Débats économiques et financiers N 1

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1 Débats économques et fnancers N 1 How dfferent s the regulatory captal from the economc captal: the case of busness loans portfolos held by the major bankng groups n France Mchel Detsch * et Henr Frasse ** [*] Autorté de Contrôle Prudentel- Drecton des Etudes et Unversté de Strasbourg. Contact : [email protected] (**) Autorté de Contrôle Prudentel- Drecton des Etudes

2 SECRETARIAT GENERAL DE L AUTORITE DE CONTROLE PRUDENTIEL DIRECTION DES ÉTUDES How dfferent s the regulatory captal from the economc captal: the case of busness loans portfolos held by the major bankng groups n France Mchel Detsch and Henr Frasse February 2013 Les ponts de vue exprmés dans ces Débats Economques et Fnancers n engagent que leurs auteurs et n exprment pas nécessarement la poston de l Autorté de Contrôle Prudentel. Ce document est dsponble sur le ste de l Autorté de Contrôle Prudentel : The opnons expressed n the Economc and Fnancal Dscusson Notes do not necessarly reflect the vews of the Autorté de Contrôle Prudentel. Ths document s avalable on Drecton des Études - SGACP 1

3 How dfferent s the regulatory captal from the economc captal: the case of busness loans portfolos held by the major bankng groups n France Abstract: There s a growng concern about the dfferences between rsk weghted assets (RWAs) amounts across banks and across countres. Ths paper provdes new evdence on ths ssue by usng French Credt Regster data and frms ratngs hstores of more than French frms, ncludng a large proporton of Small and Medum szed frms, to compute captal requrements n busness loans portfolos of French bankng groups. Usng Credt Regster nformaton and ratngs provded by the Banque de France ratng system allows computng captal requrements by usng a sngle common credt rsk metrcs and actual emprcal rates of default at the bank s exposure level. Usng ths nformaton, captal ratos are computed for each bankng group operatng n the French busness loans market. The paper addresses the ssue of the ablty of Basel 2 Internal Ratng Based (IRB) formulas to hedge portfolo s credt rsk. Here, the analyss reles on an extenson of the asymptotc sngle rsk factor model (ASFR), whch was used for the calbraton of Basel II regulatory formulas. Therefore, a multfactor portfolo s credt rsk model s mplemented to compute economc captal requrements takng account of potental credt rsk concentraton n busness loans portfolos. The paper compares the requred captal ratos provded wth ths model wth the one requred by the regulaton. Our man fndngs are frstly that regulatory captal ratos do not underestmate credt rsk: Basel II regulatory captal requrements are larger than the economc captal requrements. Secondly, the sngle rsk factor regulatory model does not capture potental dversfcaton effects n busness loans portfolos. In the regulatory model, frms heterogenety s only captured by ther ratngs. The ntroducton n the portfolo credt rsk modelng of addtonal systematc rsk factors - whch are here sze and sector show that managng large portfolos composed of borrowers of dfferent sze or sector helps to produce dversfcaton effects. Such stuatons lead n most cases to a decrease of the captal level requred to cover future unexpected losses. Keywords: Credt Rsk, economc captal, regulatory captal, busness loans JEL Classfcatons : G21, G28, G32 De comben le captal réglementare s écarte t-l du captal économque: le cas des prêts aux entreprses par les grands groupes franças Résumé : Il exste aujourd hu un débat sur la qualté des actfs des banques européennes et une éventuelle sousévaluaton du rsque de crédt dans l utlsaton des formules réglementares par les banques. Ce paper apporte des éléments de réponse à ce débat en mesurant les exgences en captal réglementares sur les portefeulles de crédts aux entreprses résdentes des sx premers groupes bancares opérant en France et en les comparant aux exgences en captal économque mesurées notamment en utlsant un modèle multfacteur de rsque de crédt de portefeulle. Ce modèle permet de tenr compte de l hétérogénété des entreprses emprunteuses en les dstnguant selon la talle ou le secteur. Il permet auss de détecter d éventuels effets de concentraton de portefeulle lés à l exstence de stuatons de défauts corrélés. Le paper explote les encours de crédt et l hstorque des ratngs de quelques entreprses, ncluant une fracton mportante de PME, dsponbles va la Centrale des Rsques et le système de cotaton des entreprses de la Banque de France pour la pérode Ces données permettent de calculer les exgences en captal en utlsant le même crtère objectf de défaut et le même système d évaluaton du rsque de crédt ndvduel des entreprses pour toutes les banques analysées. Le premer résultat est que les formules d exgences réglementares ne sous-estment pas les le rsque de crédt de portefeulle. Les exgences réglementares calculées sont supéreures aux exgences économques dans une très large majorté de segments de portefeulle construts à partr des crtères de talle et de secteur. Le paper montre auss que le modèle réglementare surestme la sensblté des emprunteurs au cycle et sousestme le potentel de dversfcaton de portefeulle. Le potentel est ms en évdence dans ce paper dès lors que les facteurs de rsque addtonnels assocés à la dfférencaton des entreprses sont ntégrés dans l analyse. Au total, les exgences en captal économque sont trées à la basse par des effets de dversfcaton nduts par l hétérogénété des emprunteurs. Mots-clés : Rsque de crédt, captal économque, captal réglementare, prêts aux entreprses Code JEL : G21, G28, G32 Drecton des Études - SGACP 2

4 Contents 1. INTRODUCTION THE DATA THE METHODOLOGY A SHORT VIEW OF THE MULTIFACTOR MODEL THE MULTI-FACTOR MODEL AS BENCHMARK FOR CAPITAL REQUIREMENTS MEASUREMENT THE RESULTS: COMPARISONS OF REGULATORY AND ECONOMIC CAPITAL RATIOS TAKING ACCOUNT FOR BORROWERS HETEROGENEITY AND POTENTIAL CREDIT RISK CONCENTRATION: THE FIRM SIZE AS AN ADDITIONAL SOURCE OF SYSTEMATIC RISK TAKING ACCOUNT FOR BORROWERS HETEROGENEITY AND POTENTIAL CREDIT RISK CONCENTRATION: THE FIRM SECTOR AS AN ADDITIONAL SOURCE OF SYSTEMATIC RISK CONCLUSION REFERENCES APPENDIX : CREDIT RISK MODEL SPECIFICATION THE ASYMPTOTIC MULTI-FACTOR CREDIT RISK FRAMEWORK THE ECONOMETRIC ESTIMATION OF THE PORTFOLIO S CREDIT RISK PARAMETERS MEASURING POTENTIAL CONCENTRATION Drecton des Études - SGACP 3

5 How dfferent s the regulatory captal from the economc captal: the case of busness loans portfolos held by major bankng groups n France Mchel DIETSCH 1 and Henr FRAISSE 2 1. Introducton There s a growng concern about the dfferences between rsk weghted assets (RWAs) amounts across banks and across countres. EBA (2012) as well as the Basel Commttee (2012) have recently launched new workng groups to assess the extent of the dfferences and to delver explanatons of ther orgns. Some nternatonal banks have expressed doubts about the relablty of banks RWAs and the consstency and comparablty of captal requrements. Such doubt about the relablty of banks RWAs could have major consequences. In partcular, nvestors could dsregard regulatory captal ratos and requre hgher captal to compensate for the low perceved relablty of the captal rato s denomnator. Then the rsk s that they restrct lendng to banks for whch they have doubts about reported captal adequacy. Prevous papers have provded an overvew of the concerns surroundng the dfferences of rskweghted assets (RWAs) across banks and jursdctons and how ths mght undermne the Basel III captal adequacy framework. They have proposed assessments of the key drvers behnd these dfferences, drawng upon samples of mportant banks n Europe, North Amerca, and Asa Pacfc. Among the man drvers whch have been proposed to explan such dscrepances are the dfferences n regulatory envronment, accountng rules, the poston of the country n the economc cycle whch nfluence the level of the probabltes of default, and, fnally, the dfferences n banks busness models across regons of the world (Le Leslé and Avramova, 2012, Cannata, Casellna and Gud, 2012, and several notes comng from fnancal analyss departments of nternatonal banks). However, the ablty of RWAs to reflect bank portfolos credt rsk could be questoned, at least for two reasons. The frst reason deals wth the the modellng of dependency across oblgors. As emphaszed by the Basel Commttee (BCBS, 2008), ths ssue s a man challenge of portfolo credt rsk measurement. Asset correlatons quantfy ths dependency. Asset correlatons measure the common senstvty of borrowers to systematc rsk factors, whch are macroeconomc, ndustry or geographc factors. If the 1 Autorté de Contrôle Prudentel, Drecton des Etudes, et Unversté de. Strasbourg. Contact : [email protected] 2 Autorté de Contrôle Prudentel, Drecton des Etudes Drecton des Études - SGACP 4

6 correlaton s hgh, ths senstvty to systematc rsk factors s hgh, and n the case where extreme values of the systematc factors append, losses wll clmb to very hgh levels. Asset correlaton reflects the uncertanty assocated to events whch can produce extremes losses. Thus, asset correlatons are a crucal metrcs n portfolo s credt rsk measurement. As nput parameters nto a credt rsk model, correlatons affect the credt portfolo Value-at-Rsk (VaR) whch measures credt rsk n a portfolo. Thus, the modelng of ndvdual asset correlatons has a strong mpact on VaR for credt portfolos of heterogeneous borrower sze, suggestng that the omsson of ndvdual dependences can substantally reduce the VaR estmate 3. That sad, to reflect portfolo s credt rsk, asset correlatons should be computed usng nternal data. Credt ratngs alone do not reflect the uncertanty assocated wth forecastng tal credt loss events. However, n the regulatory formulas defnng RWAs, asset correlaton R s entrely determned by the PDs. The second reason deals wth potental concentraton n loans portfolos. Concentraton s another man drver of credt rsk n a portfolo. Concentraton rsk n loan portfolos could come from name concentraton (the ncomplete dversfcaton of dosyncratc borrower rsk) and sector concentraton (the exstence of multple systematc rsk factors, generally related to ndustry or geographc effects). Correlated defaults can be attrbuted to the dependency of credt exposures to common factors that are specfc to some segments of the portfolo or to partcular banks clenteles. If frm heterogenety s defned as heterogenety of rsk factor loadngs across frms, t characterzes loan portfolos due to dfferences n sze 4, sector or localzaton of borrowers. Thus, takng account for potental concentraton effect mples to consder borrowers heterogenety. BCBS (2006) underlnes that concentraton of credt rsk n asset portfolos has been one of the major causes of bank dstress. However, the calbraton of the IRB formulas was allegedly chosen to match the economc rsk n a credt portfolo that should be very-well dversfed across ndustres. Consequently, regulatory formulas do not take nto account borrowers heterogenety and potental concentraton effects comng from potentally correlated defaults across borrowers belongng to the same portfolo s segment and whose fnancal stuaton s drven by systematc rsk factors whch are specfc to ther group. Takng account explctly for concentraton phenomena mples to use multfactor framework. Departures from the underlyng assumptons of the sngle factor model,.e. perfect granularty and a sngle source of systematc rsk could result n substantal devatons of economc captal requrements from regulatory captal requrements. 3 Recent studes n the credt rsk lterature (Tarashev and Zhu, 2007, Hetfeld, 2008, Coval and al. 2009) show that credt rsk models man sources of errors generally come from a msspecfcaton of default dependences. To compute credt rsk n a loans portfolo, t s necessary to characterze the entre jont dstrbuton of payoffs for the loans pool. 4 One should note that IRB formulas actually dffer wth the turnover of the frms and the sze of the exposure. The RWA are computed dfferently whether the busness s classfed n the retal portfolo, the SME portfolo or the corporate portfolo. However, the dfferences n the formulas appled to these portfolos do not stem from the sngle rsk factor theoretcal model underlyng the regulatory framework. Drecton des Études - SGACP 5

7 The am of ths study s to evaluate the ablty of the regulatory captal requrements formula to hedge the portfolo credt rsk. To ths am, we compare the level of captal requrements computed by usng regulatory Basel 2 formula to the level of captal computed by usng a model of portfolo credt rsk whch take nto account multple sources of rsk as well as borrowers heterogenety. Therefore, we have extended the asymptotc sngle rsk factor (ASRF) model to a multfactor framework whch takes account for addtonal systematc rsk factors, such as sze or sector factors. The frst advantage of ths approach s that results obtaned n a multfactor framework are consstent wth results provded by the regulatory approach, allowng drect comparsons of economc and regulatory captal requrements. The second advantage s that takng addtonal rsk factors nto account allows detectng potental dversfcaton benefts n banks loans portfolos, or on the contrary, potental credt rsk concentraton due to correlated defaults. Indeed, credt rsk concentraton could be defned as a stuaton of strong correlated defaults n a portfolo s segment, what nduces a larger number of defaults and hgher losses. In that perspectve, takng account for concentraton mples to decompose the portfolo n segments accordng to the choce of addtonal rsk factors. For nstance, followng ths logc, n ths paper, we segment the portfolo of each bankng group n four sze segment. Then, the concentraton measurement reles on the computaton of the margnal contrbuton of each segment to the portfolo s total losses. If a segment s contrbuton to losses s hgh, that means that losses are concentrated n ths segment, requrng more captal to cover unexpected losses. On the other hand, f the contrbuton s weak, there s a great chance that ths segment contrbutes to portfolo s dversfcaton. To conduct our quanttatve study, we use nformaton about busness loans portfolos contaned n the French Credt Regster ( Fcher Central des Rsques ) on a quarterly bass over the perod. Ths database ncludes all loans of all knds (short term, long term, leasng) wth an amount over Euros provded by French banks to ther busness customers. As a matter of fact, the bulk of busness loans portfolos s bult up by loans to SME. We consder the potental for correlated defaults nsde the portfolo of large bankng groups lendng to busnesses operatng n France, takng successvely sze and sector as addtonal systematc rsk factors. We use ths nformaton to compute captal requrements n each of the sx major bankng groups operatng n the French busness loans market. We compare three captal ratos: a) the regulatory rato usng the Basel II IRB formulas, b) the economc multfactor rato computed by usng a multfactor model whch takes nto account frm sze and frm sector as addtonal rsk sources, and fnally c) an economc sngle factor rato, whch uses the standard ASRF model to compute asset correlatons, and replaces correlatons computed by usng regulatory formulas by asset correlatons computed n ths way. Results show frstly, that the sngle rsk factor regulatory model does not success to capture potental concentraton or dversfcaton effects n strongly granular busness loans portfolos. In ths model, frms heterogenety s only captured by ther ratngs. The ntroducton n the portfolo credt rsk Drecton des Études - SGACP 6

8 modelng of addtonal systematc rsk factors - whch are here frm sze and frm sector show that stuatons of strongly correlated defaults could exst n certan segments of the portfolos or, on the contrary, that some segments could produce a dversfcaton effect. Such stuatons determne an ncrease or a decrease of the captal level requred to cover future unexpected losses, as compared to the regulatory level, dependng of the case. Secondly, on average, Basel II regulatory captal requrements are larger than the economc captal requrements, ether n the sngle or n the multfactor approach. In other words, our results demonstrate that present RWAs formulas do not under-estmate portfolo credt rsk, at least when consderng French busness loans portfolos, and that whatever the bankng group. Secton 2 presents the database. Secton 3 presents and justfes the use of the multfactor credt rsk model as a benchmark. Secton 4 shows comparsons of three measures of the captal rato to treat the ssue of RWAs consstency and relablty. Secton 5 concludes. 2. The data In ths paper, we explot the dversty of the portfolos composton across bankng groups. To compute regulatory and economc captal requrements, we use two sources of nformaton. The frst one s the French Credt Regster, (Fcher Central des Rsques, FCR), whch ncludes all loans of all knds (short term, long term, leasng) wth an amount over Euros provded by French banks to busnesses. We have extracted from the FCR all loans suppled by the sx large French bankng groups. We have retaned loans to ndustral and commercal sectors, excludng fnancal sector and state or muncpal servces sector. The perod of the study covers years from 2000 to 2011, ncludng the fnancal crss perod. The FCR provdes also nformaton about frms characterstcs, such as sze, ndustry, geographcal locaton. The second source of nformaton s the Banque de France (BDF) ratngs system ( Cotaton BDF ), for whch the BDF was recognzed as an External Credt Assessment Insttuton (ECAI) 5. The BDF ratngs system provdes ratngs for qute all frms whose turnover s over 0.75 M. However, even f the system provdes ratngs for mcro-busnesses (very small frms wth turnover lower than 0.75 M ), f ther 5 The frm s credt rsk assessment provded by the Banque de France Credt Regster and ratngs system s very safe, due to the fact that the database s very representatve, the threshold level guaranteeng a quasexhaustve coverage of the French busnesses populaton. Another beneft of the ratng system s ts permanent updatng, what allows an nstantaneous evaluaton of rsk. In addton, the Banque de France operates a close montorng of frms knowng fnancal dffcultes, what provdes a soluton to the ssue of mssng accountng nformaton for such frms. Fnally, these databases provde the exclusve possblty to estmate rsk n the real portfolos of the French banks accordngly to the same ratng system. Drecton des Études - SGACP 7

9 exposure s amount s greater than 0.35 M, we do not consder the latter populaton n ths study 6. The BDF ratngs system ncludes twelve ratngs grades. Among them, two refer to default states: ) legal falure, whch s bankruptcy, and ) bank default, whch corresponds n the BDF ratngs system to severe bankng problems ncdents bancares séreux. Takng together these two crterons of default help to catch a set of default stuatons whch s qute close to the set of default stuatons usng the Basel II default crteron, especally n the small busnesses populaton. So, usng ths database, t s possble to compute annual rates of quas-bank default and to dstngush them by ratngs grade. In ths study, these default rates were computed dstnctvely for four sze classes: a) very small frms, wth turnover between M 0.75 and M 1.5, b) small frms, wth turnover between M 1.5 and M 10, c) medum-szed frms, wth turnover between M 10 and M 50, and d) ntermedate sze frms and large frms, wth turnover over M 50. These emprcal rates of default were used as proxes for probabltes of default (PDs) n the Basel II captal requrements formulas. The permeter of ths study s the populaton of French frms whch fulfll four condtons: ) they have exposures n the Credt Regster, ) the BDF ratng department gves them a ratng (ncludng default grades), ) they get loans from at least one or several of the sx major bankng groups operatng n the French loans to busnesses market, and v) ther annual turnover s over 0.75 M. The populaton contans a very large number of French frms (more than frms on average each year). Table 1 shows the number of n the sample by sze classes. All types of busness loans are ncluded n the total amount, whatever the maturty or the object of loans. The loans amount of the sample s frms represents a total of more than 650 bllon Euros on average over the perod. The market share of the sx studed groups n the French busness s around 70% durng the perod. In addton, busness loans exposures of the studed bankng groups represent on average around 40% of the groups total assets. The sample does not only reflect the realty of the busness loans market but t s also very representatve of the French busnesses populaton. 6 The man reason s that the Banque de France ratngs system covers only that part of ths populaton whch s composed of frms wth exposures amounts hgher than euros. Consequently, relable nformaton about ndvdual credt rsk s mssng for most of the mcro busnesses populaton. Another reason comes from the fundamental heterogenety n terms of credt rsk of the mcro busnesses populaton, whch mxes personal affars such as doctors, lawyers,, - wth very small frms operatng on compettve markets.. Drecton des Études - SGACP 8

10 Table 1: Number of frms by sze n the sample (6 bankng groups) Very small frms Small frms Medum-szed frms Intermedate and large frms Total number of frms Total exposures of the sx groups (bn ) Share of the sx groups n total exposures (n%) Source: ACP-BDF and Authors computatons. Table 2 shows the dstrbuton of sze and sector composton of loans portfolos across bankng groups. Sgnfcant dfferences across groups appear, especally n what concerns the share of the very small frms or the largest frms (ntermedate sze and large sze frms). Sgnfcant dfferences n ndustry composton across groups also appear. In partcular, some groups are characterzed by a hgh share of those sectors whch are closest to the fnal consumers (retalng, servces to households) whle others lend more to manufacturng and constructon and real estate sectors. Table 2: Dstrbuton of sze and ndustry composton of busness loans portfolos across bankng groups (year 2011) mn mean max Sze classes Very small 16.2% 28.1% 36.1% Small 14.5% 24.2% 28.8% Medum-szed 8.3% 19.1% 26.1% Intermedate & large 9.0% 28.6% 61.0% Industres Agrculture 0.5% 1.6% 3.8% Constructon & real estate 14.9% 32.7% 42.4% Manufacturng 14.3% 19.8% 22.9% Retal 7.3% 10.9% 21.7% Wholesale 8.0% 11.1% 15.5% Transport 4.8% 7.1% 10.4% Servce to busness 4.0% 5.2% 12.5% Servces to households 5.6% 11.7% 14.9% Source: ACP-BDF and Authors computatons Note: ths table reproduces the mnmum, the mean and the maxmum fracton of exposures across bankng groups and by portfolo segment. For llustraton, n term of exposures, the bankng group the less exposed to the Agrgulture sector hods 0.5% of ts total exposure on ths segment. Drecton des Études - SGACP 9

11 To compute captal ratos, we have used the Basel II formulas n the Internal Ratngs Based Foundaton (IRBF) approach, what means usng the other retal captal requrements formula when the exposure s amount s lower than 1 mllon Euros and the correspondng borrower turnover s below 50 mllon Euros, and usng the corporate captal requrements formula, takng account for sze adjustment when the frm s turnover s lower than 50 mllon Euros, when the exposure s amount s hgher than 1 mllon Euros. In ths paper, we do not use the banks regulatory expected PDs but nstead the observed emprcal rates of default at the one year horzon as proxes for PDs to compute regulatory as well as economc captal requrements. These default rates are computed as the number of frms gong to default state durng the year relatve to the total number of frms n safe condton at the begnnng of the same year. Table 3 presents average annual rates of default at the one year horzon by frm sze, gvng a frst vew of the credt rsk structure n the sample under study. The table shows that the level of the default rates tends to decrease wth frm sze. It shows also that credt qualty tends to vary wth the busness cycle, wth a sgnfcant downgrade n Table 3: Average observed rates of default at the one year horzon by sze n the sample (n %) Very small busnesses Small frms Medum-szed frms Intermedate and large sze frms Source: ACP-BDF and Authors computatons As mentoned before, n ths paper, we compare captal requrements n busness loans portfolos at the bankng group s level. Therefore, we compute captal requrements at the level of each large bankng group s portfolo (the French Credt Regster allows to dstngush banks portfolos) and we express captal requrements n terms of captal ratos. Now, to assess the ablty of regulatory captal requrements to cover portfolo credt rsk, we need to use other measures of captal requrements as benchmarks. As argued before, our choce s to use a structural credt rsk multfactor model to compute captal requrements n an economc perspectve, takng account for multple sources of rsk. Drecton des Études - SGACP 10

12 3. The methodology To assess the exstence of a potental bas n the estmaton of captal charges assocated wth the prevous regulatory captal requrements formulas, we compare regulatory captal requrements wth captal requrements computed by usng a more comprehensve economc approach provded by a multfactor portfolo credt rsk model. In a multfactor framework, we have to determne the rsk factors. In a frst step, we nclude frm sze as addtonal systematc rsk factor and n a second step we consder frm sector as addtonal factor. The choce of these factors rely on recent research that shows that concentraton exsts n busness loans portfolos and that credt rsk vares n portfolos accordng to ther ndustry and sze composton (Carlng, Ronnegard and Roszbach, 2004, Detsch and Petey, 2004, Duellmann and Scheule, 2003, Hetfeld, Burton and Chomssengphet, 2006, Duellmann and Masschelen, 2006). Recall that regulatory formulas do not consder such factors. However, for the comparson, we compute regulatory captal requrements and economc captal requrements at the same dsaggregated level of portfolo s sze or sector segments we use to mplement the multfactor model. In what follows, frst, we gve a short presentaton of the methodology. A more detaled presentaton s n the appendx of ths paper. Then we explan why the captal requrements measures derved from a multfactor framework can be used as benchmarks A short vew of the multfactor model The multfactor model belongs to the class of structural credt rsk models 7. It s n fact an extended verson of the standard asymptotc sngle rsk factor ASRF model. The extenson conssts to ntroduce addtonal factors varyng across groups of borrowers. We have expanded the model by addng new latent factors of systematc rsk that can be lnked to observable characterstcs of borrowers. Such an extenson to a mult-factor model mproves substantally the computaton of the dependency structure (asset correlatons) across exposures n a typcal loans portfolo. Usng ths approach permts n partcular to compare the credt rsk n groups of borrowers gettng ther loans from dfferent bankng groups. The extenson of the ASRF framework allows takng account for potental credt rsk concentraton whch s lnked to borrowers heterogenety. In small portfolos of large exposure concentraton rsk comes from name concentraton. But n large portfolos of busness loans, whch are hghly granular, 7 See appendx for a complete presentaton of the model. Drecton des Études - SGACP 11

13 concentraton rsk arses from correlated defaults among groups of borrowers. Then, measurement of concentraton rsk needs to proceed to an approprate portfolo s segmentaton able to reflect borrowers heterogenety. Here, we adopt as crteron of segmentaton the belongng of an exposure to the busness loans portfolo of one of the fve French bankng groups we consder n ths study. To compute economc captal n ths framework, we proceed n two steps. The frst step s devoted to calculus of portfolos man rsk parameters, and n partcular the dependence structure among exposures measured by asset correlatons. The second step uses Monte-Carlo smulaton to buld the probablty dstrbuton functon of losses, determne the total portfolo VaR and compute the level of captal requrements assocated to each addtonal systematc factor whch are n ths study specfc to bankng groups and ther busness model and lendng polcy. Let consder brefly the frst step. As econometrc specfcaton of the multfactor credt rsk model, followng Frey and Mc Nel, 2003, and McNel and Wendn, 2006) we use the methodology of generalzed lnear mxed models (GLMM). Thus, takngs frms credt ratngs hstores to buld tme seres of rates of default by portfolo s segment, we get estmates of portfolo s credt rsk parameters n a mult-factor context. The GLMM model mplements n a coherent way the Merton latent factor default modelng approach, n whch the default occurs when the value of the frm s assets become smaller than the value of ts debt, that s, because frm s assets values are dffcult to observe, when the value of a latent varable descrbng the fnancal stuaton of the frm - whch depends on the realzaton of a set of rsk factors - crosses an unobservable threshold whch determnes the default. In ths framework, the default rate s modeled as: P ( default γ ) = Φ[ x' μ + z' γ ] t t r n whch the default rate depends on ) a fxed effect measured by the borrower s nternal ratng (μ r ), and ) random effects (γ t ), whch are related to a general latent factor (the state of the economy), augmented by a set of factors correspondng to a gven segmentaton of the portfolo. t t The GLMM model produces estmates of default thresholds consdered as fxed effects and covarance matrxes of a set of latent random effects correspondng to the set of systematc factors. The estmaton of such parameters allows computng economc captal as buffer of losses n portfolos exposed to dfferent systematc rsk factors. Let consder now the second step. In the structural credt rsk framework, measurng concentraton rsk calls for allocatng economc captal between segments of borrowers,.e. to compute margnal contrbutons of dfferent segments to portfolos total losses. A portfolo s segmentaton s bult by dentfyng groups of borrowers wth the same observable characterstcs whch expose them to the same Drecton des Études - SGACP 12

14 rsk factors. In a mult-factor context, captal allocaton can be mplemented at the segment level such that t s possble to nvestgate the heterogenety n captal allocaton nduced by the varous rsk factors. Thus, a sngle factor homogeneous framework could nduce a msrepresentaton of the concentraton rsk even n large portfolos of retal exposures. Whle they are calbrated usng a sngle factor framework, Basel 2 IRB regulatory formulas of captal requrements could be of lmted nterest n allocatng captal. The computaton of the portfolo s value-at-rsk (VaR) and margnal rsk contrbutons are made by usng a methodology proposed by Tasche (2009), whch grounds on an mportance samplng based smulaton of expected condtonal losses. Ths methodology has the advantage to take nto account the mpact of borrowers heterogenety on economc captal charges and captal allocaton The mult-factor model as benchmark for captal requrements measurement At ths stage, t s mportant to note that there s a relatonshp between regulatory captal requrements and economc captal requrements derved from a multfactor model. In fact, the Basel II rsk weght formulas were calbrated usng a smplfed verson of a portfolo credt rsk model, the Asymptotc Sngle Rsk Factor (ASRF) model. In ths framework (see Gordy, 2003), bank s total captal requrements s computed by usng two parameters whch refer to frm s ndvdual rsk, whch are the probablty of default (PD) and the loss gven default (LGD), and a thrd parameter - the asset correlaton R whch measures the senstvty of borrowers to a common sngle systematc rsk factors, whch s a macroeconomc undetermned rsk factor. The asset correlaton reflects the fact that default rates are volatle and that ths volatlty depends on ther senstvty to a systematc rsk factor. If the correlaton s hgh, ths senstvty s strong and, n case of realzatons of extreme unfavorable value of the systematc rsk factor, losses wll clmb to hgher levels. Thus, more generally, asset correlatons reflect the potental for jont defaults n a portfolo. Followng ths approach, n Basel II rsk weghtng formulas, under the IRB approach, RWAs depend upon these three credt rsk parameters. So, regulatory RWAs are consstent measures of credt rsk. However, as mentoned before, two calbraton choces determne potental dfferences between regulatory captal requrements and economc captal requrements, what justfes to compare the two types of measures. Frstly, n the regulatory formulas of Basel II, asset correlatons R are entrely determned by the PDs. In the ASRF model, asset correlatons measure the senstvty of loans to a macroeconomc rsk factor and they should vary from one portfolo to another one, dependng on the composton of the portfolo. But, n practce, Basel II provdes banks wth the formulas to compute R, nstead to leave them computng ths rsk parameter usng nternal nformaton. Consequently, RWAs depend strongly upon the value of the asset correlatons and the man dfference between regulatory captal requrements and economc captal Drecton des Études - SGACP 13

15 requrements computed by usng banks nternal data comes from the value of assets correlatons. So, one ssue arses to know f the regulatory asset correlatons computed by usng regulatory formulas and the related regulatory captal requrements - are dfferent from the economc asset correlatons computed by usng banks nternal data. Secondly, n the ASRF model, there s a sngle general undetermned credt rsk factor whch represents the state of the economy. However, borrowers are not equally senstve to common systematc rsk factors. In addton, borrowers fnancal heath s lnked to multple sources of credt rsk whch are more or less specfc to the rsk segment to whch they belong. Takng account for borrowers heterogenety oblges to expand the standard sngle rsk factor model and to adopt a multfactor framework. In a multfactor framework, groups of borrowers are exposed to addtonal systematc rsk factors whch are specfc to ther segment. It s mportant to emphasze that these addtonal rsk factors could renforce or attenuate the nfluence of the general systematc rsk factor. Moreover, a multfactor model allows detectng potental concentraton (dversfcaton) effects comng from the strong (weak) dependence of borrowers to rsk factors whch are specfc to ther own rsk segment. In case of realzaton of unfavorable value of one systematc rsk factor, the number of defaults wll ncrease and losses wll clmb to hgher levels. In such a case, the contrbuton to the portfolo s segment whch s exposed to ths rsk factor wll rase, nducng an ncrease n total losses. More generally, f the sensblty of exposures to the systematc rsk factor whch s specfc to ther segment s hgh, the relatve contrbuton of ths segment to the portfolo s total losses wll be hgh, what corresponds to a stuaton of credt rsk concentraton n that segment. So, n a portfolo composed of several segments, usng a multfactor model allows to compute the margnal contrbuton of each segment to total losses and observe ether the mpact of ths segment on the concentraton of losses or, on the contrary, the role the segment plays n the dversfcaton of the portfolo credt rsk. In practce, ths margnal contrbuton can be expressed under the form of a captal rato by relatng captal requrements needed to cover potental unexpected losses produced to ths segment (computed at a gven percentle - for nstance 99.9% - of the probablty dstrbuton functon of losses) to total exposures of the segment. In ths way, we can assess portfolo s concentraton and dversfcaton n terms of captal rato as a common metrcs, showng how sze and sector factors could contrbute to ncrease or decrease the level of captal requrements relatve to the level gven by a sngle rsk factor model. Drecton des Études - SGACP 14

16 4. The results: comparsons of regulatory and economc captal ratos To conduct our analyss, we do not have access to complete detaled bankng groups nternal nformaton, and, n partcular, to banks nternal rate of defaults. However, usng BDF ratngs hstores of French frms as well as nformaton about ther debts provded by the French Credt Regster gves us the opportunty to use an as-f approach and to compute very consstently nternal asset correlatons and economc captal requrements. A major advantage of ths approach s to consder a sngle rsk metrc the BDF ratngs- across banks. Recall that data whch are used to mplement the econometrc analyss and compute portfolos credt rsk parameters (among them the dependence structure shown by the covarance matrxes of rsk factors) are: ) the tme seres of observed rates of default n the dfferent segments for each bankng group over the perod 8, and ) the loans amounts n the French Credt Regster. The emprcal rates of default were used as proxes for probabltes of default (PDs) n the Basel II captal requrements formulas. We use ths nformaton to compute captal requrements n each of the sx French major bankng groups. In each case, we compare three captal ratos: - the regulatory rato usng the Basel II IRB Foundaton formulas, - the economc multfactor rato computed by usng a multfactor model whch takes nto account frm sze and frm sector as addtonal rsk sources, - the economc sngle factor rato computed by usng the standard ASRF model n whch the rsk factor s a general undetermned factor e.g. not constraned by the regulatory formula. It s nterestng to compute also the sngle factor rato, because the dfference between captal requrements measures provded by the standard sngle factor model and the regulatory model reles drectly on the value of asset correlaton whch s computed usng portfolos default rates dynamc n the ASRF model whle, as mentoned before, t s gven by regulatory formulas n the regulatory model. On the other sde, the dfference between captal requrements provded by the sngle rsk ASRF model and the multfactor model llustrates the role the addtonal rsk factors play n the determnaton of portfolos losses. In the multfactor framework as well as n the sngle factor framework, total portfolo s requred captal s computed by smulaton of the rsk factors gven default thresholds and rsk factor senstvtes, whch are the outputs of an econometrc model explanng the volatlty of default rates over the perod. Gven the credt rsk parameters and a set of smulated rsk factors, defaults n each sub-portfolo defned by crossng four sze segments - or eght ndustres - wth sx ratng grades are produced by 8 As mentoned before, n the BDF ratngs system, two ratngs refer to default states: ) legal falure, whch s bankruptcy, and ) bank default, whch corresponds n the BDF ratngs system to severe bankng problems ncdents bancares séreux. We took the two forms of default to compute annual rates of default and to dstngush them by ratngs grade. Drecton des Études - SGACP 15

17 drawng from a bnomal probablty wth the number of frms n each sub-portfolo and the condtonal default probablty defned by econometrc analyss result as parameters. Exposures are then defned as the average of frms total loans amounts wthn classes crossng ratng and szes,.e. we assume at ths stage homogenety n exposures wthn portfolos segments. In what follows, frst, we wll present results we obtaned when decomposng the busness loans portfolo of each major bankng group n four sze segment, takng frm sze as addtonal systematc rsk factor. Then, we present results obtaned when takng frm sector as addtonal systematc rsk factor. 4.1 Takng account for borrowers heterogenety and potental credt rsk concentraton: the frm sze as an addtonal source of systematc rsk Here, we consder frm sze as a rsk factor. The bankng groups portfolos were dvded n the four sze classes we defned prevously, that s: - a) very small frms, wth turnover between M 0.75 and M 1.5, - b) small frms, wth turnover between M 1.5 and M 10, - c) medum-szed frms, wth turnover between M 10 and M 50, - d) ntermedate and large frms, wth turnover over M 50. Usng ths segmentaton, we compute captal ratos assocated to each segment consderng three models: the sngle factor model, the multfactor model and the regulatory model. All nformaton concerns the perod and s treated on an early bass. It s lkely that the portfolos under consderaton are hghly granular due to ther sze. Therefore, dfferences n captal requrements would come from credt rsk concentraton whch corresponds to stuatons of strongly correlated defaults. If oblgors were homogenous n terms of credt rsk, captal ratos should not dffer across oblgors and/or portfolos segments. On the contrary, f oblgors are heterogeneous, a hgher captal rato n a gven segment would ndcate potental rsk concentraton. A straghtforward source of heterogenety s credt ratng whch s accounted for n the econometrc analyss by the estmaton of default thresholds. Sze could be an addtonal source of credt rsk heterogenety n SME portfolos. If there s concentraton rsk, captal ratos should vary along ths source of rsk. Thus, the heterogenety n captal charges wll manly come from the rsk factors affectng the dfferent portfolos segments. Drecton des Études - SGACP 16

18 However, table 2 has shown that, apparently, there s not a consderable potental for credt rsk concentraton when consderng the sze of the portfolos and the dstrbuton of exposures nto sze classes n the large bankng groups operatng n France. Indeed, there are patent dfferences n terms of portfolos sze and the composton of the portfolo vares from one group to another one. Notce that one excepton could come from the hgher share of ntermedate and large frms segment. But, all n all, these observatons suggest that the evoluton of potental credt losses n large portfolos mght be sustanable when consderng concentraton rsk. It s what our results tends to verfy. Table 4 shows the covarance matrxes of random effects n the sze model at the aggregate level of a global portfolo composed of all exposures of the sx bankng groups. The covarance matrxes show average values of covarances over the tme perod, takng account for default rates volatlty over tme. Table 4 shows also the mnmum and maxmum values of covarances across bankng groups 9. There s a consderable systematc component drvng the volatlty of default rates. Indeed, the varance assocated to the "general" factor, whch s the random ntercept n the GLMM model, has very hgh values. Secondly, the sze class wth the largest random effect s the very small busnesses class, wth an order of magntude hgher to the general factor. The random effects assocated to the other sze classes are generally small or equal to zero. Moreover, the general and the sze specfc rsk factors are negatvely and qute strongly correlated. Ths reflects lower rsk levels, ths negatve correlaton dampenng the fluctuatons of the general rsk factor. Sze factors and general factors tend to compensate to generate a lower level of credt rsk. Thus, results suggest a very low potental for rsk concentraton on most of sze segments. Fnally, the estmated covarance matrces of random effects do not suggest a contnuous and convex relatonshp between rsk and sze at the aggregate level. 9 Results at the ndvdual bankng group level not presented here - show that all covarance matrces share qute the same pattern when consderng the sze factors. However, dfferences across banks may exst, manly n what concerns the medumszed frms segment.such dfferences mean ether that banks are makng dfferent portfolo s choces or that they encounter dfferent envronmental condtons Drecton des Études - SGACP 17

19 Table 4: Covarance matrces results A: portfolo composed of all exposures Very small Small Medum-szed Intermedate & large General Very small Small Medum-szed Intermedate & large General B: mnmum values Very small Small Medum-szed Intermedate & large General Very small Small Medum-szed Intermedate & large General C: maxmum values Very small Small Medum-szed Intermedate & large General Very small Small Medum-szed Intermedate & large General Source: ACP-BDF, Drectorate Research Notes: for llustraton : n the top panel A, corresponds to the correlaton of borrowers belongng to the very small busnesses portfolo to the systematc rsk factor related to ths sub portfolo. A hgh level of correlaton corresponds to a hgh level of concentraton wthn the segment corresponds to the correlaton between the general systemc factor and the sze specfc systemc factor. A large negatve value captures a dversfcaton effect mtgatng the rsk wthn the portfolo. Panel B and C reproduce the mnmum and maxmum values of the components of the correlaton matrx across bankng groups. Table 5 shows the dstrbuton of: () the rato of regulatory captal requrements over captal requrements gven by a multfactor model, and () the rato of regulatory captal requrements over captal requrements gven by a sngle factor model across bankng groups 10. A rato hgher than 1 means that regulatory captal requrements are larger than economc requrements. Another vew s to consder that a rato hgher than 1 n one gven segment demonstrates that dversfcaton effects comng 10 Here, captal ratos represent average values of ratos over the perod. They are computed usng the average rsk parameters (rate of default, covarances) values over the perod. Ths perod ncludes two downturn epsodes, such that these average values could be consdered as through-the-cycle values. These values and especally correlatons, can change n case of realzaton of extreme events. However, two remarks could be made on ths ssue. Frstly, because our methodology uses mportance samplng technques, values of probablty dstrbuton of losses could be consdered as stressed values. Secondly, t s possble to obtan stressed values of credt rsk parameters by changng the observaton wndow n order to measure the nfluence of bad years or bad realzatons of the addtonal sze or sector rsk factors. Drecton des Études - SGACP 18

20 from the dependence structure n ths segment are hgh. The result could come ether from low value of covarance of random effect n ths segment or from compensaton between the rsk factor specfc to the sze segment and the general factor. Notce that the regulatory captal ratos are computed usng the IRB formulas of Basel II. The other retal formulas were used for the segments of mcro-frms, very small frms and small frms, whle the corporate formula was used for the medum-szed frms segments. Man results come as follows. Table 5: Dstrbuton of the ratos of regulatory to economc captal dstrbuton across bankng groups, by sze of frms Sze of frms n the portfolo Multfactor model Sngle factor model mn mean max mn mean Max very small 2,0 3,4 6,5 1,5 1,8 2,0 small 1,2 1,5 2,7 1,3 1,6 1,9 medum-szed 0,9 2,1 4,6 2,2 2,7 3,5 ntermedate & large 1,4 1,8 2,0 1,4 1,8 2,0 Notes 1. Left panel: rato between, on the one hand, regulatory captal requrements, and, on the other hand, captal requrements derved from the multfactor model; rght panel: rato between, on the one hand, regulatory captal requrements, and, on the other hand, captal requrements derved from the sngle factor model model. 2. Regulatory captal ratos are computed accordngly to the other retal basel formula for the very small and small busnesses and accordngly to the corporate Basel 2 formula for the other frm sze classes. Source: ACP-BDF, Drectorate Research Frstly, at the aggregate level for each group, the regulatory captal rato does not underestmate credt rsk. It remans true when lookng at specfc portfolo segments. Indeed, the level of captal requrements computed usng the multfactor model are lower than the regulatory captal requrements computed usng the Basel II formulas, except for the portfolo s segment composed of loans to medum-szed frms n some bankng groups where regulatory captal rato s lower than economc captal ratos (see for nstance the mnmum value of 0.9 n that sze segment). Ths result shows that, n practce, even f sze concentraton effects, whch the regulatory approach of captal requrements does not take nto account, could exst, the aggregate level of regulatory captal requred to cover total portfolos unexpected losses do cover de facto potental concentraton effects. However, n the (rare) cases where the level of the regulatory captal rato s lower than the level provded by the sze multfactor model, regulatory formulas could nduce dstortons n the captal allocaton across sze segments. The soluton to ths problem would be to complement the regulatory captal requrements to take nto account results provded by a multfactor approach. Drecton des Études - SGACP 19

21 Secondly, the comparson of the ratos of regulatory requrements to economc captal requrements computed by usng the sngle rsk model shows that the regulatory formulas overestmate asset correlaton. In other terms, regulatory approach overestmates the senstvty of exposures to the busness cycle approxmated by the general systematc rsk factor. Thrdly, the comparson of the ratos of regulatory captal requrements to economc captal requrements gven by a multfactor factor and a sngle model shows that the latter s n most cases lower than the former. Thus, takng nto account addtonal factors specfc to sze segments shows tends to lower n most case the economc captal requrements. Therefore, credt rsk concentraton due to a sze effect appears to be lmted. Moreover, the value of the ratos are close from one sze to another one, what means that, on average, t seems better to manage portfolos composed of exposures of all szes than to manage portfolos concentrated on one or a lttle number of szes. Ths result whch holds for qute all bankng groups n all sze segments vares n ntensty across groups. Indeed, the dsperson of the ratos shows clearly the exstence of dvergences between bankng groups when consderng the level of economc captal. The potental for dversfcaton gven by the sze rsk factor vares across bankng groups and banks are not equally effcent n managng the composton of ther portfolos by frm sze or, at least, they do not have the same opportuntes to extract dversfcaton benefts. Table 2 does show dfferences n the allocaton of credt by frm sze across bankng groups. Therefore, ths result s partly the consequence of dfferent bankng groups polcy n terms of suppled loans amount by frm. But, another mportant factor explanng the dfferences of captal rato comes from the dfferences n the dependence structure between oblgors by sze Takng account for borrowers heterogenety and potental credt rsk concentraton: the frm sector as an addtonal source of systematc rsk The prevous secton tred to detect potental concentraton lnked to frm sze n bankng groups busness loans portfolos. Here, we present results consderng potental concentraton comng from sector systematc rsk factors. To proceed, we segment portfolos exposures by ndustry and use the same methodology we used to consder sze effects. If concentraton s strong wthn a sector, ths sector wll strongly contrbute to the potental losses and hgher level of captal wll be requred. Table 6 shows the covarance matrces of random effects n the sze model at the aggregate level of a global portfolo composed of all exposures of the sx bankng groups. Table 4 shows also the mnmum Drecton des Études - SGACP 20

22 and maxmum values of covarances across bankng groups11. Here, the varance assocated to the "general" factor, whch s the random ntercept n the GLMM model, s not so hgh. The systematc component drvng the volatlty of default rates s not so hgh when one consders the sector segmentaton. However, the values of the random effects assocated to ndustry are often sgnfcant and very close from one ndustry to anoter one. In some case, such as n the servces to busness sector, the largest random effect s qute hgh. Results also show that ndustry random effects could ether compensate or renforce the general factor random effect. The general and the ndustry specfc rsk factors are negatvely or postvely correlated dependng of the ndustry. For nstance, n the constructon and real estate sector, they renforce each other, nducng larger captal requrements, whle n the servces to busnesses they compensate. In total, results suggest a low potental for rsk concentraton on most ndustry segments. 11 Results at the ndvdual bankng group level not presented here - show that all covarance matrces share qute the same pattern when consderng the sze factors. However, dfferences across banks may exst, manly n what concerns the medumszed frms segment. Such dfferences mean ether that banks are makng dfferent portfolo s choces or that they encounter dfferent envronmental condtons Drecton des Études - SGACP 21

23 Table 6: Covarance matrces results A: portfolo composed of all exposures Agrculture Manufacturng Constructon Retal Wholesale Transport Servces to Servces to General ndustry & real estate busnesses households Agrculture Constructon & real estate Manufacturng Retal Wholesale Transport Servces to busnesses Servces to households General B: mnmum values Agrculture Manufacturng Constructon Retal Wholesale Transport Servces to Servces to General ndustry & real estate busnesses households Agrculture Constructon & real estate Manufacturng Retal , Wholesale , Transport Servces ,1321 to busnesses Servces to households General , C: maxmum values Agrculture Manufacturng Constructon Retal Wholesale Transport Servces to Servces to General ndustry & real estate busnesses households Agrculture 0, Constructon & real estate Manufacturng Retal Wholesale Transport Servces to busnesses Servces to households General Drecton des Études - SGACP 22

24 Table 7 shows ratos of regulatory captal to economc captal. Man results come as follows. Table 7: Dstrbuton of the ratos of regulatory captal to economc captal by ndustry across bankng groups Regulatory captal requrements over captal requrements gven by a multfactor model Regulatory captal requrements over captal requrements gven by a sngle factor model mn mean max mn mean max agrculture 0,9 6,8 8,6 1,6 2,0 2,7 constructon & real estate 2 11,4 14,3 1,3 1,7 2 manufacturng 0,9 4,8 6,5 1,3 1,7 1,9 retal 3,9 13,6 14,4 1,4 1,9 2 wholesale 2,3 8,7 13,9 1,3 1,7 2 transport 1,4 9,3 9,1 1,5 2,0 2,8 servce to busness 1,9 19,8 26 1,6 2,1 2,8 servces to households 1,4 9,0 10,5 1,4 1,9 2,6 Source: ACP-BDF, Drectorate Research Note: regulatory captal ratos are computed as the weghted average of requrements computed usng the other retal Basel 2 IRBF formula when the loan amount s lower than 1 mllon and accordngly to the corporate formula when t s hgher. Weghts are the respectve amounts of the two borrowers populatons. Frstly, when comparng regulatory captal ratos and multfactor or sngle factor captal ratos, we come agan to the concluson that regulatory captal rato does not underestmate credt rsk. The level of captal requrements computed usng the multfactor framework are most of the tme lower than the regulatory captal requrements computed usng the Basel II formulas, except n the manufacturng sector (see mnmum value n the table). Ths result shows that even f the regulatory approach of captal requrements does not take nto account sector credt rsk concentraton, the aggregate level of regulatory captal cover total portfolos unexpected losses. However, as mentoned above for sze concentraton, regulatory formulas could nduce dstortons n the captal allocaton across ndustry. Secondly, the comparson of the ratos of regulatory requrements to economc captal requrements computed by usng the sngle rsk model shows as n the sze segmentaton case that the regulatory formulas overestmate asset correlaton. Thus, regulatory approach overestmates the senstvty to the busness cycle. Thrdly, the comparson of the ratos of regulatory captal requrements to economc captal requrements gven by a multfactor factor and a sngle model shows that the latter s n most cases sgnfcantly lower than the former. Multfactor captal rato s hgher than sngle factor captal rato n a Drecton des Études - SGACP 23

25 large number of sectors. Therefore, credt rsk concentraton due to an ndustry effect appears to be lmted. However, the value of the ratos are dfferent from one ndustry to another one, what could be the consequence of the ndustry composton n the real world but also means that, on average, t seems possble to manage portfolo s ndustry composton to modfy the portfolo s credt rsk level. It s more effcent to manage ndustry dversfed portfolos than portfolos concentrated on one or a lttle number of sectors. 5. Concluson Ths paper provdes results amng to answer to the ssue of the ablty of regulatory captal requrements to hedge busness loans portfolo credt rsk. The paper consders the case of busness loans portfolos held by the sx major French bankng groups. The paper uses a unque data combnng nformaton provded by the French Credt Regster and the Banque de France frms ratngs system. One of the man benefts of ths source of nformaton s t allows treatng real French banks busness loans portfolos. Another beneft comes from the use of objectve measures of credt rsk (rates of default) whch are common to the portfolos of all major French bankng groups. The detecton and measurement of credt rsk n large portfolos needs the extenson of the regulatory framework n order to ntroduce addtonal sources of systematc rsk n the modelng of credt rsk. In partcular, to measure consstently credt rsk n busness loans portfolos, we have to take nto account potental credt rsk concentraton. Therefore, ths paper uses a multfactor portfolo s credt rsk model, whch s an extenson of the standard asymptotc sngle rsk factor model, to compute economc captal requrements takng account for such concentraton phenomena. Here, addtonal rsk factors are assocated to sze and ndustry portfolos segmentaton. Our fndngs demonstrate that the heterogenety captured by credt ratngs, the only source of heterogenety n the asymptotc one factor framework, fals to descrbe the effectve heterogenety n default rates wthn large portfolos. Other factors mght be at play. Indeed, sze and ndustry rsk factors appear to have sgnfcant effects on the heterogenety of credt rsk n busness loans portfolos. Results show that addtonal ndustry factors tend to lower the captal requrements due to rsk mtgaton or dversfcaton effects. Our fndngs also demonstrate that regulatory captal rato does not underestmate credt rsk. The level of captal requrements computed usng the multfactor framework are most of the tme lower than the regulatory captal requrements computed by usng the Basel II formulas. A possble caveat s that we use a quas-bank default ndcator, whch s a mx of judcary defaults and a restrcted set of bankng defaults nstead of the Basel II defnton of default, to Drecton des Études - SGACP 24

26 assess the probablty of default. However, the dfference s sgnfcantly larger not to overturn our concluson. Moreover, ths rsk metrc s mmune to the crtque that the parameters of the banks s nternal models can be msspecfed and be the source of an underestmaton of the rsk taken. Ths result shows that even f the regulatory approach of captal requrements does not take nto account sector or sze credt rsk concentraton, the aggregate level of regulatory captal cover total portfolos unexpected losses. However, n very rare cases where the level of the regulatory captal rato s lower than the level provded by the sze or sector multfactor models, regulatory formulas could nduce dstortons n the captal allocaton across sze or sector segments. The soluton to ths problem would be to complement the regulatory captal approach to take nto account results provded by a multfactor approach. In that sense, the new supervsory gudelnes callng for the use of economc captal models are welcome. Addng new rsk factors allows controllng for potental rsk concentraton whch mght affect the level of captal requred to protect the banks solvency aganst unexpected losses. Even f busness loans portfolos are hghly granular, correlated defaults may generate credt concentraton rsk whch pushes the level of economc captal above the level of regulatory captal. But, n fact, at least n the portfolos we have consdered n ths study, our fndngs show that addtonal rsk factors do not contrbute to ncrease economc captal requrements. Our results show that concentraton effect s qute lmted on average n most bankng groups portfolos. On the contrary, strong dversfcaton effects seem to be at play n French busness loans portfolos. Drecton des Études - SGACP 25

27 References BCBS, 2006, Basel II: Internatonal Convergence of Captal Measurement and Captal Standards: A Revsed Framework - Comprehensve Verson, Basel Commttee on Bankng Supervson. BCBS, 2006, Studes on credt rsk concentraton, Bank for Internatonal Settlements, Basel Commttee on Bankng Supervson, WP No. 15. BCBS, 2008, Range of practces and ssues n economc captal modellng, Consultatve document, Basel Commttee on Bankng Supervson. Breslow N.E., X. Ln, 1995, Bas correcton n generalzed lnear mxed models wth a sngle component of dsperson, Bometrka 82, Cannata F., S. Casellna and G. Gud, 2012, Insde the labyrnth of RWAs: how not to get lost, Bank of Italy w.p, March 27th. Carlng K., T. Jacobson, J. Lnde, K. Roszbach, 2007, Corporate credt rsk modellng and the macroeconomy, Journal of Bankng and Fnance 31, Carlng K., L. Ronnegard, K. Roszbach, 2004, Is frm nterdependence wthn ndustres mportant for portfolo credt rsk?, Sverges Rksbank workng paper seres, n 168. Coval, J. J. Jurek, and E. Stafford, (2009), The economcs of structured fnance, Harvard Busness School, wp Detsch M., J. Petey, 2004, Should SME exposures be treated as retal or corporate exposures? A comparatve analyss of default probabltes and asset correlatons n French and German SMEs, Journal of Bankng and Fnance 28, Detsch M., H. Frasse, and J. Petey, Is SME credt rsk overestmated?, note DE/ACP-BDF, June Duellmann K., H. Scheule, 2003, Asset correlaton of German corporate oblgators: Its estmatons, ts drvers and mplcatons for regulatory captal. Paper presented at Bankng and Fnancal Stablty: A Workshop on Appled Bankng Research, Banca d ltala, Rome, March. Duellmann, K., N. Masschelen, 2006, The Impact of Sector Concentraton n Loan Portfolos on Economc Captal, Fnancal Stablty Revew, ed. Natonal Bank of Belgum, Duellmann K., M. Schecher, C. Schmeder, 2007, Asset correlatons and credt portfolo rsk: An emprcal analyss, Workng paper, Deutsche Bundesbank. EBA, 2012, Prelmnary results of the analyss concernng observatons n RWA and other regulatory outcomes, March 26th. Egloff D., M. Leppold, S. Jöhr, C. Dalbert, 2005, Optmal mportance samplng for credt portfolos wth stochastc approxmaton, workng paper. Frey, R., A. McNel, 2003, Dependent defaults n models of portfolo credt rsk. Journal of Rsk 6 (1), Glasserman P., J. L, 2005, Importance samplng for portfolo credt rsk. Management Scence, 51(11), Drecton des Études - SGACP 26

28 Gouréroux C., J.P. Laurent, O. Scallet, 2000, Senstvty analyss of Values at Rsk, Journal of Emprcal Fnance 7, Hetfeld N., S. Burton, S. Chomssengphet, 2006, Systematc and dosyncratc rsk n syndcated loan portfolos, Journal of Credt Rsk 2 (3), Hetfeld E (2008), Lessons fom the crss n mortgage-backed securtes: where dd credt ratngs go wrong? n Rethnkng Rsk Measurement and Reportng, pp Hgham N., 2002, Computng the nearest correlaton matrx: a problem from fnance, IMA Journal of Numercal Analyss 22, Klebaner F. C., Introducton to Stochastc Calculus wth Applcatons. Imperal College Press, second edton, Le Leslé Vanessa and Sofya Avramova, 2012, Revstng Rsk-Weghted Assets: Why Do RWAs Dffer Across Countres and What Can Be Done About It? IMF workng paper 12/90. Ln X., 1997, Varance component testng n generalsed lnear models wth varance component, Bometrka 84, Ln X., N.E. Breslow, 1996, Bas correcton n generalzed lnear mxed models wth multple components of dsperson, Journal of the Amercan Statstcal Assocaton 91, Lopez J.A., 2004, The emprcal relatonshp between average asset correlaton, frm probablty of default and asset sze, Journal of Fnancal Intermedaton 13, Lopez J.A., M.R. Sadenberg, 2000, Evaluatng credt rsk models, Journal of Bankng and Fnance 24, Lucas A., P. Klaassen, P. Sprej, S. Straetmans, 2001, An analytc approach to credt rsk of large corporate bond and loan portfolos, Journal of Bankng and Fnance 25, McNel A., J. Wendn J, 2007, Bayesan nference for generalzed lnear mxed models of portfolo credt rsk, Journal of Emprcal Fnance 14, Pesaran M.H., T. Schuermann, B.J. Treutler, 2005, The role of ndustry, geography and frm heterogenety n credt rsk dversfcaton, IEPR workng paper Tarashev, N., and H. Zhu, (2007), Modellng and calbraton errors n measures of portfolo credt rsk, BIS wp. N 230. Tasche, D., 1999, Rsk contrbutons and performance measurement, Workng paper, Technsche Unverstät München. Tasche D., 2006, Measurng sectoral dversfcaton n an asymptotc mult-factor framework, Journal of Credt Rsk 2 (3), Tasche D., 2009, Captal allocaton for credt portfolos wth kernel estmators, Quanttatve Fnance 9(5), Wolfnger R., M. O Connell, 1993, Generalzed lnear mxed models: A pseudo-lkelhood approach, Journal of Statstcal Computaton and Smulaton 4, Drecton des Études - SGACP 27

29 Appendx : Credt rsk model specfcaton 12 The asymptotc mult-factor credt rsk framework Losses at the portfolo level can be defned as the sum of ndvdual losses on defaultng loans n the portfolo, adjusted for the severty of ndvdual losses; n other words, portfolo-level losses may be regarded as the sum of the losses gven default for each ndvdual loan n the portfolo that goes unpad. Thus, f u s defned as the loss gven default (LGD) of an oblgor and f 1 D s defned as the default ndcator varable of oblgor, then the total portfolo losses L may be computed as follows: L = n u 1 D = 1 In structural credt rsk models, such as the model devsed by Merton (1974), default occurs f the value of an oblgor s assets s smaller than the value of the oblgor s debt that s due. Because asset and debt values may be dffcult to observe, ths framework has been extended by generalzng the modelng of default as the crossng of an unobservable threshold. Thus, the fnancal health of oblgor s represented by a latent (unobservable) varable U, and the level of U s determned by the realzatons of rsk factors that satsfy the followng condtons: U = w ' s + 1 w' R w ε (1) where S s a vector of systematc rsk factors wth realzaton s, w s the vector of senstvtes (or factor loadngs) of the -th borrower to the systematc factors, and ε s a specfc rsk factor for borrower. In the above equaton, R s the correlaton matrx of the rsk factors. Assumng that the rsk factors are multvarate Gaussan, the senstvty to specfc rsk n equaton (1) ensures that U s standard normal. Specfc rsk factors are assumed to be uncorrelated among oblgors and also ndependent from the systematc factors. In ths framework, default occurs f the latent varable U falls below a default threshold that s calbrated n accordance wth the statonary (long term) default probablty p of oblgor. In other words, f the standard normal cdf s denoted by Φ, then default occurs when the followng condtons are satsfed: 1 D = 1 U = w ' s + 1 w ' R w ε < Φ 1 ( p ) 12 Ths appendx reles on Detsch, Frasse, and Petey (2012). Drecton des Études - SGACP 28

30 Moreover, assumng specfc rsk can be entrely dversfed away, then the realzed losses can be approxmated by ther expected value condtonal to s. Condtonal portfolo losses can then be defned as follows: L () s n = 1 ( p ) 1 Φ w ' s uφ (2) 1 w ' R w Ths framework s known as the asymptotc mult-factor framework of credt rsk (e.g., Lucas et al., 2001). Equaton (2) assumes that each oblgor can be characterzed by hs ndvdual default threshold and factor senstvtes. However, n retal loan portfolos, default rates are generally computed based on ratng classes, and senstvtes to rsk factors cannot be computed on an ndvdual bass. Thus, assumptons are requred to reduce the number of parameters of the loss varable. In partcular, we assume that oblgors who belong to the same ratng notch r wll share the same default threshold. We further assume that the vector of rsk factor senstvtes wll be the same for oblgors who belong to the same segment of a portfolo. Hence, assumng the exstence of a portfolo that s composed of K segments, losses can be rewrtten as follows: L () s n = 1 ( p ) 1 Φ r uφ 1 w k w k ' s ' R w k Thus, the adopton of a mult-factor structural approach of credt rsk requres not only the specfcaton of the dependence structure of rsk factors but also the approprate estmaton of default thresholds and factor senstvtes. The econometrc estmaton of the portfolo s credt rsk parameters In ths study, we extend the sngle factor model by consderng new latent factors that can be lnked to the observable characterstcs of borrowers. To estmate default thresholds and factor senstvtes, we use an econometrc model that belongs to the class of generalzed lnear mxed models (GLMMs) that combnes fxed and random effects for observable and (latent) unobservable factors. Detaled presentatons of GLMM models n credt rsk modelng can be found n Frey and McNel (2003) and McNel and Wendn (2007). If Y s defned as the (N 1) vector of observed default data and f γ s defned as the (K 1) vector of random effects, then the condtonal expected default probablty of oblgor may be expressed as follows: [ Y = 1 γ ] = g( Xβ Zγ ) E + Drecton des Études - SGACP 29

31 where g( ) s a dfferentable monotonc lnk functon, Y s the default ndcator varable for oblgor (Y takes a value of 1 f there s a default and equals 0 otherwse), X s a (N P) matrx that contans the (observed) fxed effects, and Z s the (N K) desgn matrx for the random effects. In the followng applcatons, we wll focus on the probt lnk functon because the normal dstrbuton s the underlyng lnk functon that s assumed by the Basel 2 framework of credt rsk; thus, g(x) = Φ(x). The random effects are assumed to follow a multvarate standard normal dstrbuton wth covarance matrx G. In the equaton above, β s the vector of parameters that s assocated wth fxed effects. Consderng a portfolo of N oblgors who are categorzed nto r = 1,, R (non-default) ratng classes and gven a vector γ t of random effects, the default probablty of borrower at tme t may be expressed as follows: P ( Y = 1 γ ) = Φ( x' μ + z' γ ) t t where μ r denotes the vector of parameters from the fxed effect of the borrower s ratng class. If the ratng scale s properly bult, we expect these thresholds to be ordered and ncreasng as credt qualty decreases. In the above equaton, ' = [ 0,..., 1,...0] t t r t x s a (1 R) vector of dummes that defnes the ratng of borrower durng tme perod t. Because we assume that borrowers wthn segments are nterchangeable, the estmaton of ths vector does not nvolve ndvdual borrowers but nstead uses the quarterly default rates wthn segments. Ths approach leads to an assumpton of borrower homogenety for each credt ratng that s examned. Extendng the one factor model also calls for a specfcaton of the rsk factors dependence structure. By assumng that the general rsk factor (the rsk factor of the one factor model) represents the mpact on default rates of varatons n general economc condtons, t seems straghtforward to consder that addtonal rsk factors can renforce or weaken the senstvty of a gven subset of frms n the portfolo to general economc condtons. Ths corresponds to the dea that a gven sector or regon can be ether procyclcal, cycle neutral or countercyclcal. In order to capture these effects, we estmate the correlaton between the general rsk factor and a set of addtonal factors assocated to a gven segmentaton of the portfolo. In order to keep the model tractable, we further assume that the addtonal factors,.e. shocks that affect subgroups of the portfolo, are ndependent. Ths specfcaton mples n partcular that the nter-segment correlaton s not drectly attrbutable to the segments rsk factors but rather to the dependence between these latter factors and the general economc factor. The covarance structure we wll focus on s of the form: 2 σ1 0 G = 0 σ1, q + 1 σ 0 O 0 2, q σ 2 q K σ σ 1, q+ 1 2, q+ 1 σ M 2 q+ 1 Drecton des Études - SGACP 30

32 consderng q latent segment factors and one systematc factor (denoted q+1, thus q+1= K). Ths last random effect defnes a factor common to all oblgors and reflects the heterogenety n default rates related to tme. Ths random effect corresponds to the heterogenety n default rates attrbutable to tmeheterogenety, whch s assumed to be related to general economc condtons. In ths specfcaton, the lnear predctor n the logstc regresson contans an ntercept term that randomly vares at the year level, the hghest level n the modellng, where all other effects are nested n. In other words, a random ntercept s drawn separately and ndependently for each year. Ths structure mples that a gven oblgor s affected by two factors: the factor representatve of general economc condtons and ts ndustry rsk factor or sze rsk factor. Measurng potental concentraton To assess the credt rsk of a gven type of borrower wthn the portfolo, we compute the economc captal contrbuton of each borrower type. Ths calculaton requres the portfolo-wde economc captal to be allocated to sub-portfolos or ndvdual assets. From the fndngs of Tasche (1999) and Gouréroux et al. (2000), the margnal contrbutons to a portfolo VaR can be expressed as the expected loss on a gven exposure, condtonal on losses reachng ths VaR: RCVAR = E [ L 1 ] VaRα ( L) = L [ L L = VaRα ( L) ] = P[ L = VaR ( L) ] Equaton (3) ndcates that f there s a postve probablty for losses to reach a porfolo s VaR, then the computaton of margnal contrbutons wll rely heavly on the ablty to estmate ndvdual losses as aggregate losses approach ths VaR. Thus, n the context of a Monte Carlo smulaton, the condtonal mean may be based only on a lmted number of smulatons, producng unrelable estmates. To mprove the estmaton procedures, certan authors (Tasche, 2009, Glasserman and L, 2005, Egloff and Leppold, 2010) have used mportance samplng. Importance samplng conssts of shftng the parameters of a dstrbuton n ways that ncrease the lkelhood of observng certan desred realzatons of the rsk factors. The man dffculty wth respect to ths approach relates to the choce of the alternatve dstrbuton F*. In ths study, we follow the methodology of Tasche (2009) and shft only the rsk factor (S) means n the followng manner: S * [ S ] + μ μ = E S L = VaR ( L) = S E, F E α [ ] The next step s the computaton of condtonal expectatons (equaton 3). Because the computaton of VaR s accomplshed through Monte Carlo smulatons, one has both the realzatons of the rsk factors and the resultng credt losses. Ths nformaton permts the utlzaton of the non-parametrc Naradaya- Watson estmator for condtonal expectatons. If the standard normal densty s used as the kernel and f h s used to denote the bandwdth of the kernel, then the estmator of the condtonal expectaton for rsk factor k may be defned as follows: α (3) Drecton des Études - SGACP 31

33 Eˆ [ S L = VaR ( L) ] k α = T t = 1 T t = 1 h = 1.06σ T VaR Lt kφ h VaR Lt φ h To lmt the computatonal burden nvolved n the smulatons of margnal contrbutons, we rely on the sze of the portfolo under consderaton and assume the complete dversfcaton of dosyncratc rsk. Ths assumpton allows for losses to be smulated usng condtonal probabltes nstead of requrng the smulaton of defaults (and ther assocated losses). Thus, gven the assumed homogenety of exposures wthn sub-portfolos, t s possble to compute a sngle margnal contrbuton based on the ratng/modalty of the segmentaton varable rather than by proceedng at the asset level. Losses are then approxmated by the followng expresson: S ( p ) N 1 Φ r w sk L u Φ rk j= 1 1 w Σw Once the shfts n the means are computed for all of the rsk factors, the next step n the analyss s to obtan realzatons of the rsk factors under the new dstrbuton to once agan compute the aggregate losses for the portfolo and the ndvdual losses wthn each sub-segment and each ratng grade. Tasche (2009, proposton 4.2), who refers to the work of Klebaner (2007), establshes that condtonal on VaR, the expected losses under the natural dstrbuton can be defned as follows: E F [ L L = VaR ( L) ] α = E F * E L 1 5 [ L R L = VaRα ( L) ] R L = VaR ( L) F * [ ] As dscussed above, these condtonal expectatons can be computed wth the Naradaya-Watson estmator, and smulatons of rsk factors and losses can be obtaned under the shfted dstrbuton. Fnally, these expected losses can be aggregated across ratngs for each modalty of the segmentaton varable to compute segment-wde economc captal requrements. α Drecton des Études - SGACP 32

34 Débats Economques et Fnancers 1. M. Detsch et H. Frasse «De comben le captal réglementare dffère t-l du captal économque: le cas des prêts aux enterprses par les grands groups franças», Févrer 2013 Drecton des Études - SGACP 33

35 Economc and Fnancal Dscusson Notes 1. M. Detsch and H. Frasse, How dfferent s the regulatory captal from the economc captal: the case of busness loans portfolos held by major bankng groups n France, February 2013 Drecton des Études - SGACP 34

36 Drecton des Études - SGACP 35