Forecasting and Stress Testing Credit Card Default using Dynamic Models

Transcription

1 Forecastng and Stress Testng Credt Card Default usng Dynamc Models Jonathan Crook, Unversty of Ednburgh - Busness School 50 George Square, Ednburgh, Md Lothan EH8 9JY Unted Kngdom J.Crook@ed.ac.uk Tony Bellott Imperal College - Mathematcs and Statstcs London Unted Kngdom Abstract: We present dscrete tme survval models of borrower default for credt cards that nclude behavoural data about credt card holders and macroeconomc condtons across the credt card lfetme. We fnd that dynamc models that nclude these behavoural and macroeconomc varables gve statstcally sgnfcant mprovements n model ft whch translates nto better forecasts of default at both account and portfolo level when appled to an out-of-sample data set. By smulatng extreme economc condtons, we show how these models can be used to stress test credt card portfolos. 1. Introducton Applcaton consumer credt scorng models use detals about oblgors or potental customers that are statc. Such models are used to determne whether an applcant should be granted credt, based on data whch are collected at the tme of applcaton and then reman fxed. Behavoural consumer credt scorng models use both nformaton collected at the tme of applcaton and behavoural varables, the values of whch have changed over tme, but whch are fxed at the tme of estmaton. Both are cross-sectonal models and allow the predcton of a probablty of default wthn a specfed tme wndow, lke eghteen months. However, models that answer more specfc questons can also be estmated from credt portfolos, snce they provde panel 1 of 41

2 data (Crook & Bellott, 2009) for a sample of oblgor accounts. Panel data allow one to estmate hazard models whch predct the probablty of an event (such as a default) occurrng n the next nstant of tme, condtonal on the event not havng occurred before that tme, for any future tme perod one chooses. Unlke cross-sectonal models, n a panel model one can nclude varables whose values change over the estmaton perod. Of partcular relevance here are common economc rsk factors that affect all oblgors n a portfolo n generally the same way. For example, we would expect that a large ncrease n nterest rates would cause, ceters parbus, a general ncrease n the probablty of default (PD). Tme varyng behavoural varables may also be ncluded. We call these sorts of models that nclude tme varyng covarates (TVCs) dynamc models. Furthermore, statc models typcally only have value n assessng the rskness of applcants and oblgors. However, f we want a complete pcture we should be lookng at the return alongsde rsk, whch requres the use of dynamc rather than statc models (Thomas, Ho, & Scherer, 2001; Ma, Crook, & Ansell, 2009). In ths paper, we present dynamc models of default whch nclude tme varyng behavoural varables (BVs) and macroeconomc varables (MVs), n addton to applcaton varables (AVs). The ncluson of MVs also enables us to perform stress tests, snce extreme economc condtons can be smulated and ncluded n the model n order to generate a measure of stressed loss (default rates). Accurate stress tests are becomng ncreasngly mportant n evaluatng the rsk to banks, as s evdent from the evaluaton of US banks (Board of Governors, 2009) and the recognton by the Fnancal Servces Authorty (2008) that stress testng s a key tool n helpng fnancal nsttutons to make busness strategy, rsk management and captal plannng decsons. 2 of 41

3 Our paper contrbutes to the lterature n four ways. Frstly, for a large portfolo of credt card accounts, we show that ncludng behavoural varables mproves the model ft n a dscrete tme hazard model, and that ther ncluson mproves the forecast accuracy. Secondly, we fnd that, whle several MVs are statstcally sgnfcant explanatory varables of default, ths does not translate nto mproved forecasts at the account level. Thrdly, we show that ncludng MVs can mprove the estmaton of loss (default rate) at the portfolo level. Fourthly, usng account level data, we demonstrate the use of MVs for stress testng and report the dstrbuton of expected default rates based on Monte Carlo smulaton of economc condtons. In Secton 2, we provde a lterature revew. In Secton 3, we outlne the methods we use, descrbng the dscrete survval model, our test procedures and the stress testng methodology. We then descrbe our data n Secton 4, and present some results n Secton 5. Fnally, we offer conclusons and a dscusson n Secton Lterature Revew Several modellng technques have been proposed to develop a dynamc model of credt (see Crook & Bellott, 2009, for a revew). Thomas et al. (2001) descrbe how a Markov chan stochastc process can be used as a dynamc model of delnquency. However, the approach they descrbe does not allow for model covarates, although models can be bult on separate segments to allow the modellng of dfferent rsk groups. They also descrbe survval analyss as a means of buldng dynamc models, snce ths readly allows the ncluson of BVs and MVs as tmevaryng covarates (TVCs). Bellott and Crook (2009) follow ths path, usng the Cox 3 of 41

4 proportonal hazard survval model to model the tme to default for a large database of credt cards. They nclude MVs, but not BVs, as TVCs, and fnd a modest mprovement n predctve performance n comparson to a statc logstc regresson. We take a smlar approach usng a survval model here, but wth dscrete tme survval analyss. Dscrete survval analyss can also be understood as a logstc regresson on a panel data set, wth the data arranged so that default s condtonal on no pror default havng occurred on that account. Snce credt data are usually n the form of panel data, and n partcular account records are dscrete (e.g. monthly records), ths s a more natural choce than contnuous tme survval analyss. It also has the advantage of beng more computatonally effcent, snce probablty forecasts nvolve smple summatons over tme perods, rather than an ntegraton whch may be complex when TVCs are ncluded n the model. Dscrete survval models have been appled successfully to the analyss of personal bankruptcy and delnquency n the USA (Gross & Souleles, 2002), mortgage termnatons (Calhoun & Deng, 2002), and competng rsks of foreclosure and sales n the US subprme market (Gerard, Shapro, & Wllen, 2008). Gross and Souleles (2002) used several dfferent BVs and MVs. In partcular, they ncluded the outstandng account balance and repayments, and found that the former had a postve affect on bankruptces, whle the latter had a negatve affect. They also found that the local unemployment rate had a statstcally sgnfcant postve affect on bankruptcy, whch s what we would expect, snce an ncrease n unemployment s lkely to affect some oblgors adversely. Calhoun and Deng (2002) derved dynamc varables measurng the probablty of negatve equty and the mortgage premum. Both change over tme and have a postve affect on default. They also ncluded the rato of 10-year to 1-year Constant Maturty 4 of 41

5 Treasury yelds, and found t to be statstcally sgnfcant for models of early repayment. For fxed-rate mortgages, the coeffcent ncreases for hgher ratos, wth the ratonale that mortgagors wll be movng to adjustable-rate mortgages, n order to take advantage of the shortterm relatvely low nterest rates. Gerard et al. (2008) found that nterest rates (the 6-month lbor rate) and unemployment rate are statstcally sgnfcant explanatory varables for both mortgage default and sales, wth a postve affect on default, as we would expect, and a negatve affect on sales. These studes have shown that both BVs and MVs are useful explanatory covarates for consumer credt rsk. We therefore extend ths work by usng these dynamc models for forecasts and stress testng. Ultmately, fnancal nsttutons and regulators are nterested n consumer credt rsk models for the estmaton of future losses at both the account and portfolo levels, ether n normal (expected) crcumstances or when consderng adverse condtons. For ths reason, we focus prmarly on usng the models for forecastng PD and the default rate. The lterature on stress testng s growng rapdly. In an early paper, Berkowtz (2000) proposed a stress testng methodology n whch two separate forecast dstrbutons are generated usng a rsk model: one for normal condtons and another reflectng stressed condtons, based on changes n an underlyng factor. Our approach s rather dfferent to that of Berkowtz (2000), n that we do not choose an ntal dstrbuton under stressed condtons; nstead, we generate a sngle dstrbuton of expected default rates and focus on the upper percentles for stress testng. Stress-testng models may be dvded nto macro stress-testng models and mcro stress-testng models. Macro stress-testng models concern the mplcatons of stressed states of the economy 5 of 41

6 for the captal of groups of nsttutons, and the am s to examne the captal adequacy of the fnancal sector of an economy n the event of adverse shocks. In contrast, a mcro stress test relates to a specfc portfolo of one lender. Sorge and Vrolanen (2006) dvde macro stress tests nto () those that relate aspects of banks balance sheets to macroeconomc actvty, and () value at rsk models where macroeconomc factors are related to aggregate default rates (not the probabltes of default of ndvdual oblgors). For examples of balance sheet models, see Drehmann, Sorensen, and Strnga (2010) and Delgado and Surna (2004). Macro Value at Rsk (VaR) stress tests and mcro stress tests follow smlar methodologes. Frst, a model that relates macroeconomc varables to each other, often a Vector Autoregressve Regresson model, s estmated. Second, a default rate (macro) or probablty of default (PD) (mcro) model that ncorporates macroeconomc varables s parametersed. Thrd, a macroeconomc scenaro s chosen and ts mplcatons for the dstrbutons of PD (or expected loss) are predcted, or mpulse response functons are estmated. For examples of such macro stress tests, see Breuer, Jandacka, Menca, and Smmer (2012), Jokvoulle and Vren (2011), Castren, Dees, and Zaher (2010), and Huang, Zhou, and Zhu (2009). Our work dffers from that of others n a number of ways. We are concerned wth mcro stress testng rather than macro stress testng, and we consder account level observatons rather than aggregate default rates. The data used n our analyss are therefore more granular than the data used n almost all other studes. In addton, the vast majorty of publshed macro stress tests relate to corporate loans, whereas n ths paper we consder credt card loans. 6 of 41

7 Few stress test results for retal loan portfolos have been publshed. Breeden and Ingram (2009) dscuss the ssues whch are nvolved n the generaton of scenaros usng a model where the default rate of a portfolo over tme s explaned n terms of a functon of the duraton tme, a functon of calendar tme, and a functon of vntage, but they do not present the results of a stress test for a portfolo. Rösch and Scheule (2004) assume a Merton one-factor model and estmate loss (default rate) dstrbutons for credt cards, mortgages and other consumer loans n the US. Unlke our paper, they use aggregate default rate data and do not nclude varables whch are specfc to the oblgor. In more recent work, Rösch and Scheule (2008) consder estmatng and stressng the correlaton between PD and LGD. They present an ntegrated approach to stress testng usng a rsk model wth unobserved systematc rsk factors and observed economc varables, and show that ths model can be used to produce plausble stress testng estmates of PD, LGD and loss rates for Hong Kong mortgage loans. Ther methodology s smlar to ours. Pesaran, Schuermann, Treutler, and Wener (2006) and Drehmann, Patton, and Sorensen (2005) are the only studes we know of that use a methodology smlar to ours, but ther work relates to corporate portfolos and ther data are pooled across dfferent lenders. 3. Methods 3.1. Dscrete survval model for dynamc credt scorng We treat tme as beng dscrete and adopt the followng notaton. We denote calendar tme by c and the date that an account was opened by a. Let t be the number of months snce an account was opened (duraton tme). The term account openng (0 = non-default, 1 = default). The term d t ndcates whether account defaults at tme t after w s a vector of statc AVs collected at 7 of 41

8 the tme of account applcaton and x t s a vector of BVs collected across the lfetme of the account. The term z t s a vector of MVs, whch s the same for all accounts on the same date; that s, for any two accounts, j havng records for duraton tmes t and s respectvely, f a t a j s then z t z js. We model the probablty of default (PD) for each account at tme t as P t Pr d t F φ 1 d s 0 for all s t; w, x, z tk tl T T T t β1 w β 2 xtk β3 z tl T β 4, (1) where k and l are fxed lags on BVs and MVs respectvely; and φ s a vector transformaton functon of duraton that s used to buld a parametrc survval model; specfcally, we use the 2 transformaton t t, t,log t, log t 2 φ. s an ntercept, and β 1, β2, β3, β4 are vectors of coeffcents to be estmated. F s a gven cumulatve dstrbuton functon. We use the logt (. The vntage effect s ntroduced by usng ndcator varables n w. x functon F x) 1 1 e We ensure that the underlyng panel data are constraned by the condton n Eq. (1): that s, no observatons are recorded after the frst default on any account. Gven ths condton, the model s a proportonal odds dscrete survval model, wth the falure event defned as a default. It can be estmated usng standard maxmum lkelhood estmaton for logstc regresson (Allson, 1995). Coeffcent estmates on the duraton φ t gve a baselne hazard. If ndcator varables are ncluded for each dscrete tme, then ths would parallel the commonly used Cox proportonal 8 of 41

9 hazard model. However, havng so many ndcator varables would make the model dffcult to estmate effcently, even wth a large sample. Publshed studes have suggested that there s a common shape to the dstrbuton over duraton tme of default hazard rates: they rse sharply wthn the frst few months, then fall steadly over the remanng duraton of the account (Gross & Souleles, 2002, Fgure 1; and Andreeva, 2006, Fgure 1). We use a parametrc form for φ whch allows us to capture ths structure of the hazard over tme. Log terms are ncluded n order to allow ths structure to take a skewed shape. The estmated survval probablty of an ndvdual at some tme t s gven as the product of the probablty of not falng at each tme perod, condtonal on not havng faled prevously. That s, t t 1 Sˆ. (2) P s s1 The falure probablty 1 Sˆ t then gves the PD at tme t, whch corresponds to the usual measure of PD, and can be used n further analyss at ether the account or portfolo level, both for credt scorng and for computng captal requrements. To compare the performances of dfferent model components such as BVs and MVs, we consder the followng specal cases of Eq. (1): 1. duraton only: fx β 2, β3, β4 to zero; 2. AV only: fx β, β 3 4 to zero; 3. AV and BV only: fx β 4 to zero; 4. AV, BV and MV: all coeffcents are estmated. 9 of 41

10 The lag k on the BVs restrcts the range of forecasts that can be made by the model, snce a perod k after our observaton date, there wll no longer be any behavoural data avalable to make estmates. For example, f the lag s 6 months, then we can only forecast usng the BV model up to 6 months ahead. Clearly, the longer the perod we can forecast forward, the better. However, we would expect that f longer lags were used, the forecast performance would deterorate. Therefore, we have a trade-off. We expect forecasts of 3 to 12 months ahead to be useful, so we consder lags of 3, 6, 9 and 12 months. It s also possble that some BVs may be endogenous varables. For example, there may be a common underlyng factor whch causes both an ncrease n account balance and default. Then, a hgh balance s not a cause of default, but t may be found to be an mportant drver of default n the model. The shorter the lag perod, the more lkely ths connecton wll be, whch s a further reason for preferrng longer lags. For ths reason, we focus on a BV model wth 12 month lag. Nevertheless, we note that, although endogenety affects the dentfcaton of the cause, t does not affect the forecasts, whch are the man concern of ths paper. The mplcatons of the lag term l for the MVs are dfferent. The MVs can be estmated usng standard autoregressve methods (Hamlton, 1994), or may be used wth smulated values durng stress testng. For ths reason, we can use MV values at the tme of default. In partcular, snce we defne an account as beng n default when t s recorded as havng faled to make the mnmum payments for three consecutve months or more, where the tme wndow over whch the mssng payments were recorded changes over tme as payments are made, we use a 3-month lag on MVs, to correspond to the begnnng of mssed payments leadng to default. 10 of 41

11 3.2. Forecastng procedure Credt rsk models can be used to explore causal hypotheses of consumer credt behavour; for example, Calhoun and Deng (2002) explore the dynamcs and causes of mortgage termnatons. However, for fnancal nsttutons and regulators, these models typcally have value for estmatng the rsk to ndvdual accounts or losses on credt portfolos. In ths way, banks can assess possble future losses and calculate captal requrements as buffers aganst adverse loss (Board of Governors, 2009). It s n ths forecastng capacty that we assess these models. We dvde the panel credt data set nto an n-sample tranng data set T, and a post, out-of-sample test data set S. Models of default are bult on T and forecasts of default are evaluated on S. The procedure s as follows: 1. Gven a rato r, accounts are sampled randomly, ensurng that the rato of the number of accounts n T to S s approxmately r. 2. Gven an observaton date, accounts n T are rght censored so that only account records pror to are ncluded (.e., t a ). 3. Accounts n S are ncluded only f they were opened pror to (.e., a ), but they are left-censored so only post-observaton date records are ncluded (.e., t a ). Ths procedure s ntended to mmc the operatonal stuaton, where a model s bult at a partcular tme (correspondng to the observaton date) usng past tranng data, and forecasts ahead for accounts that already exst on the books. A sngle model s bult based on the fxed tranng data T, even though the forecast perod may be long enough to allow us to refresh wth 11 of 41

12 new data, snce ths reflects the ndustry practce of usng a credt scorng model for several years before renewal; t also allows us to report consstently on a sngle fxed model structure. Ths procedure does result n a large number of records beng removed, but the random samplng ensures that no bas s ntroduced when generatng the out-of-sample test set, whlst the censorng ensures that all predctons are forecasts. As a practcal pont, fnancal nsttutons could also use post, n-sample data sets for forecasts, and these may well gve more accurate results. However, for ths exercse, n order to avod the ntroducton of bas, the forecasts are restrcted to an out-of-sample data set (Granger & Huang, 1997) Performance measures Snce we are usng survval models whch model the tme to default, the usual predctve performance measures for classfcaton algorthms, such as error rates and the Gn coeffcent, do not naturally apply, and nor do the standard resduals for regresson, such as mean square errors. Therefore, we use the devance (the log-lkelhood rato) to measure the model ft for each model separately, and also to test the goodness-of-ft for nested models. The contrbuton of each ndvdual to the log-lkelhood functon for dscrete survval analyss usng the logstc functon s L * t s1 log P d s log P * s 1 d log1 P s * t 1 * 1 log1 P log1 Ps s s1, (5) 12 of 41

13 where * t s the last observaton avalable for account and P P * denotes the hazard * t probablty of ths last observaton, rememberng that only the last observaton can fal wthn the survval analyss framework. From Eqs. (2) and (5), t then follows that the devance resdual s where r C * t * * L r log P / 1 P, (6) C log Sˆ s the Cox-Snell resdual and d t * ndcates default. The devance resdual s used to assess predctons at the account level. However, our models can also be used to forecast at an aggregate level, e.g., across accounts wthn a sngle portfolo. The observed default rate for an aggregate of N accounts at a partcular calendar date c s gven by D c 1 N N 1 d ca, (7) whch mples that the estmated default rate forecast gven by a partcular model s E 1 N N Dc P ca 1. (8) The dfference ED c Dc then gves a measure of performance for aggregate forecasts Stress testng We consder a smulaton-based stress test of default rate on an aggregate of accounts usng Monte Carlo smulaton (see for example Marrson, 2002). Over m teratons, the procedure s as follows. 1. Buld a dynamc model wth MVs from a tranng data set. 13 of 41

14 2. Generate a smulaton of economc condtons usng values of MVs based on a dstrbuton of hstorc macroeconomc data. 3. Smulate default events on test data by substtutng the smulated MV values nto the model. 4. Repeat steps 2 and 3 m tmes, to buld a dstrbuton of estmated default rates (DR) over dfferent economc scenaros. 5. Use the DR dstrbuton to compute the estmated DR, gven extreme economc crcumstances. Stress tests should consder unexpected but plausble events. When m s large, a suffcent number of extreme events can be smulated to meet the frst crtera; basng the smulatons on hstorcal data ensures the second. Notce that f we assume a constant loss gven default and exposure at default for each account, the DR dstrbuton s proportonal to the dstrbuton of expected losses. Further explanatons of the steps are gven below. In step 1, we use the dscrete survval model descrbed n Secton 3.1, although ths s a general stress testng method, and other dynamc model structures are possble. In step 2, smulated values of MVs are drawn randomly from the avalable tme seres of hstorc MVs. However, f the structure of dependences between the MVs s not taken nto account, ths wll yeld mplausble scenaros and lead to msleadng results. Therefore, the Cholesky decomposton s used (Marrson, 2002) n order to preserve the covarance structure between MVs. If V s a matrx of covarances between tme seres of hstorc macroeconomc data, then t s decomposed by a lower trangular matrx L, such that T V LL. Then, f u s a sequence of 14 of 41

15 values generated ndependently from the standard normal dstrbuton, z* Lu wll follow the covarance structure of V, and so can be used as plausble economc smulatons. The Cholesky decomposton assumes that the varables are normally dstrbuted. However, ths s not usually the case for MVs, and so we apply a transformaton to MVs pror to smulaton, f requred. Therefore, a Box-Cox transformaton s used, snce ths often produces an approxmately normal dstrbuton (Box & Cox, 1964). Alternatvely, an emprcal probt transformaton s used to mpose a normal dstrbuton on the hstorcal data pror to usng a Cholesky decomposton. In step 3, gven a smulated vector of MVs, z *, and usng the latent varable model of logstc regresson (Verbeek: 2004, Secton 7.1.3), a default event for observaton s smulated as where d t T ˆ T T ˆ ˆ T t β w ˆ β2 xtk β3 z * β e ˆ φ 0, I 1 4 I s the ndcator functon, and the error e s smulated from the dstrbuton of the lnk functon F. Then, usng Eq. (7), the smulated DR for a calendar tme perod c s computed as Dˆ c 1 z, (9) N N 1 0 T T T T *, e I ˆ φc a βˆ w βˆ ˆ ca k ˆ e 2 x β3 z * β 4 1 where e e, 1,eN s a vector of N ndependent error terms, each of whch s cumulatvely dstrbuted as F. The smulaton takes nto consderaton the error n the model, e, along wth changes n macroeconomc condtons. Ths s natural, snce otherwse the pont predctons of Eq. (8) are assumed to be exactly correct. Notce also that the fxed are not smulated, snce they are known and are avalable from the account data. w and lagged x t varables 15 of 41

16 In step 4, multple smulatons of estmates of DR, Dˆ z, e c j j z j and e j are made for j = 1 to m, leadng to a seres of m, whch form an emprcal dstrbuton. For a further dscusson, suppose that the smulatons are ordered by descendng values of the estmated DR;.e., for all h j, Dˆ z, e Dˆ z, e. c h h c j j In step 5, extreme values are computed for stress testng. We consder two measures: Value at Rsk (VaR) and expected shortfall. VaR s defned as the maxmum expected loss, wthn a certan tme perod, for a gven percentle, q, and the VaR (at 100(1 q)%) of DR here s approxmated as Dˆ c z qm, e qm. However, VaR captures the worst loss n normal crcumstances, whereas stress tests should consder losses durng unusual crcumstances. Therefore, VaR may not be an approprate measure of loss durng adverse condtons, and may be too conservatve (BIS, 2005). For ths reason, we also consder the expected shortfall as a measure of loss. Ths s gven as the expected DR n the upper q percentle of the DR dstrbuton, for a gven q. Therefore, the Monte Carlo approxmaton of the expected shortfall DR s S 1 qm ˆ q D c qm j1 z, e. (10) The number of teratons m s chosen so that both the VaR and expected shortfall converge to stable values. j j 4. Data 16 of 41

17 4.1. Credt card data We have three large data sets of UK credt card data coverng the perod from 1999 to md-2006 and comprsng over 750,000 accounts. All of the data sets nclude AVs taken at the tme of applcaton, along wth monthly account behavoural records. Most of the data are collected n the same way and have the same objectve meanng between credt card products, although the dstrbutons vary, snce dfferent products wll have dfferent demographc and rsk profles. Some of the data have already been used for analyss by Bellott and Crook (2012). Varables that may be defned dfferently for each product have not been used. A lst of the varables used s gven n Table 4. Categorcal varables for employment and payment status are ncluded as a seres of ndcator varables. Age s dvded nto a seres of age category ndcator varables, snce age has a non-lnear relatonshp wth default. All of the monetary values, such as ncome and balance, are gven as log values, n order to normalze ther dstrbutons. There s a small proporton of mssng values n the monthly payment amount, so an ndcator varable for these values s also ncluded. Also, there s a large proporton of zero values for some BVs, such as payment amount, sales amount and APR, so ndcator varables are ncluded for those cases too. In these experments, we defne an account as beng n default when t s recorded as havng faled to make the mnmum payments for three consecutve months or more, where the tme wndow over whch the mssng payments were recorded changes over tme as payments are made. Ths s a common defnton n the ndustry, and follows the Basel II conventon of 90 days delnquency for consumer credt (BCBS 2006). The data whch we use for our analyss are commercally senstve, and therefore we cannot provde further detals or data descrpton statstcs, or report the observed default rates. It s possble that two or more credt card accounts 17 of 41

18 may relate to the same person. However, we expect ths to be very rare, and unlkely to affect model specfcaton sgnfcantly. To assess the forecasts, an observaton date of 1 January 2005 s set. Snce the dataset runs to md-2006, ths provdes up to 18 months of test data, whch s a good perod for forecasts, whlst allowng for a long run of tranng data. After censorng, usng the procedure descrbed n Secton 2.2, ths gves over 400,000 and 150,000 accounts n the tranng and test sets respectvely, thus provdng suffcent observatons for tranng whlst leavng a good number of accounts out-of-sample for forecastng Hstorc UK macroeconomc data We consder several UK MVs whch we expected to have an effect on the PD. These are lsted n Table 1. Bellott and Crook (2009) found that bank nterest rates, earnngs, the producton ndex and the house prce were statstcally sgnfcant explanatory varables of UK defaults, so we nclude these. The producton ndex s used nstead of GDP because t s avalable monthly, whereas the GDP fgures are only provded quarterly. Gerard et al (2008) also found the unemployment rate to be sgnfcant for US defaults, and n a study of a number of stressed credt markets worldwde, usng a dynamc model, Breeden and Thomas (2008) found that varables for consumer sales and prces were correlated wth default and bankruptcy. Therefore, we also nclude MVs for these rsk factors. In addton, the FTSE ndex and a consumer confdence ndex are also ncluded, snce they may be good ndcators of confdence n the economy. TABLE 1 HERE 18 of 41

19 To remove seasonalty, we took twelve-month dfferences of the MVs. In addton, log values are taken for the MVs wth clear exponental growth: earnngs, FTSE and house prces. For stress testng, hstorcal values of MVs from 1986 to 2004 are taken;.e., only MV data pror to the observaton date are ncluded. The observaton date of 1 January, 2005, gves tranng data between 1999 and Durng ths perod, there were moderate changes n the economy: the producton ndex rose n 1999 and peaked n 2000 (104.7), after whch tme t dropped (to 97.3), then stayed steady over the remanng perod (endng at 101); the unemployment rate was generally fallng n 1999 (from 1,800,000 to 1,431,000), after whch t rose slghtly n 2003 (to 1,561,000), before fallng agan (to 1,400,000); bank nterest rates fell from 6% n 1999 to 3.5% n 2003, before rsng back to 4.75% n Table 2 summarzes the MV statstcs for ths model buld perod only. Inevtably, n contrast to Table 1, the standard devatons on the varables are smaller (by approxmately half), but stll show some movement n values over the model tranng perod. TABLE 2 HERE 5. Results We present our results n fve subsectons. Frstly, we present the underlyng hazard rate for default. Secondly, we dscuss coeffcent estmates from the model buld. Thrdly, we present the model ft and forecastng results at the account level. Fourthly, we gve forecast results at the aggregate level. Lastly, we present results for stress testng. 19 of 41

20 5.1. Hazard rate for default The duraton-only model provdes ntal baselne hazards. Fgure 1 shows the shape of the hazard probablty over tme. It has the typcal survval profle for consumer credt: PD peaks early, at 8 months, then slowly declnes over tme, as those who are hghly lkely to default drop out. Ths structure has been reported prevously by others; see for example Gross and Souleles (2002) and Andreeva (2006). Fgure 1 also shows a second small ncrease n the hazard, peakng at around 36 months. Ths s because, for all credt card products, accounts wth no recent usage are removed from the portfolos after two years. Snce these are typcally low rsk accounts, ther removal leads to a small overall ncrease n the default rsk. It s therefore mportant to realze that the hazard rate s not just an ndcaton of the oblgor s propensty to default, t wll also be nfluenced by perodc operatonal decsons made by portfolo managers. The hazard probablty estmate n Fgure 1 s based on a sample sze of over 60,000 observatons for all months of account age (3 to 48) gven on the horzontal axs. For the sake of commercal confdentalty, the precse number of observatons for each account month cannot be gven. FIGURE 1 HERE 5.2. Model and coeffcent estmates Many AVs and BVs and several MVs are statstcally sgnfcant explanatory varables. We focus on the model for the BV lag 12 months, snce ths s the most practcally valuable model n terms of the forecast range. Table 3 shows coeffcent estmates for ths model. We fnd the followng key outcomes. 20 of 41

21 1. The sgns on the current balance (log) and ts square are opposte, but the postve sgn on the square term domnates. Therefore, the balance outstandng on the account has an ncreasng postve effect on the default hazard. Ths s unsurprsng, snce a larger balance wll be more dffcult to clear. Second, an ncrease n the credt lmt reduces the hazard. Intally, ths may appear surprsng, snce we mght argue that a hgh credt lmt encourages a hgher balance, and therefore a greater rsk. However, at least n the short term, a hgh credt lmt provdes the oblgor wth a buffer, enablng the oblgor to buld up debts before reachng default. 2. The bank sets the credt lmt based on ther own assessment of the oblgor s behavour, so the credt lmt acts partly as a proxy for a behavoural score. 3. The amount pad back each month, ndcated by the payment status and amount, has a negatve effect on default. Ths s to be expected, snce a greater ablty to repay mples that default s less lkely. 4. The number of transactons has a postve effect on default. Ths s to be expected, snce t ndcates a greater card usage, and hence a rsng balance. However, nterestngly, the transacton sales amount has a negatve effect. A possble explanaton s that the sales amount acts as an ndcator of wealth, when taken together wth the number of transactons. That s, people who make a few bg purchases are more lkely to be wealther, and therefore more able to repay, than those who make many small purchases. 5. When behavoural data are mssng, PD decreases consderably. However, snce no duraton tmes up to 12 months wll have BVs (because of the lag), ths s manly a jont effect wth duraton. 21 of 41

22 6. Indcator varables have been added for vntage, ndcatng the year of account openng. These are sgnfcant, mplyng that cohorts explan some of the effects over tme for default rates wthn the data set. Ths s natural, snce lenders wll allow new accounts wth hgher or lower levels of rsk onto ther books at dfferent tmes, dependng on ther changng atttude to rsk at dfferent tmes n the busness cycle. 7. Concernng macroeconomc varables, the nterest rate has a postve effect on default. Ths s to be expected, snce rsng nterest rates mply a greater demand for repayment on outstandng loans and mortgages, whch wll have an adverse effect on people who are more hghly ndebted. The unemployment rate also has a postve effect on default. The unemployment rate s an ndcator of the drect economc stress on ndvduals. In partcular, oblgors who become or reman unemployed wll fnd t more dffcult to repay ther debt. Conversely, f unemployment decreases, then we would generally expect unemployed oblgors to fnd jobs, makng t easer for them to repay. Therefore, the effect of ths MV on default s as expected. Only the nterest and unemployment rates are statstcally sgnfcant MVs n the model, so the model can be rebult wthout the other MVs, to determne how they are affectng the parameter estmates of the former. We fnd smlar results: the nterest rate remans sgnfcant (p < ), although ts effect sze s smaller (0.0615), and the unemployment rate s also sgnfcant (p < ), but ts effect sze s approxmately unchanged ( ). The frst three fndngs corroborate the results of Gross and Souleles (2002), who bult dynamc models of default for US credt card data. They found that the rsk of default rses wth the balance and falls wth repayments. They used utlzaton (the outstandng balance dvded by the 22 of 41

23 credt lmt) nstead of the raw value of the balance, whch s sensble, gven the relatonshp dscussed n pont 2. The shape of the relatonshp whch we fnd between the baselne hazard and the duraton of holdng the card s consstent wth that of other studes, but we fnd the baselne hazard to be at a maxmum at a duraton tme of 8 months, whereas Gross and Souleles (2002), who also consder credt cards, found t to be around 18 months (see ther Fgure 1). Calhoun and Deng (2002, Exhbt 2) and Wllen et al (2007, Fgure 12) both consder mortgages and fnd that the maxmum hazard occurs at between 16 and 28 quarters. One possble reason for the dfference between our results and those of Gross and Souleles may be that Amercan card holders hold more cards than UK card holders, and so spread ther outstandng balance over more cards. The fndngs for the MVs are generally consstent wth those of other comparable studes. As n our study, sgnfcant postve effects of the unemployment rate have been found by both Gross and Souleles (2002) when modellng the probablty of bankruptcy, and Wllen et al. (2008) when modellng the probablty of mortgage default. Bellott and Crook (2009), usng a dfferent data set, also found a postve sgn for unemployment, but t was not sgnfcant. Gross and Souleles (for bankruptcy), Wllen et al., and Bellott and Crook all found a postve and sgnfcant effect for nterest rates, consstent wth our result n pont 7. However, when Gross and Souleles (2002) modelled delnquency by credt card holders usng a defnton very smlar to that whch we use for default, they dd not fnd that ether the unemployment rate or house prces were sgnfcant predctors. There are a range of dfferences between our study and that of Gross and Souleles that mght account for ths dfference n fndngs. Gross and Souleles had a 23 of 41

24 dfferent set of covarates, and ther estmaton perod was much shorter than ours. Ther tranng data covered only two years, whereas ours extended over at least fve years, and so potentally ncluded more nformaton about the relatonshp between the default probablty and the unemployment rate (they dd not nclude an nterest rate). Our tranng data also covers a dfferent perod ( ) to ther data (md-1995 to md-1997). The correlatons we fnd wth the unemployment rate are also consstent wth the study by Breeden and Thomas (2008), who use several world-wde data sets, although they also found the GDP to be sgnfcant n many cases. We dd not nclude ths varable because we were usng monthly data and the GDP s not avalable monthly. They also ncluded vntage effects n ther models. Unvarate assocatons for each MV are also reported n Table 4. Ths ndcates that almost all MVs have an assocaton wth default. However, these MVs are correlated wth one another, so they do not all come out as sgnfcant varables n the multvarate model. These results are for the model wth BVs lagged 12 months. We found smlar results for models wth shorter lag perods, and the comments made above also hold n these cases, except that the effects and statstcal sgnfcance tend to be stronger for models wth shorter lags. TABLE 3 HERE TABLE 4 HERE 24 of 41

25 5.3. Model ft and forecasts of tme to default Model fts for several alternatve models are shown n Fgure 2. Ths shows a general mprovement n model ft as we move from the smple duraton-only model to the AV-only model to the AV and BV model. In addton, we also observe that the model ft mproves wth shorter lags on the BVs, wth a relatvely large mprovement at a 3-month lag. However, as we have dscussed, ths mprovement comes at the prce of a much shorter range of forecasts. We can see n Fgure 2 that, although some of the MVs are statstcally sgnfcant, ther contrbuton to model ft s weak. The ft of the nested model s also assessed, wth the results shown n Table 5. Ths shows that all model extensons lead to a statstcally sgnfcant mprovement n ft, partcularly the basc duraton tme only model and the ncluson of BVs and MVs. FIGURE 2 HERE TABLE 5 HERE Fgure 2 also shows the results of the forecasts. These follow the model ft results very closely. They show a marked mprovement n ft for the BV models, mprovng wth shorter lags. However, there s no notceable change n forecast accuracy when MVs are ncluded Estmaton of default rates Fgure 3 shows the estmated monthly default rates for varous dfferent models, along wth the observed (or true) monthly default rates for each month of the test data set. The observed default rates have hgh varance, but there s a general trend of hgh rates begnnng n 2005, fallng durng 2005, then rsng agan n The AV model s able to model the general fall n default rates, but BVs are requred to forecast the overall trend, ncludng the rse n However, the 25 of 41

26 best forecasts ndcatng a hgh DR n early 2005 and md-2006, whlst also forecastng the dp n default rates at the end of 2005, are only made when MVs are ncluded n the model. The BV model, lag 3 months, also performs well, but ths s not surprsng, gven the short forecast perod, usng behavoural data just one month before accounts begn mssng payments. Overall, the BV lag 12 month model wth MVs performs best at forecastng the aggregate DR, achevng better results than even BV models wth shorter lags, as demonstrated n Table 6. FIGURE 3 HERE TABLE 6 HERE 5.5. Stress test results We ran Monte Carlo smulatons usng the MV model gven n Table 3. The estmated DR was smulated on the test data set, 12 months after the observaton date;.e., for December A stable default rate dstrbuton was generated after m = 25,000 smulatons, and s shown n Fgure 4. Notce, frstly, that the observed DR s postoned centrally wthn the dstrbuton, close to the medan. Ths demonstrates that ths perod dd not represent extreme default rates. The rght-hand tal shows the rsk for more adverse condtons. In partcular, we have ncluded the fgure for expected shortfall at the 99 th percentle. Ths shows that, for the worst 1% of economc scenaros we consder, the expected DR s 1.73 tmes greater than normal condtons (.e., the medan estmated DR). The VaR s also shown for comparson. We see that ths gves a lower estmate of DR (1.59 tmes the medan), whch may not reflect extreme crcumstances suffcently. These fgures are slghtly hgher than those suggested as part of the US stress testng exercse by the Board of Governors (2009). In partcular, the Board of Governors study estmates a more modest rse n DR of between 20% and 55%, when contrastng a normal 26 of 41

27 baselne fgure to more adverse condtons. 1 However, our results appear lower than those of Rösch and Scheule (2004), who found the VaR for US credt cards to be 2.31 tmes the mean, although they used aggregate data, not account level data. FIGURE 4 HERE 6. Concluson Dynamc models form a flexble approach to modellng and forecastng consumer credt rsk. They have a number of well-known advantages over statc models, ncludng modellng the condtonal PD n a specfc tme perod rather than n a tme wndow, and enablng the predcton of the proftablty of specfc loans (Bellott & Crook, 2009). In partcular, we have used dscrete-tme survval analyss to model credt card rsk. Ths has two man advantages. Frstly, t s a prncpled means of buldng dynamc models of default usng accountng records; and, secondly, modellng and forecastng are computatonally effcent when compared to contnuous-tme survval models. Ths s mportant when model bulders use large databases of credt accounts. We have used a large data set of UK credt card accounts to test the effectveness of dscrete survval models wth BVs and MVs as models of default. Unlke the prevous lterature, we explore the use of these models as tools for rsk measurement, forecastng and stress testng. We 1 The Board of Governors (2009) gves baselne two-year loss rates as 12 17%, and more adverse rates as 18 20%. Takng the lower and upper bounds of each range and convertng to an average monthly DR gves a 20 55% expected ncrease n loss. Takng a mean value for the baselne and more adverse rates (14.5% and 19% respectvely) gves a mean ncrease of 34%. 27 of 41

28 conclude that many BVs are statstcally sgnfcant explanatory varables of default, and ncludng them gves mproved model ft. Important BVs are the account balance, repayments, the number of transactons wthn each month, and the credt lmt. We fnd that an mproved model ft translates nto mproved forecasts of the tme to default. The performance mproves wth shorter lags on BVs. Ths s to be expected, snce shorter lags mply that the model s usng more recent nformaton about the oblgors. However, we also note that shorter lags mply shorter ranges of forecasts and greater levels of endogenety between BVs and the default event. For ths reason, we focus on lag 12 month BVs. Ths gves an mproved performance relatve to the AV only model, and also allows for useful forecasts up to 12 months ahead. We found that bank nterest rates and the unemployment rate affect the hazard sgnfcantly. Whlst ther ncluson gave only a modest mprovement n model ft and no notceable mprovement n forecasts of the tme to default at the account level, ther ncluson mproves forecasts of default rate at the aggregate level. Ths s understandable, snce MVs affect all predcted PDs, rather than at the ndvdual account level. Hence, ther effect wll only become notceable at the aggregate level where accounts are taken together. Where they are comparable, our results corroborate the results obtaned by others (Gross & Souleles, 2002; Calhoun & Deng, 2002; Gerard et al., 2008). Thrd, the ncluson of MVs enables us to use stress tests, whch generate credble results, ndcatng that adverse condtons may rase the default rate by around 79%. We used a smulaton-based approach for our experments, but scenaros could also be desgned and used wth these models. 28 of 41

29 The followng further ssues are suggested as developments from ths study: 1. The data cover a perod of relatvely bengn economc condtons ( ), although there s some evdence of UK default rates ncreasng toward the end of ths perod. It would be nterestng to follow up ths study usng new data through the perod of the credt crunch ( ) and beyond. Nevertheless, ths study shows that credble models wth macroeconomc varables can be bult usng data from less extreme economc condtons. It may well be that a model change occurs after the credt crunch perod, n whch case t would be benefcal to have separate models bult across the pre- and post- credt crunch perods. 2. The ncluson of MVs may explan some of the correlatons between accounts over calendar tme. However, t s possble that not all borrower correlatons are explaned, and a latent varable over tme may be of beneft. Ths was suggested by Bellott and Crook (2012), who mplement an asset correlaton model and show that ncludng MVs explans some of the correlaton, but not all. 3. In ths paper, we have nvestgated the problem of forecastng and stress testng default rates. However, economc captal s often defned as the dfference between the VaR and the mean of the Expected Loss dstrbuton. The calculaton of the Expected Loss for an oblgor nvolves a model of loss gven default (LGD) and exposure at default (EAD), as well as PD, along wth ther correlatons. In effect, ths paper assumes a constant LGD and EAD. An nterestng development of ths work would be to explore the role of a dscrete survval model wthn the scope of ths larger problem. 29 of 41

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