discussion esablished ha, in small samples, 1 ~ R is biased esimaor of and so he individual errors, vˆ, are biased as well. As he sample lengh increases, he sample mean converges o he rue uncondiional series average, and so he vˆ esimaes converge o he rue macro facor innovaions in equaions (12) nowihsanding he auocorrelaion in defaul raes. While his discussion formally esablishes he consisency of vˆ for an AR 1 process, i can be shown ha he resul also holds for any higher order saionary auoregressive process. hus far I have shown ha OLS provides consisen esimaes of he ransformed uncondiional defaul raes and scaled macro facor shocks from a long ime series of observaions when common facor realizaions are auocorrelaed, bu he discussion hus far does no provide any evidence on he lengh of he sample ha is needed o obain reasonably accurae esimaes when auocorrelaion is ignored in he esimaion. able 3 provides some evidence on he properies of he small sample disribuion of he sample mean esimae from wo auoregressive process ha could be represenaive of defaul rae dynamics. One process examined is he esimaed AR 1 process for Baa raed credis repored in able 2. he oher process is he empirical AR 1 model for he 18

19 Ba defaul rae also repored in able 2. o analyze he small sample properies of he sample mean esimae from a ime series of defaul raes, he AR process is simulaed 100 imes wih 121 observaions in each sample. he firs 21 observaions from each sample are omied o remove he effec of iniial condiions. able 3: Sampling Disribuion for he Simple Average Esimaed from a Sample Generaed by Alernaive Auoregressive Processes sample size Baa process: R = R -1 +e, e ~Φ(0,0.4191) average sd dev minimum maximum Ba process: R = R -1 +e, e ~Φ(0,1.40) average sd dev minimum maximum Sampling disribuion for he simple sample mean of wo auoregressive processes based on 100 boosrap replicaions of he indicaed sample size. he auoregressive process are he empirical AR (1) models for he Baa and Ba defaul rae processes wih parameer esimaes given in able 2. he rue uncondiional sample averages for he AR (1) process are for he Baa defaul rae process and, for he Ba defaul rae process. Each boosrap sample begins afer 21 excluded ieraions aenuae he effecs of iniial condiions. For each of he100 samples, esimaes of he sample mean from alernaive subsample lenghs are calculaed and he characerisics of he sampling disribuions are repored in able 3 for differen sample sizes. he sampling disribuions of he sample mean esimaor converge oward he rue uncondiional mean of each process, bu he rae of convergence is slow. Even wih 100 observaions in a ime series sample, he sample mean esimae sill exhibis bias and significan variabiliy. A comparison of he alernaive processes shows ha convergence is faser for mean esimae of he Baa 19

20 process. his resul is inuiive as he Baa process has weaker auocorrelaion and a smaller sandard error in is independen Gaussian innovaion. o summarize he discussion hus far, hese resuls demonsrae ha i is possible o generae consisen esimaes of each credi grades ransformed uncondiional defaul rae as well as consisen esimaes of he ime series of scaled common facor realizaions using a long ime series of realized defaul raes on a cross secion of porfolios of consisenly-raed credis. While hese parameers can be consisenly esimaed, esimaes derived from as much as 100 years of daa are likely o have subsanial variabiliy given he degree of auocorrelaion in observed Moody s Raed Corporae defaul rae daa. In general, he characerisics of he raing agency defaul rae daa are no conducive o producing highly accurae esimaes of each credi grade s uncondiional probabiliy of defaul or for producing highly accurae esimaes of he common macro facor realizaions. hus daa consrains place pracical limis on our abiliy o esimae hese parameer values wih a high degree of resoluion even wih 100 years of daa. V. PORFOLIO DEFAUL RAE DAA he parameers of he Vasicek asympoic porfolio defaul rae disribuion will be esimaed using annual defaul rae daa on six differen credi raing caegories for Corporae bonds over he period as repored by Moody s Invesors Corporaion (2009) (Moody s). For each of is credi raing grades, Aaa, Aa, A, Baa, Ba, B, and CaaC, Moody s publishes annual defaul rae performance daa. Defaul raes are calculaed as he number of issuers ha were in a credi grade a he beginning of a year 20

21 and defauled wihin he year, divided by he number of issuers ha were in he credi grade a he beginning of he year. 10 Moody s raes a large number of issuers in each year of he sample and mos of he raing caegories include a large number of bonds in each annual cohor alhough some defaul rae observaions are associaed wih relaively few bonds. 11 In he analysis, I assume ha each annual defaul rae observaion is an approximaion for he annual defaul rae on an asympoic porfolio of credis and I inerpre a Moody s credi raings as an indicaor of he issue s uncondiional probabiliy of defaul. 12 According o his inerpreaion, each credi raing grade represens a group of obligors ha have approximaely he same uncondiional defaul rae over an annual horizon. I assume ha he argeed uncondiional defaul raes associaed wih each individual credi grade are fixed over he sample period. While he resuls of his sudy only provide direc evidence on he asympoic Vasicek model (Basel II AIRB) fi relaive o long erm corporae bonds raed by 10 Moody s makes some numerical adjusmen for issuers whose raings were wihdrawn wihin he year. Moody s has argued ha his adjusmen has lile effec on he repored defaul rae daa, and for purposes of his analysis I will ignore any issues creaed by raings wihdrawals. 11 Moody s does no publish daa on he number of bonds in each raing grade and cohor for he enire sample period. Moody s (2009) does provide parial informaion on he number of bonds in a raing grade and his daa indicaed ha in some sample years, he CaaC grade included relaively few bonds. In he analysis ha follows, I will assess he imporance of he small porfolio size for CaaC credis by reporing esimaes excluding his caegory. I am indebed o Ma Prisker for calling my aenion o his issue when he discussed an earlier draf of his paper. 12 Moody s would argue ha a credi raing reflecs an assessmen of he expeced performance of an issue along muliple (unspecified) dimensions and does no represen a ranking based only on he probabiliy of defaul over a fixed horizon. his issue nowihsanding, i is common o inerpre a credi agency raing as an implici esimae of an issue s probabiliy of defaul. 21

22 Moody s, hese daa play an imporan role in he Basle II framework. No only did raing agency daa play an imporan role in he developmen of Basle II, bu under Basel II implemenaion sandards, banks ha lack a long ime series of daa on he defaul rae performance of heir own inernal credi raing sysems are permied o map or benchmark heir inernal sysems o agency raings, and use defaul rae daa published by he credi raing agencies o calibrae he probabiliy of defaul inpus for heir Basle AIRB regulaory capial calculaions. 13 Raing agency daa on corporae bonds, moreover, provide he longes daa series available on he defaul rae performance of issues ha were raed over ime according o a consisen se of crierion. he Moody s annual defaul raes for Aa, A, Baa, Ba, B and CaaC raed credis are ploed in Figure 2 in wo separae panels o accommodae differences in defaul rae scales. he Aaa-raing grade is excluded from he analysis because, according o he Moody s daa, no Aaa-raed credis defauled wihin he firs year afer being raed Aaa, and so he daa provide no informaion on he 1-year uncondiional defaul rae associaed wih an Aaa raing his raing agency mapping approach is described in he Basel Commiee on Banking Supervision (June 2004), page 94, paragraph An uncondiional defaul rae of 0 is no a realisic esimae as hese corporae credis cerainly have some associaed defaul risk even if defaul is a remoe even. his issue is discussed in more deail below. here are alernaive echniques ha can be used o infer he 1-year probabiliy of defaul on hese issues using ransiion marix esimaors. See for example, Lando and Skødeberg (2002). 22

23 Figure 2: Moody's Corporae Issuer-raed Defaul Raes: Aa A Baa Year Ba B Caa-C Year he plos in Figure 2 show ha he defaul raes on hese raings classes are posiively correlaed. able 4 repors he sample correlaion esimaes along wih he sample average annual defaul raes. While he defaul raes show evidence of reasonably srong posiive correlaion, he sample correlaions are no nearly as srong as hose ha are implied by he Vasicek asympoic model for porfolio defaul raes. While measuremen error (discussed below) is expeced o lower observed defaul rae 23

24 correlaions, he correlaion esimaes from he Moody s daa are far smaller han hose ha would be expeced under he ideal correlaions repored in able 1. able 4: Correlaion Among Moody's Corporae Bond Annual Defaul Raes for Alernaive Raings Grades, Aa A Baa Ba B CaaC Aa A Baa Ba B CaaC 1 average defaul rae (%) VI. DAA ISSUES Before esimaing he model, here are a number of daa issues ha meri discussion. he plos in Figure 2 show many observaions for which he repored annual defaul raes for a credi class are 0, and indeed here are 12 years of daa on which here are no recorded defauls in any of he credi raing classes wihin ha year. 15 Under he assumpions of he Vasicek model, here is virually no probabiliy ha an asympoic porfolio wih a posiive uncondiional probabiliy of defaul should experience 0 defauls, and ye in almos 14 percen of he sample years, here are no recorded defauls on any raed credis. Similarly, a 100 percen defaul rae should be an exremely rare occurrence and, according o he Vasicek model, mus coincide wih very high defaul raes on all porfolios conemporaneously, a paern which is no exhibied in he sample. he prevalence of zeros in he Moody s defaul rae daa (as well as he 100 percen defaul rae repored for CaaC credis in 1984) may no be inconsisen wih he 15 here are no recorded defauls in any of he credi raing grades in 1946, 1948, 1950, 1952, 1953, 1956, 1958, 1959, 1964, 1965, 1967, and

25 Vasicek-AIRB model if he Moody s raing grade porfolios are no ruly asympoic porfolios, and surely hey are no. he number of credis in each raing class is limied and so he observed defaul raes include measuremen error. able 5: Poenial Measuremen Error and Porfolio Size number of obligors in a credi grade upper bound on he magniude of measuremen error bps bps bps bps bps bps o undersand he measuremen error issue, consider a porfolio of 1000 independen obligors in a single credi grade ha did no experience a defaul wihin a year. his porfolio is cerainly no an asympoic porfolio even hough i is likely o be well-diversified by any pracical sandard. Consider he measured defaul rae on his porfolio when we add a single credi and he new credi subsequenly defauls. his hough experimen provides he upper bound on his porfolio s defaul rae, or roughly 10 basis poins. While he observed defaul rae is 0, he rue unobserved defaul rae could be as large as 10 basis poins given he informaion in he porfolio. able 5 illusraes he relaionship beween he number of independen obligors in a credi grade and he magniude of he upper bound on he poenial measuremen error associaed wih he credi grade observed defaul rae. Concepually, rue asympoic defaul raes of zero happen wih zero probabiliy, and ye he observed defaul raes may be zero simply because he porfolios we observe 25

26 do no include enough credis. he number of credis in a zero defaul rae porfolio can be used o esimae he upper bound on ha porfolio s defaul rae for he year, bu his esimae almos surely oversaes he rue unobserved asympoic porfolio defaul experience. Defaul raes of zero are problemaic for esimaion purposes as well. he inverse normal ransformaion in equaion (6) will no accommodae defaul raes of 0 (he ransformaion resuls in a value of ) or 100 percen ( ) and so hese exreme defaul rae observaions mus be runcaed for esimaion. here are many repored defaul raes of 0 in he sample and so he runcaion value assigned o 0 defaul rae observaions could have a measurable effec on he model esimaes. I repor he esimaion resuls using alernaive lower bound values for porfolio defaul raes. In conras o he 0 defaul rae problem, here is only one defaul rae of 100 percen in he sample. 16 A 100 percen defaul rae is also he likely resul of measuremen error because he porfolio is no ruly asympoic. Because here is only one observaion wih a 100 percen defaul rae, he runcaion value ha is seleced for ha observaion (wihin reasonable bounds) has lile effec on he resuls I repor. Regardless, I also runcae his defaul rae using he rule: 100 percen minus he lower bound used o runcae 0 defaul raes in he sample. 17 In addiion o he issue of selecing a lower bound for observed defaul raes, anoher poenially imporan issue for esimaion is how o handle he 12 years of daa for which here are no observed defauls in any raing caegory. hese years almos 16 he defaul rae repored for he CaaC grade in 1984 is 100 percen. 17 For example, if 0 defaul raes are runcaed o.0001, hen he single 100 percen defaul rae observaion is runcaed o or percen. 26

27 cerainly represen years when here was a very srong economy (i.e., a large posiive draw from he common Gaussian facor) which is imporan informaion, bu his informaion alone is no very informaive as i does no idenify how good he economy was, nor can i be used o idenify uncondiional defaul rae differences among he credi raing grades. While he daa can be included in he esimaion sample wih an addiional resricion ha he ime dummy variable akes on he same value for each of hese years (since hey are all equally good according o he daa), including hese daa wih uniformly runcaed defaul raes may cause some addiional bias issues. For example, including all hese daa poins a a common lower defaul rae boundary will no only aler he esimae for he uncondiional probabiliy of defaul, bu i also may impar an upward bias o he esimae of he Vasicek correlaion parameer. o address his poenial source of bias, one migh be emped o exclude hese 12 years of daa from he esimaion sample. he Vasicek-AIRB model does no include any ime dependence in he Gaussian facor srucure, so he omission of hese daes does no cause any dynamic issues in he model. However, esimaion using a censored sample would impar an upward bias on he esimaes of he uncondiional defaul raes and a downward bias on he esimaes of he model correlaion parameer. So while he censored sample approach canno be expeced o produce unbiased esimaors of he model parameers, i sill may be useful o assess he sensiiviy of parameer esimaes o he inclusion/exclusion of he 0 defaul years. VII. MODEL ESIMAION AND ESING Panels A hrough D of able 6 repor model parameer esimaes under alernaive assumpions regarding he lower bound on porfolio defaul raes. In Panels A hrough C 27

28 he model coefficien esimaes ha correspond o credi grade covariaes are saisically significan and monoonically increasing (from grade Aa o grade CaaC), a paern ha is expeced under he Vasicek model if he probabiliy of defaul for a credi grade increases monoonically as raings progress from Aa o CaaC. he monoonic paern does no hold for he esimaes in Panel D when defaul raes ha repored o be 0 are runcaed a 50 bps probabiliy of defaul. he esimaes in Panels A, B, and C of able 6 show, unsurprisingly, ha as he lower bound on he porfolio defaul rae is increased, he esimaed implied uncondiional probabiliy of defaul increases across all credi grades. A he same ime, he esimaes in Panels A hrough C show ha progressive increases in he lower bound on porfolio defaul raes resuls in a reducion in he esimaed value of he Vasicek model correlaion parameer. he overall effec on correlaion can be subsanial; he correlaion parameer esimae falls from almos 20 percen when he runcaion value is 1 basis poin, o 5.5 percen when he runcaion value is assumed o be 20 basis poins. he resuls in able 6 demonsrae ha he imporance of he reamen accorded 0 defaul rae observaions when esimaing he parameers of an asympoic porfolio risk model. he issue of how bes o selec an opimal lower bound for porfolio defaul raes remains an imporan open issue. My earlier discussion suggess ha he runcaion rae should be relaed o he poenial measuremen error in he daa which in urn will depend on how many credis are in each raing grade in each annual cohor. 18 Based on he number of credis in he individual Moody s raing caegories in mos years, a runcaion value of 20 basis poins or larger is probably necessary o achieve a conservaive esimae 18 his daa is no publicly available for he enire sample, and furher analysis of his issue, while imporan, is beyond he scope of his sudy. 28

29 of a raings grade uncondiional probabiliy of defaul as few cohor porfolios in he sample rouinely include as many as 500 separae obligor raed credis. able 6: Asympoic Vasicek Model Esimaes based on Moody's Corporae Bond Raing Annual Performance Daa Panel A: 0 defaul raes runcaed o 1 bps Moody's implied raing parameer sandard saisic * uncondiional grade esimae error PD in bps Aa A Baa Ba B CaaC ρ Panel B: 0 defaul raes runcaed o 10 bps Aa A Baa Ba B CaaC ρ Panel C: 0 defaul raes runcaed o 20 bps Aa A Baa Ba B CaaC ρ Panel D: 0 defaul raes runcaed o 50 bps Aa A Baa Ba B CaaC ρ Parameer esimaes are generalized leas squares esimaes of equaion (12) based on 89 years of Moody's annual defaul rae daa. All repored es saisics are significanly differen from zero a he.0001 level of he es. 29

30 able 7: Asympoic Vasicek Model Esimaes based on Moody's Corporae Bond Raing Annual Performance Daa Excluding Years No Raed Bond Defauls Panel A: 0 defaul raes runcaed o 1 bps Moody's implied raing parameer sandard saisic * uncondiional grade esimae error PD in bps Aa A Baa Ba B CaaC ρ Panel B: 0 defaul raes runcaed o 10 bps Aa A Baa Ba B CaaC ρ Panel C: 0 defaul raes runcaed o 20 bps Aa A Baa Ba B CaaC ρ Panel D: 0 defaul raes runcaed o 50 bps Aa A Baa Ba B CaaC ρ Parameer esimaes are generalized leas squares esimaes of equaion (12) based on 77 years of Moody's annual defaul rae daa. Years in which here are no defauls among he bonds raed by Moody's are excluded from he esimaion sample. All repored es saisics are significanly differen from zero a he.0001 level of he es. 30

31 able 7 repors model parameer esimaes when he esimaion daa se excludes he 12 years of daa for which here are no defauls recorded in any credi raing class. hese esimaes also imply a monoonic (inverse) relaionship among credi qualiy (Aa highes qualiy, CaaC lowes qualiy) and he annual uncondiional probabiliy of defaul on asympoic porfolio of credis provided he runcaion value assigned o zero defaul rae observaions is less han 50 basis poins. he implied Vasicek model correlaion parameer varies from a high of 15.8 percen when he upper bound on measuremen error is assumed o be 1 basis poin, o a low of 2.6 percen when he runcaion value for zero defaul rae observaions is se o 50 basis poins. Moody s (2008) repors he number of issuers in each annual raing caegory cohor from hese daa show ha he CaaC raings caegory includes relaively few issuers in many years of he sample, and so hese CaaC defaul rae daa include relaively large measuremen errors relaive o he rue asympoic porfolio defaul raes for CaaC-raed porfolios and oher credi grade defaul raes repored by Moodys. o assess he imporance of his source of measuremen error, I re-esimae he model excluding he CaaC raings grade. When he CaaC daa are excluded from he model, here are addiional years in which here are no recorded defauls in he Aa, A, Baa, Ba, or B raing grades he addiional years are 1945, 1947, 1951, 1954, and

32 able 8: Asympoic Vasicek Model Esimaes based on Moody's Corporae Bond Raing Annual Performance Daa Excluding CaaC Raing Grade Panel A: 0 defaul raes runcaed o 1 bps Moody's implied raing parameer sandard -saisic uncondiional grade esimae error PD in bps Aa A Baa Ba B ρ Panel B: 0 defaul raes runcaed o 10 bps Aa A Baa Ba B ρ Panel C: 0 defaul raes runcaed o 20 bps Aa A Baa Ba B ρ Parameer esimaes are generalized leas squares esimaes of equaion (12) based on 89 years of Moody's annual defaul rae daa. All repored es saisics are significanly differen from zero a he.0001 level of he es. able 8 repors esimaes of he Vasicek-AIRB model parameers when he CaaC raings daa are excluded from he esimaion daa se. he esimaes repored in able 8 are no maerially differen from he full sample esimaes repored in able 6, and consequenly I conclude ha he measuremen error bias inroduced by including CaaC credis in he esimaion is no of firs-order imporance relaive o all he oher esimaion issues one mus face when aemping o esimae his model. Since here are benefis o 32

33 be gained from having an esimae of he CaaC uncondiional probabiliy of defaul, I include he CaaC daa in he remaining analysis. Sandard Errors of Vasicek Model Parameer Esimaes he Vasicek model parameer esimaes are nonlinear ransformaions of he resriced OLS parameer esimaes, and so he sandard error of hese esimaes mus be obained from an auxiliary analysis. In order o esimae he criical values and esimaes of he variabiliy of he Vasicek model parameer esimaes, I consruc a boosrap sampling disribuion for he parameer esimaes when zero defaul raes are runcaed o wo differen values: 10 basis poins and 20 basis poins. 20 I draw 5000 paired samples (wih replacemen) from he underlying esimaion sample of 89 observaions, and for each boosrap sample, I esimae he Vasicek model parameers and hereby build he sampling disribuion for he esimaes based on 89 observaions. By using paired draws, sampling boh he dependen as well as he independen variables simulaneously, I preserve heeroskedasiciy feaures of he daa and incorporae he consequences hereof in he sampling disribuion (and sandard deviaions) of he parameer esimaes Efron (1979) and oher paper ha develop boosrap and jackknife echniques appear in he References. 21 In oher words, he boosrapped sampling disribuions are robus o any heeroskedasiciy in he daa. 33