THE ISSUE OF PD ESTIMATION A PRACTICAL APPROACH

Transcription

1 M A H E M A I C A L E C O N O M I C S No. 7(14) 2011 HE ISSUE OF PD ESIMAION A PRACICAL APPROACH Abstract. he ssue of estmatng the probablty of default consttutes one of the foundatons of rsk systems appled n modern bankng. he Basel Commttee pays a lot of attenton to ways of ts estmaton and valdaton. hs paper dscusses statstcal methods enablng PD estmatons wth consderaton of the retal character of a credt portfolo. he author refers to the ssue of defnng default and to the way of calculatng the number of days n arrears. hs paper presents the results of research studes obtaned on the bass of retal credt portfolo. For selected sub-portfolos, the author makes a comparson of the probablty of default, whch enables the explct rsk assessment. Keywords: credt rsk, probablty of default, credt scorng. JEL: C51, C53, E58, G2, G Introducton Credt actvty s nseparably connected wth credt rsk. Credt rsk always comes nto beng where losses caused by cessaton of credt servcng by a clent are covered from own captal of a company takng the responsblty for rsk. In order to assess credt rsk, practtoners use a seres of statstcal methods allowng for dagnosng the current level of rsk and estmatng ts level n the upcomng future. he multplcty of statstcal models results from varous assumptons accepted n the course of ther constructon. It results also from varous ams set for analysts responsble for the area of rsk. he process of credt rsk management s such a complcated ssue that even ts quantfcaton offers many dffcultes. If somebody wants to get to know the real credt rsk of a fnancal nsttuton, a seres of parameters should be defned. he most mportant of them nclude the loss forecast horzon, value of the probablty of default, value of a recovery rate because of credt recovery proceedngs, correlaton of borrower assets, prce of overdue recevables obtaned on the market of purchase and sale of re-, Wrocław School of Bankng, Fabryczna Street 29/31, Wrocław, Poland. E-mal: pawel.sarka@4fs.pl

2 200 cevables. Only cognton of all elements beng ncluded n the full model of credt rsk assessment allows for a comprehensve vew of the studed ssue. It s also mpossble to get to know the rsk scale wthout understandng the prncples of calculatng partcular elements of the appled statstcal model. One of the essental elements of the maorty of models concernng credt rsk assessment s the determnaton of the borrower s value of the probablty of default (PD). hs standardsed measure of rsk from the <0, 1> range allows for a synthetc comparson of rsk levels especally n the area of portfolos referrng to the same credt product. Wthn the framework of Basel II (Basel Commttee..., 2006), the Basel Commttee ntroduced a global standard referrng to the loss forecast horzon, ncludng the PD probablty. Currently, banks are oblged to create forecasts n the tme horzon of one year. Such an approach enables the rsk analyss n the background of a full reportng year of a fnancal nsttuton. he estmaton of rsk n the tme horzon of one year has the advantage of allowng for testng the senstvty of the annual fnancal result, whch takes nto account provsons coverng the rsk. Borrower probablty of default s one of the fundamental measures of credt rsk assessment. Its am s to fnd an answer to the queston about the forecast of proporton of loans (expressed n percent), whch are not gong to be servced any more n the tme horzon of one year. Constant montorng of the PD enables the current analyss of factors that stmulate the rsk level. herefore, t supports the process of qualty assessment of the conducted rsk polcy. Based on PD changes, t s possble to analyse the effects of the changes n nternal rsk management polces. In ths way, early warnng systems are created whch allow, much n advance, drawng bank analysts attenton to the negatve results of decsons made before. he probablty of default (PD) consttutes the basc component of a model appled n order to calculate the captal requrements n accordance wth Basel II. Mnmum captal requrement s constructed n such a way that t nforms on the level of unexpected losses losses for whch the probablty of occurrence s low but stll possble. he possblty of gettng to know potental losses developed as a result of credt actvty and mantenance of captal on ths level ams at the mnmsaton of the danger of the bank s bankruptcy. he ssue of estmatng the probablty of default s the subect of many studes. Usually they concern companes assessed by ratng agences. Among others, Blochwtz and Hohl (2001), as well as Jafry and Schuermann (2004), have been dealng wth relatons between the obtaned ratng

3 he ssue of PD estmaton a practcal approach 201 and the probablty of default. Cantor and Falkensten (2001), as well as Cantor and Mann (2003) n later studes, presented the analyss of the nfluence of macroeconomc factors on PD. Hetfeld (2005) descrbed n hs paper the ssue of senstvty of ratng assessments to changes of economc envronment. In reference to retal loans, Allen et al. (2003) dealt wth the ssue of estmatng PD. In addton, Sabato (2006) n hs study drew attenton to the ssue of estmatng PD n the case of retal portfolo wth low rsk (low-default probablty). In the studes of Duffe and Sngelton (2003), one may fnd a descrpton of methods appled n the process of estmatng PD probablty. Haste (2001), n turn, presented extensvely the classfcaton methods. Gruszczynsk (2001) descrbed a seres of useful approaches to the ssue of estmatng the probablty of default. In addton, Saunders (2002) presented n hs book the wde range of methods used by practtoners n the process of estmatng PD. he am of ths paper s the presentaton of methods enablng the estmaton of the probablty of default n the context of portfolos ncludng retal credts. hus, a great part of ths paper concerns PD models n whch ndvdual borrower characterstcs play a key role. he author s am s also to present the results of analyss conducted wthn the framework of a retal credt portfolo. Selected methods were used as comparson of rsk levels n two separate groups. It allowed for the verfcaton of the hypothess about the presence of hgher rsk n group A than n group B. hs paper conssts of an ntroducton, after whch one presents the defnton of default, consttutng the bass of settng PD. In the further part, the author reports a seres of methods amed at the estmaton of the probablty of default. hen, one can fnd an example llustratng the practcal applcaton of a method based on mortalty tables n the process of assessng rsk of retal loan portfolos. Next, the author ncludes conclusons from the conducted research studes. 2. Borrower default defnton In a further part of ths paper, the event of default s understood n accordance wth the defnton accepted by the Basel Commttee. We assume that default takes place n relaton to a gven borrower f there has occurred one of two followng events: 1. Any exposure havng the character of credt s more than 90 days overdue, whle exposure s clamed as overdue f the overdue amount exceeds the threshold amount.

4 Due to nternal analyss, one clams that probably the borrower does not completely fulfl ther credt oblgaton. he threshold amount, n excess of whch one recognzes the gven credt as default, s accepted at the level of 50 EUR for use of further consderatons. he second condton allows for the classfcaton of gven recevables as overdue although delay n payment may come to less than 90 days. hs condton n the case of retal portfolo ams at recognton as default of all these loans for whch one has not noted repayment of the frst two nstalments. he experences of many banks ndcate that the vast maorty of these borrowers have taken loans wth the am of not repayng them. hus, qucker classfcaton of these credts as default enables the earler begnnng of credt recovery proceedngs. herefore, a key element of the assessment of default s the number of days n arrears of repayment. In the 1990s, many banks accepted the methodology descrbng the moment of occurrence of arrears as a moment n whch one observed arrears on the gven loan, and after that tme, one dd not make full repayment of arrears. Such an approach causes the number of days n arrears to be determned from the frst date of arrears n the case when the borrower pays ther nstalments wth a sgnfcant delay (that s, e.g. two months of delay), hus, a borrower who had a problem wth repayng only the frst and second nstalment, and then repays the full nstalment every month, accordng to ths method after 12 months was delayed n repayment for one year. It s worth payng attenton to the fact that the debtor had arrears to the bank equal to two nstalments. A dfferent approach to the process of establshng the date of occurrence of arrears s the method consstng n determnng the date of arrears of the oldest unpad nstalment. In other words, loan nstalments are treated as separate recevables, and the oldest overdue credt nstalment ndcates the moment from whch we calculate arrears. It s assumed n ths method that every payment s booked to cover the oldest arrears. In practce, one takes nto consderaton not only the prncpal maturty date, but also the balance of overdue nterests, overdue penalty nterests and calculated costs. he frst approach was appled for many years and t was a result of I systems lmtatons and the lack of computersaton. he second approach, much more reasonable, s currently domnatng n modern bankng. Actually, ths prncple of determnng the number of days n arrears s appled n the further part of ths paper.

5 he ssue of PD estmaton a practcal approach Methods of estmatng the probablty of default Among the methods appled n the process of estmatng PD, so-called probablty models are partcularly sgnfcant from the practcal pont of vew. Wthn the framework of ths approach, the PD s a functon of the arguments whch are the characterstcs of borrowers. hus, the probablty of the event consstng n that -th borrower becomes nsolvent s modelled accordng to the general rule: F( x ), (1) where P s the probablty of default of -th borrower, F() s a functon of vector x (borrower characterstcs). One of the smplest approaches to the ssue of estmatng the probablty of default s the lnear model, presented by the followng formula: F( x β) x β, (2) where β s a vector of model parameters. he subect of modellng here s the probablty of default, thus the followng condton shall be met: 0 x β 1. (3) Because the values of the functon of lnear regresson often exceed a range relevant to the measure of probablty, the followng form of lnear model of probablty s accepted: 0 F( xβ) xβ 1 x β 0 0 x β 1. (4) x β 1 An nterestng way of releasng from troublesome assumpton referrng to the narrow scope of the obtaned results due to the specfcty of the measure of probablty s the approach wthn the framework of the probt model. It enables the estmaton of the probablty of default through the use of the normal cumulatve probablty dstrbuton functon, whose values are wthn the [0, 1] range. In ths model, the probablty of default determned on the bass of the above-mentoned functon of borrower comes to: x β 2 1 t F( xβ ) exp( ) dt. (5) 2 2 -

6 204 Fndng the nverse functon to the cumulatve dstrbuton functon, we receve expressons called probts n the followng form: x β F -1 ( P) G( P). (6) Another bnomal model, whch s seen the most often n bankng practce, s the logt model. Accordng to ths approach, functon F() determned on a product of vectors xβ s the logstc cumulatve dstrbuton functon n the followng form: 1 exp( xβ ) F( xβ) 1 exp(1 x β) 1 exp( x β ). (7) Based on the nverse functon for F(), one sets logts n the followng form: P 1 xβ F ( ) ln. (8) 1 P he obtaned logts are natural logarthms of a quotent of probablty of default n relaton to the probablty of repayng the loan. A dfferent approach to modellng the probablty of default s the use of the logarthmc-lnear model, n whch functon F() s as follows: F( x β) exp( x β ). (9) Smlarly to the lnear model, there occurs a problem wth the scope of functon values, whch may take values above one. In order to prevent t, t s essental to keep ths condton xβ 0. An alternatve method of estmaton of PD, whch takes nto account a vector of borrower characterstcs, nvolves the use of the Burr dstrbuton functon (Burr, 1942). Accordng to ths approach, the probablty of default may be modelled by the followng relaton: 1 F( xβ) 1 (1 ( c x β )] k, (10) where parameters c, k are postve and expresson xβ s not negatve. Analogously to the above-presented approaches, the Urban model (Aldrch, Nelson, 1984) can also be used. he proposed functon F() for gven borrower characterstcs s as follows: 1 1 F( xβ) arctan( xβ ). (11) 2

7 he ssue of PD estmaton a practcal approach 205 he above-presented methods consttute a very sgnfcant support for analysts estmatng the level of the probablty of default. hese approaches may be used both to estmate PD for a specfc borrower and n reference to separate rsk classes. he latter applcaton has partcularly sgnfcant meanng n the context of determnng credt rsk captal requrements. By dvdng credt portfolo nto several classes ncludng loans wth smlar probablty of default, banks may estmate PD values n partcular groups based on the average values of borrower characterstcs n a class. In ths way, the obtaned probabltes may serve the estmaton of a value of unexpected losses of partcular borrower groups, whch determne the level of captal requrements (Basel Commttee..., 2006). However, such an actvty requres an adequately large set of hstorcal data, whch ncludes not only nformaton on default, but frst of all takes nto regard borrower characterstcs. hs last condton causes the constructon of ths type of models to be mpossble n many banks. hus, bankers often use methods that allow estmatng the probablty of default for selected portfolos based only on the repayment hstory. In ths case, the borrower characterstcs mentoned before are not known or they have been already used on the stage of groupng loans n homogenous portfolos. One of the methods used to calculate the probablty of default of portfolos based on hstorcal data s the method based on mortalty tables. It conssts n the analyss of hstorcal observatons wthn a coherent portfolo. he dea of ths approach conssts n the determnaton of annual rates MMR (Margnal Mortalty Rate), based on whch one constructs PD forecasts. Rates MMR, are set n accordance wth the followng formula: LD, MMR,, (12) L where LD, s a number of loans granted n perod, whch defaulted n year, 1,..., n (n s last year of credt lfe). L, s the number of loans whch were granted n perod and were not classfed as default cases n -th year snce ther grantng. Index means a perod n whch loans were granted. Usually one assumes annual perods for the analyss. However, one may assume a shorter perod, that s, sx months or quarters, whch has a sgnfcant meanng n the case of a dynamcally developng portfolo of loans wth a relatvely short perod of ther lfe. he best example s a portfolo of cash loans. In such a case, groupng portfolos accordng to the year of ther start may cause a relatvely hgh estmaton error. It appears, how-,

8 206 ever, that dynamcally growng sales of loans causes the average value of outstandng to grow rapdly every month. hs may cause a sgnfcant underestmaton of rsk rated. In such a case, one recommends groupng loans n quarters, whch should elmnate ths knd of estmaton error. Based on rate MMR,, one determnes one aggregated ndcator desgnatng the probablty of default of a portfolo n year snce ts grantng. Rates MMR are set n accordance wth the followng formula: k MMR MMR w, (13),, 1 where k s the last number of perod for whch one conducts the MMR analyss, w, s a weght for -th perod determned n accordance wth the followng formula: L, w, k. (14) L he above-used weghts certanly meet the assumpton: w, 1. 1 Rate MMR presents the value llustratng the proporton of loans whch defaulted n -th year snce the moment of ther grantng. Based on these rates, t s possble to set PD forecast for a portfolo of loans wth a varous age structure. Based on MMR rates, t s also possble to set SR rate (Survval Rate), consttutng the complement to the unty of MMR n the followng form:, k 1 MMR 1 SR. (15) Due to settng SR rates, t s possble to change the forecast horzon of the probablty of default. hus, nstead of analysng the nearest year wthn the framework of rsk assessment system, bank analysts may concentrate on the long-term results of the conducted rsk polcy lastng for several years. he probablty of default n the horzon of m perods (years) s marked by CMR m (Cumulatve Mortalty Rate) and set n accordance wth the followng formula: m CMR 1 SR 1 ( SR SR,..., SR ). (16) m 1 2 m 1

9 he ssue of PD estmaton a practcal approach Practcal example of estmaton of PD In order to llustrate the above-mentoned method based on mortalty tables, the author presents the results of the conducted research studes. he author analysed data obtaned from a fnancal nsttuton specalsng n car loans. All loans were granted to ndvduals between he purpose of the loans was the purchase of cars, whch consttuted collateral. In ths analyss, two groups of loans were taken nto account. he frst of them ncluded loans granted for the purchase of used cars (marked as A), the second one ncluded loans for new cars (marked as B). In the frst case, the transacton took place between two ndvduals, n the second case usually at a car dealer. In the course of analysng borrower characterstcs, the author notced that persons takng loans for new cars were on average older had a hgher net ncome and longer work experence. In addton, the fracton of sngle persons n ths group was on a relatvely low level. hus, t was reasonable to conduct the analyss of the probablty of default for both portfolos n order to verfy the hypothess about a sgnfcantly dfferent rsk level. hose results were supposed to serve the process of assessng portfolo proftablty. able 1 and able 2 present the values of parameters MMR, for portfolos observed n the course of the frst four years of lfe. In the analyss, we took nto account the dvson of portfolos nto years of ther start, from 1998 to year of lfe Source: author s own work. able 1. MMR analyss for car loan portfolo (used cars) MMR year of grantng MMR SR 1 3.5% 3.4% 3.7% 3.6% 96.4% 2 3.1% 3.1% 3.9% 3.5% 96.5% 3 3.1% 3.4% 3.3% 96.7% 4 2.5% 2.5% 97.5% he MMR column presents the average values of rates MMR, for the subsequent years of credt lfe. hus, 1 MMR refers to the average value of the probablty of default observed for loans n the course of the frst year of ther lfe. Based on the obtaned results, we can notce that the probablty of

10 208 default s the hghest n the frst year of credt lfe and came to 3.6% for the used car loan portfolo. able 2, whch refers to the portfolo of loans granted to the purchase of new cars, presents analogous data arrangement. year of lfe able 2. MMR analyss for car credt portfolo (new cars) MMR year of grantng MMR SR % 1.61% 1.63% 1.70% 98.30% % 2.52% 1.69% 1.83% 98.17% % 1.58% 1.80% 98.20% % 1.85% 98.15% Source: author s own work. Fgure 1 presents n a graphcal way values of ndcators MMR, set for the portfolo of loans granted to the purchase of used cars. he four-year tme of observng a portfolo means that only for loans granted n year 1998, we observe all values (bars), whose amount reflects hstorcal frequences of default cases. 4% 2% % Fg. 1. Graphcal presentaton of MMR ndcators for car credts (used cars) Source: author s own work. Fgure 2 llustrates the hstorcal frequency of default observed for the portfolo of loans granted to purchase new cars. Values decdedly lower

11 he ssue of PD estmaton a practcal approach 209 than those obtaned for prevous portfolo ndcate the occurrence of the sgnfcant dfference n credt rsk levels. 3% 2% 1% 0% Fg. 2. Graphcal presentaton of MMR ndcators for car credts (new cars) Source: author s own work. 4% used car portfolo new car portfolo 3% 2% 1% 0% Fg. 3. Comparatve analyss of MMR ndcators for car credt portfolo (used cars vs. new cars) Source: author s own work MMR

12 210 Fgure 3 presents the comparson of values of ndcators MMR for both portfolos. Based on the obtaned results, we can observe the sgnfcant dfference between the values of default rates n both portfolos, regardless of the year they concern. Values MMR determned for used car loan portfolo on average come to 3.2%, whch exceeds sgnfcantly the average value obtaned for new car credt portfolo comng to 1.8%. On ths bass, we may clam that the rsk level of car loans granted to purchase new cars s charactersed by a much lower rsk level. In every year of credt portfolo lfe, ths dfference s sgnfcant, thereby confrmng the thess set before. he obtaned results may be appled to the rsk assessment of a portfolo wth any age structure. Let us consder for example that the management board of a fnancal nsttuton expects the dynamc development of the market of sale of used cars. hus, t was assumed that as a result of credt sale, the balance of the bank would contan loans whose amount s presented n able 3 (lne 2). Consderng the above-mentoned assumptons, we decded to estmate the level of the probablty of default of a portfolo. akng nto regard the tme structure of loans and rates MMR set for partcular years of credt lfe, the author carred out the estmaton of the probablty of default. hat probablty came to 3.29%. Knowng ts value and the value of credt outstandng, as well as the expected recovery rate, the management board of the bank may determne the expected value of losses. he value of the expected loss shall be consdered n the proft and loss account as the value of provsons for coverng credt rsk. able 3. Probablty of default of a credt portfolo analyss current portfolo (date of grantng) age of loans [years] number of good loans weght of loans n portfolo 19% 22% 27% 32% MMR 2.52% 3.31% 3.5% 3.55% Portfolo PD 3.29% Source: author s own work. he analyss of the probablty of default may consttute also a sgnfcant element of a model of proftablty assessment of partcular loan portfo-

13 he ssue of PD estmaton a practcal approach 211 los. he recognton of rsk costs at the assessment of net cash flows generated by gven credt portfolo shall be consdered at each change of credt offer, as well as t shall be conducted perodcally n order to assess bank proftablty. However, PD values obtaned n ths way shall be verfed wthn the framework of back-testng procedure. hs analyss shall be conducted ex-post, and ts am s the assessment of accuracy of the created forecasts. One should not forget that banks functon n a dynamc envronment, where a seres of macroeconomc factors has an nfluence on the average default rate. hus, the constant control of forecast qualty ams at the detecton of the possble underestmaton of the level of losses. 5. Conclusons he ssue of credt rsk consttutes a key area for modern bankng, wthn the framework of whch quanttatve methods measurng t are dynamcally developng. Regulatons such as Basel II motvate banks to develop new tools, as well as to use the exstng ones. Models presented n ths paper may be appled n the process of estmatng the probablty of default of partcular borrowers and of homogenous credt portfolos. However, ther use s condtoned by the possesson of suffcent sets of hstorcal data. hus, t s so mportant to create n banks data warehouses, whch collect systemcally nformaton on borrower characterstcs and ther credt hstory. he results of research studes presented n ths paper llustrate one of the more popular among bankers methods appled n the process of credt rsk assessment. Based on the obtaned results, the author establshed that a hgher probablty of default s present n the credt group marked as A. Comparatve analyss of the probablty of default n both groups enables determnaton of captal requrements wth consderaton of the estmated PD. We should also remember that the allocaton of captal for coverng rsk s one of the key ssues n the process of valuaton of rsk assets. Lterature Aldrch J.H., Nelson F.D. (1984). Lnear probablty, logt, and probt models. Newgury Park. London. Allen L., DeLong G.L., Saunders A. (2003). Issues n the Credt Rsk Modelng of Retal Markets. NYU Stern School of Busness Workng Paper No. FIN Basel Commttee on Bankng Supervson (BCBS) (2006). Internatonal Convergence on Captal Measurement and Captal Standards. Bank for Internatonal Settlements. Basel.

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