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1 Ednburgh Research Explorer A new mxture model for the estmaton of credt card exposure at default Ctaton for publshed verson: Leow, M & Crook, J 206, 'A new mxture model for the estmaton of credt card exposure at default' European Journal of Operatonal Research. OI: 0.06/j.ejor gtal Object Identfer (OI): 0.06/j.ejor Lnk: Lnk to publcaton record n Ednburgh Research Explorer ocument Verson: Peer revewed verson Publshed In: European Journal of Operatonal Research General rghts Copyrght for the publcatons made accessble va the Ednburgh Research Explorer s retaned by the author(s) and / or other copyrght owners and t s a condton of accessng these publcatons that users recognse and abde by the legal requrements assocated wth these rghts. Take down polcy The Unversty of Ednburgh has made every reasonable effort to ensure that Ednburgh Research Explorer content comples wth UK legslaton. If you beleve that the publc dsplay of ths fle breaches copyrght please contact openaccess@ed.ac.uk provdng detals, and we wll remove access to the work mmedately and nvestgate your clam. ownload date: 4. Jul. 208

2 A New Mxture Model for the Estmaton of Credt Card Exposure at efault Mndy Leow a,b & Jonathan Crook c a Credt Research Centre, Unversty of Ednburgh usness School, 29 uccleuch Place, Ednburgh EH8 9JS, Scotland, Unted Kngdom b Contact detals: mndy.leow@ed.ac.uk c Correspondng author: ; j.crook@ed.ac.uk

3 A New Mxture Model for the Estmaton of Credt Card Exposure at efault Abstract Usng a large portfolo of hstorcal observatons on defaulted loans, we estmate Exposure at efault at the level of the oblgor by estmatng the outstandng balance of an account, not only at the tme of default, but at any tme over the entre loan perod. We theorze that the outstandng balance on a credt card account at any tme durng the loan s a functon of the spendng by the borrower and s also subject to the credt lmt mposed by the card ssuer. The predcted value s modelled as a weghted average of the estmated balance and lmt, wth weghts dependng on how lkely the borrower s to have a balance greater than the lmt. The weghts are estmated usng a dscrete-tme repeated events survval model to predct the probablty of an account havng a balance greater than ts lmt. The expected balance and expected lmt are estmated usng two panel models wth random effects. We are able to get predctons whch, overall, are more accurate for outstandng balance, not only at the tme of default, but at any tme over the entre default loan perod, than any other partcular technque n the lterature. Keywords: rsk management, forecastng, panel models, survval models, macroeconomc varables, tme-varyng covarates 2

4 . Introducton Predctons of Exposure At efault (EA) are useful to banks for at least two reasons. Frst, the asel Accords defne expected loss as the product of Probablty of efault (P), Loss Gven efault (LG) and EA, so predctons of EA are needed to compute Regulatory Captal. Second, predctons of EA are needed for the predcton of Economc Captal that a bank beleves t needs to protect ts depostors n the event of severe unexpected events. Snce the credt crss of 2008, there has been ncreased awareness of the models for these components, and n partcular, for retal loans. However, these have been manly focused on P and LG models, and how they should and can be mproved (see Thomas (200) for a revew). The analyss and modellng of EA at account level has so far been relatvely neglected. For loans wth fxed loan amounts over fxed terms and pre-agreed monthly repayment amounts, t s possble to estmate at least a reasonable range for EA should the loan be expected to default n the followng tme horzon, e.g. n the next 2 months. However, n the case of revolvng loans, the subject of ths paper,.e. loans wth no fxed loan amount or term, debtors are gven a lne of credt, wth a credt lmt up to whch they can draw upon at any tme (as long as they have not gone nto default). Ths could make t dffcult for fnancal nsttutons to predct account level outstandng balance should an account go nto default, especally f accounts deterorate nto default quckly and draw heavly on the card just before default. Another ssue assocated wth the analyss and modellng of EA s the measurement of EA. EA s smlar to LG n that ts value s only of nterest n the event default occurs (although ts value stll needs to be estmated for the calculaton and preparaton of economc captal). However, unlke LG, where loss s predcted to be at some tme pont after default, EA s known the very nstant the account goes nto default. Therefore, although default-tme varables could be used n the modellng of LG, they cannot be used for EA models. As such, practtoners and the lterature create varous ndcators to be estmated nstead of EA, takng nto account the current balance and avalable lmt. Unfortunately, each method has lmtatons. 3

5 Our am s to propose a new method to predct EA for each loan n a portfolo and to demonstrate ts accuracy by comparsons wth methods currently n use and n the lterature. Unlke conventonal cross secton methods, our proposed approach explots the panel nature of a typcal credt card dataset to model the values of balance and lmt over tme n a way that allows extrapolaton from the tme of predcton to the tme of default. To evaluate our model, we use a large portfolo of defaulted loans and ther hstorcal observatons, to drectly estmate EA at the level of the oblgor by estmatng the outstandng balance of an account, not only at the tme of default, but at any tme over the entre loan perod, up to the tme of default. Our methodology has several advantages over current methods. Frst for revolvng credt loans, balance typcally approaches the lmt as an account moves over tme towards default. We explot ths observaton, to the extent that t s true, and the observaton that modellng an account s lmt at each tme n ts hstory can be done more accurately than the balance to more accurately predct the balance at default (that s EA) than f ths nformaton s not used. Second we avod several of the problems assocated wth current methods of modellng EA whch we descrbe n secton 2, for example the consderable senstvty to very small values of a denomnator. Thrd by usng panel models we can more accurately nclude the effects of macroeconomc varables and so enable EA estmates to be fxed as n a downturn scenaro than cross sectonal models. Further our method yelds predctons of balance at any tme n an account s hstory and a bank would beneft from such predctons to estmate expected future nterest ncome and so a component ofn expected proft from an account. The development and valdaton of the new Mxture model contrbutes to the lterature n two ways. Frst, ths s the frst paper to predct the outstandng balance for defaulted loans at any tme durng the lfe of a revolvng loan. Second, we ncorporate macroeconomc varables nto the model and so provde a framework sutable for stress testng later. The rest of ths paper s structured as follows. Secton 2 revews the lterature and Secton 3 explans the model. In Secton 4, we llustrate the use of the method and compare ts performance to methods n the lterature. Secton 5 shows an emprcal applcaton and Secton 6 concludes. 2. EA n the lterature 4

6 Only a few papers have examned EA and usually for corporate loans (see e.g. Araten and Jacobs (200), Jacobs Jr. (2008), Jménez and Mencía (2009), Jménez et al. (2009), Yang and Tkachenko (202) and arkova and Pathasarathy (203)). Few consder account level models, and they do not model EA drectly (e.g. see Tapln et al. (2007), Rsk Management Assocaton (2004)). Instead, they typcally model the Loan Equvalent Exposure (LEQ) Factor, the Credt Converson Factor (CCF) or the Exposure At efault Factor (EAF), and then transform them back to an estmate of EA (a more comprehensve revew can be found n Moral (2006)). Thus, Jacobs Jr. (2008), usng corporate data and a GLM modellng framework, models all three factors. arakova and Parthasarrathy (203) model four ratos usng four algorthms appled to corporate level varables for large syndcated loans over 2007 to Yang and Tkachenko (202) model EAF usng eght account level varables and compare seven estmators appled to 500 commercal borrowers. The closest to our work s Q (2009), who used unsecured credt card data, to model LEQ by lookng at the level of credt drawn at one year before default. No macroeconomc varables were ncluded n the above models. All come to the concluson that EA plays an mportant part n the calculaton of the provson of captal and should be more carefully ncorporated nto rsk and loss calculatons. To defne these terms, we adopt the defntons as n Jacobs Jr. (2008), Q (2009) arakova and Parthasarathy (203) and Yang and Tkachenko (202). In terms of nomenclature from here on, outstandng balance of account at duraton tme s represented by, and lmt of account at duraton tme s represented by L. We also construct a bnary varable that takes on the value f account defaults at tme and d d that takes on the value f account defaults at some tme n the future. To smplfy the notaton, the subscrpt representng account s dropped for the equatons n ths sub-secton. The three varables EAF, CCF and LEQ are defned n Table. Note that LEQ, CCF and EAF are not unversally defned. asel II refers to a Credt Converson Factor, CCF, but does not defne t except to state that t s a factor of any further undrawn lmt (see ASEL COMMITTEE ON ANKING SUPERVISION Internatonal Convergence of Captal Measurement and Captal Standards: A Revsed Framework., Paragraph 36, ), so t s not clear that there s a standard ndustry practce towards EA modellng. 5

7 Table : efnton of EA measures n use and n the lterature Varable EAF CCF LEQ L 0 l L 0 l l l l for L f f f L f L l l l l l l l Explanaton Rato of the balance at default tme, over the lmt at observaton tme l ; lmt s usually the lmt at the tme of applcaton and s known once account s opened Rato of the balance at default tme over the balance at some observaton tme l ; ths tres to get better predctons for balance by takng nto account the outstandng balance of an account at some observaton tme before default. A more sophstcated predcton for balance by not only takng nto account balance at some observaton tme before default, l, but also the undrawn lmt at that tme,.e. the remanng amount of credt the debtor s able to draw upon. However, modellng EA n terms of these ratos nvolves a number of dffcultes, some of whch are rehearsed by Jacobs Jr. (2008) and Q (2009). In the case of EAF, although we expect ts value to range between 0 and, t s possble and qute common to see outstandng balances greater than the assgned lmts, perhaps due to accumulated nterest or banks allowng borrowers to go over ther lmts, gvng values much greater than. Ths makes the choce of dstrbuton slghtly more challengng. A further problem noted by Q (2009) s that as an account moves towards default and ts balance ncreases, lenders may respond 6

8 dfferently between accounts; n some cases ncreasng the lmt, n others reducng the lmt. Ths may ntroduce unexplaned heterogenety n a cross sectonal model of EAF. Consderng CCF, t s possble that the outstandng balance at the selected observaton tme happens to be 0, or even negatve (the account s n credt), whch would gve CCF 0, and ths rases the ssue of the treatment of these accounts. It s also possble that some of these accounts then deterorate quckly nto delnquency and default. Also, should the account have a very low balance durng observaton tme and defaults wth a large balance, CCF could become an extremely large value, causng dffcultes wth data analyss and model estmaton. Although on the one hand, t s lkely that accounts that go nto default have large balances on ther account pror to default (for example, debtors who default due to behavoural ssues), t s also possble that accounts go from a low or zero balance to default wthn a short perod of tme (for example, debtors who default due to unexpected crcumstances), whch could then mply a dfferent set of predctors for each group. From the pont of predcton, a value of 0 for CCF does not make any sense as ths would mean a predcton of 0 for balance at some tme n the future, and possbly at default. Our method does not suffer from the theoretcal nablty to deal wth zero or negatve values of balance or the dffculty n modellng a dependent varable whch s composed of a rato where ts value s very senstve to dfferent values of the denomnator. In our approach we use panel data that ncorporates unexplaned heterogenety unlke cross sectonal models that have been used for the above ratos. The dfferent values that the stuatons and whch would gve dfferent mplcatons. LEQ can take could arse due to a number of dfferent Should the account have zero undrawn lmt,.e. outstandng balance equal to lmt, at the tme of observaton, we get an LEQ value of 0. Ths s a group of debtors who have used ther maxmum avalable lmt and are lkely to default, but would be dffcult to nclude and handle n the modellng because the LEQ value computed does not have the same mplcatons as the other computed for when balance and lmt are not equal. LEQ values 7

9 The majorty of accounts would have a postve LEQ, whch could be due to one of two stuatons: (a) when balance at default s greater than balance at observaton, and balance at observaton s below the credt lmt at observaton, whch would be the most common progresson nto default; or (b) when balance at observaton s greater than balance at default, and balance at observaton s already greater than the lmt at observaton. The latter would represent debtors who are actually recoverng from a large balance (and where perhaps extendng the credt wthout puttng the account nto default mght gve lower loss). Although these two groups of debtors would have LEQ n the same range, we expect ther characterstcs and crcumstances to be qute dfferent. It s also possble to have negatve LEQ agan n dfferent stuatons 2. The possble range of dfferent types of borrowers and crcumstances could gve make t dffcult to estmate and model LEQ. LEQ, coupled wth the fact that LEQ n the same range, would One weakness of several of the above methods s that accordng to how they are defned, these varables could become unstable 3 f the denomnator s very small, so some restrctons have to be mposed on the range of values. Q (2009) ncluded only accounts at default tme where undrawn lmt s greater than 50 US; Jacobs Jr. (2008) restrcted the values of LEQ to between 0 and and replaced outlers wth the maxmum and mnmum values of hs selected range. In hs CCF model, he restrcted the range of CCF to between the and 99 percentles, and replaced outlers wth these maxmum and mnmum values. oth authors effectvely gnored accounts that go from up-to-date to default suddenly or wthn a short tme perod, but ths was the only way to get plausble results. arakova and Parthasarathy (203) wnsorse LEQ and CCF at the 99 th percentle. Yang and Tkachenko (202) capped EAF at and floored t at 0. Tapln et al. (2007) dd not attempt to estmate LEQ (referred to as CCF n ther paper) as they would have to exclude about 50% of ther observatons. They proposed regresson models that estmate EA as a functon of balance and lmt, but dd not gve any 2 These are: (a) when balance at observaton s larger than lmt at observaton and balance at default s larger than balance at observaton, whch would represent debtors who are sprallng further nto debt and default; or (b) when balance at observaton s larger than balance at default, but both are below the lmt at observaton. Agan, we have two groups of debtors wth negatve LEQ values but where they have arrved va dfferent crcumstances. 3 These varables could have large volatlty over short perods of tme, most lkely concdng wth the perod just before default occurs as balance on accounts go from small to large n a short perod of tme. 8

10 9 ndcaton of covarates used or any performance measures. Note also that predctve results from most papers n the lterature that used these dependent varables have generally been poor. 3. The new Mxture model We propose the predcton of outstandng balance usng a Mxture model. The random varable, balance of account at duraton tme could be above, equal to or below the account lmt. The expected balance for account at tme s therefore gven n Equaton : d L E d L P d L E d L P d L E d L P d E,,,. () Typcally, as an account moves towards default, the balance ncreases towards and may exceed the lmt. Often, borrowers stop ncreasng the balance when t reaches the lmt. We explot ths occurrence n our method. alance s less systematcally governed by a model than s the lmt, whch s the result of a model. Instead of modellng, d L E drectly, we assume, as an approxmaton, that such accounts have an expected balance equal to ther lmt and replace Equaton by Equaton 2. d L E d L P d L L E d L P d E,,. (2) We therefore propose the parametersaton of three models. Frst, a model of the probablty that the outstandng balance of an account s larger than the credt lmt, condtonal on default; second, a model to predct the outstandng balance, condtonal on default; and thrd, a model to predct the credt lmt condtonal on default, where the parameters to predct balance and lmt are allowed to dffer. There are cases where the lmt may not ncrease and may even decrease as balance ncreases (see Q (2009) for a good dscusson of ths). ut our method s robust to ths stuaton n that for such cases the survval model would be expected to predct a hgher probablty that n

11 the next month the predcted balance wll exceed the lmt and so the weght on predcted balance n that month wll be correspondngly lower and the weght on the predcted lmt correspondngly hgher, as Equaton 2 shows. From a tranng dataset based on only default accounts,.e. accounts that eventually go nto default, we propose the estmaton of the probablty that the outstandng balance at any duraton tme s equal to or greater than the lmt at duraton tme. Ths s done by defnng the event overstretched, S, for account at tme whch takes the value f outstandng balance s greater than the lmt at tme ; 0 otherwse, gven n Equaton 3: f L S. (3) 0 otherws e Gven ths defnton, t s possble for an account to experence the event more than once (at dfferent tmes of the loan), so a dscrete-tme repeated events survval model, gven n Equaton 4, s estmated. S PS P log ( ) X 2Y, l 3Z l, (4) where s the ntercept term; ( ) s a functon of tme snce the last event; X are accountdependent, tme-ndependent covarates,.e. applcaton varables; Y l, are accountdependent, tme-dependent covarates, lagged l months,.e. behavoural varables; Z l are account-ndependent, tme-dependent covarates, lagged l months,.e. macroeconomc varables; and, 2, 3 are unknown vectors of parameters to be estmated. To predct ether balance or lmt, we propose the estmaton of two sub-models usng two separate tranng datasets (where we use entre hstores of the accounts n each tranng set). The datasets consst of accounts that at some tme n ther hstory defaulted as shown n Fgure. The tranng dataset s segmented accordng to whether accounts ever had balance exceedng lmt (but not necessarly n default) at any pont n the loan, or accounts that never had balance exceedng lmt throughout the lfe of the loan. The subset consstng of accounts (represented by subscrpt a) where balance exceeded credt lmt at some pont durng the 0

12 loan s the lmt tranng set, and used to estmate the lmt at tme, condtonal on default. y structurng a sample n ths way, our method nvolves parametersng the dstrbuton of L gven a La and gven default. The other subset consstng of accounts (represented a by subscrpt b) where balance never exceeded lmt throughout the observaton tme of the loan s the balance tranng set, and used to estmate the balance at tme. Hence, our method parameterses the b gven the b b L dstrbuton. y segmentng the accounts n ths way, we use the full hstory of each account n the estmaton of ether balance or lmt as t changes over tme and over the course of the loan perod. Ths methodology, as well as the tranng and test sets created (detals n the next secton), s represented n Fgure.

13 Portfolo of loans efaults Non-efaults Observatons of accounts at tme of default * used to estmate EAF, LEQ and CCF models, wth some outlers removed Tranng set: all observatons for accounts opened pre 2009 * used to estmate P L d Lmt tranng set: accounts that have balance lmt at any pont n the loan * used to estmate L a alance tranng set: accounts where balance < lmt throughout the loan * used to estmate b Predcted balance = P ˆ L d L d P L d ˆ d Test set I: efaults; all observatons for accounts opened from 2009 Test set II: Observatons of accounts at tme of default Fgure : Flowchart of methodology and tranng and test set splts, where dotted lnes represent subsets of the test and tranng sets that only consst of observatons at default tme. The lmt, L a, and balance, b, for accounts a and b, respectvely, at tme could be estmated usng panel models wth random effects gven n Equatons 5 and 6 (see Cameron and Trved (2005), Gujarat (2003) and Verbeek (2004) for detals). L L L L L a da Xa 2Ya, l 3 Z l a a ˆ (5) 2

14 b db Xb 2Yb, l 3 Z l b b ˆ (6) where L, are the ntercept terms, a X b X, are account-dependent, tme-ndependent covarates,.e. applcaton varables; covarates,.e. behavoural varables, lagged l months; Ya, l, Yb, l are account-dependent, tme-dependent Z l are account-ndependent, tmedependent covarates,.e. macroeconomc varables, lagged l months;, 2, 3 are unknown vectors of parameters to be estmated; and 2 2 a, b ~ II0, and, b ~ II0, a., are the error terms, wth a a b b The Mxture model could then be used to predct balance at any gven tme durng the loan. Ths s done by frst applyng the survval model to all accounts to predct the probablty of beng overstretched at each duraton tme. Then, regardless of the estmated probablty, one apples the balance panel model and the lmt panel model onto all observatons of all accounts to get an estmated balance and estmated lmt, agan at each tme 4. ecause the models would be estmated for the subsets descrbed above, these predcted values, and Lˆ, are the values of gven b Lb and ˆ L gven L respectvely, n both cases gven default. The fnal predcted value for balance of an account at duraton tme, gven default, ~, s then a combnaton of the repeated events survval model estmatng d the probablty of balance exceedng lmt at tme, and the panel models estmatng ether balance or lmt at tme. Ths s the expected value of balance and lmt, gven the probabltes of the balance exceedng the lmt at tme, and the assumed approxmaton, as defned n Equaton 7 (whch s just Equaton 2 rewrtten n a more effcent form): ~ d P S Lˆ d PS ˆ d where PS P L d, (7) and s the estmated probablty that account s overstretched at tme,.e. that the balance for account at tme exceeds the lmt for account at tme ; and Lˆ and respectvely, from ther respectve panel models. a a ˆ are the estmated values for lmt and balance 4 When predctng balance and lmt we set the random effect term at ts mean (zero) n every case snce ts value s unknown for every case that s not n the tranng sample. 3

15 4. ata and varables 4.. ata ata s suppled by a major UK bank and conssts of a large sample of credt card accounts, geographcally representatve of the UK market. The accounts were drawn from a sngle product, and opened between 200 and 200. Accounts were observed and tracked monthly up to March 20 or untl t was closed, whchever s earler. A mnmum repayment amount s calculated n each month for each account and accounts progress through states of arrears dependng on whether they are able to make the mnmum repayment amount. We set the mnmum repayment amount at 2.5% of the prevous month s outstandng balance or 5, whchever s hgher, unless the account s n credt, n whch case the mnmum repayment amount s 0, or the account has an outstandng balance of less than 5, n whch case the mnmum repayment amount would be the full outstandng amount. It s also possble for accounts to recover from states of arrears should the borrower make repayment amounts large enough to cover accumulated mnmum repayment amounts that were prevously mssed. An account s then sad to go nto default f t goes nto 3 months n arrears (not necessarly consecutve). For more detals on the movement of accounts between states, see Leow and Crook (204), but note that the percentage used here s dfferent. Accounts that have a credt lmt of 0 at any pont n the loan are removed, based on the assumpton that these accounts would have been sngled out as problem loans by the bank. It s possble for accounts to be n credt, such that balance s negatve, so balance s constraned such that observatons that have negatve balance have 0 balance. We expermented wth varous lags on the tme-dependent covarates n all of the models and report results for lags of 2 and of 6 months. ecause of these lags, and the mnmum tme requred for accounts to go nto default, we also removed accounts that have been on the books less than 5 and 9 months respectvely. 4

16 Fgure 2: strbuton of rato of balance over lmt at tme of default (for ratos less than 3) From the data, we see that some accounts go nto default wth an outstandng balance greater than ther credt lmt. Ths s llustrated n Fgure 2, whch gves the dstrbuton of the rato of balance over lmt at the tme of default (only for ratos less than 3 for a clearer pcture of the dstrbuton). The peak n the graph corresponds to borrowers defaultng wth a balance equal to ther credt lmt, but we also do see a szeable proporton of borrowers who default wth balances on ether sde of ther credt lmts Explanatory and macroeconomc varables Common applcaton varables are avalable, ncludng age, tme at address, tme wth bank, ncome, presence of landlne and employment type. ehavoural varables are also avalable on a monthly bass, ncludng repayment amount, credt lmt, outstandng balance and 5

17 number and value of cash wthdrawals or card transactons. From these, further behavoural ndcators can be derved, for example, the number of tmes an account oscllates between states of arrears and beng up-to-date, the proporton of tme the account has been n arrears and the average card transacton value. Any behavoural varables used n the model are lagged 2 (or 6) months. The macroeconomc varables consdered here are lsted n Table 2. The man source of macroeconomc varables s the Offce of Natonal Statstcs (ONS), supplemented by data from ank of England (OE), Natonwde and the European Commsson (EC) where approprate. We use the non-seasonally adjusted seres unless unavalable because the balance and lmt data are also not seasonally adjusted. Any macroeconomc varables used n the model are also lagged 2 (6) months. Table 2: escrpton of macroeconomc varables Varable Source (d) escrpton AWEN ONS (KA5Q) Average earnngs ndex, ncludng bonus, ncludng arrears, whole economy, not seasonally adjusted CIRN OE (CFMHSG) Monthly average of UK resdent monetary fnancal nsttutons (excl Central ank) sterlng weghted average nterest rate, credt card loans to households (%) not seasonally adjusted CLMN ONS (CJ) Clamant count rate, UK, percentage, not seasonally adjusted CONS EC Total consumer confdence ndcator, UK, seasonally adjusted (CONS.UK.TOT.COF.S.M) HPIS Natonwde House prce ndex Aall houses, seasonally adjusted IOPN ONS (K24V) Index of producton, all producton ndustres, not seasonally adjusted IRMA OE Monthly average of ank of England s base rate LAMN ONS (E) Log (base e) of total consumer credt, amounts outstandng, not seasonally adjusted LFTN ONS Log (base e) of FTSE all share prce ndex, month end, not seasonally adjusted 0/4/62=00 RPIN ONS (CHAW) All tems retal prce ndex, not seasonally adjusted, January 987=00 6

18 UERS ONS (YCNO) Labour Force Survey unemployment rate, UK, all, ages 6 and over, percentages, seasonally adjusted ONS denotes Offce of Natonal Statstcs. OE denotes ank of England. Natonwde s Natonwde uldng Socety. d denotes the data source s dentfer for the varable Tranng and test set splt Although we are nterested n the predcton of outstandng balance of an account n each tme step, these predctons of balance only become EA values f and when accounts go nto default. We also beleve that balances of defaulted and non-defaulted accounts behave dfferently, and we see from Fgure 3 that balances of non-default accounts are on average lower, and have more occurrences of 0 than the balances of default accounts. As such, we only use accounts that do (eventually) go nto default. ecause we only use observatons from accounts that do go nto default for the development of the EA model, we do not need to be concerned wth accounts that are nactve, e.g. have zero transactons and zero balance on the card for an extended perod of tme, but reman n the portfolo. In the Introducton we explaned that the mxture model wll both predct balance at each tme n the hstory of a defaulted account as well as at the tme of default. The former s useful because a lender does not know when, or f, an account wll default. We compare the performance of the establshed and mxture model n these two settngs by usng two dfferent test sets as follows. The dataset s dvded to gve the tranng set consstng of all accounts that do go nto default at some tme n ther hstory and were opened on or before 3 ecember 2008, gvng about 94,000 unque accounts. Test set I s an out-of-sample test set and s created usng the remanng default accounts, consstng of all observatons of all accounts opened on or after 0 January Test set I conssts of about 2,000 unque accounts, gvng more than 66,000 month-account observatons. Test set II s created as a subset of Test set I, where only observatons at the tme of default are ncluded. Test set I would gve an ndcaton of how well the model s able to predct balance for accounts that are lkely to be delnquent but may not yet have gone nto default at each tme n ther account hstory, whlst Test set II would be an ndcaton of how well the model s able to predct at default-tme, regulatory EA. The relatonshp of the tranng and test sets are represented 7

19 n Fgure. We calculate several performance measures ncludng r-squaredr-squared values, for the two test sets: Test set I, for all accounts, for all observaton tmes; and Test set II, for all accounts, only at tme of default. The portfolo of non-default accounts s not used n ether the modellng or the testng as we estmate balance gven default. Applyng the Mxture model to observatons of non-default accounts would gve us the predcted balance should the account go nto default, whch s dfferent to the observed balance, as seen n Fgure 3, whch would mean that we wll not be able to score how well the model s predctng. Fgure 3: strbutons of observed balance, for default and non-default accounts, for balance less than 20, Model estmaton 8

20 oth panel models were estmated usng Generalsed Least Squares (GLS) estmators. We estmated models wth lags of 2 months and of 6 months lags. Covarates nclude applcaton varables, lagged behavoural varables, and lagged macroeconomc varables, defned n Equatons 5 and 6. We ntally estmated the survval model and models for balance and lmt, separately, usng a very large number of applcaton, behavoural and macroeconomc varables wth 2 month lags. Covarates were then retaned or deleted based on ther level of statstcal sgnfcance ncludng that of other varables, ther relevance and the predctve accuracy of the overall model. So for example n the lmt and balance equatons Tme at address (TAAdd) and a bnary varable ndcator for mssng or unknown tme wth bank (TWank_MU) were not sgnfcant n the balance and lmt equatons and so were not ncluded n the fnal equaton (see the Appendx) whereas they were sgnfcant n the survval model and so were ncluded n that 5. Thus dfferent sets of parameters are used n each model and between the lagged models. The survval model dd not nclude utlsaton or credt lmt because although they were very statstcally sgnfcant, the overall accuracy of the model at lag 2 months was slghtly lower when the combnaton of varables that ncluded these two was used. At lag 6 months, ncluson or excluson of these two varables actually made lttle dfference to predctve accuracy. We found the greatest predctve accuracy was ganed when the tranng set of the balance model was restrcted to cases when the mnmum balance was over 200. Snce each account typcally has multple observatons (month-account observatons), we adjusted for seral correlaton by usng a clustered sandwch estmator (on account I) to estmate varance and standard errors rukker (2003). To compare the predctve accuracy of the Mxture model wth establshed methods, we use the tranng set wth observatons only at tme of default to estmate the EAF, LEQ and CCF cross-sectonal regresson models (represented by the dotted square from the tranng set n Fgure ). For all observatons at tme of default, EAF, CCF and LEQ are predcted based on observed covarates lagged 2 (6) months before default, accordng to the equatons n Table 5 The omsson of lagged utlsaton n the lmt and balance equatons allows more flexblty n the estmated parameters concernng lagged balance and lmt. 9

21 . Smlar to the EA papers mentoned n the lterature, some observatons were further excluded from ths subset due to some very extreme observatons of CCF and LEQ. The fnal number of accounts and observatons used n each tranng set when covarates were lagged 2 months s gven n Table 3. Table 3: Number of observatons for balance and lmt subsets, lag 2 months Model Number of Number of Mnmum Maxmum Average accounts observatons observatons observatons observatons for any account for any account per account alance 3,859 84, Lmt 36, , CCF 43,686 43,686 EAF 68, ,479 LEQ 3,82 3,82 EAF, LEQ and CCF were regressed on the same covarates as those used n the survval model 6. A varety of expermentaton n terms of modellng functons and technques was done for these competng models to mprove the predctons for these varables. For the modellng of EAF, we tred several functonal forms ncludng a beta functon and logt lnk functons but found that a lner model wth OLS estmators gave the greatest predctve accuracy. For LEQ, varous values of outlers were deleted but the greatest predctve accuracy was ganed when we took only values n the range 0 < LEQ < and adopted a generalsed lnear model wth a logt lnk functon wth a maxmum lkelhood estmator. For the CCF model we took a loge transformaton to transform the dstrbuton to be close to normal, then deleted varous szes of outlers and used an OLS estmator. The predctve accuracy was very poor untl we deleted all observatons above the 80 th percentle. These models are then appled onto the test sets (Test set I and Test set II) and performance measures are calculated. These three regresson models are not further documented n ths paper. 6 Except for tme varyng duraton tme snce last event, duraton tme squared and number of tmes event has happened whch are all survval model specfc and tme on books that was ncluded n the competng models but not the survval model. 20

22 5. Results 5.. Survval model for beng overstretched The parameter estmates for the dscrete-tme repeated events survval model predctng for the event overstretched s gven n the appendx, Table A2. We fnd that the sgns of the parameter estmates are ntutve: for example, the probablty of beng overstretched decreases wth age as well as wth hgher ncome. In terms of behavoural varables, we fnd that the probablty of beng overstretched reflects how well borrowers manage ther accounts, so borrowers who move n and out of arrears frequently (see rate of total jumps) or are frequently n arrears (see proporton of months n arrears) tend to have a hgher probablty of beng overstretched. In terms of macroeconomc varables, an ncrease n housng or fnancal wealth, for example, an ncrease n the House Prce Index (HPI) would decrease the probablty of beng overstretched; but easer access to credt (ndcated by an ncrease n credt amount outstandng) ncreases the probablty of beng overstretched Panel models for balance and lmt The parameter estmates for both panel models are gven n Table A2 n the appendx. We acknowledge that the balance from 2 months prevous s ncluded as a varable n the balance model, and credt lmt from 2 months prevous s ncluded as a varable n the lmt model. Although ths would rase the ssue of endogenety n econometrc nterpretaton, t s not an ssue n ths case as we are usng the model solely for the purpose of predcton. Although the panel models are developed wth random effects, these random effects are not known for accounts n the test set(s). The random effects assocated wth each account n the test set s assgned to be the mean values of and t, that s zero n both cases. The goodness of ft statstcs for the panel models for balance and lmt, based on the tranng set wth tme varyng covarates lagged 2 months are gven n the appendx, Table A. We expect t to be easer to predct the lmt, as ths would be based on a combnaton of applcaton tme and behavoural ndcators, and s reflected n the mpressve r-squaredr- 2

23 squared value for the lmt model. The panel model for balance does not predct as well as that for the lmt, as factors affectng outstandng balance of an account would nclude borrower crcumstances whch would be mpossble to take nto account gven the nformaton we have Overall performance After applyng the Mxture model onto the test sets, we compute overall r-squaredr-squared, Mean Absolute Error (MAE), Mean Error (ME) and the symmetrc Mean Absolute Percentage Error (smape) for the predcted versus the observed balance,.e. we transform the predcted CCF, LEQ and EAF nto predcted balances, gven n Table 4. The smape s able to crcumvent the problem of havng 0 balance that would mean dvdng by 0 n the calculaton of MAPE. 22

24 Table 4: Performance measures for Mxture, LEQ, EAF and CCF models, for test sets, based on predcted balances Model Test set Lag 2 Lag 6 No.of obs R- squared MAE ME:Obs- Pred smape No. of obs R- squared MAE ME:Obs- Pred smape Mxture model developed on default accounts, mn balance > 200 Test set I Test set II 8,584 4, ,460, LEQ model developed on default accounts at tme of default, 0 < LEQ<, Test set I Test set II 8,584 4, ,460, EAF model developed on default accounts at tme of default Test set I Test set II 8,584 4, ,460, CCF model (ln CCF) developed on default Test set I Test set II , ,460,

25 accounts at tme to default, CCF>0 and truncated at 80 th percentle 24

26 We omt cases where both observed and predcted balance are 0 (.e. the predcton s accurate and there s 0 error) from the calculaton of smape as they do not contrbute to the error. We see that when consderng predctons 2 months n advance (left hand panel) the Mxture model s able to acheve an r-squaredr-squared of 0.56 when predctng for balances for accounts that are lkely to be delnquent, at all tmes they are observed. Ths s an mprovement from the r-squaredr-squared values of between 0.54 for EAF, 0.49 for LEQ and for CCF 7. The Mxture model also has the lowest ME. ut n terms of MAE and smape the EAF method gves lower errors. When consderng balance at the tme of default, the Mxture model has a lower ME at than the EAF ( ) and LEQ ( 293); n terms of r-squaredr-squared and MAE ts performance s nferor to the EAF though better than the other two methods, and n terms of smape, ts performance s below those of EAF and LEQ. The Mxture model gves a predcton at each duraton tme snce the openng of the account. When we consder the performance at a predcton horzon of, say, 6 months (rght hand panel) we see that for accounts at all observaton tmes, the r-squaredr-squared of the Mxture model at 0.58 s consderably above those of the other methods, the largest of whch s LEQ at 0.48 wth EAF at In terms of ME, the Mxture model s also consderably more accurate than the other methods, wth a ME of whlst the closest of the other methods s for LEQ. In terms of MAE, the Mxture model s more accurate than EAF but less so than LEQ. At the tme of default, the Mxture model has the hghest r-squaredrsquared at 0.66 althoughand t s far more accurate than the other methods n terms of mean error wth a mean error of just 0.65 compared wth that of EAF of It s more accurate n terms of MAE as well, although less accurate on smape.less accurate than the LEQ and EAF methods n terms of the error metrcs. 7 r-squaredr-squared s computed as -(sum of squared errors/total sum of squares). The predcted values are values of EA predcted by the relevant model and the observed values are the values observed n the data. R-squared can be negatve when predcted and observed values are compared and the mpled model does not have a constant as s the case when predctng balance from the CCF model. 25

27 It s dffcult to compare our results wth those of the lterature because many other studes quote only statstcs relatng to the regresson model and not for values of predcted EA. Thus the regresson models developed for credt cards LEQ by Q (2009) acheved adjusted r- squaredr-squared values of between 0.06 to 0.37, on a sample of default tme observatons dependng on whether the accounts were current or delnquent, and whether outlers were excluded from the model development. Jacobs Jr. (2008), workng on corporate data, acheved pseudo medan r-squaredr-squared values of 0.5, 0.9 and 0.3 for LEQ, CCF and EAF respectvely. arakova and Parthasarathy (203) fnd adjusted r-squaredr-squared values for dfferent models for corporate loans of between % and 33% dependng on the model and treatment of outlers. In contrast, Yang and Tkachenko quote an r-squaredrsquared of 0.9 for EA usng EAF wth a least squares logt algorthm but that s for a sample of corporate borrowers and we do not know f ths apples to a testng sample. Fgure 4 compares the dstrbutons of predcted and observed balances for Test set II,.e. only default tme observatons for all default accounts. The values of balance are lmted to between 0 and 20,000 for clearer representaton of the dstrbutons and all values of balances are ndexed on some value of observed balance. The Mxture model predcts the mean wth consderable accuracy (a dfference n ndexed value of ) compared wth the EAF and LEQ models (wth dfferences of and respectvely). The CCF s agan the least accurate by a large margn. 26

28 Fgure 4: Comparatve hstogram of predcted and observed balances, ndexed on observed balance, for Test set II, only observatons at tme of default (where observed balance les between 0 and 20,000). It s nterestng to note that, although wth varables lagged 2 months, the EAF model has a hgher r-squaredr-squared and lower MAE and smape than the Mxture model at default tme (Table 4), when we plot the dstrbutons (Fgure 4), the Mxture model yelds more accurate predctons compared to the EAF model n terms of the mean. Ths suggests that whlst the MAE value for the Mxture model shows the devatons from the observed values are, on average, larger for the Mxture than for the EAF model at default tme, the net value s closer to the observed value for the Mxture than for the EAF model. Lookng at the dstrbutons, the Mxture model s less accurate than EAF for the smaller values of balance but more accurate for the larger values. Arguably, the larger values are the balances that a portfolo manager would be most concerned about. Overall then, we beleve that the Mxture 27

29 model s a more accurate and useful model to use to predct EA and outstandng balances for accounts lkely to default at pre-default tmes than are conventonal models. 6. Concludng Remarks We propose a Mxture model to predct for credt card balance at any tme, gven that an account has defaulted. We explot the advantage that ths model has over conventonal crosssecton models of ncorporatng the movement n balance and n lmt over tme as the account moves towards default. Specfcally the method nvolves frst estmatng a dscretetme repeated events survval model to estmate the probablty of an account beng overstretched,.e. havng a balance greater than ts lmt, at any tme. Next, two panel models wth random effects are developed to estmate balance and lmt separately, at any tme. The fnal predcton for balance at duraton tme s then taken as the sum of two products, all at tme : the probablty of beng overstretched multpled by the estmated lmt; and the probablty of not beng overstretched multpled by the estmated balance n both cases gven default (c.f. Equaton 7). Applyng ths Mxture model to a large portfolo of default loans and ther hstorcal observatons, we fnd that we are able to get good predctons for outstandng balance for accounts that at some tme default, not only at the tme of default, but at any tme over ther entre loan perod. Ths would allow us to make predctons for outstandng balance and hence EA before default occurs, for delnquent accounts. Consderng predctons 2 months nto the future Wewe fnd that at the tme of default, the EAF model gves results that are, on three some measures, more accurate and on one measureothers less accurate than the Mxture model. However the Mxture model s more accurate n terms of the mean and mean error and has the added advantage of gvng more accurate predctons for larger balances than EAF. Turnng to predctons before the tme of default, the Mxture model has the hghest r-squaredr-squared and smallest mean error of any of the methods. If one wshes predctons a mere 6 months nto the future the Mxture model has the hghest r-squaredrsquared at both default tme and at earler tmes and the lowest mean error by a consderable margn. However, Overall, whlst we beleve the Mxture model s a compettve methodology better methodologcal choce for the predcton of balance for accounts that are lkely to 28

30 default, especally f a predcton 6 months nto the future s requred, further research s desrable to explore ts accuracy n other datasets. It s approprate to remark that some types of portfolos, such as corporate portfolos wll dffer n the proporton of cases where the balance at default exceeds the lmt and so the samplng varance of the estmated parameters of the survval model would dffer between portfolos. Followng ths work, we plan to ncorporate stress testng nto our rsk models. We plan to combne P, LG and EA models, and to stress test each component model ndependently yet retan the knock-on effects n an adverse economc stuaton, f any. The obvous covarates to stress test wthn the models would be the macroeconomc varables; however, we would also lke to consder methods whch would allow us to stress the behavoural varables as well. It s not always clear how behavoural varables are affected by the economy, especally n the case of retal loans where the economy s expected to affect ndvduals dfferently and to varyng degrees. The dfferent combnatons of P, LG and EA computed would enable us to get a dstrbuton for loss, from whch we expect to be able to predct for expected and unexpected losses better. References ARATEN, M. & JACOS, M. J Loan Equvalents for Revolvng Credts and Advsed Lnes. The RMA Journal. ARAKOVA, I. & PARTHASARATHY, H Modelng Corporate Exposure at efault. Offce of the Comptroller of the Currency Workng Paper. ASEL COMMITTEE ON ANKING SUPERVISION Internatonal Convergence of Captal Measurement and Captal Standards: A Revsed Framework. CAMERON, A. C. & TRIVEI, P. K Mcroeconometrcs: Methods and Applcatons, Cambrdge Unversty Press. RUKKER,. M Testng for Seral Correlaton n Lnear Panel-ata Models. The Stata Journal, 3,

31 GUJARATI,. N asc Econometrcs, McGraw Hll. JACOS JR., M An Emprcal Study of Exposure at efault. Offce of the Comptroller of the Currency Workng Paper. JIMÉNEZ, G., LOPEZ, J. A. & SAURINA, J Emprcal Analyss of Corporate Credt Lnes. Revew of Fnancal Studes, 22, JIMÉNEZ, G. & MENCÍA, J Modellng the dstrbuton of credt losses wth observable and latent factors. Journal of Emprcal Fnance, 6, LEOW, M. & CROOK, J. N Intensty Models and Transton Probabltes for Credt Card Loan elnquences. European Journal of Operatonal Research, 236, MORAL, G EA Estmates for Facltes wth Explct Lmts. In: ENGELMANN,. & RAUHMEIER, R. (eds.) The asel II Rsk Parameters. Sprnger erln Hedelberg. QI, M Exposure at efault of Unsecured Credt Cards. Offce of the Comptroller of the Currency Workng Paper. RISK MANAGEMENT ASSOCIATION Industry Practces n Estmatng EA and LG for Revolvng Consumer Credts - Cards and Home Equty Lnes of Credt. TAPLIN, R., TO, H. M. & HEE, J Modellng Exposure at efault, Credt Converson Factors and the asel II Accord. Journal of Credt Rsk, 3, THOMAS, L. C Consumer Fnance: Challenges for Operatonal Research. Journal of the Operatonal Research Socety, 6, YANG,. H. & TKACHENKO, M Modellng Exposure at efault and Loss Gven efault: Emprcal Approaches and Techncal Implementaton. Journal of Credt Rsk, 8(2), 8-0. VEREEK, M A Gude to Modern Econometrcs, John Wley & Sons. 30

32 APPENIX Table A: Performance ndcators for panel models, for tranng set Model Overall R-squared (tran) u e alance Lmt Table A2: Parameter estmates of survval model for event overstretched and panel models for balance and lmt lag 2 Code Parameter screte-tme repeated events survval model for P(>=L) Panel model wth random effects for balance Panel model wth random effects for lmt Estmate WaldChSq ProbChSq Estmate z P> z Estmate z P> z Intercept Intercept <.000 -, , <.000 Applcaton varables ageapp_ Age at applcaton group ageapp_2 Age at applcaton group < ageapp_3 Age at applcaton group < <.000 ageapp_4 Age at applcaton group < <.000 ageapp_5 Age at applcaton group < <.000 ageapp_6 Age at applcaton group < <.000 ageapp_7 Age at applcaton group < <.000 ageapp_8 Age at applcaton group < <.000 ageapp_9 Age at applcaton group < <.000 3