Loss given default modeling: a comparative analysis

Transcription

1 Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213 (25 59) Loss gven default modelng: a comparatve analyss Olga Yashkr Yashkr Consultng, Wood Street, Toronto, M4Y 2P9, Canada; emal: olga.yashkr@gmal.com Yur Yashkr Yashkr Consultng, Wood Street, Toronto, M4Y 2P9, Canada; emal: yur.yashkr@gmal.com (Receved September 25, 212; revsed January 9, 213; accepted February 8, 213) In ths study we nvestgated several of the most popular loss gven default (LGD) models (least-squares method, Tobt, three-tered Tobt, beta regresson, nflated beta regresson, censored gamma regresson) n order to compare ther performance. We show that for a gven nput data set the qualty of the model calbraton depends manly on the proper choce (and avalablty) of explanatory varables (model factors), but not on the fttng model. Model factors were chosen based on the ampltude of ther correlaton wth hstorcal LGDs of the calbraton data set. Numercal values of nonquanttatve parameters (ndustry, rankng, type of collateral) were ntroduced as ther LGD average. We show that dfferent debt nstruments depend on dfferent sets of model factors (from three factors for revolvng credt or for subordnated to eght factors for senor secured ). Calbraton of LGD models usng dstressed busness cycle perods provde better ft than data from total avalable tme span. Calbraton algorthms and detals of ther realzaton usng the R statstcal package are presented. We demonstrate how LGD models can be used for stress testng. The results of ths study can be of use to rsk managers concerned wth complance wth the Basel Accord. 1 INTRODUCTION The goal of loss gven default (LGD) modelng s to produce smulated LGDs close to and correlated wth hstorcal LGDs. Dffcultes wth modelng depend drectly on the specfcs of the data used and on the lmtatons of the models. In recent years, the mportance of modelng LGD has ncreased sgnfcantly. The development, calbraton and mplementaton strateges of LGD modelng have been analyzed and summarzed n several publcatons (Gupton 25; Schuermann 24). 25

2 26 O. Yashkr and Y. Yashkr The predctve power of any LGD model depends, frst of all, on the proper choce (and avalablty) of the model nput parameters obtaned from oblgor nformaton. These (predctve) varables were analyzed and used for LGD model calbraton n many publcatons. For example, the nne-factor model was analyzed n Gupton and Sten (22), the survey of LGD model factors s presented n Fredman and Sandow (23). A case study of the modelng of bank loan LGDs where the prmary factors (the perod of loan orgnaton, qualty of the collateral, the loan sze and the length of the relatonshp wth the oblgor) were dentfed s gven n Chalupka and Kopecsn (29). The lnk between default and recovery rates was nvestgated n Altman et al (23, 24) and Altman (26, 21). The ncorporaton of the dependence between probabltes of default and recovery rates nvestgated by Bade et al (211) demonstrated some mprovement of the LGD model. A sgnfcant mpact of the uncertanty of model parameters on estmated LGDs was demonstrated by Luo and Shevchenko (21). The nfluence of the length of the LGD workout process on the level of estmated LGD can be sgnfcant, as shown n Gurtler and Hbbeln (211). The LGD models based on the lnear regresson approach can be found n McDonald and Mofftt (198) (the Tobt model), Huang and Oosterlee (212) (beta regresson model), Perera and Crbar-Neto (21) (nflated beta regresson), Sgrst and Stahel (21) and Sgrst and Stahel (211) (censored gamma regresson). Altman and Kalotay (21) used a mxture of dstrbutons to model LGD. The beta-component mxture for modelng LGD and credt default swap rates as model varables was successfully used n Baxaul and Alvarez (21). The bmodal structure of the LGD dstrbuton was modeled by a mxture of two beta dstrbutons n Hlawatsch and Ostrowsk (211). The LGD model n the Merton-structured credt rsk framework was also nvestgated n Jacobs (211). The portfolo credt rsk model dependent on LGD was developed n Hllebrand (26) and compared wth several alternatve LGD models. Calbraton methods for LGD models appled to mortgage markets can be found n van der Weja and den Hollandera (29). The results n Bellott and Crook (212) contan comparson of several models (Tobt, decson-tree model, beta transformaton, fractonal logt and the least-squares method). Bellott and Crook (212) demonstrated the mportance of the ncluson of macroeconomc condtons (nterest rates, unemployment levels and earnng ndex) for LGD model stress testng. Yang and Tkachenko (212) propose some emprcal approaches for exposure at default/loss gven default modelng and provde techncal nsghts nto ther mplementaton. Valdaton technques and performance metrcs for loss gven default models were ntroduced by L et al (29). An attempt to develop analytc formulas for downturn LGD estmaton was made by Barco (27). The downturn LGDs were consdered as a one-n-a-thousand-years event takng nto account correlated probablty of default and LGD. Rösch and Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213

3 Loss gven default modelng: a comparatve analyss 27 Scheule (27) developed a framework to stress senstvtes of rsk drvers, and therefore the credt portfolo losses. Gven the results of all the above research publcatons, the man queston for a practtoner remans: what s the best model for LGD estmaton? The goals of our research are to provde comparatve tests of popular LGD estmaton models, to analyze ther performance, to calbrate the models on dfferent data subsets and to provde recommendatons on how test results can be used for the stress testng of LGDs. We do not nclude data manpulaton technques. Based on some examples, we show how models can become senstve to the choce of data. 2 METHODOLOGIES The LGD models analyzed and compared n ths paper are based on several dfferent lnear regresson algorthms. A short summary of the models s gven n ths secton. 2.1 Censored least-squares method Gven known hstorcal LGD values LGD, coeffcents x k are derved usng the least-squares method (LSM) by mnmzng the followng object functon: X mn.y.x; r/ LGD x /2 ; (2.1) wth y.x; r/ D x C nx x k r k ; (2.2) where r k s the kth predctve parameter for the th counterparty, x k s the coeffcent for the kth predctve parameter and x s a constant ( ntercept ). The LGD for debt faclty s estmated as kd1 LGD.x; r/ D maxœ; mnœ1; y.x; r/ : (2.3) 2.2 Censored lnear regresson (Tobt) model The Tobt LGD model s based on the latent loss parameter z for each debt faclty (see McDonald and Mofftt 198): z D y C./: (2.4) Here./ s a normally dstrbuted random drver wth standard devaton, y.x; r/ s the lnear combnaton of explanatory varables r k as n (2.2). Research Paper

4 28 O. Yashkr and Y. Yashkr The latent loss varable z s a normally dstrbuted random value wth expected value of y and standard devaton of. Therefore, the probablty of realzaton of LGD D s can be expressed through the standard Gaussan functon g./ as follows: 1 p 2 exp 1 s 2 y 1 s 2 g y : (2.5) Assumng that all s 6 correspond to LGD D and that all s > 1 correspond to LGD D 1, we can defne the probablty functon P.x/ for LGD as follows: 8 P./ f x 6 ; ˆ< P.x/ D 1 x g y f <x<1; (2.6) ˆ: P.1/ f x > 1; where P./ P.1/ D 1 D 1 Z 1 Z 1 1 z y g z y g dz D N dz D N y ; (2.7) 1 y ; (2.8) where N./ s a standard normal cumulatve dstrbuton functon. The probablty functon (2.6) can be also presented n a more convenent form: P.z/ D P./ ı.z/ C P.1/ ı.z 1/ C 1 z g y.1 ı.z/ ı.z 1//; (2.9) where ı./ s the standard delta functon. Note that the probablty (2.6) s also the functon of model coeffcents x k (k D ;:::;n) and of the LGD volatlty ; these are the subjects of the model calbraton. The expected LGD for the th debt faclty s calculated as The result s EŒLGD D P.1/ C.1 P./ EŒLGD D P.1/ Z 1 /y C zp.z/ dz: g y g 1 y : (2.1) Because Q.LGD / D Z LGD P.z/ dz; Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213

5 Loss gven default modelng: a comparatve analyss 29 the cumulatve LGD probablty can be calculated usng (2.6): 8 P ˆ<./ f LGD D ; LGD y Q.LGD/ D N f <LGD <1; ˆ: 1 f LGD D 1: (2.11) The th percentle of the modeled LGDs (LGD./ ) s the soluton of the equaton Q.LGD/ D : 8 f <P./ LGD./ ˆ< D y C N 1./ f P./ <<1 P.1/ ˆ: 1 f >1 P.1/ ; : ; (2.12) The calbraton of the Tobt model conssts n fndng model coeffcents x k (k D ;:::;n) and LGD volatlty by best ft of the model (wth predctve parameters r k ) to hstorcal data LGD (k D 1;:::;n and D 1;:::;J). If we consder the nput data sample as a set of ndependent measurements, then the best model ft s obtaned by maxmzng the total probablty of gettng the nput data set O P.x;/D JY P.LGD /; (2.13) whch s equvalent to mnmzaton of the followng objectve functon: D1.x;/D JX log P.LGD /: (2.14) D1 For numercal optmzaton we employ the Broyden Fletcher Goldfarb Shanno (BFGS) method (for solvng nonlnear optmzaton problems wthout constrants). The lnear dependence of the functon (2.2) on explanatory varables r kj may not be suffcent to descrbe the cause effect lnk of LGD j to r kj. It s possble to ncrease the flexblty of the model by ncludng a quadratc term, such that y.x; r/ D x C nx.x 2k 1 r k C x 2k r 2 k /: (2.15) kd1 Note that the number of model coeffcents for the nonlnear Tobt model s 2n C Censored lnear regresson three-tered Tobt model Snce processes causng LGD to be zeros or ones may have a dfferent nature from processes where <LGD <1, we ntroduce n ths secton a three-tered model. Research Paper

6 3 O. Yashkr and Y. Yashkr The LGD estmator s ntroduced n ths case n the followng lnear form: y./.x; r/ D y.c/.x; r/ D y.1/.x; r/ D nx x k r k C x nc2 ; (2.16) kd1 nx x kcnc2 r k C x 2nC4 ; (2.17) kd1 nx x kc2nc4 r k C x 3nC6 ; (2.18) kd1 where the components of the model coeffcent vector x have the followng meanng: 8 coeffcents for the LGD D model; ˆ< x.nc3/.2nc2/ coeffcents for modelng <LGD <1; x D for the <LGD <1model; ˆ: x 1;:::;n x nc1 x nc2 x 2nC3 x 2nC4 x.2nc5/.3nc4/ x 3nC5 x 3nC6 for the LGD D model; ntercept for the LGD D model; ntercept for the <LGD <1model; coeffcents for the LGD D 1 model; 1 for the LGD D 1 model; ntercept for the LGD D 1 model: (2.19) Ths LGD model s based on the followng probablty functon for an th faclty: 8 p./ f z D ; ˆ< z y.c/ P.z/ D g f <z<1; (2.2) ˆ: p.1/ f z D 1; where p./ p.1/ D N y./ D N D N y./ j ; 1 y.1/ 1 N ; Here s a normalzaton factor. 1 y.1/ C N 1 1 y.c/ N y.c/ 9 >= 1 : >; (2.21) Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213

7 Loss gven default modelng: a comparatve analyss 31 The model coeffcents x can be found as a result of the maxmzaton of the followng loglkelhood functon: H.x/ D X log P.LGD /: (2.22) Usng calbrated model coeffcents, we can estmate expected LGDs: LGD D p.1/ C y.c/ N 1 y.c/ N 2.4 Inflated beta regresson model C N y.c/ y.c/ N 1 y.c/ : (2.23) The nflated beta regresson LGD model (Perera and Crbar-Neto 21) s based on the followng probablty functon for an th faclty: 8 ˆ< P f z D ; P.z/ D.1 P ˆ: P 1 /f.zi j ; / f <z<1; (2.24) P1 f z D 1; where f.zi j ; / D. /. /..1 / / z 1.1 z/.1 / 1 (2.25) wth < <1and >( beng the mean value). Gven P, P 1, and, we can calculate (usng (2.24)) the probabltes P.LGD / of obtanng LGD values. In order to establsh a connecton between explanatory varables r of debt facltes and expected faclty LGDs, the followng k four lnear predctors are ntroduced: P log D a 1 P C P 1 log D a 1 P1 1 C log log j 1 j D a C 1 D a C nx x k r k ; (2.26) kd1 nx x 1 k r k ; (2.27) kd1 nx x k r j k ; (2.28) kd1 nx x k r k : (2.29) kd1 Research Paper

8 32 O. Yashkr and Y. Yashkr Here vectors x.;1;;/ are model coeffcents and a.;1;;/ are the ntercepts. The model calbraton conssts of fndng coeffcents and ntercepts by maxmzng the followng loglkelhood (objectve) functon: H D X log.p.lgd //: (2.3) Usng the calbrated model, we can estmate the expected LGD for an th debt faclty: Z 1 EŒLGD D zp.z/ dz D P1 C 1 P P 1.1 C exp..a C P n kd1 x k r (2.31) ///: k The loglkelhood functon (2.3) can be splt as follows: H D H 1 C Hˇ : (2.32) Here H 1.x./ ; x.1/ / and Hˇ.x./ ; x./ / can be optmzed ndependently: H 1 D Hˇ D LGD D X C <LGD <1 X log.p.x./ // C <LGD <1 X LGD j D1 X log.p 1.x.1/ // log.1 P.x./ / P 1.x.1/ //; (2.33) log f.lgd I x./ ; x./ /: (2.34) The nflated beta regresson model was tested usng the R-coded functon developed by Yashkr Consultng. 2.5 Beta lnear regresson model If < LGD < 1, then P D P 1 D, reducng the problem to a general beta regresson model (2.34). 1 Ths model can be also used f all LGDs are scaled as LGD D LGD.ˇ / C, calbraton s performed usng LGD, and values of LGD est (estmated on the bass of ths calbraton) are scaled back as LGD est D.LGD est /=.ˇ /. The beta regresson model was tested usng the BetaReg lbrary functon of the R statstcal package. 1 We can replace, for example, LGD s D wth LGD s D and LGD s D 1 wth LGD s D 1, where 1. Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213

9 2.6 Censored gamma lnear regresson model Loss gven default modelng: a comparatve analyss 33 The censored gamma LGD model (Sgrst and Stahel 21) s based on the followng probablty functon for an th faclty: 8 ˆ<.; ; / f z D ; P j.zi ; ; / D.z C ; ; / f <z<1; (2.35) ˆ: 1.1C ; ; / f z D 1; where.ui ; / D 1. / u 1 e u=.ui ; / D Z u.xi ; / dx (gamma dstrbuton); 9 >= (cumulatve gamma dstrbuton); >; (2.36) wth u>, >and >. Gven, and, usng (2.35) we can calculate the probabltes P.LGD / of obtanng LGD values. In order to establsh a connecton between explanatory varables r of debt facltes and expected faclty LGDs, the followng lnear predctors k are ntroduced: 9 log D ; log D ; log D x C nx x k r k : >; kd1 >= (2.37) Here x k are model coeffcents (ncludng the ntercept x ). The model calbraton nvolves fndng coeffcents and parameters, by maxmzng the followng loglkelhood (objectve) functon: H. ; ; x/ D X log P.LGD I ; ; /: (2.38) Usng the calbrated model, we can estmate the expected LGD for an th debt faclty: EŒLGD D Z 1 zp j.zi ; ; / dz D..1 C ;1 C ; /.;1C ; // C.1 C /.1.1C ; ; //.1.; ; j //: (2.39) The censored gamma regresson model was tested usng the R-coded functon developed by Yashkr Consultng. Research Paper

10 34 O. Yashkr and Y. Yashkr FIGURE 1 Total debt outstandng and total number of defaults. Total debt outstandng (US$ bn) Number of defaults 3 DATA, EXPLANATORY VARIABLES AND CORRELATION ANALYSIS 3.1 used for LGD model calbraton The data set All represents all avalable data n an nternal or an external database used for the LGD model development and calbraton. In our analyss, All s the S&P LossStats data (211 update, 4275 cases) of defaulted facltes. The Peaks s the LGD data related to the tme perods of the busness cycle when the number of defaults and losses s sgnfcantly hgher than the average default and losses values. We chose years as Peak 1, years 21 2 as Peak 2 and years 28 9 as Peak 3. All three peaks have dstnctly hgh levels of defaults and losses (shown n Fgure 1, based on the recent report from S&P (Standard & Poor s 212)). Durng the peaks of the cycle, global market and credt condtons are dfferent from durng the quet perods of the cycle, and therefore the most mportant predctve factors are correlated at a hgher level wth the hstorcal LGD data collected for these tme perods. Methods dscussed n ths paper were tested on four data sets: All, Bankruptcy, Peaks and Bankruptcy Peaks. All represents all Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213

11 Loss gven default modelng: a comparatve analyss 35 avalable LGD data n the data set. The Bankruptcy ncludes only bankruptcy data cases from All. We have consdered and have used separately All and Bankruptcy. The Bankruptcy Peaks represents the bankruptcy data from the peak perods. Explanatory varables/factors The explanatory varables were chosen based on how they are correlated wth the LGDs based on the collected hstorcal data. The proper choce of data and nstrument types are very mportant for good performance of the models, therefore, the model calbraton was tested for several groups based on nstrument types. The followng man fve factors (explanatory varables) were used: Rankng (defnes rank n the captal structure; the more senor the nstrument, the hgher the recovery rate), Debt Cushon (amount per percentage of debt below a defaulted nstrument), Prncpal Above (amount of debt above a defaulted nstrument), Effectve Interest Rate (prepetton rate at the tme the last coupon was pad) and Spread. We ntroduce also three addtonal factors (dependence on ndustry, on the type of collateral, and on the faclty rankng): Industry Mean LGD, Collateral Mean LGD and Rankng Mean LGD. The choce of addtonal factors makes the model dependent on ndustry, collateral type, and rankng, for whch no numercal predctve parameters are avalable. These addtonal factors were calculated as the mean of all LGD values for a gven ndustry, for a gven collateral type, and for a gven rankng. For example, Industry Mean LGD s the mean value of all LGDs for the cases related to a specfed ndustry. Ths value s added as an addtonal factor to all cases belongng to the specfed ndustry. The same was done for Collateral Mean LGD and Rankng Mean LGD. The Collateral Mean LGD depends on the type of the collateral and t defnes the mean of all cases for ths type of collateral. Correlaton analyss To dentfy factors that affect LGDs the most, ther correlatons were nvestgated and the results are presented n Table 1 on the next page and Table 2 on page 37 (where correlaton coeffcents wth absolute values exceedng 1% are n bold). If all nstrument types are consdered together, the correlaton between nstrument type mean LGDs and all LGDs s equal to :513. In the case of Peaks, f all nstrument types are consdered together, the correlaton between nstrument type mean LGDs and all LGDs s equal to.47. The comparson of the correlaton level when usng All and when usng Peaks demonstrates the followng. Research Paper

12 36 O. Yashkr and Y. Yashkr TABLE 1 Correlaton between factors and LGD: All, for dfferent nstruments. Effectve Industry Debt Prncpal nterest mean Rankng cushon above rate Spread LGD Senor unsecured Revolvng credt Term loan Senor subordnated Subordnated Senor secured Junor subordnated Other All nstruments Collateral Rankng Prncpal mean mean Orgnal default Acclamed Total LGD LGD Amount amount amount debt Senor unsecured Revolvng credt N/A.76 Term loan N/A.65 Senor subordnated Subordnated Senor secured Junor subordnated Other N/A.84 All nstruments Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213

13 Loss gven default modelng: a comparatve analyss 37 TABLE 2 Correlaton between factors and LGD: Peaks (years , 21 2, 28 9). Effectve Industry Debt Prncpal nterest mean Rankng cushon above rate Spread LGD Senor unsecured Revolvng credt Term loan Senor subordnated Subordnated Senor secured Junor subordnated Other All Instruments Collateral Rankng Prncpal mean mean Orgnal default Acclamed Total LGD LGD Amount amount amount debt Senor unsecured Revolvng credt N/A.148 Term loan N/A.52 Senor subordnated Subordnated Senor secured Junor subordnated Other N/A.586 All nstruments Research Paper

14 Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213 TABLE 3 Marked cells: absolute values of correlatons (factor, LGD) exceed 1%. Effectve Industry Collateral Rankng Debt Prncpal nterest mean mean mean Rankng cushon above rate Spread LGD LGD LGD Revolvng credt X X X Term loan X X X X X X Senor unsecured X X X Senor secured X X X X X X X X Senor subordnated X X X X X X Subordnated X X X Junor subordnated X X X X X 38 O. Yashkr and Y. Yashkr

15 Loss gven default modelng: a comparatve analyss 39 (1) Correlatons of hstorcal LGDs wth Industry Mean LGD (the mean of hstorcal LGDs for a specfed ndustry) are hgh for all nstruments (from 14% to 68%). (2) Changes n correlaton level are clearly seen when comparng All correlaton results and Peaks correlaton results. The absolute values of correlaton are hgher for the Peaks results. For example, Revolvng Credt correlaton wth Debt Cushon s equal to :299 when usng All, and t s equal to :345 when usng Peaks. (3) The sgnfcance of Spread, Effectve Interest Rate and Total Debt factors ncreases durng cycle peaks (as expected) due to the nfluence of macroeconomc condtons ncreasng n cycle peaks. The correlaton level analyss demonstrates that sets of sgnfcant factors (explanatory varables) are dfferent for dfferent nstruments (Table 3 on the facng page). In ths table, the most sgnfcant explanatory varables are marked for each nstrument. They were chosen based on the crteron that absolute values of correlatons exceed 1%. The Spread data was not always avalable. Therefore, we dd not nclude Spread n the calbraton of models. Based on the smlarty of the factor sets, there are three groups of nstruments that should be calbrated together. Group A: term loan and revolvng credt. Group B: senor unsecured. Group C: senor secured, senor subordnated, subordnated and junor subordnated. Note that, for testng purposes, we consdered senor secured separately, but the obtaned results dd not show vsble mprovement n calbraton crtera. 4 COMPARATIVE MODEL ANALYSIS 4.1 Crtera for the methodology analyss The goodness-of-ft and the model LGD correlaton were chosen as the crtera for the methodology performance analyss. As a measure of goodness-of-ft (G) weuse the followng parameter (often called the coeffcent of determnaton ): G D 1 MSE varlgd ; (4.1) where MSE s the mean square error (model versus hstorcal LGD) and varlgd s the varance of the nput data. The nterpretaton of the parameter G s as follows. Research Paper

16 4 O. Yashkr and Y. Yashkr For a nave model, where predcted values of the model LGD are equal to the mean hstorcal LGD, we would have MSE D varlgd and G D (the model ft s not better than the nave model). On the other hand, f a model provdes predcton such that MSE varlgd (deal case), then G 1 (a very good ft). Usng MSE as a crteron of the ft qualty mght be msleadng. In the followng sectons we wll use the parameter G as a crteron for comparson of dfferent models. Values of MSE and/or values of mean absolute error (MAE) are presented for convenence. The model LGD correlaton (), defned as the correlaton between hstorcal and smulated LGDs, s also used for model comparson. 4.2 Calbraton detals Ths secton descrbes the procedures, the R functons used and specfc calbraton approaches. Codes for Tobt, nflated beta and gamma regresson models were developed by Yashkr Consultng usng the statstcal R package. Addtonal applcatons were developed n Python. Least-squares method The lbrary functon n R provdes a soluton of the problem (2.1): Q D lsft.r; L;:::/; (4.2) where r s the matrx of explanatory varables for a gven set of defaulted cases and L s the vector of observed LGDs. From the output object (lst) Q we fnd the followng: the coeffcent vector x D Q 1 (ncludng ntercept x ), the array of resduals ı D Q 2 and the modeled LGD L ı for the th case. Object functon mnmzaton The lbrary functon n R provdes a soluton of (2.14) (Tobt model): Q D optm.z;.z/;:::/: (4.3) From the output object (lst) we fnd the coeffcent vector z D.x;/D Q 1. Maxmzaton of the loglkelhood functon The lbrary functon n R provdes a soluton of (2.22) (three-tered Tobt model): Q D optm.x; H.x/;:::/: (4.4) From the output object (lst) we fnd the coeffcent vector x D Q 1. Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213

17 Loss gven default modelng: a comparatve analyss 41 TABLE 4 Input data. N 1 N 2 LGD V 11 V 12 LGD 1 V 21 V 22 LGD 2 : : : :: : :: Maxmzaton of the loglkelhood functon for beta regresson The lbrary functon n R provdes a soluton of (2.34): Q D betareg.formula; lnk D logt ; data D DATA/; (4.5) where FORMULA s the strng LGD N 1 C N 2 C and DATA s the table contanng nput data n the format n Table 4, where N (predctve varable names) and LGD are column headers, V j and LGDj are correspondng numercal values for every j th transacton. 2 From the output object (lst) Q we fnd the followng: the coeffcent vector x D Q 1 (ncludng ntercept x ) and the array of resduals ı D Q 2. The modeled LGD for the th case s LGD mod D LGD ı : ˇ In general, an LGD of any j th transacton s estmated as wth the predctor LGD j D y j D x C X k 1 1 C e y j ; (4.6) x k r jk : (4.7) 4.3 Term loan and revolvng credt (Group A) Results of the varous methods performances are presented n Table 5 on the next page for Instrument Group A (term loan and revolvng credt). 3 2 LGDj are scaled values of real LGDs as follows: LGD j D LGD j.ˇ / C to ensure that < 6LGDj 6 ˇ<1. 3 Revolvng lne of credt s an agreement by a bank to lend a specfc amount to a borrower, and to allow that amount to be borrowed agan once t has been repad. Term loan s a bank loan to a company, wth a fxed maturty and often featurng amortzaton of prncpal. If ths loan s n the form of a lne of credt, the funds are drawn down shortly after the agreement s sgned. Otherwse, the borrower usually uses the funds from the loan soon after they become avalable. Research Paper

18 42 O. Yashkr and Y. Yashkr TABLE 5 Instrument Group A (factors: Debt Cushon, Industry Mean LGD, collateral mean LGD). [Table contnues on next page.] (a) Tobt G MAE (b) Least squares G MAE (c) Inflated beta G MAE The best ft for these nstruments was obtaned usng the LSM and BetaReg. Two mportant observatons can be made for the case of term loans and revolvng credts: (1) the best ft for all sets of data was acheved by usng the LSM; (2) the best ft was acheved on Peaks usng the LSM and BetaReg (goodnessof-ft s approxmately :23 and correlaton s approxmately :47). In peak condtons of the cycle, the predctve power of the chosen faclty parameters ncreases, whch results n hgher values of goodness-of-ft and correlatons Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213

19 Loss gven default modelng: a comparatve analyss 43 TABLE 5 Contnued. (d) Beta regresson G MAE (e) Gamma regresson G MAE for Peaks. Ths outcome of comparatve tests for dfferent calbraton models clearly ndcates that the success of LGD modelng depends manly on the avalablty (and proper choce) of explanatory varables and on data qualty, but not on fttng technques. 4.4 Senor unsecured (Group B) Senor unsecured transactons (Instrument Group B) do not have any collateral. Accordng to the correlaton matrces (Table 1 on page 36 and Table 2 on page 37), man parameters that are hghly correlated wth the hstorcal LGDs are Industry Mean LGD and the Effectve Interest Rate 4. Therefore, n the case of senor unsecured transactons the man factor defnng the LGD level at default s the ndustry cluster to whch the faclty belongs. 4 Effectve Interest Rate, accordng to the S&P defnton, s the prepetton rate at the tme the last coupon was pad; t has a fxed rate for fxed-coupon nstruments, whle a floatng rate s used at the tme of default for floatng-rate nstruments. Research Paper

20 44 O. Yashkr and Y. Yashkr TABLE 6 Instrument Group B (factors: Industry Mean LGD, Effectve Interest Rate (EIR)). [Table contnues on next two pages.] (a) Tobt (Industry Mean LGD) G (b) Tobt (Industry Mean LGD and EIR) G (c) Least squares (Industry Mean LGD) G (d) Least squares (Industry Mean LGD and EIR) G The case of senor unsecured transactons (Group B) s dffcult to model because only two parameters (Industry Mean LGD and EIR) have substantal correlaton wth hstorcal LGDs. Results are presented n Table 6. Two mportant observatons can be made for the case of senor unsecured: Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213

21 Loss gven default modelng: a comparatve analyss 45 TABLE 6 Contnued. (e) Inflated beta (Industry Mean LGD) G (f) Inflated beta (Industry Mean LGD and EIR) G (g) BetaReg (Industry Mean LGD) G (h) BetaReg (Industry Mean LGD and EIR) G (1) the best ft for all sets of data s done agan usng the LSM; (2) the best ft was acheved on Bankruptcy Peaks usng the LSM and BetaReg (goodness-of-ft s approxmately :32 and correlaton s approxmately s :57). Research Paper

22 46 O. Yashkr and Y. Yashkr TABLE 6 Contnued. () GammaReg (Industry Mean LGD) G (j) GammaReg (Industry Mean LGD and EIR) G The dependency on Bankruptcy Peaks shows that for bankruptcy cases n peak condtons of the cycle, the ndustry becomes even more mportant. It should be noted that for ths group of nstruments, calbrated on Bankruptcy Peaks, the Tobt and GammaReg models also provde suffcently good ft (goodness-of-ft s approxmately :3, and correlaton s approxmately :56). For contracts wth fxed nterest rates (f default data contans ths rate) the Effectve Interest Rate can also be used for calbraton. 4.5 Senor secured, senor subordnated, subordnated, and junor subordnated (Group C) For Instrument Group C (senor secured, senor subordnated, subordnated, and junor subordnated ), accordng to the correlaton matrces (Table 1 on page 36 and Table 2 on page 37), the man parameters that are hghly correlated wth the hstorcal LGDs are: Debt Cushon, Prncpal Above, Effectve Interest Rate, Industry Mean LGD, Collateral Mean LGD, Rankng Mean LGD. The Instrument Group C model strongly depends on Debt Cushon, Prncpal Above, Industry Mean LGD, Collateral Mean LGD and Rankng Mean LGD. Results are presented n Table 7 on the facng page. The EIR can also be used f avalable. Two mportant observatons can be made for ths case: Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213

23 Loss gven default modelng: a comparatve analyss 47 TABLE 7 Instrument Group C (Factors: Debt Cushon, Prncpal Above, Effectve Interest Rate, Industry Mean LGD, Collateral Mean LGD, Rankng Mean LGD). [Table contnues on next page.] (a) Tobt G (b) Least squares G (c) Inflated beta G (1) the best ft for all sets of data s done agan usng the LSM; (2) the best ft was acheved on Bankruptcy usng the LSM and BetaReg (goodness-of-ft s approxmately :39, and correlaton s approxmately :63). The dependency on chosen factors and Bankruptcy shows the mportance of the proper choce of man factors. It should be noted that, for ths group of nstruments, the Tobt model also provdes suffcently good ft (goodness-of-ft s approxmately :38, and correlaton s approxmately :62) on Bankruptcy. The results from nflated beta also provde suffcently good ft (goodness-of-ft s approxmately :37, and correlaton s approxmately :62). The results show that the factors were well chosen. The ncluson of the EIR does not change the goodness-of-ft or correlaton. Research Paper

24 48 O. Yashkr and Y. Yashkr TABLE 7 Contnued. (d) BetaReg G (e) GammaReg G CALIBRATION EXAMPLES WITH THE BEST FITTING RESULTS The results presented n ths secton are the best ft as descrbed above. The marker n Table 8 on the facng page, Table 9 on the facng page and Table 1 on page 5 ndcates the most mportant factors. 5 Our results show that the factors for all three groups were properly chosen and are the mportant factors for the smulaton. 5.1 Term loans plus revolvng credt (Group A) The best ft was obtaned wth Peaks usng the LSM and BetaReg (Table 8 on the facng page). It should be noted (n addton to the crtera used) that the mean of the hstorcal LGDs (equal to :247) and the mean of the smulated LGDs (equal to :25 for LSM, and equal to :258 for BetaReg) are very close. Ths s a supportng factor for the models results. The hstorcal LGDs and the model smulated LGDs (LSM model) are presented n part (a) of Fgure 2 on page 51. In ths fgure the hstorcal LGDs (sold lne) and 5 Markers n table columns wth the header Pr(> z ) denote *** (hghest mportance), ** (lower mportance), * (some mportance) and (the lowest mportance). Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213

25 Loss gven default modelng: a comparatve analyss 49 TABLE 8 Results for the best ft for cases of term loans and revolvng credt. BetaReg, Peaks Least squares, Peaks Parameter Coeffcent Pr.> jzj/ Parameter Coeffcent (Intercept) 1.4 <2e 16 (Intercept).19 Debt cushon e 16 Debt cushon.35 Industry 2.57 <2e 16 Industry 1.25 Mean LGD Mean LGD Collateral e 4 Collateral.39 Mean LGD Mean LGD G MAE TABLE 9 Results for the best ft for senor unsecured cases (Bankruptcy Peaks ). BetaReg Least squares Parameter Coeffcent Pr.> jzj/ Parameter Coeffcent (Intercept) 1.38 < 2e 16 Intercept.21 EIR e 7 EIR 2.94 Industry 2.34 <2e 16 Industry 1.22 Mean LGD Mean LGD G MAE smulated LGDs (dots) are plotted as functons of case numbers (all cases are sorted by ther hstorcal LGD values). In case of a perfect model the estmated LGDs (dots) would follow the hstorcal LGDs (sold lne). In the realty, the lnear regresson model used provdes a lower fttng qualty as seen n part (a) of Fgure 2 on page 51. The advantage of ths presentaton of results s that one can clearly see what the lmtatons of the chosen set of explanatory varables are: LGD s overestmated for low hstorcal LGDs and s underestmated for hgh hstorcal LGDs. Another method of vsual representaton of calbraton results s shown n Fgure 3 on page 51 (estmated LGDs versus hstorcal LGDs). The better model fttng would correspond to model LGDs concentrated around the dagonal lne. The hstogram of the smulated LGDs s presented n part (b) of Fgure 2 on page 51. The smulated LGD values are concentrated around the mean LGD value. Research Paper

26 5 O. Yashkr and Y. Yashkr TABLE 1 Results for the best ft for Instrument Group C (Bankruptcy ). BetaReg Least squares Parameter Coeffcent Pr.> jzj/ Parameter Coeffcent (Intercept).98 <2e 16 Intercept.6 Prncpal above Prncpal above.16 Debt cushon.72 8.e 14 Debt cushon.36 EIR EIR.24 Industry e 16 Industry.76 Mean LGD Mean LGD Collateral 1.47 <2e 16 Collateral.81 Mean LGD Mean LGD Rankng.19 Mean LGD G MAE Senor unsecured (Group B) The best ft was obtaned for Bankruptcy Peaks usng the LSM and BetaReg (Table 9 on the precedng page). In addton to the fttng qualty crtera used, t s worth mentonng that the mean of the hstorcal LGDs (equal to.5617) and the mean of the smulated LGDs (equal to.5616 for LSM, and s equal to.5583 for BetaReg) are very close. Ths s a supportng factor for the models results. Lmted flexblty of modelng of senor unsecured LGDs s due to the fact that only Industry Mean LGD has sgnfcant mportance for these nstruments. The chosen factors are all shown to be mportant. The hstorcal LGDs and the model smulated LGDs (model LSM) are presented n part (a) of Fgure 4 on page 52, and the smulated LGD hstogram s presented n part (b). There s concentraton of the smulated values around the mean value as expected. Smulated LGDs (dots) reflect the general trend of hstorcal LGDs (sold lne). 5.3 Senor secured, senor subordnated, subordnated and junor subordnated (Group C) The best ft for Bankruptcy was obtaned usng the LSM and BetaReg models (Table 1). Note that the mean of the hstorcal LGDs (equal to :6178) and the mean of the smulated LGDs (equal to :619 for LSM, and to :6123 for BetaReg) are very close. Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213

27 Loss gven default modelng: a comparatve analyss 51 FIGURE 2 Modelng results for calbraton on the Term Loans plus revolvng credt data (G D.2341, MAE D.2262). 1. (a) 3. (b) LGD.6.4 Densty Faclty number LGD (a) LGD nput (lne); LGD model (dots). (b) Model LGD densty. FIGURE 3 Model LGD versus hstorcal LGDs (All, LSM, G D.1658, D.472) Model E[LGD] LGD (nput) All chosen factors, except EIR, are shown to be mportant. The hstorcal LGDs (sold lne) and the model smulated LGDs (dots, LSM model) are presented n part (a) of Fgure 5 on the next page, and the hstogram of smulated Research Paper

28 52 O. Yashkr and Y. Yashkr FIGURE 4 Fttng results for senor unsecured data set (G D.328, MAE D.257). 1. (a) 3. (b) LGD Densty Faclty number LGD (a) LGD nput (lne); LGD model (dots). (b) Model LGD densty. FIGURE 5 Fttng results for Instrument Group C (G D.394, MAE D.2335) 1. (a) 3.5 (b) LGD.6.4 Densty , Faclty number LGD (a) LGD nput (lne); LGD model (dots). (b) Model LGD densty. Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213

29 Loss gven default modelng: a comparatve analyss 53 TABLE 11 Summary of calbraton examples on the All set. Root mean Mean Goodness- Sample Number of Group square error absolute error of-ft sze varables A B C Mxture N/A 5 8 Model test from Yang and Tkachenko (212). LGDs s shown n part (b). The results show good agreement between the smulated and the hstorcal LGDs. The cloud of smulated values follows the hstorcal LGDs. 5.4 Summary of calbraton examples on the All set The results of our tests for nstrument Groups A C based on the All set are presented n Table 11. In the last row, the results of the LGD mxture model test by Yang and Tkachenko (212) are presented for comparson. In ther tests, Yang and Tkachenko explored several models (logt raw, HL logt, logt, least-squares logt, nave Bayes, mxture and neural network) and found that the mxture model has the lowest fttng error. The results of our tests provde a smlar or better level of fttng errors. Fnally, as an example, we compare calbraton results for Groups A and C wth a nave model, where the model LGDs are defned as hstorcal average LGD. Results presented n Table 12 on the next page demonstrate clearly advantages of usng parameter G as a crteron for model comparson. For example, n the case of Group A, the MAE only changes from 23.65% to 2.21%, but the goodness-of ft changes dramatcally from.6% to 16.58%. 6 DATA SENSITIVITY AND STRESS TESTING 6.1 senstvty test The example for the data set of senor unsecured cases was tested to demonstrate the senstvty of results to adjustments of the ntal data. The ntal hstorcal data was adjusted by excludng all cases where LGDs are lower than 15% (takng nto account that LGDs for ths nstrument, n general, could not be low). After performng the model calbraton (BetaReg) and the smulaton of the LGDs, we obtaned the followng results (see Fgure 6 on the next page). The ft s better than for the case descrbed for the general senor unsecured case (correlaton ncreased from :52 to :54). Ths example shows that even small flterng of the data made based on the Research Paper

30 54 O. Yashkr and Y. Yashkr TABLE 12 Comparson of the lnear regresson model results wth the nave model. (a) Group A Mean absolute error G Lnear regresson Nave model (b) Group C Mean absolute error G Lnear regresson Nave model FIGURE 6 Results for senor unsecured wth adjusted data, Bankruptcy Peak ntal data (G D.2963, MAE D.1796). 1. (a) 3. (b) LGD.6.4 Densty Faclty number LGD (a) LGD nput (lne); LGD model (dots). (b) Model LGD densty. Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213

31 Loss gven default modelng: a comparatve analyss 55 FIGURE 7 Stress testng of LGDs LGD Faclty number All calbraton (sold lne). Bankruptcy calbraton (dots). reasonable busness assumptons could mprove calbraton results even for dffcult cases such as senor unsecured. 6.2 Stress testng for LGDs The approach of the LGD stress testng comes naturally from the results of our analyss of models and data. The stress-test procedure s as follows. (1) Derve model coeffcents for peak perods usng the Peaks and/or data for each peak separately. These coeffcents emphasze the peak of crss perod n the busness cycle condtons. (2) Run smulatons of All LGDs usng these peak-related coeffcents. The resultng smulated LGDs provde predctons of the LGD levels n crss (stress) condtons. The smulaton results wth Bankruptcy coeffcents are presented n Fgure 7. Research Paper

32 56 O. Yashkr and Y. Yashkr FIGURE 8 Stress testng of LGDs Faclty number All calbraton (grey dashed lne); Bankruptcy calbraton (black lne); Bankruptcy Peaks calbraton (open dots). The smulaton results for Bankruptcy coeffcents and Bankruptcy Peaks coeffcents are presented n Fgure 8. The mean of smulated LGDs based on All coeffcents s equal to :55; the mean of smulated LGDs based on Bankruptcy coeffcents s equal to :62; and the mean of smulated LGDs based on Bankruptcy Peaks coeffcents s equal to approxmately :66. Therefore, the average LGD ncrease, compared wth the All LGD level, s equal to 7% (calbraton on Bankruptcy ), and t s equal to 11% (calbraton on Bankruptcy Peaks ). The Bankruptcy results and Bankruptcy Peaks results are shown n Fgure 8. Our approach for the estmaton of downturn LGDs does not requre any addtonal model assumptons such as the analytc approach by Barco (27) or the parameter senstvty approach by Rösch and Scheule (27). It naturally follows the chosen model calbraton procedure and the data choce. The downturn LGDs are estmated based on the chosen data subset consstent wth the downturn condtons n the bus- Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213

33 Loss gven default modelng: a comparatve analyss 57 ness cycle. If a fnancal nsttuton does not have enough data for the Peaks data set, then the external data for peaks perods can be used (followng specfc Basel II regulatons). The external data set that contans all avalable cases provdes the data for the peak/stress LGD calbraton. 7 CONCLUSIONS Several of the most popular LGD models (LSM, Tobt, three-tered Tobt, beta regresson, nflated beta regresson, censored gamma regresson) were tested on real data n order to compare ther performance. Model factors were chosen based on the ampltude of ther correlaton wth hstorcal LGDs of the calbraton data set. Numercal values of nonquanttatve parameters (ndustry, rankng, type of collateral) were ntroduced as ther LGD averages. It was shown that for a gven nput data set, the model calbraton qualty depends manly on the proper choce (and avalablty) of explanatory varables (model factors), but not on the model used for fttng, dfferent debt nstruments depend on dfferent sets of model factors (from three factors for revolvng credt or for subordnated to eght factors for senor secured ), calbraton of LGD models usng dstressed busness cycle perods provdes a better ft than the data from the total avalable tme span, calbraton parameters obtaned usng dstressed busness cycle perods can be productvely used for stress testng. Calbraton algorthms and detals of ther realzaton usng the R statstcal package are presented. The results of ths study can be of use to rsk managers concerned wth complance wth the Basel Accord. REFERENCES Altman, E. (26). Are hstorcally based default and recovery models n the hgh-yeld and dstressed debt markets stll relevant n today s credt envronment? Stern School of Busness, Specal Report. Altman, E. (21). Default recovery rates and LGD n credt rsk modelng and practce. Workng Paper, URL: ealtman/updatedrevewoflterature. pdf Altman, E., and Kalotay, E. (21). A flexble approach to modelng ultmate recoveres on defaulted loans and. Workng Paper, URL: ealtman/ FlexbleRecovery_v1.1.pdf. Research Paper

34 58 O. Yashkr and Y. Yashkr Altman, E., Brady, B., Rest, A., and Sron, A. (23). The lnk between default and recovery rates: theory, emprcal evdence, mplcatons. Workng Paper S-CDM-4-7, Seres Credt & Debt Markets Research Group Altman, E., Rest, A., and Sron, A. (24). Default recovery rates n credt rsk modelng: A revew of the lterature, emprcal evdence. Economc Notes 33, Bade, B., Rösch, D., and Scheule, H. (211). Emprcal performance of loss gven default predcton models. The Journal of Rsk Model Valdaton 5(2), Baxaul, J. S., and Alvarez, S. (21). The role of market-mpled severty modelng for credt var. Annals of Economcs and Fnance 11(2), Barco, M. (27). Gong downturn. Rsk 2(9), Bellott, T., and Crook, J. (212). Loss gven default models ncorporatng macroeconomc varables for credt cards. Internatonal Journal of Forecastng, Specal Secton 2: Credt Rsk Modellng and Forecastng 28(1), Chalupka, R., and Kopecsn, J. (29). Modelng bank loan LGD of corporate and SME segments: a case study. Czech Journal of Economcs and Fnance 59(4), Fredman, C., and Sandow, S. (23). Ultmate recoveres. Rsk 16(8), Gupton, G. M. (25). Advancng loss gven default predcton models: how the quet have quckened. Economc Notes by Banca Monte de Pasch d Sena SpA 34(2), Gupton, G. M., and Sten, R. M. (22). Losscalc: model for predctng loss gven default (LGD). Workng Paper, Federal Reserve Bank of New York. Gurtler, M., and Hbbeln, M. (211). Ptfalls n modelng loss gven default of bank loans. Workng Paper, Technsche Unverstät Braunschweg. Hllebrand, M. (26). Modelng and estmatng dependent loss gven default. Workng Paper, CRC. Hlawatsch, S., and Ostrowsk, S. (211). Smulaton and estmaton of loss gven default. The Journal of Credt Rsk 7(3), Huang, X., and Oosterlee, C. W. (212). Generalzed beta regresson models for random loss gven default. The Journal of Credt Rsk 7(4), Jacobs, M., Jr. (211). A two-factor structural model of ultmate loss-gven-default: captal structure and calbraton to corporate recovery data. Journal of Fnancal Transformaton 31(4), L, D., Bharok, R., Keenan, S., and Santll, S. (29). Valdaton technques and performance metrcs for loss gven default models. The Journal of Rsk Model Valdaton 3(3), Luo, X., and Shevchenko, P. V. (21). LGD credt rsk model: estmaton of captal wth parameter uncertanty usng MCMC. Workng Paper, CSIRO Mathematcs, Informatcs and Statstcs. McDonald, J. F., and Mofftt, R. A. (198). The uses of Tobt analyss. Revew of Economcs and Statstcs 62(2), Perera, T. L., and Crbar-Neto, F. (21). A test for correct model specfcaton n nflated beta regressons. Workng Paper, Insttuto de Matemátca, Estatístca e Computação Centífca Unversdade Estadual de Campnas. URL: default/fles/reset_beo.pdf Rösch, D., and Scheule, H. (27). Stress-testng credt rsk parameters: an applcaton to retal loan portfolos. The Journal of Rsk Model Valdaton 1(1), Journal of Rsk Model Valdaton Volume 7/Number 1, Sprng 213

35 Loss gven default modelng: a comparatve analyss 59 Schuermann, T. (24).What do we know about loss gven default? Workng Paper, Federal Reserve Bank of New York. Sgrst, F., and Stahel, W. A. (21). Censored gamma regresson models for lmted dependent varables wth an applcaton to loss gven default. In 28th European Meetng of Statstcans, August 17 22, 21, Praeus, Greece. Sgrst, F., and Stahel, W. A. (211). Usng the censored gamma dstrbuton for modelng fractonal response varables wth an applcaton to loss gven default. ASTIN Bulletn 41(2), Standard & Poor s (212). Default, transton, and recovery: 211 annual global corporate default study and ratng transtons. Report. van der Weja, W., and den Hollandera, M.(29). Improvng PD and LGD Models: Followng the Changes n the Market. SNS Reaal, Utrecht. Yang, B. H., and Tkachenko, M. (212). Modelng exposure at default and loss gven default: emprcal approaches and techncal mplementaton. The Journal of Credt Rsk 8(2), Research Paper

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