Modeling a distribution of mortgage credit losses Petr Gapko 1, Martin Šmíd 2

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1 Modeling a disribuion of morgage credi losses Per Gapko 1, Marin Šmíd 2 1 Inroducion Absrac. One of he bigges risks arising from financial operaions is he risk of counerpary defaul, commonly known as a credi risk. Leaving unmanaged, he credi risk would, wih a high probabiliy, resul in a crash of a bank. In our paper, we will focus on he credi risk quanificaion mehodology. Generalizing he well known KMV model, sanding behind Basel II, we build a model of a loan porfolio involving a dynamics of he common facor, influencing he borrowers asses, which we allow o be non-normal. We show how he parameers of our model may be esimaed by means of pas morgage deliquency raes. We give a saisical evidence ha he non-normal model is much more suiable han he one assuming he normal disribuion of he risk facors. Keywords: Credi Risk, Morgage, Delinquency Rae, Generalized Hyperbolic Disribuion, Normal Disribuion JEL Classificaion: G21 In our paper, we will focus on credi risk quanificaion mehodology. The minimum sandards for credi risk quanificaion are ofen heavily regulaed. The curren recommended sysem of financial regulaion is formalized in he Second Basel Accord ( Basel II, Bank for Inernaional Selemens, 2006). Basel II is a documen describing minimum principles for risk managemen in he banking secor. I is applicable all over he world, and in he European Union i is implemened ino European law by he Capial Requiremens Direcive (CRD) (European Commission, 2006). The regulaion is designed in he way ha banks are required o cover wih a sock of capial a cerain quanile loss from a cerain risk (i.e. from a risk ha counerpary wouldn pay back is liabiliies). Banks usually cover a quanile ha is suggesed by a raing agency, bu wih he condiion ha hey have o observe he regulaory level of probabiliy of 99.9% a minimum. The regulaory level may seem a bi excessive, as i can be inerpreed as meaning ha banks should cover a loss which occurs once in a housand years. The fac is ha such a far ail in he loss disribuion was chosen because of an absence of daa. The quanile loss is usually calculaed by a Value-a-Risk ype model (Saunders & Allen, 2002; Andersson e al., 2001). In his paper, we will inroduce a new approach o quanifying credi risk wih a focus on a morgage porfolio which can be classed wih he Value-a-Risk models. Our approach is differen from he regulaory mehod in he assumpion of he loss disribuion. In he general version of our model, we assume ha risk facors ha drive losses can be disribued no only sandard normal assumed by he regulaory framework bu can follow a more general disribuion in ime, he disribuion of he common facor possibly depending on is hisory (allowing us o model a dynamics of he facor which appeared o be necessary especially during periods like he presen financial crisis). To es our model, we will demonsrae is goodness-of-fi on a naionwide morgage porfolio. Moreover, we will compare our resuls wih he regulaory approach. The paper is organized as follows. Afer he inroducion we will describe he usual credi risk quanificaion mehods and Basel II-embedded requiremens in deail. Then we will derive a new mehod of measuring credi risk, based on he class of generalized hyperbolic disribuions and Value-a-Risk mehodology. In he las par, we will focus on he daa descripion and verificaion of he abiliy of he class of generalized hyperbolic disribuions o capure credi risk more accuraely han he regulaory approach. A he end we summarize our findings and offer recommendaions for furher research. 2 Credi risk measuremen mehodology The Basel II allows wo possible quanificaion mehods for he credi risk: he Sandardized Approach (STA) and he Inernal Raing Based Approach (IRB). The IRB approach is more advanced han STA and is based on a Vasicek-Meron credi risk model (Vasicek, 1987) The main difference beween STA and IRB is ha under 1 Insiue of Informaion Theory and Auomaion, Academy of Sciences of he Czech Republic, Insiue of Economic Sudies, Faculy of Social Sciences, Charles Universiy, Prague 2 Insiue of Informaion Theory and Auomaion, Academy of Sciences of he Czech Republic 150

2 IRB banks are required o use inernal measures for boh he qualiy of he deal (measured by he counerpary s probabiliy of defaul PD ) and he qualiy of he deal s collaeral (measured by he deal s loss given defaul LGD ). The PD is he chance ha he counerpary will defaul (or, in oher words, fail o pay back is liabiliies) in he upcoming 12 monhs. A common definiion of defaul is ha he debor is more han 90 days delayed in is paymens (90+ days pas due). LGD is an esimae of how much of an already defauled amoun a bank would lose. PD is usually obained by one of he following mehods: from a scoring model (Moody's KMV, JP Morgan CrediMerics, ec.), from a Meron-based disance-o-defaul model (mainly used for commercial loans; Meron, 1973 and 1974) or as a long-erm sable average of pas 90+ delinquencies. Two basic measures of credi risk are expeced and unexpeced losses. The expeced loss is he mean loss in he loss disribuion, whereas he unexpeced loss is he difference beween he expeced loss and a chosen quanile loss in he loss disribuion. The expeced (average) loss ha could occur in he following 12 monhs is calculaed as follows: where EAD is he exposure-a-defaul 3 and EL is he abbreviaion for Expeced Loss. For calculaions of unexpeced losses, i is usually assumed ha losses follow a cerain disribuion in ime. The regulaory IRB framework uses for his purpose a mix of disribuions of wo risk facors, one individual for each borrower and one common for all borrowers. Boh facors are assumed o follow a sandard normal disribuion and o be correlaed wih a cerain assigned value of he correlaion coefficien. 3 Our approach The usual approach o modelling he loan porfolio value is based on he famous paper by Vasicek (2002) assuming ha he value or he -h's borrower's asses a he ime one can be represened as where is he borrower's wealh a he ime zero, and are consans and is a (uni normal) random variable, which may be furher decomposed as (1) (2) where is a facor, common for all he borrowers, and is a privae facor, specific for he borrower (see Vasicek (2002) for deails). The generalizaion We generalize he model in wo ways: we assume a dynamics of he common facor and we allow nonnormal disribuions of boh he common and he privae facors. Similarly o he original model, we assume ha where is he wealh of he -h borrower a he ime, is a random variable specific o he borrower and is he common facor following a general (adaped) sochasic process (such an assumpion makes sense, for insance, if models a macroeconomic variable or a price on a capial marke). We assume all o be muually independen and idependen of, such ha all,, are idenically disribuied wih,,, having a sricly increasing coninuous cummulaive disribuion funcion (here, n is he number of borrowers). Noe ha we do no require incremens of o be cenered (which may be regarded a compensaion for he erm presen in (1) bu missing in (2). (3) 3 Exposure-a-defaul is a Basel II expression for he amoun ha is (a he momen of he calculaion) exposed o defaul. 151

3 Percenage loss in he generalized model Denoe he hisory of he common facor up o he ime Analogously o he original model, he condiional probabiliy of he banrkupcy of he -h borrower a he ime given equals o where are he borrower's debs. Denoe he percenage loss of all he porfolio of he borrowers a he ime. Afer aking he same seps as Vasicek (1991) (wih condiional non-normal c.d.f. s insead of he uncondiional normal ones), we ge, for a very large porfolio wih homogeneous (in he sense ha, borrowers ha which furer implies ha hence (4) he laer formula deermining roughly he dynamics of he process of he losses, he former one allowing us o do saisical inference of he common facor based on he ime series of he percenage losses. To see ha he Meron-Vasicek model is a special version of he generalized model, see he Appendix. In our version of he model we assume Z o be normally disribued and he common facor o be muliplicaively defined by Y = ( 1+ ) Y where,, 1 2 are i.i.d. (possibly non-normal) variables (noe ha our choice of he dynamics corresponds o he assumpion of i.i.d. reurns if he common facor sands for prices of a financial insrumen). Since he equaion (3) may be rescaled by he inverse sandard deviaion of Z wihou loss of generalliy, we may assume ha is he sandard normal disribuion funcion. By (4), we ge ha Y Y = Y = ψ ( L ) ψ ( L = ψ ( L ) Y which allows us o use he sample 1, 2, o esimae parameers of 1 (and consequenly he disribuion of he losses). As i was already said, we assume he disribuion of 1 o be generalized hyperbolic and we use he ML esimaion o ge is parameers. Moreover, we compare our choice o several oher classes of disribuions. 4 Daa and resuls 4.1 The class of generalized hyperbolic disribuions Our model is based on he class of generalized hyperbolic disribuions firs inroduced in Barndorff-Nielsen e al. (1985). The advanage of his class of disribuions is ha i is general enough o describe fa-ailed daa. I has been shown (Eberlein, 2001, 2002, 2004) ha he class of generalized hyperbolic disribuions is beer able o capure he variabiliy in financial daa han he normal disribuion, which is used by he IRB approach. Generalized hyperbolic disribuions have been used in an asse (and opion) pricing formula (Rejman e al., 1997; Eberlein, 2001; Chorro e al., 2008), for he Value-a-Risk calculaion of marke risk (Eberlein, 2002; Eberlein, 1995; Hu & Kercheval, 2008) and in a Meron-based disance-o-defaul model o esimae PDs in he banking porfolio of commercial cusomers (e.g., Oezkan, 2002). To verify ha our model based on he class of generalized hyperbolic disribuions is able o beer describe he behavior of morgage losses, we will use daa for he 0 (5) ) (6) 152

4 US morgage marke. The daase consiss of quarerly observaions of 90+ delinquency raes on morgage loans colleced by he US Deparmen of Housing and Urban Developmen and he Morgage Bankers Associaion.4 The rae used is he bes subsiue for losses ha banks faced from heir morgage porfolios, relaxing he LGD variable. The daase begins wih he firs quarer of 1979 and ends wih he hird quarer of The developmen of he US morgage 90+ delinquency rae is illusraed in Figure 1. We observe an unprecedenedly huge increase in he 90+ delinquency rae beginning wih he second quarer of Resuls Figure 1 Developmen of US 90+ delinquency rae We considered several disribuions for describing he disribuion of 1 ( L ) (hence of 1 ), namely loglogisic, logisic, lognormal, Pearson, inverse Gaussian, normal, gamma, exreme value, bea and he class of generalized hyperbolic disribuions. The daase used for disribuion fiing was consruced from he above described daa by using formula (6) from he previous par. In he se of disribuions compared, we were paricularly ineresed in he goodness-of-fi of he class of generalized hyperbolic disribuions and heir comparison o oher disribuions. For he fiing procedure we used he R package ghyp. We used he chi-square goodness-of-fi es. In general, only five from he considered disribuions were no rejeced o describe he daase based on he chisquare saisic (on a 95% level). Beside he chi-square saisic, we used a differen saisic o compare all he esed disribuions and sor hem by heir values: he Anderson-Darling saisic (Anderson & Darling, 1952) and he Wassersein disance. All he saisics are measures of he disance beween he original sample and he esed disribuion. The following able summarizes our resuls. I includes disribuions ha were no rejeced based on he chi-square saisic. The able is sored by he Anderson-Darling saisic: Disribuion Wassersein meric Anderson Darling Chi-square saisic P-value of chi-square Generalized hyperbolic LogLogisic Logisic Inverse Gaussian Normal Table 1 Comparison of goodness-of-fi of esed disribuions According o he Table 1, boh he chi-square and Anderson-Darling saisics show ha he generalized hyperbolic disribuion (GHD) has he bes fi. Our calculaions show ha he class of generalized hyperbolic disribuions is able o describe he behavior of delinquencies much beer han he oher disribuions widely used for risk assessmen (normal, lognormal, logisic, gamma), even if we considered he dynamics of he common facor when using hem. This fac can have a large impac on he economic capial requiremen, as he class of generalized hyperbolic disribuions is heavy-ailed and hus would imply a need for a larger sock of capial o cover a cerain percenile delinquency. We will now demonsrae he difference beween he economic capial requiremens calculaed under he assumpion ha morgage losses follow a generalized hyperbolic disribuion and under he Basel II IRB mehod (assuming sandard normal disribuions for boh risk facors and a 15% correlaion beween he facors 5 ). 4 The Morgage Bankers Associaion is he larges US sociey represening he US real esae marke, wih over 2,400 members (banks, morgage brokers, morgage companies, life insurance companies, ec.). 5 The correlaion 15% is a benchmark se for he morgage exposures in he Basel II framework and hus we will use his benchmark for our compuaions. 153

5 4.3 Economic capial a he one-year horizon: implicaions for he crisis In his secion, we compare he capial requiremen calculaed by he IRB regulaory approach (assuming ha boh risk facors are driven by he sandard normal disribuion) and our dynamic framework wih a generalized hyperbolic disribuion. To show he difference beween he regulaory capial requiremen (calculaed by he IRB mehod) and he economic capial requiremen calculaed by our model, we will perform he economic capial requiremen calculaions a he 99.9% probabiliy level as well. When consrucing loss forecass, we faced he following problem: we esimaed he generalized hyperbolic disribuions on quarerly observaions and hus we needed o ransform he quarerly changes obained o yearly figures. In paricular, o forecas a yearly loss, we may repeaedly use (4) o ge L ϕ ϕ ε + 4 = ( ( L ) + + 1) 1 i 4 which leads o complicaed inegral expressions. We herefore decided o use simulaions o obain yearly figures. We were paricularly ineresed in he following: he capial requiremen based on average loss and he capial requiremen based on las experienced loss. The average loss is calculaed as a mean value from he original daase of 90+ delinquencies and serves as a hrough-he-cycle PD esimae. This value is imporan for he regulaory-based model (Basel II) as a hrough-he-cycle PD should be used here. The las experienced loss is, on he second hand, imporan for our model wih GHD disribuion due o he dynamical naure of he model. The nex Table summarizes our findings. To illusrae how our dynamic model would predic if he sandard normal and he normal disribuions were used, we added his version of he dynamic model as well. Model Basel II IRB (hroughhe-cycle PD) Our dynamic model wih normal disribuion Our dynamic model wih GHD Disribuion used for Sandard Normal Sandard Normal Sandard Normal he individual facor Disribuion used for Sandard Normal Normal Generalized Hyperbolic he common facor 99.9% loss % % % Table 2 Comparison of Basel II, Dynamic Normal and Dynamic GHD models ail losses The firs column in he Table 2 relaes o he IRB Basel II model, i.e. a model wih a sandard normal disribuion describing he behavior of boh risk facors and he correlaion beween hese facors se a 15%. The second column conains resuls from he dynamic model where a sandard normal disribuion of he individual risk facor is supplemened by he normal disribuion, which describes he common facor and is parameers were esimaed in he same way as hose of GHD. The las column is relaed o our dynamic model where he GHD is assumed for he common facor. The resuls in he Table 2 show ha he dynamic model, based on he las experience loss, predics much higher quanile losses in boh cases. However, heavy ails of he GDH disribuion furher evoke higher quanile losses, which a he end lead o a higher capial requiremen. 5 Conclusion We have inroduced a new model for quanificaion of credi losses. The model is a generalizaion of he curren framework developed by Vasicek and our main conribuion lies in wo main aribued: firs, our model brings dynamics ino he original framework and second, our model is generalized in ha sense ha any saisical disribuion can be used o describe he behavior of risk facors. To illusrae ha our model is able o beer describe pas risk facor behavior and hus beer predics fuure need of capial, we compared he performance of several disribuions common in credi risk quanificaion. In his sense, we were paricularly ineresed in he performance of he class of Generalized Hyperbolic disribuions, which is ofen used o describe heavy-ail financial daa. For his purpose, we used a quarerly daase of morgage delinquency raes from he US financial marke. Our suggesed class of Generalized Hyperbolic disribuions showed much beer performance, measured by he Wassersein and Anderson-Darling merics, han oher classic disribuions like normal, logisic or gamma. In he nex secion, we have compared our dynamic model wih he curren risk measuremen sysem required by he regulaion. The curren banking regulaion uses he sandard normal disribuion as an underlying disribuion ha drives risk facors for credi risk. Our resuls show ha he mix of sandard normal disribuions used in he Basel II regulaory framework is, a he 99.9% level of probabiliy, underesimaing he poenial unexpeced loss on he one-year horizon. Therefore, inroducing he dynamics may lead o a beer capuring of ail losses. Wihin our dynamic model we have 154

6 furher compared he predicions based on he normal and he class of generalized hyperbolic disribuions. Our resuls show ha he heavy-ailed generalized hyperbolic disribuion predics he bigges ail loss. Acknowledgemens Suppor from he Czech Science Foundaion under grans 402/09/H045 and 402/09/0965, and GAUK is graefully acknowledged. References [1] Abramowiz, S. (1968). Handbook of Mahemaical Funcions New York: Dover publishing. [2] Andersson, F., Mausser, H., Rosen, D., & Uryasev, S. (2001). Credi Risk Opimizaion wih Condiional Value-a-Risk Crierion Mahemaical Programming, [3] Anderson, T. W., & Darling, D. A. (1952). Asympoic heory of cerain "goodness-of-fi" crieria based on sochasic processes. Annals of Mahemaical Saisics 23, [4] Bank for Inernaional Selemens (2006). Basel II: Inernaional Convergence of Capial Measuremen and Capial Sandards: A Revised Framework. Rerieved from hp:// [5] Barndorff-Nielsen, O. E., Blæsild, P., & Jensen, J. L. (1985). The Fascinaion of Sand. A Celebraion of Saisics, [6] Chorro, C., Guegan, D., & Ielpo, F. (2008). Opion Pricing under GARCH Models wih Generalized Hyperbolic Innovaions (II): Daa and Resuls. Paris: Sorbonne Universiy. [7] Eberlein, E. (2001). Applicaion of Generalized Hyperbolic Lévy Moions o Finance. Lévy Processes: Theory and Applicaions, [8] Eberlein, E. (2002). The Generalized Hyperbolic Model: Financial Derivaives and Risk Measures. Mahemaical Finance-Bachelier Congress 2000, [9] Eberlein, E. (2004). Generalized Hyperbolic and Inverse Gaussian Disribuions: Limiing Cases and Approximaion of Processes. Seminar on Sochasic Analysis, Random Fields and Applicaions IV, Progress in Probabiliy 58, [10] Eberlein, E., & Keller, U. (1995). Hyperbolic Disribuions in Finance. Bernoulli, 1, [11] European Commission (2006). DIRECTIVE 2006/49/EC OF THE EUROPEAN PARLIAMENT AND OF THE COUNCIL of 14 June European Commission. [12] Hu, W., & Kercheval, A. (2008). Risk Managemen wih Generalized Hyperbolic Disribuions. Florida Sae Universiy. [13] JP Morgan. (n.d.). CrediMerics. [14] Meron, R. C. (1974). On he Pricing of Corporae Deb: The Risk Srucure of Ineres Raes. Journal of Finance 29, Chaper 12. [15] Meron, R. C. (1973). Theory of Raional Opion Pricing. Bell Journal of Economics and Managemen Science, 4, Chaper 8. [16] Moody's. (n.d.). Moody's KMV. Rerieved from [17] Oezkan, F. (2002). Lévy Processes in Credi Risk and Marke Models. Universiy of Freiburg. [18] Rejman, A., Weron, A., & Weron, R. (1997). Opion Pricing Proposals under he Generalized Hyperbolic Model. Sochasic Models, Volume 13, Issue 4, [19] Saunders, A., & Allen, L. (2002). Credi Risk Measuremen: New Approaches o Value a Risk and Oher Paradigms. John Wiley and Sons. [20] Vasicek, O. A. (1987). Probabiliy of Loss on Loan Porfolio. KMV. [21] Vasicek, O. A. (1991). Limiing loan loss probabiliy disribuion KMV [22] Vasicek, O. A. (2002). The disribuion of loan porfolio value, RISK 15,