STRUCTURAL MODELS IN CONSUMER CREDIT

Transcription

1 STRUCTURAL MODELS IN CONSUMER CREDIT 1 Fabio Wendling Muniz de Andrade EAESP-FGV and Serasa fandrade@serasa.com.br Lyn Thomas Universiy of Souhampon l.homas@soon.ac.uk ABSTRACT We propose a srucural credi risk model for consumer lending using opion heory and he concep of he value of he consumer s repuaion. Using Brazilian empirical daa and a credi bureau score as proxy for crediworhiness we compare a number of alernaive models before suggesing one ha leads o a simple analyical soluion for he probabiliy of defaul. We apply he proposed model o porfolios of consumer loans inroducing a facor o accoun for he mean influence of sysemic economic facors on individuals. This resuls in a hybrid srucural-reducedform model. And comparisons are made wih he Basel II approach. Our conclusions parially suppor ha approach for modelling he credi risk of porfolios of reail credi. 1 INTRODUCTION Srucural models for credi risk assessmen were inroduced by Meron (1974). In his approach he sochasic behaviour of he value of a firm s asses is modeled and if he value becomes lower han a hreshold, usually a proporion of he firm s deb value, he company is considered o be in defaul. Meron s model assumes a diffusion process for a firm s asse value and ha he firm will defaul if is asse value is lower han is deb on he mauriy dae of he deb. Following his work he model was developed in many ways, (Saunders, 1999), including variaions in he iming of when defaul occurs (Black and Cox, 1976) and in he sochasic process ha drives he value of a firm asses (Zhou, 1997). Srucural models for corporae credi give a heory of he causes of defaul based on financial opion reasoning a a micro economic level. The shareholders of he firm have a call opion on he firm s asses wih a srike price equal o he firm s deb. If he value of he firm decreases below he value of he deb i will no be worh exercising he opion and he firm will defaul. As alernaives he reduced-form and inensiy-based models (Duffie & Singleon (1999); Jarrow, Lando & Turnbull (1997); Jarrow & Turnbull (1995)) consider defaul o be a random exogenous even and ry o model he iming or inensiy of occurrence of defaul evens wihou worrying abou is causes. Applicaion of srucural models in reail and specifically in consumer credi is more of a challenge, since i is difficul o measure a consumer s asses (even for he consumers hemselves) nor is i necessarily he case ha defaul occurs when a consumers debs exceed heir asses. So o develop a srucural model for his segmen i is necessary firs o propose a defaul heory for consumer credi ha can use available informaion on consumers. Perli and Nayda (2004) propose a srucural model for revolving reail credi ha uses he exacly he same approach of he corporae models, considering ha a

2 2 consumer is in defaul if his asses are lower han a hreshold. Then, following Vasicek (1991), hey generae an analyic soluion for he cumulaive disribuion of losses in he porfolio. However, as a grea deal of consumer credi is unsecured and i is no he case ha a consumer in defaul will lose he righs over all his asses, jus ransposing he corporae defaul models o consumer defaul can lead o some aspecs of consumer defaul being missed. The New Basel Accord uses a formula for capial requiremen in reail porfolios ha is derived from Meron s model for corporae credi. Academics and pracioners ha work wih consumer credi had srong doubs concerning he applicabiliy of ha framework o consumer credi as poined ou by Thomas (2003). There are also some works in srucural modelling in consumer credi ha are no relaed o individual risk assessmen or porfolio modelling. Examples are Longhofer and Peers (2004), who sudies lending discriminaion and self-selecion and Ahreya (2004) who analyse he relaion beween he imporance of he sigma of bankrupcy and bankrupcy raes. Our objecive is o develop a srucural approach for consumer credi and o compare he Basel II capial requiremen formula for consumer credi wih such an approach. In secion 2 we esablish an opion-based reasoning for consumer defaul and propose a heory for defaul in consumer credi. In secion 3 we develop a defaul predicion model for consumer credi using he srucural approach. We also propose he use of a behaviour or credi bureau score as proxy of crediworhiness ha will be he consumer s equivalen o corporae asse values in srucural models. Based on Brazilian empirical daa we es differen alernaives for modelling he sochasic behaviour of he crediworhiness proxy and compare he resuls of hese approaches o risk discriminaion wih he radiional scoring approach for risk assessmen. In secion 4 we use he srucural approach o model he disribuion of defaul rae in a porfolio. We propose a mehodology o inser a sysemic risk facor ha will accoun for join movemens of defauls due o economic condiions. This procedure makes our approach for porfolio modelling an hybrid srucural-reduced-form model. We compare resuls obained by he proposed model wih he ones obained using Vasicek s model (1991) ha is he basis of he Basel II formula for capial requiremen. 2 PROPOSITION OF A THEORY FOR CONSUMER DEFAULT The model of consumer defaul proposed here is based on he following premises: 1. There is a unobservable sochasic quaniy Q i, he crediworhiness of consumer i ha comprises all informaion abou consumer i ha is relevan for credi risk assessmen. 2. Alhough Q i is no direcly observable, he marke (lending insiuions), use informaion inernally or exernally available abou he consumer, such as pas credi experiences, financial condiions of he consumer, credi repors o esimae

3 i or a surrogae for i. Mos of he ime, he lending insiuion uses a proxy of Q i for credi assessmen such as a credi or behavioural scoring. 3. The probabiliy of he consumer being acceped as a clien by a lending insiuion, P ai, is a sricly increasing funcion of Q i, P ai = f(q i ). 4. There are credi agencies, credi bureaus and oher mechanisms of making defaul informaion available o any lending insiuion. If a consumer is in defaul all lending insiuions will know i and he consumer will lose his repuaion and have no more access o credi in he immediae fuure. We say ha when his happens, he consumer loses his repuaion. 5. Access o credi has a value for he consumer. This value, which is relaed o he exen of access o credi ha he consumer has, is called he value of he consumer s repuaion, R i. This value is a sricly increasing funcion of P ai, R=h(P ai ). Using hese assumpions, i is possible o provide an opion-based reasoning for he process of defaul in consumer credi ha is similar o Meron s approach for corporae credi. The consumer has a call opion on his repuaion wih a srike price equal o he value of he consumer s deb under consideraion. If he value of his repuaion is lower han he value of his deb, D i, he consumer will defaul. The lending insiuion will repor he non-paymen o he credi agencies or credi bureaus ha will make i public o all he marke. The consumer will lose he residual value of his repuaion and access o credi. On he oher hand, if he value of his repuaion is greaer hen he value of he deb, hen i is worh he consumer paying off he deb (possibly in insallmens) or servicing a revolving credi deb and keeping his repuaion. As f and h are sricly increasing funcions of Q i and P ai respecively, R i is a sricly increasing funcion of Q i, R i = g(q i ). I means ha values of R i can be mapped o unique values of Q i and he srike price, K Ri, can be mapped o a corresponding hreshold of crediworhiness, K Qi. So we can say ha he consumer will defaul if his crediworhiness is lower han K Qi. As g( ) is an individual specific funcion unknown funcion and D i is a dynamic quaniy, K Qi (D i ) varies over ime and from individual o individual. In his work we use he simples firs passage approach o srucural credi models which says ha a consumer will defaul as soon as his crediworhiness his he barrier K Qi Cash flow consideraions I could argued ha he proposed heory does no consider he consumer s cash flow ha can play an imporan role in he consumer s defaul process. Bu his effec can be insered ino i hrough K Qi. To do ha we make an addiional assumpion: 6. If on he mauriy of a deb obligaion a consumer does no have enough cash he will raise cash hrough addiional deb. I means ha a consumer wih a deb D i, requiring Z i o service his deb and wih C i in cash, will raise he deb o D i + Z i - C i, or more likely D i + Z i o cover he repaymen.

4 4 This is consisen wih realiy as far as i is common for a consumer wih cash resricions o raise more credi o pay his acual credi obligaions. This behaviour also leads o a decrease of he consumer s crediworhiness or a leas his perceived crediworhiness. So wih lower crediworhiness and higher srike price, he consumer wih cash flow resricions will be increasingly more likely o defaul. As i is very difficul o rack C i because of is sochasic naure, he effec of cash flow resricions can be refleced in he model by seing K Ri and consequenly K Qi as sochasic quaniies. 3 A MODEL FOR DEFAULT PREDICTION 3.1 Modelling he sochasic behavior of crediworhiness Having esablished a heory for he process of defaul in consumer credi i is necessary o describe he sochasic behavior of Q i. Srucural models for corporae credi usually use diffusion processes, like he Meron Model, or jump-diffusion processes (Zhou, 1997) for he log of he asse value. Taking he laer more complex model for crediworhiness, we have for a individual s crediworhiness: dq = µ + σdw + a dy (1) Where: dq is he variaion of crediworhiness in period ; µ is he drif parameer; σ is a volailiy parameer; W is a sandard Brownian moion; a is he jump ampliude in period, where a is a i.i.d. variable wih disribuion N(µ a,σ a ); dy is a Poisson process wih inensiy λ; dw, dy and a are muually independen. The hree erms on he righ side of equaion 1 accouns respecively for he drif, volailiy and jump effecs of crediworhiness. These effecs can be aribued o evens ha affec he consumer s crediworhiness. The drif effec can be aribued o increase in age or in sabiliy (e.g., a person who has been longer in a job is less likely o lose ha job, ageing and experience lead o less irresponsible credi behavior). The volailiy effec can be due o evens relaed o he consumer credi behaviour and day-o-day changes in he financial condiion of he consumer ha can cause a flucuaion in he consumer s crediworhiness. Jump effecs are due o sporadic low probabiliy evens ha can cause a

5 5 sudden drop in he consumer s crediworhiness such as loss of job, divorce, serious disease or oher evens ha can cause serious financial disress. One oher characerisic of he sochasic behavior of crediworhiness is ha, he evens ha drive i are somewha infrequen, so here can be periods of consan crediworhiness. This suggess using a zero-inflaed process for dq as is shown in equaion 2: ( µ + σdw a dy ) dq C + = (2) where C is a random erm ha follows a Bernoulli disribuion wih probabiliy p c. If C is one hen here is a variaion in Q value in period oherwise Q will remains consan in ha period. The parameer p c is he probabiliy of change in Q value in one period and could vary from consumer o consumer. Thus some consumers are more likely o have movemens in heir crediworhiness (movers) and oher are more likely o have periods wih consan crediworhiness (sayers). For a overview and references abou zero-inflaed models refer o Tu (2002). Eigh differen alernaives for he crediworhiness sochasic model are presened in Table 1. These are he models presened in equaions 1 and 2 and simplificaions of hese models go by dropping he drif and/or jump effecs. Each one of hese eigh models was esed wih he empirical daa which will be described in secion 3.2. Table 1 Alernaive sochasic models for crediworhiness. Model Drif effec Volailiy effec Jump Effec Zero-Inflaed 1 Yes Yes Yes Yes 2 Yes Yes Yes No 3 Yes Yes No Yes 4 Yes Yes No No 5 No Yes Yes Yes 6 No Yes Yes No 7 No Yes No Yes 8 No Yes No No 3.2 A proxy for crediworhiness Alhough Q i is a non-observable quaniy here is available informaion boh wihin lending insiuions and in credi bureaus or credi agencies ha can be used o evaluae an individual s crediworhiness. This informaion is mainly relaed o he credi behaviour of he consumer wihin he insiuion or, hrough credi bureau repors, heir general credi behaviour.

6 6 Acually, lending insiuions already make wide use of his informaion for credi assessmen by building and using behavioural scores and credi bureau scores ha can be inerpreed as proxies of he consumer crediworhiness. Using a behavioural or credi bureau score as a proxy of Q i we can implemen in pracice he proposed model for consumer defaul. From now on we will subsiue Q i by S i where S i is a behavioural or credi bureau score which is a proxy for crediworhiness. Similarly, he hreshold K Qi will be replaced by K Si. To selec he mos appropriae sochasic model we esimae he parameers of each one of he alernaive models using ime series daa of S i. The empirical daa used in his aricle were monhly observaions of Credi Bureau scores supplied by Serasa for he Brazilian marke. Serasa is he leading credi bureau company in Brazil. The daa comprised 37 observaions of he individual s scores from January 2000 o January 2003 of 1,000 consumers randomly seleced from he oal number of consumers ha had credi aciviy regisered a Serasa. Besides scoring daa, informaion was available on he occurrence of defaul for each consumer in he welve monhs period afer he las score observaion (February 2003 o January 2004). This informaion will be used o validae he models developed. The definiion of defaul is any negaive repor regisered in he publicly available files and he privae files managed by Serasa. The negaive repor could relae o any credi operaion of he consumer in he marke. Usually a consumer is lised in he negaive files when hey are beween 30 and 60 days pas due. Due o his wide definiion of defaul and o characerisics of he Brazilian marke he defaul proporion in he sample was high, namely 35.9%. Noe ha a consumer in defaul is a consumer ha had any credi problem in any insiuion wihin one year. If we had analyzed defauls only in he credi operaions of a porfolio he defaul rae would have been considerably lower. Typical defaul raes in Brazilian consumer credi porfolios are around 10% o 20%. Besides he Brazilian marke has quie high ineres raes for consumer credi allowing financial insiuions o keep profiable porfolios even wih relaively high defaul raes. According o daa available a he Brazilian cenral bank he average annual ineres raes for unsecured personal loans and revolving credi in financial insiuions were respecively 85.3% and 159.6% in Serasa s credi bureau scores have a scale of 0 o 1000 and can be inerpreed as (1 probabiliy of defaul) x This original daa was ransformed ino he naural log of he odds relaion: probabiliy of no defauling/ probabiliy of defauling. We used his ransformaion so ha we could work wih a quaniy ha is no resriced o lower and upper bounds ha would make he model consrucion more difficul. Parameers for each alernaive model were esimaed using MCMC (Markov Chain Mone Carlo) echniques. The volailiy and drif parameers are consumer specific and were esimaed in he forma of vecors of parameers (one elemen for each consumer). All oher parameers were relaed o he whole porfolio. MCMC is a simulaion echnique ha uses Bayesian approach and is suiable for parameer esimaion of complex non-linear sochasic models. For a descripion of MCMC echnique, algorihms and examples applied o financial economerics refer o Johannes and Polson (2002). The

7 7 MCMC mehod is a suiable choice for esimaing parameers of he alernaive models since oher alernaive mehods have drawbacks for our models. Sandard maximumlikelihood esimaion has inconsisencies in models wih jumps (Honoré, 1998) and Kalman filering echniques are no suiable for non-linear non-gaussian models such as our zero-inflaed jump diffusion models. Comparing he proposed defaul model wih he radiional approach of using a score for risk discriminaion, we see ha our proposal uses he score and addiional informaion on he sochasic behavior of ha score. So we should ge beer risk discriminaion using hese opion based defaul model as we are using addiional informaion. To compare he alernaive sochasic models for S i we evaluaed how much was he increase in risk discriminaion when compared wih he radiional scoring approach. The esimaed parameers were used in Mone Carlo simulaions of score pahs. If he simulaed score pah of a consumer reaches he barrier K he consumer is considered as in defaul. Doing many simulaions runs, he probabiliy of defaul of a consumer is jus he proporion of he runs when he simulaed score pah reached he barrier K. Figure 1 shows how he proposed model can be used for defaul predicion. In his run he model predics defaul because he simulaed score pah goes lower han he hreshold K wihin he simulae 12 monh period. S Figure 1 Defaul predicion in a simulaed score pah. K defaul ime Hisorical Simulaed We assumed previously ha K was specific o each consumer, could depend on he deb and migh be sochasic. For our iniial empirical comparison of he models we simplify hese assumpions. We will no esimae K for each consumer, insead we esimae i a a porfolio level. There are wo alernaives o finding such a porfolio value of K:

8 8 Se K so ha he simulaed defaul rae of he porfolio is equal o he real defaul rae. Se K o maximize he defaul risk discriminaion. The firs alernaive is useful when i is necessary o mach he prediced probabiliy of defaul wih he empirical probabiliy as in porfolio modelling for capial requiremens applicaions. On he oher hand, if he objecive is o use he srucural model o improve he risk predicion a an individual level, wha really maers is he discriminaion beween defaulers and non-defaulers. The prediced probabiliy of defaul works as a ranking measure and can be mapped o real defaul probabiliies by abulaing he proporion of good and bad payers for differen bands of prediced defaul probabiliy in he same way credi scoring modelers do for heir scores. In his case he definiion of K should be driven by he risk discriminaion performance. Figure 2 shows he relaion beween K and he Kolmogorov-Smirnov (KS) saisic ha is used o assess risk discriminaion. The KS saisic measures he maximum difference beween he cumulaive proporions of defaulers and non-defaulers below a paricular score as his score varies. The chosen K should be he one ha leads o he highes KS value. The maximum can be reached by a numerical echnique like Newon-Raphson or, as we did in his work, simply calculaing KS along all values of K wih a specified precision in a feasible range. Figure 2 Relaion beween K and he KS saisic. KS K We esimaed K values for each one of he alernaive sochasic models. For each consumer in he sample we simulaed 10,000 score pahs, each of welve monhs. Table 2 presens he increase in KS values obained using he eigh varians of he srucural approach compared wih he radiional scoring approach for risk discriminaion. In each case we chose K as he value ha maximized he KS saisic.

9 Table 2 KS resuls for alernaive models. 9 Model KS Increase in KS Behavourial Score (a las observaion ime) Zero-inflaed jump-diffusion Jump-diffusion Zero-inflaed diffusion Diffusion Zero-inflaed jump-diffusion wihou drif Jump-diffusion wihou drif Zero-inflaed diffusion wihou drif Diffusion wihou drif The resuls presened above show ha bes performance in risk discriminaion was obained wih he simples model (diffusion wihou drif), which considers only a volailiy effec. This is a very convenien model since such diffusion models have an analyic soluion for he probabiliy of defaul. The differenial equaion for his varian of he sochasic model becomes simply: ds = σ (3) dw Using a firs passage approach for defaul occurrence, he firs hiing ime of a Brownian moion o a barrier has an inverse Gaussian disribuion. Using he resul presened by Avellaneda e Zhu (2001) and simplifying for a consan barrier (K) we ge for a zero drif diffusion model: K S P() = 2Φ S0 σ 0 > K (4) Where: P() is he probabiliy of defaul wihin he ime horizon ; K is he defaul hreshold; S 0 is he curren score of he consumer; σ is he sandard deviaion of he consumer s score; Φ( ) is he cumulaive sandard normal disribuion funcion. We can see from equaion 4 ha for a fixed defaul horizon, (S 0 K)/σ gives an equivalen risk ranking o ha given by he probabiliy of defauling wihin. The quaniy (S 0 K)/σ can be inerpreed as he disance o defaul in sandard deviaions. Table 3 shows he acual defaul percenages for differen bands of prediced probabiliy of defaul. The ordering of he prediced bands does reflec he acual risks, bu here is a bias in he prediced values of probabiliy of defaul for high and low credi qualiy consumers. The bias can be aribued o he simplificaion of a unique consan K for he

10 10 overall porfolio. I also suggess ha high credi qualiy consumers have higher K han he average while low credi qualiy consumers have lower values of K han he average. Table 3 Table of acual defauls by prediced defaul probabiliy bands. Prediced Defaul probabiliy band Good payers Bad payers Percenage of defauls [0; 0.1[ % [0.1; 0.3[ % [0.3; 0.5[ % [0.5; 0.7[ % [0.7; 0.9[ % [0.9; 1.0] % Toal % To es he saisical significance of he difference in risk discriminaion beween he proposed model and he radiional scoring approach we used a boosrap procedure o generae he empirical disribuion of he difference of KS saisics beween boh approaches. The Boosrap mehod was proposed by Efron (1979) and is a resampling procedure ha allows saisical inference of saisics wih unknown disribuion. We exraced from he original 1,000 consumers sample 50,000 samples wih replacemen, each one wih 1,000 elemens. For each sample we calculaed he difference of he KS measures, so obaining 50,000 values ha represen a empirical disribuion ha can be used for inference purposes. The 1% percenile of he empirical disribuion of he difference (KS srucural KS score ) is So he resuls of Table 2 imply we can rejec he null hypohesis ha hese differences are zero for all alernaive models a he 1% significance level. The proposed model requires one o esimae he volailiy parameers using hisorical ime series of scores of he individuals. The esimaion of hese parameers does no use he defaul daa. The only parameer ha is esimaed using he defaul daa is he defaul hreshold K, and so his is he only variable ha needs o be given when one has a new sample o esimae defaul predicions for. To es if he value of K is generalizable, we ran he model on a 1,000 consumer validaion sample, using he original value of K. The increase in KS for he validaion sample was slighly smaller han in he original sample, bu he improvemen was sill very saisically significan. The difference beween he KS saisic using he proposed model and he radiional scoring approach was 4.3 in he validaion sample insead of he 5.6 obained for he original sample. The value of KS saisic using he radiional scoring and proposed approaches were respecively 39.0 and Chosing he K so ha he KS saisic in he validaion sample is maximized leads o a KS value equal o 43.5.

11 4 PORTFOLIO MODELLING 11 One of he reasons o develop a new approach o he credi risk in lending o individual consumers, is ha i gives he basis o develop porfolio level credi risk models. Porfolio credi risk models seek o esimae he disribuion of value or credi loss for a specific porfolio. Srucural models supply a heoreical framework for many corporae credi risk porfolio models, including he popular Credimerics (Gupon e al., 1997) and Moody- KMV s model (1993a, 1993b). In his work we concenrae on he disribuion of he number of defauls in a porfolio (he defaul rae). Corresponding loss disribuions could be achieved by adding in a model for recovery raes. Modelling he credi risk of a porfolio of loans means using he mulivariae version (each variae corresponding o one loan or one loan class) of he differenial equaion ha underlies he srucural model. In he simples mulivariae exension of our model, we use a mulivariae normal disribuion wih correlaion marix Σ o describe join movemens of crediworhiness proxies. Mulivariae normal disribuions can be simulaed by he use of he Cholesk ransformaion and we use Romano s (Romano 2001) algorihm for generaing such simulaions. In one simulaion run of he pahs for all he behavioural scores in a porfolio of loans, he number of defauls in he porfolio is obained by couning how many consumers had score pahs ha reached he defaul barrier K. Running many ieraions of he simulaion leads o a disribuion of he number of defauls for ha porfolio. Since i is very imporan in porfolio modelling o esimae he unbiased values of he probabiliy of defaul in he porfolio i is recommended ha he porfolio level K is chosen so ha he empirical and simulaed probabiliy of defauls are mached. Simulaing join score movemens for a consumer credi porfolio is inensive in ime and compuaional power. We used a relaively small porfolio (1,000 consumers) and 100,000 join simulaions of he pahs of he nex 12 monhs for all he elemens of he porfolio ook approximaely 10 hours in a Penium4 2.5 Ghz deskop using SAS IML sofware. We recognize ha such an approach for a large porfolio of 1 million consumers is currenly impracical. 4.1 Correlaions One key elemen in he porfolio approach is he esimaion of he correlaion since his is he main driver for increasing he variance of he defaul rae in a large porfolio. In he limiing case of a porfolio wih infinie elemens, he only source of variance in he porfolio defaul rae is he correlaion amon is elemens. As consumer credi porfolios can have millions of elemens he correlaion assessmen is very imporan. We calculaed a correlaion marix for he porfolio using he ime series of monhly behavioural scores for he 1,000 consumers. We used 36 observaions of score variaions for each consumer. Surprisingly, he mean of pairwise correlaions was very close o zero, , when we

12 12 had expeced a more posiive mean correlaion. One possible reason for such a low value is ha we used monhly ime inervals. One monh migh be oo shor a period o capure he join influence of exernal facors in individuals. Bu even using 6-monhly basic ime periods insead of a monhly ime period we obained a mean pairwise correlaion of , which is sill low. The low average correlaion is due o he fac ha he scores used as crediworhiness proxies were no able o capure he influence of economic sysemic facors, bu only he influence of idiosyncraic characerisics of he individuals. There are wo possible alernaives o solve ha problem: Use as crediworhiness proxy scores ha ake ino accoun economic facors; Include he economic sysemic influence in he model by an add-on model for sysemic risk. The firs alernaive is ineresing as i could make he resuls for risk discriminaion beer and parly correc he bias in values of prediced defaul probabiliy. Thomas (2003) and Avery e al. (2004) recognize he imporance of incorporaing economic facors in he credi score. However o build such a scorecard, one requires daa on individual s characerisics and heir defauls hrough various economic cycles, which a presen is very scarce in he consumer case. The requiremens of Basel II will of course evenually provide such daa. So our curren opion was he use of a simple add-on model for sysemic risk, which does have a compaibiliy wih our original inerpreaion. 4.2 A simple model for sysemic risk Define: S ui as he uncondiional crediworhiness of individual i. S ci (s) as he crediworhiness of individual i condiional on he sae of economy being s. Assume S ui and S ci (s) have he following relaionship: S ci (s) = S ui + f i (s) (5) where f i (s) is a facor ha accouns for he influence of he sae of he economy, s, on he crediworhiness of individual i. If we consider he influence of he sae of economy as homogenous along he individuals we have: S ci (s) = S ui + f(s) (6) So he influence of an economic scenario can be summarized by a homogeneous addiive facor o he individual s crediworhiness. An alernaive explanaion is o consider crediworhiness o be an inrinsic characerisic of he individual ha is no affeced direcly by economic facors and consider he influence of sysemic risk as movemens in he defaul hreshold K. So we would have a condiional hreshold K c ha is relaed o he uncondiional hreshold K u by:

13 13 K c (s) = K u f(s) (7) This suggess ha as economic condiions worsen, he value of a consumer s repuaion drops, because i is more onerous now o service he deb, since cash flow problems increase and here is more uncerainy abou he fuure cash flow of he consumer. If i is more cosly o keep he repuaion hen is value o he consumer will drop. Obviously as far as he model of he defaul process is concerned, he reasoning ha underlies equaions 6 and 7 lead o he same resul. Since f(s) is now par of he porfolio model, i is no longer a pure srucural approach, bu a kind of hybrid srucural-reducedform approach. The facor f(s) can be esimaed empirically by a Marke Defaul Index (MDI). To do his, we used a ime series of balances of consumer credi operaions ha are available a he cenral bank of Brazil. The daa included monhly balances classified by risk caegories and includes all privae financial insiuions in Brazil. The balances did no include residenial morgage credi operaions, which correspond only o 9.4% of he oal balance of consumer credi operaions in Brazilian privae financial insiuions. We used balances on loans which were he equivalen of 60 days pas due or more as a measure of he balance in defaul and consruced a ime series of defaul raes ha represen our MDI. The period used was from July 1994 o April The values of he MDI were sored by heir values and classified in four saes of he economy in he following way: Table 4 Classificaion of saes of he economy. Sae (s) Inerpreaion Observaions 1 Very favorable 1 s quarile (25% of observaions wih lowes defaul raes) 2 Favorable 2 nd quarile 3 Unfavorable 3 rd quarile 4 Very unfavorable 4 h quarile (25% of observaions wih highes defaul raes) The facor f(s) for each of he four saes was calculaed by: 1 1 MDI 1 1 MDI = f (s) ln ln (8) ns s MDI n MDI where n s is he number of observaions of he MDI in sae of he economy s and n is he oal number of observaions of he MDI. Thus f(s) is he difference beween he averages of he naural log of he no defaul/defaul odds in sae s compared wih he odds averaged over all he saes. This ransformaion is necessary o make f(s) compaible wih he scale of S or K. I is srongly relaed o Shannon s definiion of enropy (Shannon, 1948). The facors obained for he four saes of he economy are presened in Table 5.

14 Table 5 Addiive sysemic facors for each sae of economy. 14 Sae of Economy f(s) The evoluion of he economy over hese four saes were modeled as a firs-order Markov process using monhly ime inervals. Table 6 presens he ransiion marix esimaed from he 118 monhs of empirical daa. Table 6 Transiion marix among saes of economy. s() s(+1) % 10.3% 0.0% 0.0% % 69.0% 17.2% 3.5% 3 0.0% 16.7% 73.3% 10.0% 4 0.0% 6.7% 10.0% 83.3% To es if a firs-order Markov-chain is suiable for he evoluion of saes of he economy we used he es proposed by Anderson and Goldman (1957). We esed he null hypohesis ha he ransiion probabiliy o sae s condiional on s -1 (p(s s -1 ) is equal o he ransiion probabiliy o s condiional on saes s -1 and s -2 (p(s (s -1,s -2 )) for all saes of economy. The saisic used for he es is: 1 2 [ p s p ( s,s )] 2 2 s 1 s 1 2 X = ns 1,s (9) 2 p s,s,s s s 1 where n s 1, s 2 is he number of imes ha he pah s -2 s -1 occur in he series of saes no considering he las period. The saisic X 2 has a Chi-square disribuion wih m(m 1) degrees of freedom, where m is he number of saes. The criical value for rejecing he null hypohesis ha he process is a firs order Markov chain is 51.0 wih 5% significance level. The markoviy es applied o our daa of saes of economy evoluion resuled in a value of X 2 equal o 10.6, meaning ha we canno rejec he null hypohesis and so we are no far from realiy in assuming a firs order Markov chain model of he economy. The sysemic facor can be easily insered in he simulaion process o produce he disribuion of defauls in he porfolio. For each period a migraion from he curren sae of economy (s ) o he sae of economy in he following period (s +1 ) is simulaed. The variaion in scores simulaed in ha period have a facor (f(s +1 )-f(s )) added o hem, or, equivalenly, he defaul hreshold is lowered by (f(s +1 )-f(s )). In each run of he

15 15 simulaion (one run is he join simulaion of he score pahs for all elemens of he porfolio) a new migraions of he economy are simulaed. 4.3 Capial requiremen under Basel II The Basel II formula for capial requiremen o cover credi risk in reail exposures (BIS, 2004) is based on he work of Vasicek (1991), where he derives an analyic soluion for he disribuion of defaul rae of a porfolio of corporae credi. Basel II applies his formula o reail credi. Vasicek uses Meron s diffusion model and, assuming an infinie number of exposures of equal amouns in he porfolio and equi-correlaion among he asse value of he borrowing companies, shows ha he cumulaive defaul rae disribuion a defaul rae x is given by: F(x) = Φ 1 ρ 1 1 ( 1 ρ) Φ (x) Φ (PD)) (10) Where: PD is he mean probabiliy of defaul of he porfolio; ρ is he correlaion among firm s asses value; Φ and Φ -1 are he cumulaive sandard normal disribuion funcion and is inverse funcion respecively. I can be shown (Smihson, 2003) ha using he inverse of F(x) he j h -percenile (j =100 x α) of he disribuion of defaul rae is: x( α) = Φ 1 1 ρ 1 Φ (PD) + Φ ( α) (11) 1 ρ 1 ρ Basel s formula for capial requiremen in reail credi uses his x(0.999), he defaul rae in he percenile 99.9, muliplies i by he loss given defaul (LGD) and subracs he expeced loss o ge he required capial: CR = LGD Φ 1 1 ρ 1 Φ (PD) + Φ (0.999) PD LGD (12) 1 ρ 1 ρ According o Basel II he correlaion parameer is se o 15% for residenial morgages exposures and 4% for revolving exposures. For oher reail credi exposures he correlaion has o be calculaed as a weighed average of wo exreme values by he following equaion: b PD b PD 1 e 1 e ρ = ρ + ρ min max 1 b b 1 e 1 e (13) Where: b = 35, ρ min = 0.03 and ρ max = 0.16.

16 16 To es he Basel formula we ry firs o esimae he correlaion ha is implici in he Brazilian consumer credi marke o check if i is compaible wih he value of correlaion obained hrough equaion 13. Again we use macroeconomic daa from he Brazilian cenral bank, namely he ime series of marke defaul index (MDI) described in secion 4.3. The 118 monhly observaions of defaul rae (his ime beween July 1994 o April 2004) in he MDI represen an empirical disribuion of defaul in he marke porfolio of consumer credi in Brazil. This marke porfolio follows closely he Vasicek s assumpion of an infinie porfolio. This daa supplies values of x and F(x) ha can be used in equaion 10. The probabiliy of defaul (PD) is he average defaul rae of he MDI, which urns ou o be 14.8% for he period considered. So he only remaining unknown quaniy in equaion 9 is ρ, and ha can be esimaed by non-linear regression. Using he same daa se again, Table 7 displays he leas square analysis for he regression. The parameer ρ was significan wih an esimaed value of 2.28%. Table 7 Leas square analysis for implici correlaion esimaion. Sum of Mean Approx Source DF Squares Square F Value Pr > F Regression <.0001 Residual Uncorreced Toal Correced Toal Parameer Esimae Sd Error Approximae 95% Confidence Limis RHO Using he average defaul rae of he MDI, 14.8%, in he Basel correlaon formula for oher reail exposures we go a correlaion of 3.07%. Alhough we did no have available daa on he Brazilian marke o es he relaionship beween PD and correlaion in equaion 13, a oher values of he PD, he Basel esimae for correlaions in revolving and oher reail exposures seem o be slighly higher han he correlaion implici in he heoreical model of Vasicek. If he Basel commiee has sayed wih he correlaion formula ha was proposed in Consulaive Paper 3 (BIS, 2003) he value of he correlaions would have been 2.01% for revolving credi and 2.08% for oher reail exposures. These values are considerably closer o he implici correlaion in he Vasicek model for he Brazilian consumer credi marke. To compare he resuls of he Vasicek models wih our proposed model we made poin esimaes of he 99% and 99.9% perceniles of he disribuion of defaul rae for he porfolio of 1,000 consumers used in our empirical work. The resuls are shown in Table 8. We esed he Vasicek model using he correlaion ha is proposed by he Basel s commiee for oher reail credis (since his is he predominan ype of credi produc in he empirical daa) and by using he esimaed implici correlaion from he Brazilian consumer credi marke.

17 Table 8 Exreme perceniles derived from Vasicek and proposed models. 17 Model Percenile 99% of defaul rae disribuion Percenile 99.9% of defaul rae disribuion Proposed model 46.3% 47.9% Vasicek model wih Basel correlaion formula 51.7% 57.0% Vasicek model wih esimaed implici correlaion 49.6% 54.2% Comparing he resuls of our proposed model and he Vasicek model we see ha boh he laer are more conservaive for high perceniles of he defaul rae disribuion. This difference is smaller when we use Vasicek model wih he esimaed implici correlaion insead of he Basel correlaion. Figure 3 shows he 99.9% percenile poin esimae of he defaul rae disribuion using he Vasicek model as a funcion of differen values of he probabiliy of defaul and of differen correlaion values. I immediaely shows ha he correlaion value is he key parameer in he Vasicek model and ha even small changes in he correlaion can cause significan differences in he required capial. Thus using an arbirary value of correlaion for calculaing he required capial can be very misleading as i ignores he specific characerisics of differen markes, producs and porfolios. Figure 3 Effec of he correlaion and PD on he value of 99.9% percenile prediced by Vasicek model. 80% Percenile 99.9% 70% 60% 50% 40% 30% 20% PD: 3% 5% 10% 15% 20% 25% 10% 0% 0% 2% 4% 6% 8% 10% 12% 14% 16% Correlaion 5 CONCLUSION In his work we have suggesed a srucural approach o modelling he credi risk of reail lending boh a he individual and porfolio level. Our model indicaes a way of generalizing he srucural corporae credi models o reail credi by subsiuing for he

18 18 value of a firm s asses a behavioural score ha is a proxy of he individual s crediworhiness. We have shown ha his approach could add significan predicive power o he radiional approach based on behavioural scoring models. Our resuls for Brazilian credi bureau daa revealed ha a simple diffusion model wih no drif erm and no zeroinflaion has he bes performance in modelling he sochasic behavior of he scores. This surprising resul means ha here is a simple analyic soluion for an individual s probabiliy of defaul and here is no need for compuer inensive simulaions. Much research sill needs o be done on srucural models for reail credi risk. One promising field is o sudy more closely exensions of he simple defaul barrier considered here, by finding ways o differeniae he barrier among segmens or elemens of a porfolio and by sudying he sochasic behaviour of he barrier. Anoher area worh more sudy would be o develop more sophisicaed models of how o incorporae economic facors ino he score so as o improve is use as a proxy for crediworhiness, or alernaively o develop more sophisicaed models of how o include hem in he dynamics of he defaul barrier. Clearly here is also he need o furher es and validae srucural models in reail credi as he resuls obained for one specific porfolio and marke may no be generalizable. Tess on porfolios in oher markes (especially ones where he level of defaul is hisorically much lower) and in oher economic periods in order o allow empirical validaion of he disribuion of defaul sill have o be done before one can consolidae he srucural modelling approach for reail credi. Concerning porfolio modelling and he Basel II Accord, our resuls suppor he Basel approach in he sense ha, by finding ha he simple diffusion models are adequae for consumer credi modelling, i suppors he use of a diffusion based srucural model like Vasicek model for reail porfolio modelling. Moreover some of he assumpions of he Vasicek model large numbers of relaively small and equal loans seem o be more plausible for reail credi hen for corporae credi. On he oher hand we indicaed how he capial required under he Vasicek model is very dependen on he correlaion coefficiens chosen and ha he use of a fixed value or fixed formula for hese may no be appropriae for all he produc ypes and economic condiions ha occur in he world. Comparing he resuls of he Basel approach wih he resuls of he proposed srucural model buil on Brazilian empirical daa, showed ha he Basel approach leads o more conservaive resuls increasing he amoun of required capial. One half way house beween hese wo exremes would be if he financial insiuions could esimae he correlaion parameer of heir loan porfolios in he same way ha hey do for PD, LGD and EAD in he Inernal Raings Based approach. This would lead o capial requiremen more closely adjused o he insiuion s real disribuion of defaul.

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