Pitfalls in Modeling Loss Given Default of Bank Loans

Ptfalls n Modelng Loss Gven Default of Bank Loans by Marc Gürtler * and Martn Hbbeln ** * Professor Dr. Marc Gürtler Braunschweg Insttute of Technology Department of Fnance Abt-Jerusalem-Str. 7 3806 Braunschweg Germany Phone: +49 53 39 895 Fax: +49 53 39 899 E-mal: marc.guertler@tu-bs.de ** Dr. Martn Hbbeln Braunschweg Insttute of Technology Department of Fnance Abt-Jerusalem-Str. 7 3806 Braunschweg Germany Phone: +49 53 39 898 Fax: +49 53 39 899 E-mal: martn.hbbeln@tu-bs.de Electronc copy avalable at: http://ssrn.com/abstract=75774

Ptfalls n Modelng Loss Gven Default of Bank Loans Abstract The parameter loss gven default (LGD) of loans plays a crucal role for rsk-based decson makng of banks ncludng rsk-adusted prcng. Dependng on the qualty of the estmaton of LGDs, banks can gan sgnfcant compettve advantage. For bank loans, the estmaton s usually based on dscounted recovery cash flows, leadng to workout LGDs. In ths paper, we reveal several problems that may occur when modelng workout LGDs, leadng to LGD estmates whch are based or have low explanatory power. Based on a data set of 7,463 defaulted bank loans, we analyze these ssues and derve recommendatons for acton n order to avod these problems. Due to the restrcted observaton perod of recovery cash flows the problem of length-based samplng occurs, where long workout processes are underrepresented n the sample, leadng to an underestmaton of LGDs. Wrte-offs and recoveres are often drven by dfferent nfluencng factors, whch s gnored by the emprcal lterature on LGD modelng. We propose a two-step approach for modelng LGDs of nondefaulted loans whch accounts for these dfferences leadng to an mproved explanatory power. For LGDs of defaulted loans, the type of default and the length of the default perod have hgh explanatory power, but estmates relyng on these varables can lead to a sgnfcant underestmaton of LGDs. We propose a model for defaulted loans whch makes use of these nfluence factors and leads to consstent LGD estmates. Keywords: Credt rsk; Bank loans; Loss gven default; Forecastng JEL classfcaton: G; G8 Electronc copy avalable at: http://ssrn.com/abstract=75774

Introducton For the descrpton of the rsk of a loan, the most central parameters are the probablty of default (PD) and the loss gven default (LGD). Whle a decade ago the focus of academc research and bankng practce was manly on the predcton of PDs, recently substantal effort has been put nto modelng LGDs. One reason s the requrement of the Basel II / III framework, accordng to whch banks have to provde own estmates of the LGD when usng the advanced nternal ratngs-based (A-IRB) approach or the IRB approach for retal exposures. Besdes the regulatory requrement, accurate predctons of LGDs are mportant for rsk-based decson makng, e.g. the rsk-adusted prcng of loans, economc captal calculatons, and the prcng of asset backed securtes or credt dervatves (cf. Jankowtsch et al., 008). Consequently, banks usng LGD models wth hgh predctve power can generate compettve advantages whereas weak predctons can lead to adverse selecton. There exst dfferent streams of LGD related lterature. Lterature dealng wth the relaton between PDs and LGDs nclude Frye (000), Altman et al. (005), Acharya et al. (007), and Bade et al. (0). LGD models that seek to estmate the dstrbuton of LGDs for credt portfolo modelng are Renault and Scallet (004) and Calabrese and Zenga (00). Furthermore, there are several emprcal studes that analyze nfluencng factors of ndvdual LGDs. Whle most of the lterature conssts of emprcal studes for corporate bonds, a smaller fracton focuses on bank loans, whether retal or corporate, whch s manly due to lmted data avalablty. A survey of emprcal studes of LGDs wth a classfcaton nto bank and captal market data can be found n Grunert and Weber (009). There are some relevant dfferences between LGDs of corporate bonds and bank loans. Frst, LGDs of bank loans are typcally lower than LGDs of corporate bonds. Accordng to Schuermann (006), ths emprcal fndng s manly a result of the (on average) hgher senorty of loans and a better montorng. Second, LGDs of corporate bonds are typcally determned on the bass of market values resultng n market LGDs whereas the LGDs of bank loans are usually workout LGDs. If the market value of a bond drectly after default s dvded by the exposure at default (EAD), whch s the face value at the default event, we get the market recovery rate (RR). Applcaton of the equaton LGD = RR results n the market LGD. Contrary, the workout LGD s based on actual cash flows that are connected wth the defaulted debt poston. These are manly dscounted recovery cash flows but also dscounted costs of the workout process. If these cash flows are dvded by the EAD, we get the workout LGD. Even f the calculaton of workout LGDs s more complex, the advantage Electronc copy avalable at: http://ssrn.com/abstract=75774

s that the results are more accurate and that ths approach s applcable for all types of debt (cf. Calabrese and Zenga, 00). A frst step towards forecastng ndvdual LGDs of bank loans has been done by emprcal studes reportng LGDs for dfferent categores of nfluence factors (cf. Asarnow and Edwards, 995; Felsovaly and Hurt, 998; Eales and Bosworth, 998; Araten et al., 004; Franks et al., 004). More recent studes analyze nfluence factors of LGDs va lnear regressons (cf. Ctron et al., 003; Casell et al., 008; Grunert and Weber, 009), log regressons (cf. Casell et al., 008) or log-log regressons (cf. Dermne and Neto de Carvalho, 005; Bastos, 00). Belott and Crook (007) compare the performance of dfferent models, constructed as combnatons of dfferent modelng algorthms and dfferent transformatons of the recovery rate, e.g. OLS regressons or decson trees on the one hand and log or probt transformatons on the other hand. Bastos (00) proposes to model LGDs wth nonparametrc and nonlnear regresson trees. The man motvaton of ths paper s to call attenton to relevant ptfalls n modelng workout LGDs of bank loans. Moreover, we derve recommendatons for acton n order to avod these problems and demonstrate the proposed methods on a data set consstng of 7,463 defaulted loans of a German bank. In the followng, we characterze these ptfalls wthn the typcal steps of the modelng process. After collectng all payments durng the workout processes of hstorcal defaults, the realzed workout LGDs have to be calculated. Wthn the calculaton of LGDs, we observe that the emprcal lterature on LGDs gnores the effect that samples of hstorcal LGDs are usually based, whch s due to dfferences n the length of the workout process (ptfall ). Two types of default end can be dstngushed: contracts that can be recovered and contracts that have to be wrtten off. Snce wrte-offs are typcally connected wth a longer perod of the default status, the number of wrte-offs s usually underrepresented n samples of defaulted loans, leadng to an underestmaton of LGDs. On the bass of calculated workout LGDs, predcton models for non-defaulted loans can be developed. Ths s mostly done wth a drect regresson on LGDs. However, due to the dfferent characterstcs of recovered loans and wrte-offs, the estmaton of LGDs wth a sngle model performs poorly (ptfall ). We propose a two-step estmaton of LGDs: In the frst step, the probablty of a recovery/wrte-off s estmated. In the second step, the LGD of For retal loans, a default s usually assgned on contract level. Contrary, for corporate loans a default s generally determned on frm level so that several contracts default smultaneously. Ths has to be consdered n the calculaton of LGDs, too.

recovered loans as well as the LGD of wrte-offs s predcted separately. These predctons are combned nto the total LGD forecast. The exstng lterature on LGD modelng only concentrates on non-defaulted loans. Though, also for defaulted loans wth actve default status, estmates of LGDs are requred, e.g. for regulatory and economc captal calculatons. For defaulted loans, there s some addtonal nformaton avalable that can be used for LGD predctons, e.g. we fnd that the length of the default perod has a hgh explanatory power. However, f LGDs are modeled on the bass of the (ex-post known) total length of default and the model s appled usng the (exante known) current length of default, LGDs wll be sgnfcantly underestmated (ptfall 3). Thus, we show how the ex-ante nformaton of the current length of default can be used approprately. These aspects can sgnfcantly nfluence the forecasts and should be consdered when modelng LGDs to acheve reasonable results. However, to our best knowledge, these ptfalls have not been addressed n the lterature before. There are some further nterestng fndngs. Wthn the frst step of our estmaton,.e. the predcton of recovery/wrte-off probabltes, we fnd that the accuracy s lower for secured loans than for unsecured loans. However, wthn the second step,.e. the predcton of LGDs condtonal on the type of default end, the opposte s true. Furthermore, we propose a smple but well workng model for estmatng LGDs of defaulted loans, whch have up to now wdely been gnored n the LGD lterature. The remander of ths paper s structured as follows. Secton contans a descrpton of the data and descrbes the calculaton of LGDs. In ths context, we gve attenton to the frst ptfall. In Secton 3, we dscuss LGD modelng for non-defaulted loans ncludng ptfall. Secton 4 deals wth LGD modelng for defaulted loans, whch covers ptfall 3. Secton 5 concludes. Calculaton of workout LGDs and descrpton of the data set For the forecastng of LGDs, we have to calculate hstorcal workout LGDs of our modelng data. Let S be a set of loans and S an ndvdual loan. The workout LGD of loan s typcally expressed as follows: RCF C LGD = EAD, () where RCF stands for the sum of dscounted recovery cash flows of loan, C represents the sum of dscounted drect and ndrect costs of loan, and EAD s the exposure at default of 3

loan. However, a defaulted loan can have two dfferent types of default end, whch drectly nfluence the calculaton of LGDs: Some contracts can be recovered whereas other contracts have to be wrtten off. Recoveres (RCs): In the case of a recovery, the default reason s no longer exstent, e.g. the oblgor pad the amount that he was n arrears wth payments or a new payment plan has been arranged. Thus, the contract s thenceforward handled as a normal non-defaulted loan. Wrte-offs (WOs): If the chance of recoverng addtonal money from the oblgor or the realzaton of collateral s consdered to be small, the contract wll be wrtten off. Thus, there are generally no further payments for ths contract. Whle equaton () s correct for wrte-offs, we addtonally have to consder the exposure at recovery (EARC) for the case of RCs. At the tme of recovery, there s stll a sgnfcant exposure resultng from nstallments after the tme of recovery. However, snce the EARC reduces the economc loss resultng from a default but the EARC s not ncluded n the cash flows, we have to add the (dscounted) exposure at recovery EARC of loan to the correspondng (dscounted) recovery cash flows: RCF C + EARC LGD =. () EAD If the type of default end s a wrte-off, we can set the value of EARC to zero. We apply equaton () to calculate the LGDs of defaulted loans for a data set of a large German bank. The data set conssts of 7,463 loans wth default end between October st, 006, and September 30 th, 008. 3 The loans correspond to several subportfolos of the bank, whch can be dvded nto prvate and commercal clents meetng the crtera of retal portfolos, 4 as well as secured and unsecured loans. The descrpton of the data set can be found n Table. - Table about here - We used the effectve nterest rate to dscount the cash flows snce ths method has been favored by the natonal bankng supervsor. For detals regardng approprate dscount rates see Basel Commttee on Bankng Supervson (005a) and Maclachlan (005). 3 Whle most studes on LGDs present the number of loans that defaulted n a gven perod (default begn), we focus on the default end. Detals wll be descrbed subsequently. 4 See e.g. Basel Commttee on Bankng Supervson (005b), 70. 4

Wth a total of 59,44 contracts, the maor part of the data conssts of secured loans to prvate clents. The LGD frequency dstrbuton correspondng to ths subportfolo s presented n Fgure. - Fgure about here - In the emprcal lterature about LGDs t s often reported, that the dstrbuton of LGDs s bmodal wth most LGDs beng qute hgh (0-30%) or qute low (70-80%) (cf. Schuermann, 006). Whle ths seems to be true for corporate bonds or combned data of corporate bonds and corporate loans, the dstrbuton for retal loans can be qute dfferent. For our data of secured loans to prvate clents, t s strkng that the maor share of loans has a LGD whch s close to zero, whereas a smaller share of loans s concentrated at values around 50% and a small peak can be found for an LGD of 00%. Ths dstrbuton has smlartes to the data set of Bastos (00). However, n our data the fracton of LGDs close to zero s consderably hgher whereas the fracton of LGDs close to one s substantally lower. The LGD dstrbutons of the other subportfolos show some mnor dfferences to Fgure. For secured loans of commercal clents, the dstrbuton s very smlar but the small peak at LGD = s mssng. Ths mght be a result of hgher effort that s made to recover a part of the exposure n connecton wth a better cost-beneft rato due to hgher loan amounts. If the loans are unsecured, the LGDs are on average sgnfcantly hgher for both prvate and commercal clents. However, for all subportfolos there s a large amount of contracts wth LGDs close to zero. Whle these observatons manly consst of loans that have been recovered, observatons wth hgh LGDs largely belong to contracts that had to be wrtten off. The dstrbuton of LGDs for both types of default end, RC and WO, are llustrated n Fgure. - Fgure about here - Banks are manly nterested n the total LGD of contracts and not only n the loss n a predefned perod after default. For example, Bastos (00) mentons for hs study that the dates of wrte-offs were not avalable, but that LGDs calculated on the bass of recovery cash flows wthn a long tme perod after default are a good approxmaton of the demanded LGDs. Thus, f there s suffcent data avalable, only contracts wth realzed default end (RC or WO) should be consdered n the modelng data. However, f we develop LGD models on the bass of all defaults wth completed workout process that are avalable, defaults wth a 5

short workout process are overrepresented, whch s due to nterval censored data. Ths s llustrated n Fgure 3. - Fgure 3 about here - Snce LGDs and the duraton of the workout process are not stochastcally ndependent, not only the average duraton of the workout process but also average LGD s based f ths effect s gnored. If we were solely nterested n the duraton of the workout process, we could account for censorng e.g. by usng the proportonal hazard or accelerated lfetme model. 5 However, we want to determne the LGDs of censored data and not the duraton, so that we cannot apply these models. In Proposton, we show that the censored data lead to an underestmaton of LGDs. Furthermore, we propose to restrct the data set n order to get unbased results. Ptfall : Underestmaton of LGDs due to restrcted data observaton perods Proposton 6 Let S be a loan, τ s the pont n tme of default of loan, and T s the duraton of the workout process for loan. 7 Assume τ to be ndependent of LGD and of T. In addton, there exsts a barrer T max wth T T max. Furthermore, for all t t the (condtonal) random varable LGD T > t s assumed to have strct frst-order stochastc domnance over = LGD T t. Fnally, τ and τ wth τ < τ are two ponts n tme wth T max < τ τ. Then the followng statements hold: (I) LGD has strct frst-order stochastc domnance over the condtonal random varable LGD τ τ < τ + T τ. Partcularly, E( LGD ) > E( LGD τ τ < τ + T τ ). (II) The random varables LGD, LGD τ + τ + τ τ τ τ Tmax, and LGD Tmax T are dentcally dstrbuted, whch mples E( LGD ) = E( LGD τ τ τ T ) = E( LGD T τ + T τ ). max max 5 The estmaton of the survval functon for censored data usng nonparametrc and parametrc methods s descrbed n Kefer (988). 6 The proof of the proposton s presented n Appendx A. 7 Random varables are denoted by a tlde ~. 6

If we model LGDs on the bass of defaults wth completed workout process, the data set conssts of observatons where the default occurs after the begn of the observaton perod,.e. τ τ, and the pont n tme of the default end s τ + T τ. Thus, an estmaton of LGDs on the bass of the complete sample leads to an underestmaton of LGDs due to Proposton (I). The mpact of ths underestmaton s the greater, the shorter the tme perod that s covered by the data of a bank. The relevance of ths ssue becomes apparent f we look at the mnmum data requrements for own estmates of LGDs accordng to the mplementaton of the regulatory captal rules (Basel II) nto German law (Solvabltätsverordnung, SolvV). Accordng to 33 and 34(4) SolvV, LGD estmates must be based on a data observaton perod of at least 5 years for corporate and years for retal exposures, f the bank uses own estmates of LGDs for the frst tme. Subsequently, the mnmum data observaton perod ncreases to 7 and 5 years, respectvely. For these data observaton perods, the problem of uncompleted defaults can lead to a sgnfcant underestmaton of LGDs. In order to analyze the relatonshp between LGDs and default lengths further, we present the length of the default perod separately for recovered loans and wrte-offs. As can be seen n Fgure 4, the workout process s typcally sgnfcantly shorter for loans that can be recovered than for wrte-offs. Snce recoveres usually have sgnfcantly smaller LGDs than wrte-offs, as already demonstrated n Fgure, we have an essental reason for the fndng that defaults wth a short default length typcally have small LGDs. - Fgure 4 about here - As can also be seen n Fgure 4, almost all workout processes of the presented data are completed after 450 days. Hence, we set T max = 450 and restrct the data set accordng to Proposton (II). Ths means that we do not consder all avalable default data but only those that could have been recovered or wrtten off wthn 450 days, n order to avod the systematcal underestmaton of LGDs. There are two ways of assurng ths. Frst, we can apply the condton τ τ τ T, so that we reduce the data set to loans wth default begn between the begnnng of the observaton perod and 450 days before the end of the observaton perod. Second, we can apply the condton that we restrct the data to loans wth default end between 450 days after the begnnng of the observaton perod and the end of the observaton perod. We use the second alternatve snce n ths case we consder the most recent defaults and reect defaults from the begnnng of the 7 max τ + T τ + T τ, so max

observaton perod. Contrary, f we chose the frst alternatve, we would have gnored the most recent defaults. Snce our observaton perod comprses the tme perod between July st, 005 and September 30 th, 008 we restrct the analyss to loans wth default end between September 4 th, 006 and September 30 th, 008. As a consequence of ths restrcton, the relatve ncrease of LGD s 8.3%. Ths s the amount that LGDs would have been underestmated f ptfall has been gnored. Thus, ptfall can ndeed lead to a sgnfcant bas. Nevertheless, n exstng emprcal studes on LGDs there s no remark that ths potental bas s accounted for. For example, Grunert and Weber (009) analyze loans whch defaulted between 99 and 003. They note that only loans wth completed workout process are consdered, leadng to a small number of defaults n the years 00 and 003. Thus, the mentoned bas has apparently not been accounted for. The same s true for Asarnow and Edwards (995), even f the bas should be less substantal, whch s due to the long data observaton perod from 970 to 993. As mentoned before, Bastos (00) calculates LGDs on the bass of recovery cash flows wthn a recovery horzon of, 4, 36, and 48 months, where especally the recovery horzon of 48 months could be used as an approxmaton of the requred LGD. Aganst ths background, the author only consdered defaults wthn the frst out of a 6 years data observaton perod. They thus do not consder the most recent defaults. The same s true for the emprcal study of Dermne and de Carvalho (006), where only the frst 54 out of 374 defaults are consdered for the recovery horzon of 48 months. 3 LGD forecastng for non-defaulted loans 3. Methodology of LGD modelng Most of the emprcal lterature regardng nfluence factors of LGDs performs lnear regressons and sometmes log or log-log-regressons wth target varable LGD or RR. However, only few studes report out-of-sample tests of the specfed models. 8 Ths s surprsng snce t s essental for banks that the models delver a hgh accuracy of LGD estmates for unobserved data. We fnd that the predctve power of the mentoned approaches s very low for our data set. When analyzng the data n detal, we have found that the characterstcs of recovered loans are often very dfferent from loans that have to be wrttenoff. Especally, the characterstcs that lead to the bnary event recovery vs. wrte-off are often dfferent from the characterstcs nfluencng the LGD wthn the group of wrte-offs. For example, t s obvous that the LGD of wrte-offs s low f the value of collateral s hgh. 8 Ths s also notced by Bastos (00). 8

Contrary, a hgh value of collateral does not necessarly reduce the probablty of a wrte-off. As notced before, reasons for a recovery can be that the oblgor pad the amount that he was n arrears wth payments or a new payment plan has been arranged. However, there s no obvous reason that the probablty of these events should be nfluenced by the value of collateral. Thus, t seems reasonable to explctly account for the dfferences between wrteoffs and recovered loans n the methodology of LGD forecastng. Ptfall : Neglectng dfferences between wrte-offs and recovered loans n LGD forecastng In order to account for the dfferent characterstcs of wrte-offs (WO) and recovered loans (RC), we estmate the LGDs wth a two-step model. As a frst step, we estmate the probablty WO ˆλ of a wrte-off. Accordngly, the probablty of a recovery s ˆ λ ˆ RC = λwo. In the second step, we determne the LGDs for both types of default end separately, whch leads to LGD forecasts LGD WO and LGD RC. Fnally, for each credt, wth =,..., n, these estmates can be combned nto an LGD forecast, whch s gven by ( ) LGD = ˆ λ LGD + ˆ λ LGD. (3) WO, WO, WO, RC, The probablty of a wrte-off WO ˆλ s estmated usng a logstc regresson model: ( { WO }, ) k ˆ E x,..., x = λ = wth z = β + β x, + (4), k, WO, 0, exp( z ) = where { WO } s an ndcator varable, whch equals one f credt s wrtten-off and zero, otherwse. The varables x,,, x k, correspond to k dfferent characterstcs, whch can be borrower, loan or collateral specfc. In cases where t s not possble to develop a model wth suffcent predctve power, the probablty WO ˆλ s set to the hstorcal average wrte-off rate of the respectve subportfolo. In the second step, we perform lnear regressons for estmatng the LGD of loans that have to be wrtten-off: m LGD WO, = γ0 + γ y, =, (5) where y,,, y m, are m dfferent varables, whch can also be borrower, loan or collateral specfc. Snce the LGDs of recovered loans, n contrast to wrte-offs, mostly have only small 9

varatons and these varatons could not be predcted accurately, we assgn the EADweghted hstorcal average LGD for ths type of default end: RC, = N RC, = LGD w LGD,, (6) wth w : = / N EAD EAD = n. Our methodology s related to the modelng approach of Belott n and Crook (007). They apply the followng two-step approach: In the frst step, t s determned whether LGD = 0, LGD =, or 0 < LGD <. 9 In the second step, the case 0 < LGD < s modeled wth lnear regressons. However, n our settng we do not model the fnal outcome of the LGD but the recovery-/wrte-off-probablty. Even f a recovery s often assocated wth very low outcomes of LGD, the event that a loan can be recovered and the outcome LGD = 0 concde only for a part of the data. Moreover, we dd not fnd dfferent characterstcs for defaults wth LGD =. Consequently, we get more reasonable results f the target varable s the type of default end (recovery or wrte-off). The predctve power of the model can be evaluated at dfferent stages. Frst, we evaluate the performance of the logt-model on the bass of the adusted R and the recever operatng characterstc (ROC). The ROC curve plots the senstvty,.e. the true postves, on the ordnate and specfcty,.e. the false postves, on the abscssa. The value for the area under the ROC curve s abbrevated as AUC. Second, the lnear model s evaluated usng the coeffcent of determnaton R. Fnally, n order to assess the total performance of the model, we combne the predctons of the two-step model accordng to (3) and compute the R for the combned forecast. However, the statstc expressng the predctve power can be overestmated when calculated n-sample. Aganst ths background, we evaluate the models on the bass of the out-of-sample statstc. The out-of-sample statstc where R M = OS = M = ( LGD LGD ) ( LGD LGDIS ) R OS s computed as, (7) LGD IS s the average LGD of the n-sample data, LGD (wth =,, M) are the forecasted LGDs calculated out-of-sample (applyng the model whch s based on the nsample data), and LGD are the realzed LGDs of the out-of-sample data. 0 Ths statstc 9 The authors model recovery rates and not LGDs, but due to LGD = RR ths dstncton does not matter. 0 The out-of-sample R statstc s proposed by Campbell/Thompson (008) n context of equty premum predcton. 0

measures the reducton of the mean square predcton error relatve to the average LGD of the n-sample data. If R OS > 0, the forecasts are better than the n-sample average. 3. Comparson of the two-step model and the drect regresson by smulaton The followng statement reveals that the two-step model s superor to a drect LGD regresson. We formulate the statement as a hypothess that has to be tested snce an explct proof s not possble. Hypothess The out-of-sample coeffcent of determnaton (6)) s hgher than R OS, drect of a drect LGD regresson. ROS, two-step of the two-step model (formulas (3)- Test of the Hypothess by smulaton We analyze the performance of the proposed two-step model n comparson to a drect regresson on LGDs on the bass of a smulaton study. Frst, we smulate LGDs for a portfolo of 000 defaulted loans. When generatng LGDs, we use a structure whch ncorporates dfferences between wrte-offs and recovered loans, consstent to our argument and emprcal fndngs. However, we choose a model structure whch dffers from (4) and (5) to nduce some model error. We generate the event of a wrte off f some observable or unobservable nfluence factors x, y,ε lead to an excess of the barrer δ: = : Φ ρ + ρ + ρ ρ ε > δ, (8) { WO }, ( x, x y y x y ) wth x, y, ε (0,) and Ф s the standard normal CDF. Snce the argument of Ф s standard normally dstrbuted, the result Φ () s unformly dstrbuted wth Φ ( ) (0,). In our smulaton, we set δ = 0.8, leadng to a 0% probablty of a wrte-off. Smlarly, we generate the LGDs wthn the group of wrte-offs by LGD =Φ( ρx x + ρz z + ρx ρz ξ) WO,,, (9) wth x, z, ξ (0,). Thus, the LGD s bound between zero and one. Altogether, the outcome of LGD s calculated as LGD = { } LGD WO,, (0) WO, whch mples that the LGD of recoveres s set to zero.

Accordng to our argument above, the event of a wrte-off and the LGD wthn the group of wrte-offs can be nfluenced by dfferent varables. However, some varables can be relevant for both equatons. Aganst ths background, x nfluences both dependent varables but the coeffcents can be dfferent. Contrary, y and z each affect only one of the dependent varables. Moreover, we assume that x, y, z are observable whereas ε and ξ are unobservable random varables. Thus, only x, y, and z are nput varables for the regressons whch are appled subsequently. In order to compare the performance of both modelng approaches, we perform a drect regresson wth target varable LGD on the one hand and apply the two-step model on the other hand. As stated above, we combne the predctons of the two-step model accordng to (3) and compare the out-of-sample R of both modelng approaches wth formula (7). For the out-of-sample analyss, we generate 0,000 addtonal LGDs usng formula (8)-(0). The smulaton procedure from above s performed for a broad range of parameter combnatons. The coeffcents the coeffcents ρ y and z ρ x, and ρ x, are ndependently set to (0., 0.,, 0.9) and ρ are set to (0.,, ρ x,) and (0.,, ρ x, ), respectvely. Ths leads to a total number of,936 dfferent parameter combnatons. For each parameter combnaton, we repeat the smulaton procedure,000 tmes and compare the average n- and out-of-sample R of both models. The mean R OS of the two-step model s 5.% whereas the mean R OS of the drect regresson s only 3.5%, as can be seen n Table. Moreover, the dfference R = R R s postve for each ndvdual parameter combnaton, OS OS, two-step OS, drect whch confrms our hypothess. Thus, the two-step model mpressvely outperforms the drect regresson. - Table about here - The applcaton of our two-step approach to real data s presented subsequently. Due to the known LGD generatng process, we can create an arbtrary number of LGDs for testng the models out-of-sample. Wth an ncreasng number of LGDs the measured predctve power converges towards the true value.

3.3 Applcaton of the two-step model The models for estmatng LGDs are developed wth SAS Enterprse Mner. The models for forecastng the wrte-off probabltes WO ˆλ are estmated usng multvarate logtregressons accordng to (4). Snce the data base s suffcently large, we do not use a k-fold cross-valdaton lke Belott and Crook (007) or Bastos (00) but splt the data nto 70% tranng data (n-sample) and 30% valdaton data (out-of-sample). For many of the used categorcal varables, the out-of-sample performance could be mproved by aggregatng the varables to a smaller number of classes, e.g. usng the varables lmted lablty or unlmted lablty nstead of the concrete legal form of a company. The predctve power of the dfferent logt-models s manly evaluated on the bass of the recever operatng characterstc (ROC) for the valdaton data. The ROC curves for the tranng and for the valdaton data, whch correspond to the model of choce for one of the secured subportfolos, are presented n Fgure 5. The respectve values for the area under the ROC curve are AUCTran = 73.5% and AUCValdate = 7.3%. As a fnal step, the coeffcents of the model are calbrated on the bass of the full data set, leadng to an AUC value of AUCAll = 73.0%. The explanatory varables, whch are used n the models, are borrower characterstcs (e.g. the lablty of a company for commercal clents or occupatonal category and martal status for prvate customers), collateral characterstcs, and loan characterstcs (e.g. the prevous number of defaults and the collateralzaton level). 3 Interestngly, for unsecured loans t was possble to develop a model where the explanatory power s sgnfcantly hgher, wth AUCTran = 8.6% and AUCValdate = 8.% (cf. Fgure 6). - Fgure 5 about here - - Fgure 6 about here - Smlarly, we develop the lnear regresson models for estmatng LGDs n the scenaro of a wrte-off. Thus, we splt the data set of contracts whch had to be wrtten-off nto tranng and valdaton data and perform multvarate lnear regressons. The predctve power of the Interestngly, when checkng the economcal plausblty,.e. the concordance wth the workng hypotheses, the ROC curves for the tranng and the valdaton data generally become more smlar f varables wth mplausble coeffcents are dropped, resultng n a reduced performance for the tranng data but an ncreased predctve power for the valdaton data. 3 The publcaton of the concrete model ncludng the coeffcents s prohbted by the bank. 3

models s manly evaluated wth the coeffcent of determnaton for the valdaton data R Valdate applyng formula (7). For secured loans to prvate customers, the coeffcents of determnaton for the selected model are R Tran = 9.9% and R Valdate = 7.6%. 4 The fnal coeffcents are calbrated on the complete data set leadng to R All = 9.3%. Agan, the explanatory varables can be classfed nto borrower characterstcs (e.g. the occupatonal category for prvate customers), collateral characterstcs (e.g. type and value of collateral), and loan characterstcs (e.g. /EAD or down payment/ead). Remarkably, when developng LGD models for unsecured loans to prvate customers, the predctve power of wrte-off LGDs was so low that the (exposure-weghted) average wrte-off LGD s assgned n ths scenaro. Thus, we fnd that for secured loans to prvate customers the accuracy when predctng wrte-off probabltes s lower than for unsecured loans, but wthn the second step, the predcton of LGDs n the case of wrte-offs, the opposte s true. 4 LGD forecastng for defaulted loans For defaulted loans, the parameters PD and EAD are realzed values but the LGD s stll a random varable. However, we have some addtonal nformaton about the loan whch can be used for LGD forecastng. Especally, we have knowledge about the default reason and the current length of the default perod: The concrete events whch characterze the default of a loan vary from bank to bank. Some typcal reasons are () the oblgor s past due for more than 90 days, () a notce of cancellaton, (3) a court order, or (4) a sgnfcant downgradng. We fnd that the average LGD vares sgnfcantly dependng on dfferent default reasons. For example, defaults wth default reason (beng past due) on average lead to smaller losses than defaults wth default reason (notce of cancellaton). Furthermore, the average LGD of contracts wth a long default perod s usually hgher than the LGD of contracts wth a short default perod. A part of ths effect stems from the on average dfferent default perods of loans that can be recovered and loans that have to be wrtten off (cf. secton ). Addtonally, even wthn the wrte-offs, the LGDs are mostly hgher for contracts wth a long default perod. 4 After transformng the LGD estmates usng ( = ) Loss LGD EAD, t s also possble to evaluate the predctve power wth respect to absolute nstead of relatve losses. Ths leads to coeffcents of determnaton of 5.3% and 57.7%, respectvely. 4

In order to analyze whch factors are most mportant for explanng the LGD of defaulted loans, we use regresson trees wth the software SAS Enterprse Mner. 5 Regresson trees are a nonlnear and nonparametrc predctve modelng tool, whch splts the data nto several groups on the bass of a seres of bnary questons, e.g. default reason =? and default perod > 00 days?. These questons are set n a way that the nformaton about the LGD s maxmzed. 6 As notced by Bastos (00), regresson trees are well-suted for producng accurate results of LGD forecasts usng only a few mportant explanatory varables. We fnd for dfferent subportfolos that the most mportant explanatory varables are the default reason, the length of the default perod, and some segmentaton varables regardng the type of oblgor, loan, and collateral. However, we have to consder the dfferent set of nformaton about the default length of contracts wth actve and completed workout process. For modelng purposes, we have knowledge of the total length of the workout process. Contrary, when applyng the model to actve defaults, we only know the current default length, whch s obvously smaller than the total length T. In Proposton, we show that gnorng the dfference between the nformaton sets would lead to a sgnfcant underestmaton of the LGD. Furthermore, we present a consstent estmator usng the nformaton of the current default length. Ptfall 3: Underestmaton of LGDs when usng the total length of the default perod as explanatory varable Proposton 7 Let the assumptons of Proposton be fulflled and let CDL denote the current default length of loan. Furthermore, consder a sequence of loans denoted by =,,, whereby ( LGD { I T > t }) s a sequence of ndependently and dentcally dstrbuted random varables, each member of the sequence wth expectaton value E( LGD { I T > t }). 8 Furthermore, ( IT { > t }) s a sequence of ndependently and dentcally dstrbuted random varables, each member of the sequence wth expectaton value EIT ( { > t }) Fnally, the 5 The frst publshed study whch models LGDs wth regresson trees s Bastos (00). However, we apply regresson trees to forecast LGDs of defaulted nstead of non-defaulted loans. 6 For detals see Breman (984). 7 The proof of the proposton s presented n Appendx B. 8 IT { > t} takes the value one f the argument s true and zero otherwse. 5

correspondng exposures at default EAD, EAD, are assumed to be determnstc and to fulfll the followng condtons: (a) N EAD, (b), N = = Then the followng statements hold: (I) P( LGD = ) ( x CDL t P LGD x T = t ). Var( LGD { I T > t}) Var( I{ T > }) <, and (c) t <. = EADk EADk k = k = (II) N EAD LGD { I T > t} =.. N = EAD I{ T > t} as E ( LGD > ). N T t If we model LGDs usng the default length as explanatory varable and gnore the dfferent nformaton sets of the default length for the modelng and scorng data, the LGDs are underestmated as shown n Proposton (I). However, snce the length of the default perod has a hgh explanatory power for LGDs, we ntend to use the known nformaton set. The nformaton that the current default length equals t for the scorng data s dentcal to the nformaton that the total length of the default perod T s larger than t. Though, for the modelng data we can calculate the (EAD-weghted) average LGDs for all contracts wth T > t. If we proceed so for every value of t [0, T max ], we can assgn LGDs to every defaulted loan usng the nformaton of the current default length and, as shown n Proposton (II), get consstent LGDs when we apply the model. Snce these LGDs are calculated on the bass of modelng data wth a mnmum default length (MDL) of t, we call the correspondng values LGD(MDL = t). Though, we want to nclude addtonal nfluence factors,.e. the mentoned segmentaton varables and the default reason. Aganst ths background, we frst partton our modelng data nto classes whch are homogeneous regardng these varables and calculate LGD(MDL = t) for every class. Under consderaton of LGD Default, : = E( LGD CDL = t) = E( LGD T > t ) and due to Proposton (II), we are able to defne an estmator of LGD Default, as follows: N EAD LGD I{ T > t} = LGDDefault, = = LGD ( MDL = t) N = EAD I{ T > t} :, () 6

where N and =,, N stands for all contracts of our modelng data wthn a class. However, for large values of MDL, we set the LGD to a constant value n order to reduce the estmaton error resultng from the small number of observatons. Moreover, snce the emprcal LGDs exhbt some economcally mplausble umps or non-monotonous sectons, we descrbe the rest of the functon pecewse wth polynomal functons. Graphcal llustratons of the emprcal LGDs resultng from equaton (), whch correspond to one of the segments, are presented n Fgure 7. - Fgure 7 about here - There are some characterstcs of the llustratons worth mentonng. Frst, default reasons and 3 are aggregated snce one of these categores s usually almost empty dependng on whether the collateral has already been lqudated n a prevous default or not. 9 Second, for most contracts wth default reason,, or 3, the LGD ncreases wth the default length. Thrd, the average LGD of contracts wth default reason 4 decreases for small values of MDL and has a ump at MDL = 365 days. To understand ths effect, we have to consder that default reason 4 means a sgnfcant downgradng. Banks often retreve addtonal scorng nformaton from credt agences. In the presented case of retal loans, the values of the negatve scorng characterstcs are updated one year after default. If the negatve scorng characterstc s no longer exstent and f ths s the only actve default reason at ths tme, a loan recovers, leadng to a small LGD. Ths effect was already vsble n Fgure 4, where we could observe a small peak of recovered loans for a default length of 365 days. However, f default reason 4 s stll exstent, the probablty of a wrte-off s qute hgh. Thus, the LGD has a ump at a mnmum default length of one year. 5 Concluson In ths paper, we dentfy relevant ptfalls n modelng workout LGDs whch can easly lead to naccurate LGD forecasts. Furthermore, we propose methods how to deal wth these ptfalls and apply these methods to a data set of 7,463 defaulted loans of a German bank. Frst, the LGDs wthn the modelng data can be sgnfcantly based downwards f all avalable defaults wth completed workout process are consdered. Ths s manly due to length-based samplng n connecton wth a dfferent default length of recovered loans and 9 Durng the default perod, the default status can change, e.g. from to 3. However, the default reason remans unchanged. 7

wrte-offs. We show how the modelng data could be chosen n order to get unbased LGD estmates. Second, we propose a two-step approach for modelng LGDs of non-defaulted loans. Wth ths approach, we could acheve better predctons than wth other approaches proposed n the lterature, snce dfferent nfluencng factors of recoveres and wrte-offs can be consdered. We demonstrate the potental of ths approach on the bass of a smulaton study and apply the model to the data set. Thrd, we propose a model to forecast LGDs of defaulted loans on the bass of regresson trees. We fnd that both the type of default end and the default length have a hgh explanatory power when forecastng those LGDs. Snce the actual default length of scorng data and the total default length of the modelng data nclude dfferent nformaton sets of the default length, the LGDs are sgnfcantly underestmated when ths dfference s gnored. However, neglectng ths nfluence factor leads to consderable worse predctons. Aganst ths background, we have constructed the varable mnmum default length for the modelng data, whch contans the same nformaton set as the current default length of the scorng data, leadng to consstent LGD estmates. Another nterestng fndng s that the predctve power for estmatng the probablty of a recovery or a wrte-off s hgher for unsecured than for secured loans. Contrary, for the predctons of LGDs condtonal on the type of default end the opposte s true. However, t would be nterestng to verfy that ths observaton s generally vald and not specfc to the used data set. Moreover, whle we manly focused on retal loans, our models could also be benefcal for the predcton of LGDs of corporate loans. Ths s left for further research. References Acharya, V.V., Bharath, S.T., Srnvasan, A., 007. Does ndustry wde dstress affect defaulted frms? Evdence from credtor recoveres. Journal of Fnancal Economcs 85, 787 8. Altman, E.I., Brady, B., Rest, A., Sron, A., 005. The lnk between default and recovery rates: Theory, emprcal evdence, and mplcatons. Journal of Busness 78, 03 8. Araten, M., Jacobs Jr., M., Varshney, P., 004. Measurng LGD on commercal loans: An 8- year nternal study. The RMA Journal 4, 96 03. Asarnow, E., Edwards, D., 995. Measurng loss on defaulted bank loans. A 4-year-study. Journal of Commercal Lendng 77(7), 3. Bade, B., Rösch, D., Scheule, H., 0. Default and recovery rsk dependences n a smple credt rsk model. European Fnancal Management 7, 0 44. 8

Basel Commttee on Bankng Supervson, 005a. Gudance on paragraph 468 of the framework document, Bank for Internatonal Settlements. Basel Commttee on Bankng Supervson, 005b. Internatonal convergence of captal measurement and captal standards a revsed framework, Bank for Internatonal Settlements. Bastos, J.A., 00. Forecastng bank loans loss-gven-default. Journal of Bankng and Fnance 34, 50 57. Bellott, T., Crook, J., 007. Modellng and predctng loss gven default for credt cards. Workng paper, Quanttatve Fnancal Rsk Management Centre. Breman, L., Fredman, J.H., Olshen, R.A., Stone, C.J., 984. Classfcaton and Regresson Trees. Wadsworth: Belmont, CA. Calabrese, R., Zenga, M., 00. Bank loan recovery rates: Measurng and nonparametrc densty estmaton. Journal of Bankng and Fnance 34, 903 9. Campbell, J.Y., Thompson, S.B., 008. Predctng Excess Stock Returns Out of Sample: Can Anythng Beat the Hstorcal Average? Revew of Fnancal Studes, 509 53. Casell, S., Gatt, S., Querc, F., 008. The senstvty of the loss gven default rate to systematc rsk: new emprcal evdence on bank loans. Journal of Fnancal Servces Research 34, 34. Ctron, D., Wrght, M., Ball, R., Rppngton, F., 003. Secured Credtor Recovery Rates from Management Buy-Outs n Dstress. European Fnancal Management 9, 4 6. Dermne, J., Neto de Carvalho, C., 006. Bank loan losses-gven-default: a case study. Journal of Bankng and Fnance 30, 43 9. Eales, R., Bosworth, E., 998. Severty of loss n the event of default n small busness and larger consumer loans. The Journal of Lendng and Credt Rsk Management, 58 65. Felsovaly, A., Hurt, L., 998. Measurng loss on Latn Amercan defaulted bank loans: A 7- year study of 7 countres. Journal of Lendng and Credt Rsk Management. Franks, J., de Servgny, A., Davydenko, S., 004. A comparatve analyss of the recovery process and recovery rates for prvate companes n the UK, France, and Germany. Standard and Poor s Rsk Solutons, June 004. Frye, J., 000. Collateral Damage. Rsk 3(4), 9 94. Gordy, M.B., 003. A Rsk-Factor Model Foundaton for Ratng-Based Captal Rules. Journal of Fnancal Intermedaton (3), 99 3. Grunert, J., Weber, M., 009. Recovery rates of commercal lendng: emprcal evdence for German companes. Journal of Bankng and Fnance 33, 505 53. 9

Jankowtsch, R., Pullrsch, R., Veža, T., 008. The delvery opton n credt default swaps. Journal of Bankng and Fnance 3, 69 85. Kefer, N.M., 988. Economc duraton data and hazard functons. Journal of Economc Lterature 6, 649 679. Maclachlan, I., 005. Choosng the dscount factor for estmatng economc LGD. In: Altman, E., Rest, A., Sron, A. (Eds.), Recovery Rsk: The Next Challenge n Credt Rsk Management. Rsk Books: London. Petrov, V., 996. Lmt Theorems of Probablty Theory: Sequences of Independent Random Varables. Oxford Unversty Press: Clarendon. Renault, O., Scallet, O., 004. On the way to recovery: A nonparametrc bas-free estmaton of recovery rates denstes. Journal of Bankng and Fnance 8, 95 93. Schuermann, T., 006. What Do We Know About Loss Gven Default? In: Shmko, D. (Ed.), Credt Rsk Models and Management, nd Edton. Rsk Books: London. 0

FIGURES AND TABLES Fgure Frequency dstrbuton of loss gven default of secured loans of prvate clents I

FIGURES AND TABLES Fgure Frequency dstrbuton of loss gven default for recovered loans (top) and for wrte-offs (bottom) II

FIGURES AND TABLES Fgure 3 Interval censored data: Defaults wth default begn and default end wthn the data observaton perod,.e. completed workout process, are avalable n the data base (sold lnes), other defaults are not ncluded n the data base (dashed lnes) III

FIGURES AND TABLES Fgure 4 Length of the default perod for recovered loans (top) and for wrte-offs (bottom) n days IV

FIGURES AND TABLES Fgure 5 Recever operatng characterstc when forecastng wrte-off probabltes for the tranng (left) and valdaton data (rght) of a secured subportfolo V

FIGURES AND TABLES Fgure 6 Recever operatng characterstc when forecastng wrte-off probabltes for the tranng (left) and valdaton data (rght) of an unsecured subportfolo VI

FIGURES AND TABLES Fgure 7 EAD-weghted LGDs (damonds) and number of contracts (sold lne) for default reason : beng past due (top), default reason & 3: notce of cancellaton & court order (mddle), and default reason 4: sgnfcant downgradng (bottom) dependng on the mnmum default length (n days) VII

FIGURES AND TABLES Fgure 7 (contnued) VIII

FIGURES AND TABLES Table Summary statstcs Number of defaults Prvate clents 6,860 Commercal clents 8,5 Secured loans 67,40 Unsecured loans,575 Mean Std. Dev. Medan Exposure at default ( ) 9,39.34 7,563.85 7,57.5 Collateralzaton level of secured loans.04.5 0.68 IX

FIGURES AND TABLES Table The table shows statstcs for the R on the bass of,000 smulaton runs for each,936 dfferent parameter combnatons. The n- and out-of-sample R s calculated for the two-step model and the drect regresson. Obs. Mean Std. Dev. Mn. Max. R IS,two-step,936 0.590 0.3 0. 0.997 R IS,drect,936 0.346 0.077 0.07 0.506 R OS,two-step,936 0.584 0. 0.7 0.99 R OS,drect,936 0.34 0.078 0.0 0.504 R = R R,936 0.44 0.68 0.05 0.807 IS IS,two-step IS,drect R = R R,936 0.4 0.66 0.05 0.77 OS OS,two-step OS,drect X

APPENDIX Appendx A. Proof of Proposton Ad (I): Frst of all, the random varable LGD T > t has strct frst-order stochastc domnance over LGD T t for all t snce P( LGD ) = [ ( = ) ] > ( x T t Eθ P LGD x T θ θ t P LGD x T > t). () On ths bass we get > P( LGD x T > t) P( LGD ) = ( ) ( x P LGD x T τ τ P T τ τ) + P( LGD > ) ( x T > τ τ P T τ τ) P( LGD x T τ τ) = P( LGD x τ τ T τ τ ). The nequalty results from the statement that LGD T > τ τ strctly domnates LGD T τ τ accordng to frst order stochastc domnance, and the latter equalty results from the stochastc ndependence of τ to LGD and T. (3) Ad (II): Snce τ s ndependent of LGD, and T < max τ τ, t mmedately follows that P( LGD x) = P( LGD x τ τ τ T ). (4) Furthermore, snce τ s addtonally ndependent of T, and T Tmax, we have P( LGD ) = ( x P LGD x T Tmax ) = P( LGD x T T τ + T T τ τ T ). max max max (5) XI

APPENDIX Appendx B. Proof of Proposton Ad (I): For all t the (condtonal) random varable LGD T > t s assumed to have strct frst-order stochastc domnance over LGD T = t (cf. secton ). Thus, t mmedately follows: P( LGD = ) = ( > ) ( x CDL t P LGD x T t P LGD x T = t ). (6) Ad (II): By defnton we have ( ) E LGD { I T > t} E ( LGD T > t) =. EIT ( { > t}) Furthermore, under consderaton of the assumptons wth regard to the sequences ( LGD { I T > t }) and ( IT { t }), we are able to apply the strong law for weghted > averages as presented n Petrov (996), Theorem 6.7, 0 accordng to whch (7) and N k = EAD k N N (.. { > } { as EAD LGD I T t EAD E LGD I T > t} ) 0 N = = (8) N k = EAD k N N.. { > } ( { as EAD I T t EAD E I T > t} ) 0. (9) N = = Snce E( LGD { > }) = ( { I T t E LGD I T > t }) and EIT ( { > t}) = EIT ( { > t }) for all, the almost sure convergences n (8) and (9) lead to and N k = EAD k N.. { } as EAD > { LGD I T t E LGD > } N I T t = ( ) (0) N k = EAD k N EAD I T > t E I T > t = ( ) as.. { } { }. N (0) and () together wth (7) mmedately mply the statement of part (II). () 0 See Gordy 003, p. 3, for a smlar applcaton of the Theorem. XII