A framework for loss given default validation of retail portfolios

Transcription

1 The Journal of Rsk Model Valdaton (23 48) Volume 4/Number 1, Sprng 2010 A framework for loss gven default valdaton of retal portfolos Stefan Hlawatsch Department of Bankng and Fnance, Otto-von-Guercke Unversty Magdeburg, Postfach 4120, Magdeburg, Germany; emal: stefan.hlawatsch@ovgu.de Peter Rechlng Department of Bankng and Fnance, Otto-von-Guercke Unversty Magdeburg, Postfach 4120, Magdeburg, Germany; emal: peter.rechlng@ovgu.de Modelng and estmatng loss gven default (LGD) s necessary for banks that apply for the nternal ratngs based approach for retal portfolos. To valdate LGD estmatons, there are only a few approaches dscussed n the lterature. In ths paper, two models for valdatng relatve LGD and absolute losses are developed. The valdaton of relatve LGD s mportant for rsk-adjusted credt prcng and nterest rate calculatons. The valdaton of absolute losses s mportant to meet the captal requrements of Basel II. Both models are tested wth real data from a bank. Estmatons are tested for robustness wth n-sample and out-of-sample tests. 1 INTRODUCTION Accordng to Basel II, banks can choose between two approaches to measure ther credt rsk: the standardzed approach and the nternal ratngs based approach. Socalled nternal ratngs based approach banks have to underlay credts wth equty dependng on the unexpected loss (denoted as UL n equatons) accordng to Equaton (1): 1 relatve unexpected loss {}}{ [ ( 1 (PD) + R 1 ) ] (0.999) UL = LGD PD 1 R } {{ LGD} EAD (1) }{{} expected loss maxmum loss The unexpected loss equals the dfference between the so-called maxmum loss, whch s computed as the value-at-rsk of the loss, and the expected loss, whch s computed as the product of probablty of default (PD) and loss gven default (LGD). Here, R denotes the correlaton coeffcent of the PD wth the systematc rsk factor. We thank Hendrk Rtter and two anonymous referees for helpful comments. 1 ndcates the standard normal dstrbuton functon and 1 denotes the nverse of. 23

2 24 S. Hlawatsch and P. Rechlng FIGURE 1 PD and LGD elastcty of rsk-weghted assets for the retal nternal ratngs based approach PD PD elastcty (secured by resdental propertes) PD elastcty (other) PD elastcty (qualfyng revolvng exposures) LGD elastcty 4 The product of the thus computed relatve unexpected loss and the exposure at default (EAD) results n the unexpected loss n monetary unts. Accordng to Artcle 87 No. 6 and 7 of Europe s Captal Requrement Drectve, 2 banks have to estmate PD and LGD for retal clams or contngent retal clams on ther own. Furthermore, the estmaton procedures have to be valdated for robustness and accuracy of the models. Ths valdaton should transcend the smple comparson of hstorcal data wth estmated parameters, as t s mentoned n Annex VII, Part 2 No. 112, Captal Requrement Drectve. Whle valdaton technques for PD estmatons are dscussed extensvely n the lterature, research on quanttatve valdaton nstruments for LGD estmaton models s rare. Valdatng LGD estmatons s crucal because the requred captal reacts more senstvely to changes n LGD than to changes n PD. By way of llustraton, the LGD and the PD elastctes of unexpected loss are shown n Fgure 1. 3 The LGD elastcty s constant and amounts to one. However, the PD elastctes for the shown subcategores for retal clams (up to a PD of 50%) are absolutely smaller than the LGD elastcty. Therefore, the rsk-weghted assets react more senstvely to changes n LGD. 4 Thus, the hgh senstvty of the rsk-weght functon wth respect to the LGD (n the relevant PD range up to 50%) mples the necessty 2 Drectve 2006/48/EC of the European Parlament and of the Councl. 3 The rght hand sde of Equaton (1) s multpled by factor 12.5 n Fgure 1, so that unexpected loss equals the rsk-weghted assets. 4 The partly negatve PD elastcty shown n Fgure 1 results from the fact that, wth an ncreasng PD, the unexpected loss becomes smaller and the expected loss, whch reduces the captal requrement, rses. Expected loss s are consdered by deprecatons, provsons or wrte-offs and, therefore, no captal underlay s requred. The Journal of Rsk Model Valdaton Volume 4/Number 1, Sprng 2010

3 A framework for LGD valdaton of retal portfolos 25 for precse estmatons of LGD. 5 An evaluaton of the accuracy of LGD estmatons can be done by the technque of valdaton. Our paper s organzed as follows. After a bref revew of the lterature of dfferent LGD estmaton models, two valdaton models are developed n Secton 2. For an emprcal analyss, real data from a bank s used. Secton 3.1 descrbes the data and emprcal analyss and Secton 3.2 reports the results. Secton 4 concludes. 2 LOSS GIVEN DEFAULT VALIDATION 2.1 Lterature revew There are four dfferent approaches to compute the LGD: workout LGD, market LGD, mpled market LGD, and mpled hstorcal LGD. 6 The workout LGD belongs to the group of so-called explct methods of LGD estmaton; explct here refers to the data used. Explct methods use hstorcal LGDs of defaulted credts n order to derve prognoses for future LGDs. The workout LGD s cashflow-orented. To compute the workout LGD, all recoveres, as well as all costs, are consdered n the perod from the day of the credt event up to fnal recovery. In order to consder dfferent ponts n tme where costs and recoveres emerge, payments have to be dscounted to the day of the credt event. Therefore, the workout LGD s computed as follows: 7 where: LGD = EAD n j=1 E,j (r) + m k=1 K,k (r) EAD nj=1 E,j (r) m k=1 K,k (r) = 1 (2) EAD E,j (r) = dscounted recoveres j of credt K,k (r) = dscounted costs or losses k of credt r = dscount rate Typcal recoveres are collaterals or securtes. 8 Examples of costs and losses are a loss on nterest payments, opportunty costs for equty, handlng costs, and workout costs such as overhead costs of the recovery department. 5 Despte the hgher LGD senstvty wthn the relevant range, ths does not mply a less precse or less robust estmaton of PD. 6 See Basel Commttee on Bankng Supervson (2005b, pp ). 7 See Basel Commttee on Bankng Supervson (2005b, p. 66). 8 Especally for tangble fxed assets, for example real-estate or machnery, market prces can change untl the collateral utlzaton. Therefore, harcuts should be computed to consder ths loss n value. See also Basel Commttee on Bankng Supervson (2005b, p. 67). Research Paper

4 26 S. Hlawatsch and P. Rechlng The dscount rate to determne the economc loss has to be rsk-adjusted. In partcular, for parameters such as collaterals or workout costs, no market exsts. Therefore, the determnaton of the dscount rate s dffcult. If hstorcal nterest rates are used, the rsk-free nterest rate plus a loss mpact or the ntally agreed nterest rate can be used. 9 After computng the workout LGD, the estmaton model can be developed usng, for example, regressons. 10 The ndependent varables that should be used here depend on the nsttuton and branch. Commonly accepted varables nclude provsons of securty, repayment prorty, ndustry afflaton, macro-economc factors such as economc growth or ratngs. 11 A further explct method to determne the LGD s the so-called market LGD method. In ths approach, market prces of publcly traded defaulted loans or securtzed credts are used. After a default, the recovery rate can be determned by the market value of the loan because nvestors antcpate possble proceeds from realzatons as well as possble costs. Thus, loss results n the dfference between the par offerng prce and the market prce after default. For publcly traded loans, ths data s collected by ratng agences. The appeal of ths concept comes from the fact that only the recovery rate s needed for the LGD computaton. However, ths recovery rate corresponds to the market prce after default for ntally prced at par loans. In ths approach, t s crtcal that several parameters are based on subjectve estmates. It remans doubtful whch tme horzon should have been taken after the pont n tme of default of the loan, such that all nvestors antcpate possble earnngs and costs. 12 In addton, nternal workout costs of the bank are not reflected by the market prce. Moreover, market prces are nfluenced by both supply and demand. Therefore, on llqud markets, the use of market prces can lead to false estmates of LGD. After computng the market LGD, the development of the estmaton model s smlar to the workout LGD. A well-known model for LGD estmaton based on market LGD s LossCalc 2.0 from Moody s KMV. 13 The market LGD s only sutable for securtzed loans or credts due to the need for market data. Unncluded workout costs have to be ntegrated by an adjustment of the LGD. Therefore, nternal bank data s requred. Alternatvely, the workout LGD can be used. A further possblty to determne LGD s the mpled market LGD, whch belongs to the group of mplct methods. Non-defaulted securtzed loans or credts form the 9 Brady et al (2006) emprcally show that dscount rates sgnfcantly dffer for dfferent branches of ndustry and ratngs. The determned dscount rate ranges from 0.9% to 29.3%. 10 Hamerle et al (2006) use a two stage regresson. Sddq and Zhang (2004) assume that the LGD s beta dstrbuted and transform the LGD nto a normally dstrbuted random varable before estmaton. 11 For more examples and analyses of nfluencng factors see Schuermann (2005). 12 Moody s, for example, uses a tme horzon of one month after default. See also Gupton (2005). 13 Gupton (2005) gves an overvew of the model and procedure of LossCalc 2.0. The Journal of Rsk Model Valdaton Volume 4/Number 1, Sprng 2010

5 A framework for LGD valdaton of retal portfolos 27 database for ths approach. Here, t s assumed that the spread between a loan-specfc nterest rate and the rsk-free nterest rate equals the expected loss as a percentage. If the spread s known, the LGD comes from the rato of spread and default probablty. Ths concept s based on the model of Jarrow et al (1997). The value of a loan V equals the value of a loan wthout rsk V rf multpled by the probablty of nondefault plus the value of a loan wthout rsk multpled by the recovery rate (RR) and the PD. Then, the followng valuaton equaton holds for rsky bonds (see Jarrow et al (1997)): V = V rf (1 PD) + V rf PD RR (3) Impled market LGD models dffer only n the statstc modelng of the parameters of Equaton (3) and dfferent nterpretatons of the recovery rate. In general, there are three possble nterpretatons. The recovery rate s defned as a porton of the ssue prce, a porton of the current present value or a porton of the value of the loan shortly before default (see Baksh et al (2006)). Madan and Unal (2000) and Baksh et al (2006) use a hazard process n order to model the default probablty. The recovery rate, together wth the process of the rsk-free nterest rate term structure, s modeled by stochastc processes. 14 Also, the use of alternatve nterest rate spreads s dscussed. 15 For the mpled market LGD, the decomposton of the nterest rate spread nto ts components s crucal. Snce the nterest rate spread may contan a lqudty premum as well as a rsk premum for the unexpected loss, the appled asset prcng models must be able to determne the sngle components separately. The mpled hstorcal LGD s also a concept of mplct LGD determnaton. Accordng to the Basel Commttee on Bankng Supervson, the use of the mpled hstorcal LGD s only allowed for retal portfolos. 16 Accordng to ths approach, banks are allowed to determne ther LGD on the bass of PD estmatons. The database conssts of hstorcal loss data of retal portfolos. Here, the LGD of a retal credt s computed smlarly to that of the mpled market LGD: EL = PD LGD. All the mentoned LGD models possess the same shortcomng, namely that LGD and PD are estmated ndependently. Therefore, the Basel Commttee on Bankng Supervson has tred to overcome ths weakness by ntroducng a so-called downturn LGD. Downturn LGD s defned as an LGD for each faclty that ams to reflect economc downturn condtons where necessary to capture the relevant rsks. Furthermore, banks must consder the extent of any dependence between the rsk of the borrower and that of the collateral or collateral provder. 17 Banks are free to choose an approprate downturn LGD model. However, the bank has to 14 For a detaled overvew of the modelng of the parameters see Ong (1999, pp ). 15 For the use of credt default swap spreads to determne mplct recovery rates see Pan and Sngleton (2008). 16 See Basel Commttee on Bankng Supervson (2004, paragraph 465). 17 Basel Commttee on Bankng Supervson (2004, paragraphs 468 and 469). Research Paper

6 28 S. Hlawatsch and P. Rechlng dentfy the relevant rsk drvers for each debt type to verfy possble dependences between PD and LGD and to ntegrate these dependences nto the downturn LGD model. 18 Hartmann-Wendels and Honal (2006) use a lnear regresson model ncludng a downturn dummy varable. As a result, the downturn LGD exceeds the defaultweghted average LGD by up to eght percentage ponts. Mu and Ozdemr (2006) analyze the correlaton between PD and LGD wth respect to a sngle rsk drver and estmate the ncrease n the mean LGD needed to acheve the adequate economc captal. Barco (2007) and Gese (2005) ntegrate the correlaton between PD and LGD drectly nto Merton s framework under the assumpton that PD and LGD both depend on the same systematc rsk factor. Rösch and Scheule (2009) extend these approaches assumng that PD and LGD depend on dfferent systematc rsk factors, whch are correlated. 2.2 Proportonal decomposton of the credts As seen n the prevous secton, the dscussed concepts of LGD computaton are developed on defaulted credts, especally for retal clams. If realzed LGDs are known, a valdaton model should use those realzed LGDs as a benchmark for LGD estmaton models. 19 Ths dea s also used n PD valdaton. 20 Therefore, the basc dea of our valdaton models s to use well known PD valdaton technques for LGD valdaton purposes. Frstly, transformed ratos of the PD valdaton, for example area under curve (AUC) or accuracy rato, are computed for realzed LGDs. Snce the nterpretaton of these ratos s dfferent for PD valdaton, t s necessary to compare the ratos generated on realzed LGDs wth those ratos generated on estmated LGDs. As a result, the qualty of the LGD estmaton model does not depend on the rato tself but on the comparson of the ratos of realzed and estmated LGDs. The better the estmaton model fts, the more equal the compared ratos are. In order to smplfy the nterpretaton of the followng model, a sample retal portfolo of a bank s analyzed. At frst, the technque of proportonal decomposton s developed on ths sample retal portfolo. Subsequently, the model s appled to real data. Assume the portfolo conssts of 100 credts wth ndvdual EADs and losses. The number of defaulted credts amounts to 54 wth a total loss of e600,000. The total EAD adds up to e3 mllon, whch results n an average LGD of 20% accordng to Appendx E. 18 See Basel Commttee on Bankng Supervson (2005a) for further detals. 19 L et al (2009) provde an overvew of dfferent valdaton technques for LGD models, where they focus on descrptve metrcs of LGD models usng dscretzed LGD ratng scales. 20 See Engelmann et al (2003) and Sobehart and Keenan (2001) for an overvew of PD valdaton technques. The Journal of Rsk Model Valdaton Volume 4/Number 1, Sprng 2010

7 A framework for LGD valdaton of retal portfolos 29 Each EAD s dvded nto n portons of equal sze. 21 The number of portons should realgn to the EAD amount. 22 For every porton = 1,...,n of credt k = 1,...,K, a bnary varable d,k s defned as follows: d,k = { 1 f porton of the EADk s defaulted 0 f porton of the EAD k s not defaulted (4) Furthermore, a second bnary varable nd,k s defned by nd,k 1 d,k. Credt 1 of our sample retal portfolo s not defaulted. Therefore, all d possess a value of zero and all nd possess a value of one. For credt 12, the varables d 1,12 to d 462,12 possess a value of one and the varables d 463,12 to d 1,000,12 possess a value of zero. 23 After computng the varables d and nd for all credts K, the varables are added up for each porton. Therefore, D K k=1 d,k represents the number of credts where porton s defaulted. Smlarly, ND s defned as ND K k=1 nd,k and corresponds to the number of credts where porton s not defaulted. As a consequence, the sum of D and ND for each porton must be equal to the number of credts. The results for our sample retal portfolo are shown n extracts n Table 1 (see page 30). The decomposton n Table 1 s based on the assumpton that all credts exhbt an LGD smaller or equal to one. However, ths s not always ensured. For credts wth hgh workout costs and small EADs, LGDs above one are possble. In order to prevent ths, the varables d and nd can be redefned as follows: d,k = { 1 f porton of the double EADk s defaulted 0 f porton of the double EAD k s not defaulted (5) Wth ths redefnton, credts wth LGDs smaller or equal to 200% are possble. 24 The further computatons are carred out smlarly to the model wth smple EADs. Analogously to measures of the accuracy of ratng functons, ht rates and false alarm rates can be computed for each porton. The nterpretaton of these rates s, however, not equal to those of ratng functons. The ht rate hr and the false alarm 21 If n = 100 s chosen the decomposton corresponds to a percental decomposton, for n = 1,000 t corresponds to an one-tenth of a percent decomposton. 22 If the portfolo conssts of credts wth EADs below e1,000, a more precse decomposton than n = 1,000 makes lttle sense, snce every porton then corresponds to an amount of less than one euro. For our sample retal portfolo, n = 1,000 was selected. Here, the resultng porton amounts to e51.60 n maxmum. 23 The fgure 462 was rounded. For a more precse decomposton, the devaton converges to zero. In our case, the devaton of the loss due to roundng amounts to e2.80 and therefore les n a neglgble range. 24 To reduce roundng errors, the number of portons should be ncreased when rsng the permtted LGD. Research Paper

8 30 S. Hlawatsch and P. Rechlng TABLE 1 Proportonal decomposton of our sample retal portfolo. Porton n % D ND , The second column ndcates the number of defaults wthn porton and the thrd column shows the number of nondefaults wthn porton. rate far are computed as follows: hr = D ND and far = D n (6) ND The ht rate hr s the fracton of porton of all defaulted portons. The false alarm rate far s the fracton of porton of all portons that are not defaulted. If the ht rates and the false alarm rates are summed up, the cumulated ht rate HR j and the cumulated false alarm rate FAR j for porton j, respectvely, result n: 25 HR j = j hr = j n LGD j LGD where LGD j = D j K LGD = D n K 25 For the dervaton see Appendx A. The Journal of Rsk Model Valdaton Volume 4/Number 1, Sprng 2010

9 FAR j = j far = j n RR j RR A framework for LGD valdaton of retal portfolos 31 where RR j = ND j K RR = ND n K The cumulated ht rate corresponds to the rato of the average LGD of the frst j portons (denoted by LGD j ) to the average LGD over all portons (denoted by LGD), multpled by a weghtng factor. The cumulated false alarm rate corresponds to the rato of the average recovery rate of the frst j portons (denoted by RR j ) to the average recovery rate over all portons (denoted by RR), multpled by the same weghtng factor. The average LGDs LGD j and average recovery rates RR j are unweghted. Ths mples the advantage that LGDs can be valdated wthout any nfluence of the sze of EADs. Thus, t can be ruled out that banks arrange ther models so that LGDs of credts wth hgh EADs are estmated more precsely whle estmatons for credts wth smaller EADs are mprecse. The LGD for our sample retal portfolo equals 18.42%. The recever operatng characterstc (ROC) curve evolves by plottng the cumulated ht rates aganst the cumulated false alarm rates. Ths curve shows the homogenety of the credt portfolo wth respect to the LGDs. The more steeply t runs, the more homogeneous the LGDs of sngle credts are. If all credts exhbt the same LGD, the portfolo s perfectly homogeneous concernng the LGDs. Thus, the ROC curve runs vertcally along the y-axs and then horzontally at the level of one. If the credt portfolo conssts of two dsjunct quanttes, one wth credts that possess an LGD of 100% and the other wth only non-defaulted credts, the portfolo s perfectly heterogeneous and the ROC curve shows a slope of one. Snce the latter porton can only default f the frst one also defaults, D cannot be larger than D 1. Therefore, the ROC curve s lnear n sectons but concave overall. For our example, a ROC curve arses accordng to Fgure 2 (see page 32). The AUC measures the steepness of the ROC curve. It corresponds to the probablty that the rank of a defaulted porton (Rank d Por ) s hgher than the rank of a non-defaulted porton (Rank nd Por ) plus half of the probablty that the rank of a defaulted porton s dentcal to the rank of a non-defaulted porton: 26 n ( AUC = (FAR FAR 1 ) HR ) + HR 1 2 = n ( far HR ) + HR 1 2 In probablstc terms, AUC reads as: n ( AUC = Prob(Rank nd Por = ) Prob(Rankd Por ) + Prob(Rankd Por 1) ) 2 = Prob(Rank d Por < Ranknd Por ) Prob(Rankd Por = Ranknd Por ) (9) 26 See Bamber (1975) for a dervaton n the context of the accuracy of dscrmnatve power. Research Paper (7) (8)

10 32 S. Hlawatsch and P. Rechlng FIGURE 2 ROC curve of the sample retal portfolo HR ROC curve of the sample retal portfolo ROC curve of homogeneous portfolo 0 ROC curve of heterogeneous portfolo FAR The fgure shows the ROC curves of a perfect heterogeneous, perfect homogeneous and of the sample portfolo. AUC equals one for a perfectly homogeneous portfolo. For a perfectly heterogeneous portfolo AUC becomes 0.5. The AUC of our sample retal portfolo shows a value of Another well-known valdaton measure s the cumulatve accuracy profle (CAP). The CAP curve measures, analogously but not equal to the Lorenz curve, the degree of nequalty, e, how the ht rates are dstrbuted over all portons. Therefore, the cumulated ht rates are plotted aganst the cumulated portons. The CAP curve, lke the ROC curve, s lnear n sectons but concave overall. If the ht rates are equally dstrbuted over all portons, the CAP curve possesses a slope of one and the portfolo s perfectly heterogeneous. If all credts possess the same LGD, the portfolo s perfectly homogeneous and the CAP curve runs lnearly rsng up to the unweghted average LGD and then horzontally. For our sample retal portfolo, a CAP curve results n accordance wth Fgure 3 (see page 33). 27 Addtonally, an nequalty coeffcent can be formed analogously to the Gn coeffcent. In order to retan the notaton of accuracy measures for ratng functons, the coeffcent s called the accuracy rato (denoted as AR n equatons) and s computed as follows: 28 ((1/n) ((HR + HR 1 )/2)) 0.5 AR = LGD/2 = 2 n HR 1 n n (1 LGD) (10) 27 The CAP curve of the perfectly homogeneous portfolo corresponds to a portfolo wth an unweghted average LGD (LGD) of 18.42% 28 See Appendx B for the dervaton. The Journal of Rsk Model Valdaton Volume 4/Number 1, Sprng 2010

11 A framework for LGD valdaton of retal portfolos 33 FIGURE 3 CAP curve of the sample retal portfolo: HR CAP curve of the sample retal portfolo CAP curve of homogeneous portfolo CAP curve of heterogeneous portfolo Cumulated percentles The fgure shows the CAP curves of a perfect heterogeneous, perfect homogeneous and of the sample portfolo. The accuracy rato ncreases the more unequally the ht rates are dstrbuted over all portons, e, the more homogeneous the portfolo s wth respect to ts LGDs. For a perfectly heterogeneous portfolo the accuracy rato becomes zero, for a perfectly homogeneous portfolo the accuracy rato equals one. Furthermore, the relatonshp between AUC and the accuracy rato s AR = 2 AUC Our sample retal portfolo possesses an accuracy rato of The ROC curve, the CAP curve, AUC and accuracy rato of the realzed LGDs provde a benchmark for the parameters and curves of the dataset of the LGD estmaton model, whch have to be computed smlarly. Afterwards, the results of both datasets can be compared wth each other. Nevertheless, t has to be consdered that the nterpretatons of ROC, CAP, AUC and accuracy rato n our LGD framework dffer from the nterpretaton of these measures n the framework of ratng accuracy. There are no perfect curves. Rather, the parameters and curves of the LGD estmaton model should devate as lttle as possble from those of the hstorcal dataset. If AUC and accuracy rato of the realzed and estmated LGDs are almost equal, then the LGD estmaton model can be seen as a good forecastng tool for future LGDs. ROC curves can ntersect each other. In ths case, area dfferences mght be compensated. It can happen that the datasets of the hstorcal losses and of the LGD estmaton model possess the same AUC and accuracy rato but the results of the LGD estmaton model are not related wth the hstorcal loss dataset. Therefore, a modfed AUC, that we call MAUC, should be computed addtonally, where MAUC 29 Ths relaton s well-known for the accuracy measure of ratng functons. For a proof wth respect to our LGD framework see Appendx C. Research Paper

12 34 S. Hlawatsch and P. Rechlng equals the sum of the absolute values of the area dfferences for each porton : where: FAR hst FAR est HR hst HR est MAUC = n ( (FAR hst FAR hst 1 ) HRhst + HR hst 1 2 ( (FAR est FAR est 1 ) HRest + HR est 2 ) 1 ) (11) = false alarm rate of porton of the hstorcal loss dstrbuton = false alarm rate of porton of the estmaton model = ht rate of porton of the hstorcal loss dstrbuton = ht rate of porton of the estmaton model. A goal wthn valdaton should be to develop LGD estmatons that mnmze the value of MAUC. In order to create a fgure that measures AUC dfferences of the two ROC curves of the hstorcal and estmaton dataset, we compute the followng sngle AUC : AUC = (FAR FAR 1 ) HR + HR 1 2 = far HR + HR 1 2 Subsequently, we suggest the followng lnear regresson: (12) AUC est = α + β AUC hst (13) where: AUC hst AUC est = AUC of the hstorcal loss dstrbuton = AUC of the estmaton model. If the LGD estmaton model perfectly forecasts future LGDs, α should be zero and β should be one. If α s sgnfcantly dfferent from zero, there s a bas n the LGD estmaton. A further lnear regresson wth suppresson of the locaton parameter shows whether the AUC of the hstorcal dataset s, on average, systematcally underestmated (β <1) or overestmated (β >1). A measure of the qualty of the estmaton model s the coeffcent of determnaton R 2 (45 ). 30 It s computed 30 A correlaton analyss of the AUC of the hstorcal and estmated datasets s not meanngful snce a smlar movement does not mply a hgh accuracy of the estmaton model. Therefore, a correlaton factor lke R 2 (45 ) should be chosen. The Journal of Rsk Model Valdaton Volume 4/Number 1, Sprng 2010

13 A framework for LGD valdaton of retal portfolos 35 as follows: R 2 (45 ) = 1 (AUC est AUC hst ) 2 (14) (AUC est AUC est ) 2 As mentoned before, the proportonal decomposton of credts has the appeal that t valdates LGD estmatons wthout sze consderatons of the EADs. Ths s reasonable f LGDs are estmated as exactly as possble to prce credts and calculate nterest rates. Imprecse estmatons of LGDs can lead to the wrong credt nterest rates for new contracts. If the LGDs are used to compute losses n euros, addtonal valdaton nstruments should be mplemented to guarantee that losses of large credts are estmated precsely. 31 For ths purpose, a margnal decomposton of the credt should be used. Ths s the subject of the followng subsecton. 2.3 Margnal decomposton of the credt In the framework of a margnal decomposton of credts, each credt s dvded nto sngle euros. 32 Here, the number of euros dffers for each credt because each credt amount vares. For each sngle euro = 1,...,EAD k of credt k = 1,...,K,now, a bnary varable d e,k s defned as follows:33 d e,k = { 1 f sngle euro of EADk s defaulted 0 f sngle euro of EAD k s not defaulted (15) Correspondng to the proportonal decomposton, agan a second bnary varable nd e,k 1 de,k s defned.34 Subsequently, the sum of d e,k and nde,k over all credts for each sngle euro, denoted by D e and NDe, respectvely, s computed. In contrast to the proportonal decomposton, the sum of D e and ND e s not equal for every sngle euro. For the margnal decomposton, the same sample retal portfolo as for the proportonal decomposton s used n order to smplfy the nterpretaton. The results of the margnal decomposton for our sample retal portfolo are shown n Table 2 (see page 36). The ht rate hr e and the false alarm rate far e are agan computed accordng to Equaton (6). The ht rate hr e equals the fracton of the sngle euro over all defaulted 31 Clearly, for a credt wth an EAD of e100, a stronger devaton of the relatve LGD estmaton s less problematc than for a credt wth an EAD of e10, It s also possble to defne a dfferent decomposton. For example a decomposton nto sngle hundred euros s more reasonable for large credt amounts. 33 The superscrpt e denotes equatons and parameters for the margnal decomposton. For a decomposton n sngle hundred euros, les between one and arg max k (EAD k /100). 34 For losses larger than the EAD, a modfcaton of the varables d e,k and nde,k, such as that for the proportonal decomposton, s also possble. Research Paper

14 36 S. Hlawatsch and P. Rechlng TABLE 2 Margnal decomposton of our sample retal portfolo. Sngle euro D e ND e 1st nd rd th th ,595th ,596th ,597th ,598th ,599th ,600th ,601st ,602nd ,603rd ,604th ,596th ,597th ,598th ,599th ,600th 0 1 The second column ndcates the number of defaults of the sngle euro and the thrd column shows the number of nondefaults of the the sngle euro. euros. The false alarm rate far e equals the fracton of the sngle euro over all euros that are not defaulted. The functon of the ht rates s monotonc, decreasng n because the second sngle euro can only default f the frst sngle euro also defaults. The functon of the false alarm rates can ncrease or decrease n because of dfferent amounts of EADs of every credt. The sum of ht rate and false alarm rate s agan a decreasng functon n. The cumulated ht rate HR e j and the cumulated false alarm rate FAR e j are computed as follows:35 HR e j = j hr e = EAD EAD LGDe j LGD e 35 For the dervaton see Appendx D. The Journal of Rsk Model Valdaton Volume 4/Number 1, Sprng 2010

15 A framework for LGD valdaton of retal portfolos 37 where: De LGD e j = EAD LGD e = De EAD where: FAR e j = j far e = EAD EAD RRe j RR e RR e j = NDe EAD RR e = NDe EAD The varable max s computed as arg max k (EAD k ) and equals the largest EAD amount of all credts of the portfolo when assumng a decomposton nto sngle euros. Here, LGD e j, LGDe, RR e j and RRe are credt weghted averages. The credt weghted LGD of our sample retal portfolo s 0.2. The ROC e curve, the CAP e curve, AUC e and AR e are computed smlarly to the proportonal decomposton. AUC e can be nterpreted as the probablty that the rank of a defaulted sngle euro s hgher than the rank of a non-defaulted sngle euro plus half the probablty that a defaulted sngle euro ranks on the same poston of a nondefaulted sngle euro. The formula for AUC e reads as follows: max( AUC e = FAR e FARe 1 HRe + ) HRe 1 2 max( = far e HRe + ) HRe 1 2 In probablstc terms, AUC e reads as: max( AUC e = Prob(Rank nd Eur = ) Prob(Rankd Eur ) + Prob(Rankd Eur 1) ) 2 (16) (17) = Prob(Rank d Eur < Ranknd Eur ) Prob(Rankd Eur = Ranknd Eur ) (18) If all credts show the same amount of loss, AUC e assumes a value of one. If the rato of defaulted to non-defaulted euros s equal for every sngle euro, AUC e assumes a value of 0.5. In contrast to the proportonal decomposton, AUC e and AR e can assume values below 0.5. The valdaton procedure s smlar to the procedure for the proportonal decomposton. After computng the valdaton ratos for realzed and estmated losses, they are compared wth each other. If the ratos are almost equal, the estmaton model can be seen as a good forecast nstrument for future losses. In order to avod mstakes n the nterpretaton caused by ntersectons of ROC curves, a modfed AUC e, denoted Research Paper

16 38 S. Hlawatsch and P. Rechlng as MAUC e, should agan be computed correspondng to Equaton (11). For statstcal valdaton of the results, R 2 (45 ) e can be computed analogously to Equaton (14). Both approaches, the proportonal and margnal decomposton, are developed to compare the estmated LGD wth the realzed LGD. To valdate the accuracy of the downturn LGD, we suggest the comparson of the estmated downturn LGDs wth realzed LGDs of an out-of-sample dataset. Ths dataset should contan an economc downturn to verfy that the estmated downturn premum s hgh enough to cover the relatve and absolute ncrease n LGDs. If the downturn premum s not a constant but a result of an estmaton model as mentoned n Secton 2.1, the accuracy of the downturn model wth an out-of-sample dataset contanng an economc boom can also be tested. Such data enables us to consder whether or not the downturn model overestmates the downturn LGD n a healthy economc envronment. 3 EMPIRICAL ANALYSIS 3.1 Data For our emprcal analyss, real loss data from a retal portfolo of a commercal bank s used. The LGD estmaton model s based on workout LGDs. It conssts of two parts. Intally, a logstc regresson for estmatng the probablty of a recovery or a wrte-off s carred out and, subsequently, a lnear regresson for estmatng LGDs for each case s run. Afterwards, the sum of both LGDs, weghted by the rates of a recovery and a wrte-off, s computed and s used as the LGD for the credt n queston. 36 The retal portfolo s splt up nto four subportfolos, dstnct as to prvate and commercal clents and collateralzed and uncollateralzed credts at default date. The four subportfolos are analyzed wth the proportonal and margnal decomposton models of Secton 2. Both n-sample and out-of-sample tests for the robustness of the estmaton model are mplemented. For the n-sample test, the complete modelng database s used for valdaton. Therefore, the AUCs of the realzed LGDs, estmated LGDs, realzed losses and estmated losses are computed. Afterwards, the MAUC and the R 2 (45 ) are calculated. For the out-of-sample test, a rejecton level s computed for the MAUC at a 90%, 95% and 99% confdence levels and for the R 2 (45 ) at a 10%, 5% and 1% confdence level usng the bootstrappng method. For bootstrappng, a random subset of the modelng database, reduced to each sngle subportfolo, s drawn 100 tmes. Afterwards, a valdaton database s used, whch was not n the modelng set. 37 The valdaton data tme perod follows the modelng data tme perod. Therefore, the out-of-sample test also represents an out-of-unverse test. The computed MAUC and R 2 (45 ) of the valdaton database are compared wth the confdence levels of the modelng database. If MAUC and/or R 2 (45 ) of the valdaton database exceed the 36 Also Hamerle et al (2006) dvde the LGD estmaton nto two parts. 37 The sze of the random subset for bootstrappng equals the porton of the valdaton database on the modelng database, computed for each subportfolo. The Journal of Rsk Model Valdaton Volume 4/Number 1, Sprng 2010

17 A framework for LGD valdaton of retal portfolos 39 TABLE 3 Results for the proportonal decomposton of the modelng database. Subportfolo Portons MAUC R 2 (45 ) Prvate clents, collateralzed Prvate clents, uncollateralzed Commercal clents, collateralzed Commercal clents, uncollateralzed Prvate clents, collateralzed Prvate clents, uncollateralzed Commercal clents, collateralzed Commercal clents, uncollateralzed Prvate clents, collateralzed 1, Prvate clents, uncollateralzed 1, Commercal clents, collateralzed 1, Commercal clents, uncollateralzed 1, The frst column ndcates the analyzed subportfolo and the second column denotes the chosen proportonal decomposton. MAUC and R 2 (45 ) are presented n the thrd and fourth column, respectvely. rejecton levels, the analyzed estmaton model s not robust wth respect to tme or sample changes. 3.2 Results Frstly, relatve LGDs are valdated usng the proportonal decomposton. Decompostons nto 100, 500 and 1,000 portons are chosen. Because some contracts of the subportfolos possess LGDs larger than one, a modfcaton accordng to Equaton (5) s used. For each subportfolo, an EAD multple s chosen, so that at least 95% of the database does not have to be modfed. 38 After computng the valdaton ratos for the realzed and estmated LGDs for all portfolos, MAUC and R 2 (45 ) are calculated for further analyss and are shown n Table 3. The LGD estmaton model analyzed n our paper s manly desgned to meet the Basel II captal requrements. Therefore, we are nterested n precse estmatons of absolute losses. However, for rsk-adjusted credt prcng, a dfferent approach wll be used. Hence, the valdaton for relatve LGDs s only of secondary mportance n the latter case. Nevertheless, three of the subportfolos possess an R 2 (45 ) larger than Only the LGD estmaton of the subportfolo prvate clents, collateralzed exhbts an nadequate accuracy. The LGD estmaton model s robust wth respect to changes n the decomposton. Therefore, for the rejecton levels of MAUC and R 2 (45 ), a proportonal decomposton of 100 for bootstrappng s used. The rejecton levels for the out-of-unverse 38 The 95% level was chosen to avod data shortenng due to outlers. Research Paper

18 40 S. Hlawatsch and P. Rechlng TABLE 4 Confdence levels of MAUC and R 2 (45 ). MAUC subportfolo 90% c.l. 95% c.l. 99% c.l. Prvate clents, collateralzed Prvate clents, uncollateralzed Commercal clents, collateralzed Commercal clents, uncollateralzed R 2 (45 ) subportfolo 10% c.l. 5% c.l. 1% c.l. Prvate clents, collateralzed Prvate clents, uncollateralzed Commercal clents, collateralzed Commercal clents, uncollateralzed The abbrevaton c.l. means confdence level. TABLE 5 Results for the proportonal decomposton of the valdaton database. Subportfolo MAUC R 2 (45 ) Rejecton Prvate clents, collateralzed Yes Prvate clents, uncollateralzed Yes Commercal clents, collateralzed No/yes Commercal clents, uncollateralzed Yes test are presented n Table 4. For MAUC, the 90%, 95% and 99% confdence level are computed. However, a lower MAUC n the valdaton database n contrast to the modelng database s not a problem, snce the estmaton model then works even better for the valdaton database than for the modelng database. For R 2 (45 ), the 10%, 5% and 1% confdence levels are computed. In the case of a hgher R 2 (45 ) n the valdaton database n contrast to the modelng database, agan the estmaton model works even better for the valdaton database. For the out-of-unverse test, MAUC and R 2 (45 ) for the valdaton database are computed usng a proportonal decomposton wth, agan, 100 portons. Subsequently, these ratos are compared wth the correspondng rejecton levels. Table 5 presents the results for the valdaton database. For every subportfolo, the R 2 (45 ) fgures of the valdaton database are smaller than the confdence level. The results are smlar for MAUC. Only for the subportfolo commercal clents, collateralzed, the MAUC fgure would not lead to rejecton. Here, t can be seen that data qualty s mportant. Snce the valdaton data tme perod follows the modelng data tme perod, an ncrease n data computaton and data collecton qualty can be assumed. Also changes n the portfolo structure or n debtor-specfc characterstcs may lead to ths result. The Journal of Rsk Model Valdaton Volume 4/Number 1, Sprng 2010

19 A framework for LGD valdaton of retal portfolos 41 TABLE 6 Results for the margnal decomposton of the modelng database. Subportfolo Margnal MAUC R 2 (45 ) Prvate clents, collateralzed Prvate clents, uncollateralzed Commercal clents, collateralzed Commercal clents, uncollateralzed Prvate clents, collateralzed Prvate clents, uncollateralzed Commercal clents, collateralzed Commercal clents, uncollateralzed Prvate clents, collateralzed Prvate clents, uncollateralzed Commercal clents, collateralzed Commercal clents, uncollateralzed The frst column ndcates the analyzed subportfolo and the second column denotes the chosen margnal decomposton. MAUC and R 2 (45 ) are shown n the thrd and fourth column, respectvely. The next step s the valdaton of losses n euros usng the margnal decomposton. Therefore, decompostons nto sngle e100, e50 and e10 are chosen. The EAD multples reman the same as for the proportonal decompostons. Agan, AUCs of realzed and estmated losses are calculated at the begnnng. Subsequently, the comparson of both realzed and estmated losses s done by usng MAUC and R 2 (45 ). The results of the n-sample test for the margnal decompostons are presented n Table 6. The estmatons of losses n euros are more precse for every subportfolo than the estmatons of relatve losses. Every subportfolo possesses an R 2 (45 ) above 66%, two subportfolos even show an R 2 (45 ) above 90%. Therefore, the estmaton model represents an approprate model for estmatng absolute losses, whch are needed to determne the captal requrements accordng to Basel II. Thus, the am of the bank, e, the development of an estmaton model for absolute losses, s acheved. The results are robust wth respect to changes n the margnal sze. Therefore, for computng the rejecton levels va bootstrappng, a margnal decomposton nto sngle e100 s chosen. The rejecton levels for the out-of-unverse test are presented n Table 7 (see page 42). To compare the modelng database wth the valdaton database, frstly, a margnal decomposton nto sngle e100 of the valdaton database s done. Afterwards, MAUC and R 2 (45 ) of the valdaton database are compared wth the correspondng rejecton levels. If the valdaton ratos exceed the rejecton levels, the estmaton model s not robust wth respect to addtonal new data. Table 8 (see page 42) shows the correspondng results. Research Paper

20 42 S. Hlawatsch and P. Rechlng TABLE 7 Confdence levels of MAUC and R 2 (45 ). MAUC subportfolo 90% c.l. 95% c.l. 99% c.l. Prvate clents, collateralzed Prvate clents, uncollateralzed Commercal clents, collateralzed Commercal clents, uncollateralzed R 2 (45 ) subportfolo 10% c.l. 5% c.l. 1% c.l. Prvate clents, collateralzed Prvate clents, uncollateralzed Commercal clents, collateralzed Commercal clents, uncollateralzed The abbrevaton c.l. means confdence level. TABLE 8 Results for the margnal decomposton of the valdaton database. Subportfolo MAUC R 2 (45 ) Rejecton Prvate clents, collateralzed Yes Prvate clents, uncollateralzed No Commercal clents, collateralzed Yes Commercal clents, uncollateralzed Yes/no The reasons for rejectons are the same as for the proportonal decomposton. Agan, an ncrease n data qualty and possble changes n the portfolo structure may lead to dfferences. However, the magntude of msspecfcaton of absolute losses s smaller than the msspecfcaton of relatve LGDs. Though two subportfolos are rejected by both measures, the R 2 (45 ) fgures of all subportfolos are above 50% and for two subportfolos even above 75%. Thus, the estmaton model for absolute losses s stll applcable to forecast future losses. Therefore, the estmaton model can be mplemented to determne Basel II captal requrements. 4 CONCLUSION The dea of ths paper was to develop an LGD valdaton method for retal portfolos. Ths topc s mportant because banks have to estmate LGDs f they want to apply for the nternal ratngs based approach for retal portfolos. The Basel II regulatons postulate that retal portfolos have to be homogeneous concernng, at least, rsk drvers such as borrower and transacton rsk characterstcs and delnquency of exposure. 39 Ths makes t dffcult to develop LGD ratng or scorng models for retal portfolos because of ther smlar characterstcs. 39 See Basel Commttee on Bankng Supervson (2004, paragraph 402). The Journal of Rsk Model Valdaton Volume 4/Number 1, Sprng 2010

21 A framework for LGD valdaton of retal portfolos 43 In contrast, our suggested proportonal and margnal decomposton methods are applcable wthout LGD ratngs. Furthermore, there are no specfc requrements for the LGD estmaton model. Even for arthmetc mean estmatons of the LGD for retal portfolos, both methods can be used. Our methods use valdaton nstruments, that are well-known from PD valdaton, where the nterpretaton s dfferent. The used nstruments, eg, AUC and accuracy rato, are accepted by supervsory authortes. Because of the dfferent nterpretaton, the rato tself contans no nformaton about the accuracy of the estmaton model. In fact, the rato descrbes the composton and structure of the portfolo. To valdate the estmaton model, t s necessary to compare the ratos calculated on realzed LGDs or losses wth those based on estmated LGDs or losses. Therefore, AUC and accuracy rato of the hstorcal database provde a benchmark for the ratos of the estmated values. If the measures are smlar, the estmaton model can be seen as a good forecastng tool for future losses. It also turns out that the proportonal decomposton s credt sze ndependent. Thus, the method s also an nstrument to prove the functonalty of the estmaton model over all sngle credts n the portfolo. The proportonal decomposton can reveal possble weaknesses n the estmaton model. Therefore, banks can avod arrangng ther models such that LGDs for credts wth hgh EADs are estmated more precsely, whle estmatons for credts wth smaller EADs are mprecse. Ths fact s mportant f the estmaton model s to be used for credt prcng, where a precse estmaton of the LGD s mportant for the calculaton of the contract s nterest rate. Furthermore, our approaches can also be used to valdate downturn premums by usng out-of-sample datasets contanng ether an economc downturn or an economc boom. We also showed that the models work on real data and that out-of-sample and out-of-tme tests can easly be mplemented. APPENDIX A DERIVATION OF HIT RATES AND FALSE ALARM RATES In the context of LGD valdaton, the sngle and cumulatve ht rates and false alarm rates, respectvely, can be transformed as follows: hr = D D = K ND D = 1 = LGD n ND LGD n K j HR j = hr = LGD n = j (1 LGD j ) j LGD n K D ND LGD n K = = j n LGD j LGD ND D j LGD n (1 LGD j ) j LGD n Research Paper

22 44 S. Hlawatsch and P. Rechlng far = ND ND = = 1 RR n j FAR j = far = K D ND = D RR n K j RR n = j (1 RR j ) j RR n D RR n K = = j n RR j RR K ND APPENDIX B DERIVATION OF ACCURACY RATIO D ND j RR n (LGD j ) j RR n The accuracy rato, assocated wth the LGD-based CAP curve, can be rearranged as follows: ((1/n) ((HR + HR 1 )/2)) 0.5 AR = LGD/2 ((1/n) ( LGD /(n LGD)+( 1) LGD 1 /(n LGD))/2) 0.5 = 0.5 LGD/2 = ( n ( LGD ) 0.5 n LGD)/(n 2 LGD) LGD = 2 n ( LGD ) n LGD n 2 LGD n 2 LGD (1 LGD) = 2 n HR 1 n n (1 LGD) APPENDIX C LINEAR RELATIONSHIP BETWEEN ACCURACY RATIO AND AREA UNDER CURVE We prove that the well-known lnear relatonshp of AUC and accuracy rato n the context of measurng PD-based accuracy also holds n the framework of LGD valdaton: n ( AUC = FAR FAR 1 HR ) + HR 1 2 n ( RR = n RR ( 1) RR ) 1 n RR LGD /(n LGD) + ( 1) LGD 1 /(n LGD) 2 The Journal of Rsk Model Valdaton Volume 4/Number 1, Sprng 2010

23 A framework for LGD valdaton of retal portfolos 45 ( LGD ) 0.5 n LGD 0.5 n 2 LGD 2 = n 2 RR LGD HR n LGD = n (1 LGD) 2 AUC 1 = 2 n HR 1 n LGD n (1 LGD) 1 = 2 n HR 1 n LGD n (1 LGD) n (1 LGD) = 2 n HR 1 n n (1 LGD) APPENDIX D DERIVATION OF HR e AND FAR e = AR For the margnal decomposton of the analyzed credt portfolo, the ht rates and false alarm rates, respectvely, can be determned as follows: hr e = D e De j HR e j = hr e = D e l = l=1 De l=1 = EAD l EAD LGDe j LGD e far e = ND e NDe j FAR e j = far e = ND e l = l=1 NDe l=1 = EAD l EAD RRe j RR e l=1 De l De l=1 NDe l NDe EAD l=1 EAD l EAD l=1 EAD l APPENDIX E DATA OF THE SAMPLE RETAIL PORTFOLIO l=1 EAD l EAD l=1 EAD l EAD Credt EAD n e Loss n e Credt EAD n e Loss n e 1 7, , , , , , , ,200 6,500 Research Paper

24 46 S. Hlawatsch and P. Rechlng APPENDIX E Contnued Credt EAD n e Loss n e Credt EAD n e Loss n e 5 8, , , ,500 11, , , , , , ,400 11, , ,100 10, , ,700 7, ,600 4, , ,000 2, ,000 13, , , ,900 4, , ,200 2, ,800 18, , ,500 1, ,500 4, , , ,700 16, ,400 13, ,200 10, , ,800 10, ,800 13, ,200 4, , , ,700 3, ,300 2, , ,900 10, , , ,600 4, ,900 7, , ,200 17, , , ,900 8, , ,000 7, , ,000 5, ,900 23, , ,400 11, , ,600 13, , ,100 17, ,500 3, , , ,700 7, ,800 4, ,800 22, , ,200 19, ,000 2, ,300 11, , ,600 16, ,300 10, ,700 25, ,400 24, ,000 5, ,600 9, ,100 20, ,700 6, ,200 18, ,900 10, , , , ,200 8, ,400 17, ,500 7, ,500 23, , ,600 23,300 Sum 3,000, ,000 The Journal of Rsk Model Valdaton Volume 4/Number 1, Sprng 2010