Segmentation, Probability of Default and Basel II Capital Measures. for Credit Card Portfolios

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1 Segmenaion, Probabiliy of Defaul and Basel II Capial Measures for Credi Card Porfolios Draf: Aug 3, 2007 *Work compleed while a Federal Reserve Bank of Philadelphia Dennis Ash Federal Reserve Bank of Philadelphia Shannon M. Kelly HSBC* William W. Lang Federal Reserve Bank of Philadelphia William Nayda Capial One Corporaion Haining Yin Capial One Corporaion

2 Inroducion This paper examines alernaive mehods for differeniaing he likelihood of defaul among credi card borrowers and how hese alernaive mehods affec capial requiremens for he porfolio when using a varian of he Vasicek (2002) asympoic single risk facor (ASRF) model o consruc a value-a-risk (VAR) capial measure. The resuls in our paper have imporan implicaions for banking pracice. Firs, he general modeling approach discussed in he paper is an approach commonly used in he banking indusry (see Risk Managemen Associaion, 2003). Moreover, his modeling framework is embodied in he proposed Basel II regulaory framework for bank capial. In he ASRF model, accurae esimaes of he ail of he porfolio disribuion require differeniaing he porfolio ino separae homogeneous risk buckes. Credi card risk managers ofen refer o hese differeniaed risk buckes as segmens. While here are some imporan differences beween credi card and commercial loan porfolios, he segmenaion process for he consumer porfolio is in many ways analogous o he loan raing process commonly used for esimaing loss disribuions of a commercial lending porfolio (see Treacy and Carey, 998). We show ha he esimaes of VARs obained from he ASRF model are inversely relaed o he degree of homogeneiy of he porfolio segmens. Oher hings equal, a finer differeniaion of defaul risk among borrowers produces a more accurae and lower esimaed VAR. See Basel Commiee on Banking Supervision, 2004.

3 Using daa from a sample of credi card loans over he period , we examine he imporance of various cusomer aribues in differeniaing he likelihood of borrower defaul in credi card porfolios. The aribues we examine are cusomer credi score a he ime he loan was originaed, updaed or refreshed credi scores, and delinquency saus. We esimae he oneyear horizon probabiliy of defaul (PD) for various alernaive segmenaion approaches based on hese aribues. We also examine he role of loan age or seasoning in predicing credi card defauls. Our PD esimaes are hen enered ino he porfolio loss model o esimae economic capial measures using differen segmenaion schemes. This economic capial measure can be hough of as a summary indicaor of he imporance of improving segmenaion for measuring economic capial. The nex secion of he paper oulines a simple heoreical model of consumer defaul and he corresponding disribuion of porfolio losses. The following secion hen discusses segmenaion crieria for credi card porfolios. Secion III of he paper describes he daa and analysis design. Secion IV discusses PD resuls using our alernaive segmenaion crieria. Secion V discusses he affecs of loan age on PD. Secion VI analyzes he capial measures resuling from alernaive approaches o segmenaion. I. A Model of Consumer Defauls In his secion, we deail a consumer credi version of he ASRF model. The model is a special case of a Meron (974) opions-based srucural model of defaul. Consider a consumer borrower j wih ne worh (measured in naural logs) a ime of w, and ha ne worh follows a sandard geomeric random walk model wih drif: j 2 w j, + = wj, + µ + v j, ; wih v j, ~ N(0, σ v j ) () 2

4 In he model, a period is equivalen o he forecas ime horizon. Mos porfolio risk models use a one-year ime horizon and our empirical esimaes will be based on a one-year horizon. Defaul is assumed o occur a ime + when he borrower s ne worh falls below some hreshold value: w θ (2) < j, + j, + The hreshold value migh differ across individuals depending on a number of observable facors such as marial saus, deb burden, sae laws, and employmen saus. Individuals migh also differ by nonobservable rais such as heir aiude oward defaul or repuaion coss associaed wih defaul. Subracing w, from boh sides of equaion (2): j g j, + = + ν j, < m j, + µ (3) where g j, = w j, + w j, + and m j, + = θ j, + w j, The shock o borrower j s wealh is assumed o be driven by a sochasic sysemaic facor and an idiosyncraic facor. Boh he sysemaic and idiosyncraic random variable are assumed o be independen and idenically disribued. Wihou loss of generaliy, we assume ha hese shocks are sandard normal random variables. v Y ~ ε N (4) j, = ρy + ρε j and N(0,) and j ~ (0, ) 3

5 where Y is he sysemaic facor and ε j is he borrower-specific idiosyncraic shock and ρ is he correlaion of borrower wealh wih he sysemaic risk facor. Equaions (3) and (4) imply: π j + = Φ m j + µ ), (, j (5) where π j, + is he uncondiional probabiliy of defaul of borrower j a ime +, and Φ is he sandard normal cumulaive densiy funcion. Alernaively: Φ π j, + ) = m j, + µ j ( (6) The sae of defaul condiional on Y can now be wrien as: ε j, < [ Φ ( π j, + ) ρy ]/ ρ (7) This implies a probabiliy of defaul for borrower j condiional on he realizaion of Y as: π Y ) Φ([ Φ ( π ) ρy ]/ ) (8) j, + ( = j, + ρ Now consider a porfolio of loans o n borrowers wih homogeneous (ype j) risk characerisics. For simpliciy, we assume ha all loans are of equal size. 2 p Le D + Y ) be he defaul frequency of he porfolio. Then as n : ( j, 2 Allowing for differen size exposures is sraighforward as long as we assume ha here are a large number of borrowers and individual loans represen a very small share of he oal loan porfolio. 4

6 5 Φ = Φ ρ ρ π π ) ( ) ( ) (,,, j j p j Y Y Y D (9) Equaion (9) can be used for calculaing he VAR for any chosen hreshold. The hreshold chosen corresponds o some arge level of solvency. The Basel II proposal ses his hreshold a he.00 probabiliy of failure. Le ν be he chosen hreshold probabiliy of failure. Since by assumpion idiosyncraic risk is diversified away, he ν upper ail of he disribuion of ) (, p j Y D + is equivalen o he lower ν ail of Y. Le Y be he ν lower ail of he disribuion of Y. Then, Φ Φ = Φ Φ Φ ρ ν ρ π ρ ρ π ) ( ) ( ) ( ) (,,, j j p j Y Y D (0) Subracing expeced losses from equaion (0) provides an esimae for capial required for a homogeneous porfolio of credi card loans. Wha if he loan porfolio consiss of borrowers wih differen defaul probabiliies? Suppose a lender is able o discriminae beween S discree ypes of borrowers wih differen defaul probabiliies bu a common correlaion wih he sysemaic risk facor. The defaul raes beween borrower ypes are correlaed o he sysemaic facor, bu he idiosyncraic shocks for all borrowers are assumed o be uncorrelaed. The lender is assumed o have a large number of borrowers of each ype, wih α j represening he share of ype j borrowers in he porfolio. The ν upper ail of he defaul frequency disribuion for a porfolio wih S heerogeneous borrower ypes is hen: Φ Φ Φ + = = + ρ ν ρ π α α ) ( ) ( ) (,, j S j j S j p j j Y D ; = = S j j α () Suppose ha lenders differ in heir abiliy o disinguish borrower ypes, wih some lenders being able o disinguish a smaller number of borrower ypes or some lenders making

7 more errors in idenifying he borrower s ype. In eiher case, he effec will be o group borrowers wih differen defaul frequencies and esimae he model as if hey are he same ype. As discussed in Lauren (2004), he VAR measure in equaion () is concave for mos of he relevan range of PDs. From Jensen s inequaliy, i is easy o demonsrae ha reaing loans wih differen PDs as a single group resuls in overesimaing he upper ail of he defaul frequency disribuion. Since aggregaing across differen borrower ypes does no bias esimaes of expeced loss, more accuraely disinguishing borrower ypes lowers he esimaed capial for he porfolio. Thus, we expec, and our empirical analysis below confirms, ha more accurae segmenaion of a porfolio will produce lower esimaes of economic capial using he ASRF model. II. Segmenaion Crieria for Credi Card Porfolios The model discussed in he previous secion requires ha loans be grouped ino homogeneous risk classes o accuraely measure he ail of he loss disribuion and he resuling capial requiremen for he porfolio. For credi card porfolios, his grouping or segmenaion process is ypically done by assigning loans o risk groups where he groups are defined by proxy borrower risk characerisics reflecing borrowers abiliy and inen o make paymens. PD is hen esimaed using he realized defaul experience of individual segmens. In his secion, we discuss risk facors in credi card porfolios ha are ypically analyzed by risk managers of credi card lenders. Credi Scores and Delinquency Saus Unlike large commercial lenders, credi card lenders ypically devoe limied resources o analyzing he idiosyncraic risk of an individual borrower or individual loan. Raher han relying 6

8 on direc analysis and monioring of he idiosyncraic characerisics of individual consumers, large credi card lenders rely heavily on saisical models of borrower performance based on sandardized daa for credi approval decisions, risk-based pricing, deermining credi limis, and seing collecion sraegies. The primary saisical ool for making credi card lending decisions is credi scoring. The exisence of exensive credi bureau daa in he U.S. allows lenders o use a wealh of readily available daa on individuals o esimae bureau scoring models. In addiion o using sandardized bureau scoring models, banks ofen buy or develop cusomized scoring models ailored o a bank s own clien populaion. Many companies wih a sufficienly large consumer lending porfolio also employ applicaion scoring models ha allow an insiuion o incorporae addiional informaion colleced during he loan applicaion process. In addiion o scoring cusomers a he ime of applicaion, scoring is used in a dynamic way for managing accouns and for performing inernal bank analyics. Large credi card lenders ypically obain updaed or refreshed bureau scores on a monhly basis. Some lenders also esimae behavioral credi scores ha combine updaed credi bureau informaion wih informaion on he borrower s performance on accouns wih he bank. These updaed scores are hen used for a wide variey of purposes, including credi line changes, repricing, and collecion sraegies. One commonly used modeling echnique for esimaing credi scores is o build a sandard logisic regression model using credi bureau daa and possibly bank-specific daa. The discree oucome ypically modeled is he probabiliy of a borrower ever becoming seriously delinquen over some fixed ime horizon. Seriously delinquen is ypically defined as more han 7

9 60 days pas due or more han 90 days pas due. The ime horizon window can vary from six monhs o wo years. Logisic and oher scoring models can be used o generae an esimae of he probabiliy of a bad oucome. However, while he oupu of a sandard scoring model produces a probabiliy, he probabiliy esimae of he sandard scoring model is generally no used as he PD esimae in inernal economic capial models and would no produce he PD required by regulaors under Basel II. One reason is ha he bad oucome used in he scoring model is usually some measure of serious delinquency (e.g., 90 days or more pas due) raher han a probabiliy of defaul or loss. A U.S. banks, credi card loans are ypically charged off when hey are 80 days or more pas due or when he borrower declares bankrupcy. Moreover, he PD esimae enering he ASRF model is an uncondiional PD represening an average defaul rae over a long ime period for borrowers wihin a risk class. The probabiliy of defaul generaed by credi scoring is ypically a shor-run condiional esimae. While mos credi scoring models do no produce a PD ha is appropriae as a direc inpu ino he ASRF loss disribuion model, he rank ordering properies of scoring models play a criical role in he segmenaion process, since hey are srong indicaors of risk ype. A common pracice among banks for esimaing PD is o esimae he relaionship beween ex pos one-year defaul raes and credi scores, conrolling for oher risk facors. Our mehodology for incorporaing scores ino PD esimaion follows his approach. In addiion, we examine he imporance of updaing credi scores in esimaing PD and segmening he porfolio. 8

10 Delinquency saus is anoher criical risk facor for predicing loss for a credi card porfolio. When using a one-year horizon for esimaing fuure losses, a large percenage of hose losses will come from borrowers who are currenly delinquen. In our esimaes below, we examine he relaionship beween delinquency saus and PD as well as he imporance of incorporaing credi scores ino he segmenaion process conrolling for delinquency saus. Refreshed Credi Scores Some banks wih sophisicaed risk managemen sysems obain refreshed credi scores as well as updaed informaion on oher risk facors. Clearly, updaing his informaion provides a beer measure of an individual borrower s risk ype. Thus, segmenaion of borrowers ino homogeneous risk ypes will be more accurae if banks reallocae loans ino segmens based on his updaed informaion. Some credi card lenders use an alernaive risk measuremen approach ha fixes loans ino segmens a he ime of originaion. Individual loans remain wihin a single segmen for he enire life of he loan. Performance is hen racked for hese fixed segmens, and risk parameer esimaes can be esimaed based on he hisorical performance of segmens wih similar characerisics a originaion. Vinage analysis is a varian of he fixed segmen approach where all loans in he segmen are originaed during a common ime period (e.g., all loans in he segmen are originaed in he same monh). Vinage analysis is commonly used a banks for long-erm projecions of loss and profiabiliy. In he analysis below we compare wha effec using refreshed credi scores has on esimaing PD when allocaing loans o segmens versus using a fixed segmenaion approach based on scores a originaion. 9

11 Loan Age For many loan ypes wih long effecive mauriies, loan age is a predicive facor for defaul. In credi card porfolios his effec of loan age on defauls is referred o as seasoning, and plos of porfolio defaul raes agains ime on books are called seasoning curves. The ypical seasoning curve has an upward slope ha peaks ypically beween 8 monhs o hree years and hen eiher flaens ou or declines. Vinage curves are a varian of seasoning curves where each seasoning curve represens porfolio performance of loans originaed in he same ime period. Credi card lenders compare he seasoning curves for differen vinages o deermine relaive performance across porfolios originaed a differen ime periods and use exrapolaion mehods for predicing fuure defaul raes. Noe ha he upward slope ypically observed in a sandard seasoning curve for credi card loans does no necessarily imply ha age is a maerial predicor of defaul, conrolling for oher risk facors. Sandard seasoning or vinage curve analysis uses fixed segmenaion mehods where exposures do no migrae across segmens as heir risk profile changes. Thus, in he sandard seasoning or vinage analysis, loans remain in he same segmen even if he risk of he loan changes over ime. Thus, he shape of a sandard seasoning curve does no represen a pure marginal age effec, bu raher incorporaes boh loan age effecs and credi qualiy ransiions. 3 To illusrae his poin, consider he following simple example. Assume ha available informaion allows a lender o classify all non-defauled borrowers ino wo groups: good (G) or poor (P). Furher assume ha ype G borrowers have a % probabiliy of defaul and ype P borrowers have a 5% 3 I also incorporaes he effecs of economic sae variables. Tha is, he dynamic pah of vinage performance will be affeced by he dynamic pah of he economy. 0

12 probabiliy of defaul and ha hese probabiliies are independen of loan age. Now furher assume ha here is a 50% probabiliy ha a ype G borrower (excluding defauled loans) will become a ype B borrower. Consider he seasoning curve for a porfolio of ype G borrowers assuming ha acual defauls equal expeced defauls. In he firs year, he defaul frequency will be %. In he second year, on average, half of he remaining porfolio will be ype P borrowers and he porfolio will have an expeced defaul rae of 3% (.5x% +.5x5%). Thus, he seasoning curve will slope upward even hough loan age is no a risk facor afer conrolling for credi qualiy. In our empirical analysis, we examine he imporance of loan age and wheher i coninues o be a maerial risk facor afer conrolling for updaed measures of credi qualiy. III. Daa Descripion and Design of he Analysis Analysis Populaion The empirical daa used in his research are monhly observaions of credi card accouns originaed in 999 and 2000 from seleced business lines a Capial One. For he purposes of his analysis, he loans have been limied o non-rewards and non-affiniy accouns, hereby ruling ou produc-specific facors ha could oherwise disor he resuls. This analysis focuses on wo discree porfolio segmens, low risk and high risk, ha reflec he highes and lowes qualiy credis in he porfolio. Wihin each porfolio, accouns are furher grouped by cohor, i.e., he quarer in which credi card accouns were originaed. To examine loan performance over a five-year window, we have eigh quarerly cohors originaed over he period 999 o 2000 and observe heir

13 performance from originaion unil Sepember The mehod used o deermine sample sizes for he daa is discussed in he Appendix. Observaion Poin and Oucome Window For any group of accouns wih cerain common characerisics (i.e., accouns originaed in he same cohor from he same business line), monhly snapshos are aken over an exended ime period, usually hree o five years, depending on he hisory of available daa. To examine he seasoning effecs of PD and capial, we arrange hese monhly observaions on he basis of accoun ages raher han calendar monhs. Cohors are defined by he quarer of originaion and age. For example, accouns originaed in January 999 and observed in January 200 correspond o accoun age of 24 monhs, while accouns originaed in March 999 reach he age of 24 monhs in March 200. In his analysis, boh accouns would be included in a single observaion poin: cohor Q 999 a he age of 24 monhs. The oucome window represens he 2-monh window immediaely following he observaion poin. Tha is, each observaion poin corresponds o he observaion window of [+, +3]. Following he earlier example, cohor Q 999 a he age of 24 monhs corresponds o he oucome window of 25 o 37 monhs of age. Since he daa run as lae as Sepember 2004, we have a minimum of 33 observaion poins/oucome windows for he younges accouns originaed in December Char illusraes he observaions and he oucome windows for quarerly cohors of which PDs are observed quarerly. Each shaded window corresponds o an oucome period saring a an observaion poin, a which poin he hisorical one-year PD is assigned. When muliple cohors are grouped ogeher, observaions wih he same age are analyzed in aggregae 2

14 o examine seasoning effecs. The exac framework used in our analysis allows for more granulariy by aking monhly observaions of monhly originaions grouped ino quarerly cohors. Laer in he analysis we noe ha funcional relaionships and correlaions are no significanly differen beween cohors; herefore, for his paper we display graphical resuls from all cohors only in aggregae. Analysis Variables For any combinaion of cohor and business line a any observaion poin, he following measures are calculaed and analyzed, eiher a he enire cohor level or, more relevan o his paper, a he segmen level defined by differen crieria. The segmenaion crieria used in he analysis are described in deail in he nex secion. PD (probabiliy of defaul) = % of accouns charged off during he 2-monh oucome window. PD is measured as he number of accouns ha defaul over a 2-monh window divided by he oal number of open accouns a he observaion poin. EAD (exposure a defaul) = he expeced dollar losses for he segmen if all accouns in he segmen defaul wihin he 2-monh oucome window. EAD is esimaed using he acual hisorical increase in balances up o he period of defaul. The muliplicaion of a cohor s PD and EAD equals he expeced gross dollar loss rae for he cohor. LGD = expeced ne losses as a percen EAD afer accouning for recoveries. LGD is measured using broad indusry averages for recovery raes. 3

15 IV. Segmenaion Analysis This secion examines he esimaion of PD using he segmenaion crieria discussed above. For each combinaion of cohor and porfolio a any observaion poin, he populaion is divided ino segmens using various combinaions of originaion credi score, refreshed credi score, and delinquency saus. Our segmenaion facors are: ORIGSC = credi bureau FICO score assigned o a borrower a he ime of credi applicaion For each accoun, FICO scores are ypically available from all hree of he major credi bureaus - Equifax, Experian, and TransUnion. To obain ORIGSC, we use a cascaded score, i.e., use credi bureau A s score if available; oherwise use credi bureau B s score if available; and finally use score from credi bureau C if neiher of he oher wo is available. REFRSC = he updaed credi bureau score assigned on a monhly basis. DELINQ = he delinquency sae of an accoun ha has no ye defauled, updaed on a monhly basis. DELINQ can ake on one of he following discree values: curren, days pas due, days pas due, 90-9 days pas due, days pas due, days pas due. Since loans 80 days or more pas due are required o be charged off, hese are defauled accouns. AGE = he number of saemens since he accoun was originaed. 4

16 Oher variables used in he analysis are: BALANCE = ousanding dollar balances for an accoun. LINE = dollar value of an accoun s credi line. REFRSC, DELINQ, AGE, BALANCE, and LINE are updaed a each observaion poin. Regression Model Our prior discussion of segmenaion is primarily drawn from curren indusry risk managemen pracice. To confirm he imporance of he role of hese risk facors, we conduc some preliminary mulivariae regression analyses. We esimae logisic models predicing PD. The conrol variables used are ORIGSC or REFRSC, AGE, DELINQ, BALANCE, and LINE. The regressions are run separaely for he low risk and high risk business porfolios. All eigh quarerly cohors are used in he regression. Wih wo porfolios (low risk and high risk) and wo credi scores (ORIGSC and REFRSC), we esimae four models of he following form: PD log = α0 + α *AGE+ α *SC+ α *DELINQ+ α *BALANCE+ α *LINE+ ε PD () 5

17 Table and 2 repor he maximum likelihood esimaes for he four regressions. As expeced, DELINQ has a srong posiive coefficien, while he score variables are negaively associaed wih PD. In addiion, BALANCE is posiively relaed o PD, while he coefficiens on LINE are negaive. Noe ha since we are conrolling for ousanding balances, a marginal increase in LINE implies a marginal decrease in credi line uilizaion rae. AGE is a much sronger posiive predicor of PD in he low risk porfolio when using ORIGSC raher han REFRSC. This resul suggess ha much, hough no all, of he relaionship beween loan age and defaul for he low risk porfolio is due o credi qualiy deerioraion ha is capured in he updaed scores. Noe ha AGE remains a srong predicor for he high risk porfolio even when conrolling for REFRSC, bu he coefficien on AGE is negaive. This suggess ha wihin a porfolio of newly acquired high risk credi card accouns, poorer qualiy borrowers defaul quickly leaving a higher qualiy pool of residual borrowers. Analysis of Segmenaion Crieria: Credi Scores and Delinquency Saus We examine he imporance of updaing credi scores and delinquency saus for risk segmenaion for he low risk and high risk samples. Figure plos PD agains REFRSC and ORIGSC. The chars repor PDs as an index raher han using he acual esimaed PD values o avoid revealing proprieary informaion. Figure shows he expeced negaive relaionship beween credi scores and PD. No surprisingly, here is a seeper negaive slope when using refreshed PDs, indicaing ha REFRSC provides imporan addiional informaion abou borrower qualiy. Noe ha REFRSC is paricularly imporan in differeniaing credi risk for he lower score bands in he high risk porfolio. These resuls indicae ha borrowers who 6

18 defaul ypically go hrough a period of declining credi performance and migrae o lower credi scores prior o enering heir year of defaul. However, noe ha some borrowers do defaul direcly ou of he high REFRSC buckes, indicaing ha some borrowers move o he defaul sae relaively quickly wihou going hrough a proraced period of credi problems. Figure 2 plos PD raes by DELINQ for he low risk and high risk porfolios. As would be expeced, here is a monoonic posiive relaionship beween he degree of delinquency and PD for boh porfolios. For he mos par, Figures and 2 display graphically wha is already well known: credi scores provide informaion on borrower risk, more recen credi scores are beer indicaors of risk han sale credi scores, and delinquency saus is a srong indicaor of credi risk. I is less clear how well updaed credi scores differeniae risk conrolling for DELINQ. While i is reasonable o expec ha updaed credi scores will differeniae risk among curren borrowers, i is less clear wheher credi scores will help in predicing PD among delinquen borrowers. In paricular, a credi card lender generally becomes aware of a delinquency prior o is incorporaion ino a bureau credi score. In oher words, DELINQ provides updaed informaion no necessarily refleced in REFRSC. In addiion, he credi score is buil o predic he likelihood of a borrower becoming seriously delinquen and a differen model migh be relevan for predicing he ransiion from seriously delinquen o defaul. Figure 3 displays resuls from segmening he porfolios by REFRSC and DELINQ for boh porfolios. REFRSC coninues o rank risk for he curren bucke and for lower sages of delinquency. This can be seen by he negaive slope for hese caegories. However, a laer sages of delinquency, PD is no longer monoonically declining in REFRSC. These resuls sugges ha REFRSC and DELINQ are joinly imporan segmenaion facors for he curren 7

19 bucke and earlier sage delinquencies, while REFRSC may no be a useful segmenaion facor for laer sage delinquencies. V. Seasoning Effecs Figure 4 shows seasoning curves ha plo PD agains AGE for he low risk porfolio and he high risk porfolio. The low risk seasoning curve is monoonically increasing up o 33 monhs wih a concave shape. This is in sharp conras o he high risk porfolio, which has a sharp upward slope ha peaks afer seven monhs and hen is downward sloping. This is consisen wih he evidence in he preceding secion indicaing ha higher qualiy loans ofen go hrough a period of decline prior o enering heir year of defaul. I also suggess ha newly originaed high risk porfolios conain a disinc subpopulaion of borrowers ha move quickly o defaul. 8

20 The difference in seasoning paerns for he low risk vs. high risk porfolio suggess ha seasoning paerns differ depending on porfolio credi qualiy. To invesigae his issue furher, we examine seasoning curves conrolling for eiher ORIGSC or REFRSC. Figure 5 displays seasoning curves conrolling for ORIGSC. For porfolios in bands wih ORIGSC of 720 or less, he seasoning curves display an upward slope ha peaks in he 8-20 monh range and hen flaens ou. For higher score bands, he seasoning curves are monoonically increasing ou o 33 monhs and he curves are concave. These seasoning curves for highly scored borrowers have he same general seasoning paern as he seasoning curve of he pooled low risk porfolio, indicaing ha he seasoning peak in defaul raes for porfolios wih very high credi scores a originaion is a leas 33 monhs. The resuls for he high risk porfolio indicae ha ORIGSC rank orders borrower performance for high risk credis. However, here are some disinc differences in he seasoning paern of he high risk porfolio as compared o he low risk porfolio. All of he high risk porfolio credi bands display a common seasoning paern, wih PDs rising very rapidly, peaking a seven monhs, and hen falling. Ineresingly, he PDs for all of he score bands converge o a very narrow range a he end of 33 monhs. Tha is, condiional on surviving for nearly hree years, ORIGSC adds lile informaion on PD. As discussed earlier, seasoning curves based on segmenaion crieria ha reflec credi qualiy a originaion implicily include he effecs of changes in credi qualiy prior o defaul. Does loan age coninue o be a facor when we separae ou he porfolio by updaed credi qualiy informaion? To examine his quesion, Figure 6 plos seasoning curves by REFRSC wihin he low risk and high risk porfolios. 9

21 As expeced, REFRSC provides beer risk separaion compared o ORIGSC. For he low risk porfolio, here is a subsanial upward slope o he seasoning curve for REFRSC of 740 or below. These curves eiher peak a around wo years or are monoonically increasing up o 33 monhs. The seasoning curves for accouns wih REFRSC above 740 are fla. For he high risk porfolio, seasoning curves conrolling for REFRSC show he familiar seven-monh peak in he seasoning curve for accouns wih scores of 580 or less. For hese score bands, PDs fall for a few monhs afer he peak and hen quickly flaen ou. Noe ha for higher score bands he seasoning curves are fla and hen have a sligh downward slope ha evenually flaens ou. Figure 7 displays seasoning curves for he wo porfolios conrolling for DELINQ, which is an alernaive updaed measure of credi qualiy. For he low risk porfolio, condiional on he loan being curren, AGE does no appear o have an independen effec on PD afer conrolling for delinquency. There remains a monoonically upward seasoning curve for earlier sage delinquencies. However, for delinquencies beyond 60 DPD he seasoning curves become essenially fla excep for very young loans. For he high risk porfolio shown in he second graph, seasoning curves display a downward slope for he firs few monhs he loans are on he books and hen are eiher fla or downward sloping. For he high risk porfolio, loans ha become delinquen a an early age have a very high PD, bu loans ha become delinquen afer having been on he books for some ime are less likely o defaul. This evidence, combined wih he previous seasoning curves for he high risk porfolio, indicaes ha here is a subpopulaion wihin he newly booked high risk accouns ha moves quickly o delinquency saus and subsequenly defaul. Once hese accouns are purged from he sample, he remaining accouns display improved credi qualiy. 20

22 VI. Regulaory Capial and Segmenaion In his secion, we calculae Basel II regulaory capial requiremens associaed wih he alernaive segmenaion schemes discussed above. As discussed previously, capial requiremens in he model fall if he segmenaion sysem produces beer risk separaion wihin he porfolio. For regulaory purposes, his has he advanage ha banks have an incenive o improve heir abiliy o differeniae risk. Moreover, he effec on capial requiremens can be inerpreed as one measure of he relaive qualiy of risk separaion when using alernaive segmenaion sysems. The PD used for our calculaions are he same as hose we have used in our graphs: PD is averaged over our enire sample, broken down by he drivers of he paricular segmenaion. For example, in a score-only segmenaion PD is averaged for all he daa cells wihin a paricular score band wihin a porfolio. These averages are accoun weighed. For a score by delinquency segmenaion, PD is averaged over he score band by delinquency by porfolio combinaion. LGD is also averaged. Here he averages are aken over each of he porfolios, high risk or low risk. The weighing is by he number of defauls in he daa cell. EAD is he sum of he acual EADs in each segmen. The proposed Basel II capial regulaions require an upward adjusmen o PD where seasoning effecs are maerial. This reflecs he view ha for some ypes of credis, significan credi deerioraion does no show up in defauls when he credis are young and ha capial should cover hese ypes of losses. Adjusing PD upward for seasoning is an issue in our daa only when AGE is a segmening facor. For young accouns of a paricular age, we adjused PD 2

23 upward by averaging over all daa cells from ha unseasoned age up o he average mauriy of ha porfolio. Where he average mauriy exceeds he age of our sample, we exrapolaed. We used a fla exrapolaion from he las age or average of he las few ages for high risk and a linear exrapolaion for low risk segmens. These should be conservaive, since delinquencies are declining or are increasing a a falling rae as seen in our figures. The PDs, LGDs, and EADs using various segmenaion approaches are hen enered ino he curren Basel II risk-weigh formulas for credi card exposures o calculae regulaory capial for each porfolio. These regulaory capial calculaions for differen segmenaion mehods are indexed relaive o he capial requiremens wih no segmenaion o avoid revealing proprieary informaion. The resuls of hese calculaions for he low risk and high risk porfolios are shown in Figure 8. There is a relaively small reducion in regulaory capial requiremens for he low risk porfolio when segmening by ORIGSC, and his reducion is negligible for he high risk porfolio. These relaively small effecs are in par due o he limiaions of ORIGSC for differeniaing risk and in par due o he relaive homogeneiy a originaion of each of hese porfolios. There is a subsanial drop in capial requiremens for boh porfolios when using REFRSC in he segmenaion sysem. There is an addiional drop in capial requiremens when segmening by ORIGSC and DELINQ. Noe ha his drop is more subsanial for he lower credi qualiy porfolio. There is a furher drop when using REFRSC and DELINQ, bu here he drop is more subsanial for he higher credi qualiy porfolio. These resuls poin o he significan improvemens in differeniaing risk by moving o a segmenaion mehod ha includes scores and delinquency saus as segmening crieria. For higher credi qualiy porfolios, our daa sugges ha i is imporan o combine updaed credi 22

24 scores wih delinquency saus. For lower credi qualiy porfolios ha segmen by DELINQ, our evidence suggess ha here are relaively small gains from using REFRSC raher han ORIGSC in he segmenaion sysem. Our resuls for seasoning sugges negligible changes when applying seasoning adjusmens compared o a segmenaion sysem ha includes delinquency and score facors. This may no be surprising, since our previous analysis showed ha AGE was no an imporan risk facor for lower qualiy credis afer conrolling for delinquency saus, and lower qualiy credis ypically accoun for a large share of he capial calculaion. However, since our previous resuls showed ha AGE remains a risk facor for higher qualiy credis afer conrolling for scores and delinquency, his negligible effec of seasoning adjusmens on he capial calculaion depends on he composiion of he specific credi card porfolio. To summarize, our calculaions indicae subsanial capial relief incenives for improved risk differeniaion when segmening he credi card porfolio. They also indicae he significan gains in risk differeniaion from using updaed credi qualiy informaion, since he bulk of he regulaory capial relief occurs when moving o a segmenaion sysem ha provides updaed risk measures such as refreshed score and delinquency saus. VII. Summary This paper examines mehods for differeniaing he likelihood of defaul among credi card borrowers and how hese alernaive mehods affec capial requiremens for he porfolio in an ASRF model. Using proprieary daa on wo large credi card porfolios wih differen average credi qualiy characerisics, we demonsrae he imporance of updaing credi qualiy informaion in differeniaing risk among borrowers and for calculaing ail risk of a credi card 23

25 porfolio. In paricular, we find imporan benefis o conrolling for boh delinquency saus and updaed credi scores, and hese benefis are greaer for relaively high credi card porfolios. Finally, we find ha while credi card defauls are srongly correlaed wih loan age, his correlaion is significanly weak afer conrolling for updaed credi qualiy, paricularly for lower credi qualiy loans. Moreover, afer segmening he porfolio using updaed credi qualiy measures, adjusing PD for loan-age effecs had lile impac on esimaed porfolio capial requiremens. 24

26 Bibliography: Basel Commiee on Banking Supervision (2004), Inernaional Convergence of Capial Measuremen and Capial Sandards: A Revised Framework, available a Lang, William, W. and Anhony M. Sanomero (2004), "Risk Quanificaion of Reail Credi: Curren Pracices and Fuure Challenges," in Moneary Inegraion, Markes and Regulaion Research in Banking and Finance, Volume 4, 5, Elsevier Ld. Lauren, Marie-Paule (2004), "Asse Reurn Correlaion in Basel II: Implicaions for Credi Risk Managemen," Working Papers CEB RS, Universié Libre de Bruxelles, Solvay Business School, Cenre Emile Bernheim (CEB). Meron, Rober C. (974). On he Pricing of Corporae Deb: The Risk Srucure of Ineres Raes, Journal of Finance 29, Risk Managemen Associaion Repor, Reail Credi Economic Capial Esimaion Bes Pracices, February Treacy, William F., and Mark S. Carey (998). Credi Risk Raing a Large U.S. Banks, Federal Reserve Bullein 84, Vasicek, Oldrich (2002). Loan Porfolio Value, Risk 5,

27 Chars, Tables and Figures Char Cohor Observaion Poin PD (age), age in quarers Originaion Poin Q 999 Q2 999 Q3 999 Q4 999 Q 2000 Q Q Q Q 200 Q2 200 Q3 200 Q4 200 Q 2002 Q Q Q Q 2003 Q Q Q Q 2004 Q Q Q 999 PD () PD (5) PD (9) PD (3) PD (7) PD (2) PD (6) PD (0) PD (4) PD (8) PD (3) PD (7) PD () PD (5) PD (9) PD (4) PD (8) PD (2) PD (6) Q2 999 PD () PD (5) PD (9) PD (3) PD (7) PD (2) PD (6) PD (0) PD (4) PD (8) PD (3) PD (7) PD () PD (5) PD (4) PD (8) PD (2) PD (6) Q3 999 PD () PD (5) PD (9) PD (3) PD (7) PD (2) PD (6) PD (0) PD (4) PD (3) PD (7) PD () PD (5) PD (4) PD (8) PD (2) PD (6) Q4 999 PD () PD (5) PD (9) PD (3) PD (2) PD (6) PD (0) PD (4) PD (3) PD (7) PD () PD (5) PD (4) PD (8) PD (2) PD (6) Q 2000 PD () PD (5) PD (9) PD (3) PD (2) PD (6) PD (0) PD (4) PD (3) PD (7) PD () PD (5) PD (4) PD (8) PD (2) Q PD () PD (5) PD (9) PD (3) PD (2) PD (6) PD (0) PD (4) PD (3) PD (7) PD () PD (4) PD (8) PD (2) Q PD () PD (5) PD (9) PD (3) PD (2) PD (6) PD (0) PD (3) PD (7) PD () PD (4) PD (8) PD (2) Q PD () PD (5) PD (9) PD (2) PD (6) PD (0) PD (3) PD (7) PD () PD (4) PD (8) PD (2) 26

28 Table a: Maximum Likelihood Esimaes of Logisic Regressions Originaion Score Model Model Parameer Esimae Sandard Wald Pr > ChiSq Odds Error Chi-Square Raio INTERCEPT ,4 <.000 Low Risk AGE ,088 < ORIGINATION SCORE ,72 < DELINQ ,83 < BALANCE E-06 23,307 <.000 LINE E-06 6,697 <.000 INTERCEPT ,824 <.000 High Risk AGE ,252 < ORIGINATION SCORE E-06 90,275 < DELINQ ,329,955 < BALANCE E-06 42,069 <.000 LINE E-06 80,544 <

29 Table b: Maximum Likelihood Esimaes of Logisic Regressions Refreshed Score Model Model Parameer Esimae Sandard Wald Pr > ChiSq Odds Error Chi-Square Raio Low Risk INTERCEPT ,288 <.000 AGE ,362 < REFRSC ,87 < DELINQ E-03 02,87 < BALANCE E-06 7,659 <.000 LINE E-06 4,68 <.000 INTERCEPT ,523 <.000 High Risk AGE ,79 < REFRSC E-06,326,08 < DELINQ ,286,676 < BALANCE E-06 73,542 < LINE E ,96 <

30 Figure 400% Superprime PD by Credi Scores Low Risk PD by Credi Score Indexed Indexed -year -year PD PD (%) (%) 200% 200% 000% 000% 800% 800% 600% 600% 400% 400% 200% 200% 0% Refreshed FICO Refreshed FICO Originaion FICO Originaion FICO 0% ~ 660 ~ ~ ~ ~ 680 FICO Segmen 70~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 900 Indexed -year PD (%) Indexed -year PD (%) 350% 350% 300% 300% 250% 250% 200% 200% 50% 50% 00% 50% 50% FICO Segmen High Risk PD by Credi Scores Subprime PD by Credi Scores Refreshed Refreshed FICO FICO Originaion Originaion FICO FICO 0% 0% ~ 520 ~ ~ ~ ~ ~ ~ ~ ~ ~ 60~ ~ 62~ ~ 64~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 900 FICO Segmen FICO Segmen 29

31 Figure 2 Qualifying Revolving Exposure: Low Risk 6000% 4000% Indexed -year PD (%) 2000% 0000% 8000% 6000% 4000% 2000% 0% Curren ~ 30 DPD 3~ 60 DPD 6~ 90 DPD 9~ 20 DPD 2~ 50 DPD 5~ 80 DPD Delinquency Saus Qualifying Revolving Exposure: High Risk 600% Indexed -year PD (%) 500% 400% 300% 200% 00% 0% Curren ~ 30 DPD 3~ 60 DPD 6~ 90 DPD 9~ 20 DPD 2~ 50 DPD 5~ 80 DPD Delinquency Saus 30

32 Figure 3 Low Risk Indexed PD by Refreshed Score and Delinquency Qualifying Revolving Exposure: Low Risk 4000% Indexed -year PD (%) 2000% 0000% 8000% 6000% 4000% 2000% ~ ~ ~ ~ ~ ~ 760 0% 76~ 780 Curren ~ 30 DPD 3~ 60 DPD 6~ 90 DPD 9~ 20 DPD 2~ 50 DPD 5~ 80 DPD 780~ 900 FICO Segmen High Risk Indexed PD by Refreshed Score and Delinquency 600% Qualifying Revolving Exposure: High Risk Indexed -year PD (%) 500% 400% 300% 200% 00% 0% Curren ~ 30 DPD 3~ 60 DPD 6~ 90 DPD 9~ 20 DPD 2~ 50 DPD 5~ 80 DPD ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 900 FICO Segmen 3

33 Figure 4 60% Low Risk Indexed PD by Age 40% Indexed -year PD (%) 20% 00% 80% 60% 40% 20% 0% Age of Accoun (# monhs) High Risk Indexed PD by Age 60% 40% Indexed -year PD (%) 20% 00% 80% 60% 40% 20% 0% Age of Accoun (# monhs) 32

34 Figure 5 Low Risk Indexed PD by Originaion Score and Age 350% Indexed -year PD (%) 300% 250% 200% 50% 00% 50% 66~ ~ ~ ~ ~ ~ ~ 900 0% Age of Accoun (# monhs) High Risk Indexed PD by Originaion Score and Age 200% Indexed -year PD (%) 80% 60% 40% 20% 00% 80% 60% ~ ~ ~ ~ ~ ~ ~ ~ % Age of 33 Accoun (# monhs)

35 Figure 6 Low Risk PD by Refreshed Score and Age 350% Indexed -year PD (%) 300% 250% 200% 50% 00% 50% 66~ ~ ~ ~ ~ ~ ~ 900 0% Age of Accoun (# monhs) High Risk PD by Refreshed Score and Age 400% Indexed -year PD (%) 350% 300% 250% 200% 50% 00% 50% ~ ~ ~ ~ ~ ~ ~ ~ 660 0% Age of Accoun 34 (# monhs)

36 Figure % Low Risk PD by Delinquency and Age Indexed -year PD (%) 4000% 2000% 0000% 8000% 6000% 4000% 2000% 0% Curren ~ 30 DPD 3~ 60 DPD 6~ 90 DPD 9~ 20 DPD 2~ 50 DPD 5~ 80 DPD ~ ~ ~ ~ ~ ~ ~ ~ 900 FICO Segmen High Risk PD by Delinquency and Age 600% Indexed -year PD (%) 500% 400% 300% 200% 00% 0% Curren ~ 30 DPD 3~ 60 DPD 6~ 90 DPD 9~ 20 DPD 2~ 50 DPD 5~ 80 DPD ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 900 FICO Segmen 35

37 Figure 8 Capial Based on Long Term PD (Low Risk) Qualifying Revolving Exposure: Low Risk 00.0% Indexed Capial Rae (%) (%) 90.0% 80.0% 70.0% 60.0% 50.0% 40.0% 30.0% 20.0% 0.0% 0.0% No Segmenaion Originaion FICO Refreshed FICO Originaion FICO and Delinquency Refreshed FICO and Delinquency Originaion FICO and Delinquency and Age Refreshed FICO and Delinquency and Age Segmenaion Schema Capial Based on Long Term PD (High Risk) Qualifying Revolving Exposure: High Risk 00.0% Indexed Capial Rae (%) (%) 90.0% 80.0% 70.0% 60.0% 50.0% 40.0% 30.0% 20.0% 0.0% 0.0% No Originaion Segmenaion FICO Refreshed FICO Originaion FICO and Delinquency Refreshed FICO and Delinquency Originaion FICO and Delinquency and Age Refreshed FICO and Delinquency and Age Segmenaion Schema 36

38 Appendix: Sample Size Deerminaions Accouns were pulled wihin each band of ORIGSC wih a sample size sufficien o produce reliable esimaes of he PD over he lifeime of each cohor. This required an iniial analysis of he expeced lifeime PD per score band o deermine he required sample sizes. A good benchmark was o use he coefficien of variaion: CV = sandard deviaion/mean = sqr(-p)/sqr(pn), where n is he sample size. By using he CV raher han he sandard deviaion, he level of variance is wihin a cerain percenage of he average PD. This is especially imporan for very low PD segmens, since he effec on capial of small changes in he PD is he greaes in he low-pd segmens. Sample sizes for one-year PDs, wih a CV of 0% are as follows. N = Sandard Deviaion PD - 2*S.D. PD + 2*S.D. K-IRB PD (-PD)/(PD*CV 2 ) = sqr(pd*(-pd)/n) PD PD - 2*S.D. PD + 2*S.D. 0.0% 399, % 0.09% 0.%.4%.06%.23% 0.25% 59, % 0.23% 0.28% 2.8% 2.03% 2.32% 0.50% 79, % 0.45% 0.55% 3.33% 3.4% 3.5% 0.75% 52, % 0.68% 0.83% 4.0% 3.90% 4.28%.00% 39, % 0.90%.0% 4.65% 4.45% 4.83%.25% 3, %.3%.38% 5.05% 4.87% 5.2%.50% 26, %.35%.65% 5.35% 5.8% 5.50%.75% 22, %.58%.93% 5.58% 5.43% 5.7% 2.25% 7, % 2.03% 2.48% 5.90% 5.78% 6.00% 2.75% 4, % 2.48% 3.03% 6.% 6.00% 6.20% 3.25%, % 2.93% 3.58% 6.27% 6.7% 6.36% 3.75% 0, % 3.38% 4.3% 6.4% 6.3% 6.5% The upper and lower bounds on he PD above (PD 2xS.D. and PD + 2xS.D., respecively) provide a 93.4% confidence inerval, using he normal approximaion o he binomial. This ranslaes ino a confidence inerval on he capial raio, using he Basel II capial funcion for credi cards. 37

39 Capial Raio by PD wih 93.4% Confidence Inerval, assuming a CV of 5% 2% Capial Raio, LGD=90% 9% 6% 3% 0%.% 0.38% 0.00% 2.00% 4.00% 6.00% 8.00% 0.00% 2.00% 4.00% PD P P - 2*S.D. P + 2*S.D. Range However, for larger PDs he sandard deviaion and widh of he resuling confidence inerval is beyond an accepable level. For higher PD segmens he sample size was fixed a approximaely 0,000 accouns. PD N Sandard Deviaion PD - 2*S.D. PD + 2*S.D. K-IRB = sqr(pd*(-pd)/n) PD PD - 2*S.D. PD + 2*S.D. 4% 0, % 3.6% 4.39% 6.48% 6.37% 6.59% 5% 0, % 4.56% 5.44% 6.77% 6.64% 6.9% 6% 0, % 5.53% 6.47% 7.% 6.94% 7.30% 7% 0, % 6.49% 7.5% 7.5% 7.30% 7.73% 8% 0, % 7.46% 8.54% 7.95% 7.7% 8.20% 9% 0, % 8.43% 9.57% 8.4% 8.4% 8.67% 0% 0, % 9.40% 0.60% 8.87% 8.59% 9.5% 5% 0, % 4.29% 5.7%.03% 0.75%.3% 20% 0, % 9.20% 20.80% 2.73% 2.49% 2.95% Sample size reducions in he segmens due o defauls, ariion, and score migraion were also considered; his is especially rue for defauls in he higher PD segmens. The oal sample size in each segmen wihin each cohor was increased so ha he segmen (using eiher REFRSC or ORIGSC) mainained he minimum sample size hroughou he sample ime frame. 38

40 To reduce he sample required, accouns ha defaul by he end of he observaion period (hree years) were sampled a 00% wih only a 0% sample of accouns ha do no defaul by he end of he period. The under-sampling of non-defauled accouns by random sampling leads o increased variance and a biased esimae. However, aking every 0h accoun does no lead o he problem of increased variance, unless here is some cyclic naure in he order of he accouns. Over he inerim, he number of accouns ha defaul wihin each one-year ime horizon and he oal number of non-defauled accouns (from he sample imes 0) a he beginning of he year were used o produce unbiased, consisen esimaes of he one-year PD. 39

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