Modelling the Credit Risk for Portfolios of Consumer Loans: Analogies with corporate loan models

Transcription

1 Modellng the Credt Rsk for Portfolos of Consumer Loans: Analoges wth corporate loan models Lyn C Thomas Quanttatve Fnancal Rsk Management Centre School of Management, Unversty of Southampton Southampton, U.K. The Internal Ratngs Based (IRB) approach suggested n the New Basel Accord regulatons ( BIS 2005) uses a captal allocaton formula derved from a Merton style structural model of the credt rsk of portfolos of corporate loans. Yet ths formula s beng appled n the case of consumer loans as well as corporate loans. Ths has hghlghted that although there are a number of well establshed credt rsk models for portfolos of corporate loans whch are wdely used by fnancal organsatons, there are no such establshed consumer credt rsk models for portfolos of consumer loans. Ths s surprsng gven that modellng the credt rsk of ndvdual consumer loans,- credt scorng has proved the underpnnng that allowed the phenomenal expanson n consumer lendng over the last ffty years. The need for models of the credt rsk of portfolos of consumer loans s clear for captal adequacy provsons, where the Basel New Accord descrbes the regulatory captal needed, but banks themselves need smlar models for economc captal provsonng. It s also needed to support the prcng estmates when such consumer loan portfolos are securtzed. There appears to be a dsconnect between the models of credt rsk of portfolo of consumer loans needed for these more recent portfolo level decsons and the wdely used credt scorng model of the credt rsk of ndvdual loans whch have long been used for acceptance and operatng decsons. So s t possble to develop new models for estmatng the credt rsk of portfolos on consumer loans? In fact t may be easer to get a modfed or renterpreted corporate credt rsk model accepted by the ndustry than to develop completely new consumer loan credt models. In ths paper we nvestgate possble approaches to consumer credt rsk models, whch buld on the exstng credt scorng knowledge but have parallels wth the corporate models.

2 Intally there seems a number of dfferences n the two markets.. The market n corporate bonds s well establshed wth the bonds n a company beng bought and sold several hundred tmes. In the bond market the orgnal company has lttle control over changes n the debt structure ntally ssued apart from occasonal well defned dates on whch some bonds can be called or converted nto stock shares. Structural models of corporate credt rsk ( lke that n Basel) assumes that a company wll default when ts loans exceed ts assets or some proporton of ts assets. Wth consumer loans on, the other hand, there s no market mechansm for contnuously tradng n such loans and hence no contnuously avalable market prce. Even f a consumer loan portfolo s securtzed and converted nto a tradable asset, ts sale s often a one-off event. In the consumer market the ndvdual consumers who make up the portfolo can decde at any tme to cease usng that lendng product wth consequent changes to the value and composton of the portfolo. For consumers, default on a loan s much more related to cash flow and ncome beng nsuffcent to servce the loan snce few consumers can calculate, let alone realse, ther assets. The correlatons between defaults of bonds of dfferent companes s assumed to be related to the correlatons n ther asset movements and t s assumed the correlatons n the share prces of the companes reflect the correlatons n the asset movements. There s however no consumer equvalent of a share prce. So are these models approprate or the credt rsk of consumer loans and f not what models would be? Corporate credt rsk models splt nto structural and reduced form modellng and these are revewed n more detal n secton 2. Structural models were ntroduced by Merton (Merton 1974), whle reduced form models were developed from the work of Jarrow & Turnbull (1995)) Perl and Nayda (Perl and Nayda 2004) propose a structural model for revolvng retal credt that uses the exactly the same approach of the corporate models, consderng that a consumer s n default f hs assets are lower than a threshold. However, as a great deal of consumer credt s unsecured and t s not the case that a consumer n default wll lose the rghts over all hs assets, just transposng the corporate default models to consumer default can lead to some aspects of consumer default beng mssed. Musto and Souleles ( 2005) use equty prcng as an analogy for ther nvestgaton nto portfolos of consumer loans. They take the changes n default probabltes gven by the changes n the monthly behavoural scores of ndvduals as surrogates for the return on that ndvdual. They they fnd the betas of how these ndvdual return depend on the market return of the whole

3 portfolo. They fnd that the betas are heterogeneous and the lower the behavoural score, the hgher the correlaton rsk. There are also some works n structural modellng n consumer credt that are not related to ndvdual rsk assessment or portfolo modellng. Examples are Longhofer and Peters (Longhofer and Peters 2004), who studes lendng dscrmnaton and self-selecton and Athreya ( Athreya 2004) who analysed the relaton between the mportance of the stgma of bankruptcy and bankruptcy rates. As ndcated above n secton 2 we revew the dfferent models for estmatng the credt rsk of portfolos of corporate loans, n partcular the structural and reduced form models. In secton 3 we look at a structural model based on the reputaton of the ndvdual for the credt rsk of consumer lendng at both the ndvdual and portfolo level. Ths draws heavly on the work of Andrade and Thomas (Andrade and Thomas 2005) and the model bult theren usng Brazlan data. The remanng setons are much more ndcatve of possble approaches rather than detals of specfc models. Secton 4 suggests one can buld structural models based on affordablty. Sectons 5 and 6consder how one can use survval analyss and Markov chan approaches to behavoural scorng to buld credt rsk models for portfolos of consumer loans, whch are the equvalent of reduced form default mode and reduced form mark to market models n the corporate case. 2. Corporate Credt Rsk Models Corporate bonds are the way lendng to large companes s structured and losses occur wth these a) because the company defaults; b) because the ratng of the loan s lowered by ratng agences, lke Standard and Power or Moodys, and hence the loan s worth less n the market because of the ncreased rsk of default; c) the dfference between the value of a bond of a gven ratng and that of the hghest grade bond the credt spread ncreases. Ths last problem affects the whole bond market and so s consdered to be a market rsk rather than a credt rsk. The frst splt n models s between those whch only try to estmate the chance of default, called default mode (DM) models, and those whch try to estmate both the chance of default and of ratngs changed, called mark-to-market (MTM) models. The second splt s between structural models whch relate the value of a bond and ts default probablty to the captal and debt structure of the frm, and the reduced form models, whch model the default process and/or the ratng change process drectly usng hstorcal estmates. Lastly there s splt between statc models whch can only model the

4 credt over a fxed tme nterval and dynamc models whch allow one to model over any tme nterval. The Merton model (Merton 1974) provdes the foundaton for all structural models wth ts assumpton that a frm defaults when at the maturty of the loan ts debts exceed ts assets where ts assets have been followng a random process. Thus when a bank lends to a company t s equvalent to wrtng a put opton on the assets of the frm, snce f the assets of the frm are hgh enough the bank gets back the return on the loan but f the assets do not exceed the debt ths return goes down lnearly wth the assets of the frm. Hence one can apply the Black-Scholes model to prce the value of the loan ( bond). The Longstaff- Schwartz model (Longstaff and Schwartz 1995) allowed the frm to default when the assets fell below some default barrer D(t)) whch means that default can happen before maturty. Vascek ( Vascek 2002) gave condtons under whch Merton s model can be extended to the sngle rsk factor model whch s the bass of the Basel formula. The reduced form approach developed from the dea of Pye (Pye 1974) that one could use an actuaral approach and model the default rate of a group of bonds over tme as s done n mortalty tables for humans. Ths s a statc, one-perod or nfnte perod model ( f one thnks of the one perod beng nfntely repeated). Jarrow and Turnbull (Jarrow and Turnbull 1995) were able to extend the dea to a dynamc model by replacng the fxed probablty of default by a hazard rate ( or ntestty rate as t s called n ths lterature) whch descrbes the changes n the default probablty over the duraton of the loans. Ths s applyng the survval analyss models used n mortalty, relablty of equpment and mpact of medcal treatments n a fnancal context. The basc approach s a DM model but by modellng the dynamcs of the process n terms of a Markov chan model on the ratngs grades ncludng a defaulted ratngs grade, they were about to develop a MTM verson of ths approach ( Jarrow Lando Turnbull 1997). There s now a realsaton that the dfference n these models s more n the nformaton avalable, n that as Guo, Jarrow and Zeng (Guo, et al 2005) pont out structural models are based on the nformaton on asset values and labltes avalable only to the frm s management whle reduced form models are based on nformaton avalable n the market, where there s only ndrect nformaton on a frm s assets and labltes. Reconclaton models are beng developed ( see the survey by Elzalde ( Elzalde 2005)) whch do not have complete nformaton on the dynamcs of

5 the process that trggers default n the structural models and hence leads to cumulatve rates of default as n reduced form models. The reduced form approach s lke the structural approaches s a bottom-up approach n that one seeks to estmate the credt rsk for each frm separately or wthn small groups and then puts them together to estmate the credt rsk n a portfolo. The other development of the actuaral approach was to keep wth top down models whch model at a portfolo or segment of portfolo level drectly and make no assumptons concernng correlatons between ndvdual frms default rsk. ( Fnger 2000) Fnally there are the statc scorecard models whch seek to estmate the probablty of default n a fxed perod of tme as a functon of accountng ratos of frms, and now ncludng other market nformaton. These are DM models and are essentally credt scorng appled to frms rather than consumers. It was poneered by Altman( Altman 1968) who has contnued to be one of the leadng developers of the approach. More recent models ( Chava and Jarrow 2004) seek to connect the results of ths approach wth the hazard rate estmates necessary for the reduced form models. The scorng approaches such as Altman s z-scores are statc DM-approaches as opposed to all the other above models whch are dynamc. They measure the probablty of default over a gven future tme horzon usually one year. Ths approach s almost dentcal to consumer credt scorng, whch preceded t by a dozen years, but n the subsequent 40 years t has proved far less successful or wdely used. Ths s for two reasons. It s hard to get large enough populatons of homogeneous frms as one needs to segment by ndustry, sze and country. Also, the scores depend on accountng nformaton, whch, no dsrespect to the audtors, s less accurate than the transactonal nformaton used n behavoural scorecards. Thus t does not have the dynamc element these have. Moreover the scorecards gnore any correlaton n the default rate of dfferent frms, whch s exactly the problem that has been gnored n consumer credt scorng untl now. So can these credt rsk approaches be used n the context of consumer lendng portfolos? If not can they be renterpreted or modfed to make sense n ths context or can the dfferent nformaton avalable n the consumer case more detaled ndvdual characterstcs, very

6 detaled transactonal data, but no market prcng data allow new approaches to be used. These are the ssues we wll concentrate on n the rest of ths paper. 3. Consumer structural model based on reputaton Andrade and Thomas (2005) suggested that whereas the corporate structural models assumed that the shareholders have a call opton on the assets of the frms one could construct a consumer structural model by assumng a consumer has a call opton on hs reputaton. One can construct a model usng ths approach where one uses the behavoural score as a surrogate for the reputaton for credt worthness of the borrower. Assume that for borrower, there s an unobservable quantty Q, ther credt worthness. Ths ncludes all the nformaton about the borrower that s needed to assess hs credt rsk. Lenders have nformaton from both nternal and external sources, such as credt bureaus about the fnancal condton and the repayment performance of the borrower. Ths nformaton s usually translated nto behavoural scores and t seems reasonable to take these as a proxy for the credtworthness Q. One can assume that the probablty of the consumer beng accepted as a clent by a lendng nsttuton, p a, s a strctly ncreasng functon of Q, p a = f(q ). Moreover f the borrower defaults on the loan, the nformaton s quckly passed to the credt bureaus and hence to all other lenders. Thus the consumer wll have lost hs reputaton for credt worthness and wll have no access to credt n the mmedate future. Ths access to credt s clearly of value to a borrower and ths value s related to the amount of credt to whch the borrower has access. Defne ths value as R the value of the borrower s reputaton. Ths value s a strctly ncreasng functon of P a, R=h(P a ). Wth these assumptons one can construct an opton based model for the credt rsk of an ndvdual borrower. If the value of borrower s reputaton R s hgher than hs debt D,, then the borrower wll seek to repay the debt. If R < D then the borrower wll default. The lendng nsttuton wll report ths to the market va the credt reference bureaus and the borrower s reputaton wll be lost along wth access to credt. Thus we can thnk of a borrower as havng a call opton on hs reputaton wth the strke prce beng the amount of debt owed.

7 Snce t s reasonable to assume f(.) and h(.) are strctly ncreasng functons of Q and P, R s a strctly ncreasng functon of Q. R h ( f ( Q ) ) g ( Q ) Thus there s a unque one to one correspondence between the values of R and the values of Q, and vce versa. In partcular let K Q, be the credt worthness (Q) whch corresponds to havng a reputatonal value of D. Thus the opton analogy can be appled also n the credtworthness dmenson. The borrower wll default f hs credtworthness s below K Q, and contnue to repay f hs credtworthness s above K,Q, Credt worthness, t s assumed s unobservable, but lenders are used to usng behavoural scores as a surrogate for t. If that connecton s used n ths credt rsk model, default wll correspond to the behavoural score S (t) for borrow at tme t,droppng below a threshold value K S,. To complete the model we need to derve the dynamcs of the behavoural score S (t). Andrade and Thomas (2005) consdered credt worthness to be a contnuous tme dffuson process wth jumps. Thus S(t) satsfes d S ( t ) P V d W a d Y t t Where P s the drft of the process, V dw s a Brownan moton and dy t s a Posson jump process. Although the process s wrtten n contnuous tme, when t comes to estmatng the parameters, one wll need to use a dscrete tme equvalent wth tme ntervals of one month. The dea s that P corresponds to a natural drft n credt worthness caused n part by the account and the customer ageng and so mprovng. The Brownan moton descrbed the natural varaton n behavoural score whle the Posson jump term s ncluded to model jumps n behavoural scores due to major events lke the change n the economc stuaton whch could lead to the overall populaton odds changng dramatcally. (2) (1) Gven a tme seres of behavoural scores for a sample of borrowers one can then estmate the parameters P V for each consumer and a t for the whole populaton ether usng Bayesan MCMC ( Markov Chan Monte Carlo ) technques or by maxmum lkelhood estmates. All that s left s to estmate K S, the behavoural score level at whch default s trggered. Ths can be done by Monte Caro smulatons of the behavoural score paths. For each borrower the hstorcal scores are avalable and one can apply smulaton to obtan score paths for the next few perods. Ths s done a number of tmes for each ndvdual and f we consder a possble default value K S, then we can calculate the number of paths that go below that value, see Fgure 1. Ths gves the estmated default probablty for that borrower. In order to make the model tractable the same

8 value K S s used for all borrowers, and so we have a default probablty for each ndvdual for each value of K consdered. One can then choose the value of K S to ensure good calbraton or good dscrmnaton. In the former case, one sets K S so that the smulated default rate n the portfolo s equal to the actual default rate. In the latter case, K S s chosen to maxmse a measure of dscrmnaton such as Kolmogorov-Smrnov statstc or Gn coeffcent. In ths approach the estmate of K S and the estmaton of the process parameters are separate processes. The later models the dynamcs of credt worthness and the former, together wth what default threshold, gves the approprate default behavour. S tme Hstorcal Smulated Fgure 1 Default predcton n a smulated score path. Andrade and Thomas (2005) bult such a model usng Brazlan data on behavoural scores over a three year perod and chose the K S value that maxmsed the Kolmogorov-Smrnov dstance. They dd ths for fve dfferent models; the most recent behavoural core whch essentally gnores the dynamcs and the four models gven by ncludng or droppng the drft term and the jump process. Table 1 shows the results. As one would expect, and hope, all the dynamc models gave superor dscrmnaton to the statc model wth the KS statstc ncreasng from 41.0% to 44.4% or hgher. The surprse was that the best dscrmnaton was n the model wth no drft and no jump process, and so the dynamcs was purely

9 d S ( t ) V d W ( 3 ) The results do not prove that ths basc model s always better than the other models, but do suggest that t s compettve. Usng the prncple of Occam s razor, that one should choose the smplest explanaton, mples ths s the model to use. If one bult models on other data, partcularly from countres wth lower average default rates, the result could be dfferent Table 1 KS results for alternatve models. Model KS Increase n KS Behavoural Score (at last observaton tme) Dffuson wth drft and jump process Dffuson wth drft but no jump process Dffuson wth no drft but jump process Dffuson wthout drft and no jump process The advantage of usng (3) ( rather than ts dscrete equvalent (4)) P{ S S V } p; P{ S S V } 1 p (4) t 1 t t 1 t s that there s an analytc soluton for default wthn the tme horzon t, namely P t K S V t ¹ S 0 ( ) 2 ) S 0! KS (5) where K S s the default threshold. S 0 s the current behavoural score and V s the standard devaton of the borrower s score. Ths follows from the reflecton prncple for Brownan moton whch says that a stochastc process whch s Brownan moton wth a reflectng barrer the frst tme t hts the level K S s stll Brownan moton. Ths coupled wth the fact that the orgnal system has not ht the default level K S only f the orgnal Brownan moton and the reflected Brownan moton are stll the same leads to (5). Note that (S 0 K S )/ V gves the same rankng of customers as that gven by the probablty of defaultng wthn tme t, and so s akn to the dstance to default used n some propretary corporate credt rsk models.. Ths approach s not yet a portfolo credt rsk model as we need to combne the credt rsks of the ndvdual loans. The standard way of dong ths s to take a multvarable verson of the basc structural equaton ( 3) where each varable corresponds to one loan or one class of loan. The smplest such model s to use a multvarate normal dstrbuton wth a correlaton matrx 6 descrbng the jont movement of the credt worthness proxes. There are standard ways of smulatng such dstrbutons ( Cholesky transformatons for example) and for each smulaton run we can estmate the PD of the portfolo by countng the number of loans that default durng

10 that run ( ht the default barrer K S ). Repeatng the smulaton many tmes gves a dstrbuton for the default probablty PD. Snce the calbraton s most mportant n ths context, one should choose K so as the emprcal and smulated probabltes of default match. Surprsngly when ths was done usng the Brazlan data descrbed above the parwse correlaton of behavoural scores ( the credtworthness proxes) was very close to 0. Ths poses a dlemma because t s clear the default of ndvdual loans n any gven tme perod are correlated. One way to model ths s to assume the credt worthness of the ndvdual loan conssts of an dosyncratc part whch s the behavoural score and a systemc part whch s due to economc condtons and s not captured by the behavoural scores as they do not contan economc factors. The tradton n consumer credt has been to rebuld or at least recalbrate the scorecard as soon as the economc changes are suffcent to make notceable dfferences to the default rates. So defne Q t, (e t ) as the credtworthness of consumer at tme t f the economc state s e t by (6) Q ( e ) Q f ( e ) S f ( e ) t, t t, t t, t where S t, s the behavoural score of borrower at tme t and f(e) s the mpact of the economc state on the credt worthness of. Note that f we wrte the correspondng dfferental equaton that Q wll satsfy we get df ( e t ) dqt, V dw dt (7) dt whch has consderable smlartes to the Vascek corporate model ( Vascel 2002). Default occurs f Q ( e ) K œs f ( e ) K œs K f ( e ) (8) t, t Q t, t S t, S t Thus one could renterpret the factor f(.) as the way the changes n the economc condtons affect the default thresholds. An analogy would be that the behavoural score of a consumer ( ther dosyncratc credtworthness) s movng up and down accordng to a random walk, but f t falls below the water level K S the borrower drowns n hs debt. The water level tself moves up and down accordng to the economc condtons. When t s hgh, a lot of borrowers wll drown at the same tme; when t s low very few wll drown. Hence the correlaton between defaulters are gven by these changes n the economc condtons.

11 The economc states e and the related factors f(e ) model the dynamcs of the economy and ts connecton wth default rates. Central banks ( and some credt bureaus) keep data on default rates DR(t) for each three monthly quarter t and we could connect these to macro economc varable to dentfy whch combnatons of the latter affect default rates. One could then partton the economc stuaton nto a number of states each havng sets of economc condtons wth smlar default rates and use hstorc data to buld a Markov chan descrbng the transtons between these dfferent economc states. Thus one can complete the dynamcs of the model by defnng these transton probabltes P(e t+1 = e t =j)= p j The normal relatonshp between the behavoural score s and the probablty of beng good, p, s gven by s= log(p/1-p), and ths relatonshp could be used to defne the factors f(e ) whch s added to the behavour score. Suppose we have nformaton on DR(t) for t=1,..,t, then an emprcal estmate for the log odds of good aganst bad when the economy s n state e (whch we assume occurs n tmes n the tme horzon under consderaton) s D R ( t ) t e 1 n lo g D R ( t ) t e n (9) Then f we defne f(e )to be the weghts of evdence when the state of the economy s n state e we have f D R ( t ) T t e 1 D R ( t ) 1 n T t 1 ( e ) lo g lo g T D R ( t ) D R ( t ) t e T t 1 n (10) Wth ths extra factor f(e ) descrbng the state of the economy, one can smulate the dstrbuton of defaults n a portfolo over a gven tme perod n the future. Frst one smulates the path of the Markov chan whch descrbes the economy and then one can smulate the dynamcs of each ndvdual s behavour score and hence the number of defaults n the portfolo. Andrade and Thomas ( 2005) used the data avalable from the Central Bank of Brazl to create a four state Markov chan descrbng the behavour of the economy and then appled ths procedure

12 to a portfolo of 1000 personal loan accounts. They also used the Vascek model that underpns the IRB formula n the New Basel Accord usng both the correlatons specfed by the Accord and the estmated correlatons obtaned from the data n the 1000 accounts. The latter s the correlaton that the structural reputaton model s usng. The results are gven n Table 2 by descrbng the extreme values ( 99% and 99.9%) of the probablty of default dstrbuton obtaned from the models. The results suggest that the Vascek-Basel model s more conservatve than the reputaton one but that some of ths conservatsm s due to the correlaton that s mposed n that model. Model Percentle 99% of default rate dstrbuton Percentle 99.9% of default rate dstrbuton Structural reputaton model 46.3% 47.9% Vascek model wth Basel correlaton formula 51.7% 57.0% Vascek model wth estmated mplct correlaton 49.6% 54.2% Table 2 Extreme values of three credt rsk of consumer portfolo models 4. Consumer structural models based on affordablty It has been argued that consumers default more because of cash flow problems than because ther debts exceed ther assets because a consumer s assets are not easly realzable. Regulators are therefore becomng more nsstent on responsble lendng whch means applyng affordablty models. ( Australa has been n the forefront of ths for many years). These affordablty models can also be developed to gve structural models of the credt rsk of consumer loan portfolos. A smple verson of the model relates the varables at tme t+1 to those at tme t by (Realsable Assets) t+1 = ( Realsable Asset) t +(Income) t (Expendture) t ( Loan Repayment) t where default occurs f realzable asset becomes negatve. One approach to estmate ncome and expendture s to use a survey of consumers expendture and ncome and then to use ths data to buld regresson models of expendture and ncome aganst consumer characterstcs and where necessary behavour scores. These regresson of ncome and expendture models should nvolve economc factors and these as n the reputaton structural models wll gve the correlatons between the default rsk of dfferent consumers. If the lender s a bank whch has access to the detals of the consumer s current account then t could use the total value of credts n month t as an estmate of ncome and total value of debts as an estmate of expendture.

13 5. Consumer Reduced form models based on hazard functons One can develop the equvalent of reduced form credt rsk models for portfolos of consumer loans by extendng the work on buldng behavoural scorng systems usng survval analyss hazard deas ( Banask et al 1999, Stepanova and Thomas 2001, 2002) and Markov chan models ( Ho et al 2004, Hand and Tll, Trench et al). The hazard models have strong parallels wth the Default Mode ntensty based corporate credt rsk models. Recall that n the hazard-based applcaton scorng models of that secton the tme t untl a consumer defaults s gven by a s hazard functon h t exp c h t e h t w.x where x =(x 1, x 2,, x n ) s a vector the 0 0 consumer s socal-demographc characterstcs whch can be combned n the score s. The correlaton between the default dstrbutons of dfferent borrowers can be ntroduced by addng common factors to the score functon. The most obvous common factors are the varables that descrbe the external economc condtons. Thus one can extend the basc hazard based default model by allowng two types of characterstcs x 1, x 2,, x n and y 1 (s), y 2 (s),, y m (s), where x =(x 1, x 2,, x n ) s a vector of socal-demographc characterstcs descrbng the statc characterstcs ( whch wll manly be the soco-demographc nformaton of the borrowers) and y(s) = (y 1 (s), y 2 (s),, y m (s)) s the vector of external economc condton varables at tme s. The proportonal hazards assumpton s then that f t the borrower s lendng faclty was opened at tme u then the chance the loan wll default after a further t perods s gven by the hazard rate u h t f the ndvdual and envronment characterstcs are gven by x, y(.), s c u h t h0 t exp w x v y u t (11) where h 0 (t) terms represents the baselne hazard,.e., the propensty of a default event occurrng when all ndependent varables equal zero. Recall there are two approaches n defnng h 0 (.), w, v - the parametrc and the non-parametrc (or at least the sem-parametrc). In the parametrc approach the lfetme dstrbuton and hence the hazard rate s chosen to be of a certan form; e.g. f exponental then h 0 (t) = O, f Webull wth shape parameter k and scale parameter O, h 0 (t) = k( Ot) k-1. The parameters of the dstrbuton as well as w and v coeffcents are then estmated from the data. The more commonly used approach s the sem-parametrc one of Cox s proportonal hazard functon (Cox (1972), Kalbflesch and Prentce (1980)) where the coeffcents w and v can be estmated wthout havng to assume a specfc dstrbutonal form for h 0 (t), and the latter can be backed out of the data usng the Kaplan-Meer estmaton procedure. The fact the same

14 economc condtons affect all the consumers allows correlatons between the default probabltes. Of course the common varables n the score of dfferent consumers does not have to be the economc condtons. One could use geographcal area or type of employment category to ntroduce the correlaton. However wth ths basc model, consumer s relatve hazard rate ( hs hazard score ) at tme t s(,t) s a lnear combnaton of the two types of factors so that s(, t) c t w x v y where the persona characterstcs of consumer are x. Notce that ths means s(,t 1 )-s(j,t 1 )=s(,t 2 )-s(j,t 2 ) so that the rankng of the ndvdual scores stays the same n all tme perods. Thus t s only the personal characterstcs that affect the orderng of how lkely a borrower s to default and ths orderng s the same for all tme perods. If we assume the hazard rate and survval functon for borrower at tme t s h(,t) and S(,t) then D(t), the total expected number of defaults n perod t for a portfolo of loans =1,..n ( where to keep notaton easy we assume all loans start at tme 0) satsfes n n n t 1 n t ³ h(, u) du h(, u) du 0 ³ 0 D( t) S (, t 1) S (, t) e e e e e e e ¹ n t 1 t t 1 t 1 w. x v. y ( u ) n w. x v. y ( u ) n v. y ( u ) v. y ( u ) e ( ) ( ) w. x ³ e ho u du e o o ( ) o ( ) 0 ³ e h u du e h u du e h u du 0 e ³ 0 ³ (12) Thus D(t 1 )/D(t 2 ) s ndependent of the applcant characterstcs x and depends only on the economc varables. So t s these and only these that affect the varatons n the total numbers of defaults between perods. Thus each of the sets of varables has a completely dfferent role. There are parallels wth the strucutural reputaton model descrbed earler n that the behavoural score s based on the ndvdual borrowers characterstcs and ths gves ther relatve propensty to default whle the actual number of defaults was gven by the default value K whch depended on the state of the economy. A more powerful model would use both sets of varables to estmate the two dfferent measures relatve default probabltes and total numbers of defaults. There are two ways of dong ths n ths context. The frst s to make the coeffcents of the varables ( or at least the borrower characterstcs) tme dependent. Thus borrower s hazard score at tme t would be s(, t) ( w tw ) cx v y t 0 1

15 Ths can be calculated by ntroducng varable t.x as well as x nto the hazard functon and ther coeffcents are the w 1. The second approach s to ntroduce borrower-economy nteracton varables. Assumng that on has chosen the scorng approach so that all the borrower characterstcs are bnary,then one can an nteracton varable x *y j (t) to be the varable that takes the values y j (t) for the borrowers where x = 1 and s 0 for the borrowers wth x =0. An applcaton of these two extensons n the context of propensty to purchase can be found n Tang ( Tang et al 2005). 6. Consumer reduced form models based on Markov chans Another approach n corporate credt rsk reduced form modellng s to use the nformaton avalable n the market place about the current ratng of the corporate loans. These mark-to-maket models enable one to prce the credt losses whch occur not because of default but because of the drop n a loans ratng. Ths s of lmted mportance n the consumer context because such loans are rarely beng sold and so such partal losses rarely occurrng. The other aspect, though, s that they allow one to use the most recent estmate of the probablty of default of the loan and ths s exactly what behavoural scorng does n the consumer context. Thus one can seek to develop consumer equvalents of those models. In the Markov chan models ( Ho et al (2004), Tll and Hand (2005), Trench et al (2002)) one has a state space of the current state of the borrower s account. The states can be the current default status up to date, 30 days overdue, 60 days overdue, etc -, bands of behavoural scores or more complex stuatons nvolvng frequency and recency of payment and usage and current balances. The ones most drectly related to credt rsk would be the Markov chans based on default status or behavoural score bands. In both cases one needs to dentfy states whch correspond to default. In the days overdue model t s obvous what the default state should be. whereas n the behavoural score model one needs to usually add a default state. One can use the behavoural score data to develop a seres of Markov chan models by estmatng the transton probabltes p t (,j) of movng from state to state j durng perod t. Ths allows estmaton of the probablty of an ndvdual consumer defaultng n any tme nterval. The dstrbuton functons S t of the probablty of whch state a borrower wll be n at tme t can also be renterpreted va the ergodc theorem as the proporton of the borrowers n any portfolo who are n the dfferent states at that tme. Ths t seems gves estmates of the credt rsks for a portfolo of such loans.

16 The one dffculty n ths s that there s an mplct assumpton that the default status of dfferent borrowers are ndependent of one another. There s no correlaton between the processes. How can one ntroduce correlatons nto such a model? The answer s to make the transtons probabltes depend on other parameters whch hold for all borrowers at the same tme. If these parameters are not know but themselves satsfy a Markov chan, ths s a hdden Markov chan process of whch there s a large lterature ( Rabener 1989). A smpler model would be to make the Markov chans transtons p t (,j) depend on the economc parameters y(t) whch hold at that perod. There are many ways of modellng such a dependency but the approach used n other applcatons of Markov chan theory ( Bozetto et al 2005) s to estmate the transtons by a cumulatve logstc transformaton of the underlyng varables. Ignorng the tme dependence of the transton probabltes, so they are affected only by the current economc condtons ths corresponds to defnng p t (,j) by frst defnng the cumulatve transton dstrbuton by P (, k) p (, j) t k j 1 t Then connect these cumulatve transton dstrbuton to the economc varables by Pt (, k) log D, k E, m ym( t) 1 Pt (, k) ¹ m where the constants D are chosen to ensure the transton probabltes are all postve. There are many other forms whch connect the underlyng stuaton wth a dynamcal model of the state of the ndvduals borrower and thorugh some common parameters allow the model to be used at a portfolo level. 7. Concluson Apart from the reputatonal structural model the paper has only gven ndcaton of how these approaches should work. There s an obvous research agenda to develop complete models and to deal wth the complcatons that wll arse n such model developments. Only then wll t be possble to make meanngful comparsons of the dfferent approaches to estmatng the credt rsk of portfolos of consumer loans. In partcular one would be better placed to decde whether the corporate portfolo credt rsk models can be successfully renterpreted n the consumer context. References Altman, E. (1968). Fnancal ratos, dscrmnant analyss and the predcton of corporate bankruptcy. The Journal of Fnance,

17 Andrade F.Wendlng Munz de, Thomas L.C., (2004), Structural models n consumer credt, Workng paper, AF 4-19, School of Management, Unversty of Southampton. Athreya, K. (2004). Shame as t ever was: Stgma and personal bankruptcy. Federal Reserve Bank of Rchmond Economc Quarterly, v.90(2). Banask J., Crook J.N., Thomas L.C. (1999), Not f but when borrowers default, J. Operatonal Research Socety 50, , (1999) Bank of Internatonal Settlements (2005). Internatonal convergence of Captal measurement and Captal Standards, November Basel Commttee on Bankng Supervson. Bozzetto J_F,Tang.L, Thomas.L.C., Thomas S. (2005), Modellng the purchase dynamcs of nsurance customers usng Markov chans (wth Workng Paper CORMSIS 05-02, School of Management, Unversty of Southampton. Chava S, Jarrow R.A., (2004), Bankruptcy falure wth ndustry effects, Revew of Fnance 8, Cox D.R., (1972), Regresson Models and Lfe Tables, J. Royal Statstcal Socety B, 34, Elzalde A,(2005), Credt Rsk Models III; ReconclatonStructural-Reduced Models, Fnger C.C. (2000), A comparson of stochastc default rate models, Rsk Metrcs Group WP 00-02, RskMetrcs, Geneva Guo X, Jarrow R.A., Zeng Y., (2005), Informaton Reducton n Credt Rsk Models, Workng Paper, School of Operatons Research and Industral Engneerng, Cornell Unversty, Ithca. Ho J., Thomas L.C., Pomroy T.A., Scherer W.T., (20040, Segmentaton n Markov chan consumer credt behavour, n Readngs n Credt Scorng ( ed Thomas L.C., Edelman D.B., Crook J.N.), Oxford Unversty Press, Oxford. Jarrow R.A., Lando D., Turnbull S., ( 1995), A Markov model for the term structure of credt rsk spreads, Revew of Fnancal Studes 10, Jarrow, R. A., & Turnbull, S. (1995). Prcng dervatves on fnancal securtes subject to credt rsk. Journal of Fnance, 50, Kalbflesch J.D., Prentce R.L., (1980), The Statstcal analyss of falure tme data, Wley and Sons, New York. Longhofer, S., & Peters, S. (2004). Self-selecton and dscrmnaton n credt markets. Workng paper.

18 Longstaff F.A., Schwartz E.S. (1995), A smple approach to valung rsky fxed and floatng rate debt, Journal of Fnance 50, Merton, Robert C. (1974). On the prcng of corporate debt: the rsk structure of nterest rates. Journal of Fnance, 29, Musto D.K., Souleles N., (2005), A portfolo vew of consumer credt, Workng Paper 05-25, Federal Reserve bank of Phladelpha, Phladelpha. Perl, R., & Nayda, W. I. (2004), Economc and Regulatory Captal Allocaton for revolvng retal exposures. Journal of Bankng and Fnance, 28, Pye G. (1974), Gaugng the Default Process, Fnancal Analysts Journal, Jan-March, Rabner L.R., ( 1989),A Tutoral on Hdden Markov Models and selected applcatons n Speech Recognton, Proceedngs of the IEEE 77, Stepanova M. Thomas L.C., (2001), PHAB scores: Proportonal hazards analyss behavoural scores, J. Operatonal Research Socety 52, Stepanova M., Thomas L.C., (2002) Survval analyss methods for personal loan data, Operatons Research 50, , Tang L.L., Bozetto J-F, Thomas L.C., Thomas S., ( 2005),It s the economy stupd, Workng Paer CORMSIS 05-1, School of Management, Unversty of Southampton Tll R.J., Hand D.J., ( 2003), Behavoural models of credt card usage, J. Appled Statstcs 30, Trench M.S., Pederson, Lau E.T., Wang H., Nar S.K.(2003), Credt lne and Prce Management for Frst USA credt cards, Interfaces 33, ssue 5, 4-21 Vascek O. ( 2002). Loan portfolo value, Rsk, December,