Sector Concentration in Loan Portfolios and Economic Capital. Abstract

Transcription

1 Sector Concentration in Loan Portfolio and Economic Capital Klau Düllmann and Nancy Machelein 2 Thi verion: September 2006 Abtract The purpoe of thi paper i to meaure the potential impact of buine-ector concentration on economic capital for loan portfolio and to explore a tractable model for it meaurement. The empirical part evaluate the increae in economic capital in a multi-factor aet value model for portfolio with increaing ector concentration. The ector compoition i baed on credit information from the German central credit regiter. Finding that buine ector concentration can ubtantially increae economic capital, the theoretical part of the paper explore whether thi rik can be meaured by a tractable model that avoid Monte Carlo imulation. We analyze a implified verion of the analytic value-at-rik approximation developed by Pykhtin (2004), which only require rik parameter on a ector level. Senitivity analye with variou input parameter how that the model perform well in approximating economic capital for portfolio which are homogeneou on a ector level in term of PD and expoure ize. Furthermore, we explore the robutne of our reult for portfolio which are heterogeneou in term of thee two characteritic. We find that low granularity c.p. caue our model to underetimate economic capital, wherea heterogeneity in individual PD caue overetimation. Indicative reult imply that in typical credit portfolio, PD heterogeneity will at leat compenate for the granularity effect. Thi ugget that the analytic approximation etimate economic capital reaonably well and/or err on the conervative ide. Keyword: ector concentration rik, economic capital JEL-Claification: G8, G2, C Addre: Deutche Bundebank, Wilhelm-Eptein-Str. 4, 6043 Frankfurt am Main, Germany, Tel.: , e- mail: [email protected] 2 Addre: National Bank of Belgium, Boulevard de Berlaimont 4, 000 Bruel, Belgium, Tel.: , [email protected] For their comment we thank, Marc De Ceuter, Michael Gordy, Thilo Liebig, Janet Mitchell, Joël Petey, Peter Raupach, Andrea Reti, participant in the Reearch Tak Force of the Bael Committee on Banking Superviion a well a participant at the 2006 AFSE conference on 'Recent Development in Financial Economic' and at the 2006 European Banking Sympoium. We thank Eva Lütkebohmert, Chritian Schmieder and Björn Wehlert for their invaluable upport in compiling the German credit regiter data and Julien Demunyck and Jeu Saurina for providing u with data from the French and the Spanih credit regiter. The view expreed here are our own and do not necearily reflect thoe of the Deutche Bundebank or the National Bank of Belgium.

2 . Introduction Although the failure to recognize diverification within bank' credit portfolio wa a key criticim of the 988 Bael Accord, the minimum regulatory capital requirement (Pillar ) of the Bael Framework of June 2004 are baed on a ingle-factor model 3 which till doe not account for difference in diverification. However, recognizing that bank portfolio can exhibit credit rik concentration, Bael II tipulate that thi rik be addreed in the uperviory review proce (Pillar 2), thu creating a need for an appropriate methodology to meaure thi rik. Concentration rik in bank credit portfolio arie either from an exceive expoure to certain name (often referred to a name concentration or granularity) or from an exceive expoure to a ingle ector or to everal highly correlated ector (i.e. ector concentration). In the pat, financial regulation and previou reearch have focued mainly on the firt apect of concentration rik. 4 Therefore, in thi paper our focu i on ector concentration rik, although granularity i alo analyzed. Sector are defined in the following a buine ector. Although ector can alo be defined by geographical region, thi cae i not conidered in thi paper. The critical role credit rik concentration ha played in pat bank failure ha been documented in the literature. 5 Therefore, the importance of prudently managing ectoral concentration rik in bank credit portfolio i generally well recognized. However, exiting literature doe not provide much guidance on how to meaure ectoral concentration rik. Conequently, whether particular level of concentration need to be tranlated into an additional capital buffer remain an open quetion. Thi paper contribute to the literature in the following way. Firt, we meaure economic capital in a CreditMetric-type multi-factor model and evaluate how important the increae in economic capital i in a equence of portfolio with increaing ector concentration. The analyi i baed on portfolio which were contructed from German central credit regiter data. They reflect the average buine-ector ditribution of the banking ytem a well a higher ector concentration oberved in individual bank. Information on buine-ector concentration of bank i not publicly available, thu central credit regiter repreent unique ource of data on ector concentration in exiting bank. Our emphai on empirically obervable ector concentration i therefore an important contribution. Second, we evaluate the accuracy of a multi-factor adjutment propoed in Pykhtin (2004), which offer a tractable, cloed-form olution for value-at-rik (VaR) and economic capital () and, thereby, for the meaurement of concentration rik. i defined a the difference See Gordy (2003). See EU Directive 93/6/E, Joint Forum (993) and Gordy (2003). See, for example, BCBS (2004a).

3 between the VaR and the expected lo of a credit portfolio. We have applied a implified verion of the model in order to reduce the computational burden. Such a methodology could be ueful for rik manager and upervior in earch of robut, fit-for-purpoe tool to meaure ector concentration in a bank loan portfolio. In the empirical part of thi paper we how that economic capital can ubtantially increae with ector concentration acro portfolio. It increae from a credit portfolio repreenting the average ector ditribution of the German banking ytem to a portfolio that i concentrated in a ingle ector can be a high a 50%. In the theoretical part we explore whether ector concentration can be approximated by a implified verion of the model in Pykhtin (2004), which offer a tractable method to avoid Monte Carlo imulation. The methodological framework of the Pykhtin model build on earlier work by Gordy (2003) and Wilde (200) on granularity adjutment in the aymptotic ingle rik factor (ASRF) model. Wherea the granularity adjutment deal with an unbalanced expoure ditribution acro name, the Pykhtin model offer a treatment for an unbalanced ditribution acro (correlated) ector. The i given in cloed form a the um of the in a ingle rik factor model (in which the correlation with the ingle ytematic rik factor depend on the ector) and a multi-factor adjutment term. The model allow bank and upervior to approximate economic capital for loan portfolio without running computationally intenive Monte Carlo imulation. Furthermore, it poe only moderate data requirement ince it require the input parameter expoure ize and probability of default (PD) only on a ector level. We found that for portfolio with highly granular and homogeneou ector, the analytic approximation formulae perform extremely well. Moreover, the multi-factor adjutment term i relatively mall, o that in the ingle rik factor model i already cloe to the true value obtained by imulation. Our reult hold for portfolio with different level of ector concentration, a different number of ector a well a under variou ector weight and correlation aumption. Furthermore, we explore the accuracy of our model when the two aumption that the portfolio i infinitely granular within each ector and that all expoure in the ame ector have the ame PD are violated. We found that the model underetimate in cae of low granularity, wherea it overetimate in the preence of heterogeneity in individual PD, in particular if creditworthine increae with expoure ize. Which of the two effect prevail depend on the pecific input parameter. The reult eem to ugget, however, that for typical credit portfolio, the effect of PD heterogeneity i likely to be tronger than the effect of granularity. Thi implie that the analytic approximation err on the conervative ide. To our knowledge there i only one recent empirical paper that conider the impact of ector concentration rik on economic capital. Burton et al (2005) imulated the ditribution of portfolio credit loe for a number of real US yndicated loan portfolio. They found that, 2

4 although name concentration can meaningfully increae for maller portfolio (which are defined a portfolio with expoure of le than US$0 billion), ector concentration rik i the main contributor to for portfolio of all ize. Two other model that meaure concentration rik in a tractable model are preented by Cepede et al (2005) and Düllmann (2006). Cepede et al (2005) developed an adjutment to the ingle rik factor model in the form of a caling factor to the economic capital required by the ASRF model. Thi diverification factor i an approximately linear function of a Hirchmann-Herfindahl index, calculated from the aggregated ector expoure. Thi model, however, doe not allow for different aet correlation acro ector. Contrary to the approach in our paper, it cannot ditinguih between a portfolio which i highly concentrated toward a ector with a high correlation with other ector, and another portfolio which i equally highly concentrated, but toward a ector which i only weakly correlated with other ector. Düllmann (2006) extend Moody' Binomial Expanion Technique by introducing default infection into the hypothetical portfolio on which the real portfolio i mapped in order to retain a imple olution for VaR. Unlike the Pykhtin model, both model developed by Cepede et al and Düllmann require the calibration of a parameter uing Monte Carlo imulation. The paper i organized a follow. In Section 2 we preent the default-mode verion of the well-etablihed multi-factor CreditMetric model which erve a a benchmark. Furthermore, we dicu the implified verion of the Pykhtin model. The empirical part of our paper comprie Section 3 and 4. The loan portfolio on which the empirical analye are baed are decribed in Section 3. In Section 4 we the impact of ector concentration on. For thi purpoe we gradually increae ector concentration, tarting from the benchmark portfolio, and analyze it impact on. In the theoretical part, which comprie Section 5 to 7, we evaluate the performance of the Pykhtin (2004) analytic approximation for economic capital uing etimate from Monte Carlo imulation a a benchmark. Section 5 focue on highly granular portfolio which are homogeneou on a ector level and, in particular, on the enitivity of the reult to the number of elected factor. Section 6 deal with portfolio characterized by lower granularity and Section 7 introduce PD heterogeneity on an expoure level. Section 8 ummarize and conclude. 2. Meauring Concentration Rik in a Multi-Factor Model 2.. General framework We aume that every loan in a portfolio can be aigned to a different borrower, o that the number of expoure or loan equal the number of borrower. Each borrower i can uniquely 3

5 be aigned to a ingle pecific ector. In practice, (large) firm often comprie buine line from different indutry ector. However, we poe thi aumption here for practical and preentational purpoe. Let M denote the total number of borrower or loan in the portfolio, M the number of borrower or loan in ector, and S the total number of ector. Each expoure ha the ame relative portfolio weight of w,i = /M. Therefore, the weight of the aggregated ector expoure, w, alway equal M /M. The general framework i a multi-factor default-mode Merton-type model. 6 The dependence tructure between borrower default i driven by ector-dependent ytematic rik factor which are uually correlated. Each rik factor can be uniquely aigned to a different ector, o that the number of ector and factor are the ame. Credit rik occur only a a default event which i conitent with traditional book-value accounting and form the bai of traditional loan portfolio management. The unobervable, normalized aet return X,i of borrower i in ector trigger the default event if it croe the default barrier γ,i. The unconditional default probability p,i of borrower i in ector i defined a ( ) p P X γ, i =, i, i. The latent variable X,i follow a factor model and can be written a a linear function of an indutry ector rik factory and an idioyncratic rik factor ε, i : (a) X = r Y + r ε. 2, i, i The higher the value of the factor weight r, the more enitive the aet return of firm i in ector are to the ector factor. The diturbance term ε,i follow a tandard normal ditribution. The ector weight alo determine the factor weight of the idioyncratic rik factor in order to retain a tandard normal ditribution for X,i. The correlation between the ytematic ector rik factor Y are denoted by ρ, t and often referred to a factor correlation. The ector factor can be expreed a a linear combination of independent, tandard normally ditributed factor Z,,Z S. α, t, t S The matrix ( ) S S 2 (b) Y = α, jz j with α, j = for S. j= j= i obtained from a Choleky decompoition of the factor correlation matrix. The aet correlation for each pair of borrower i and j in ector and t can be hown to be given by cor( X, X ) = r r ρ = r r α α. (2), i t, j t, t t, n t, n n= S 6 See alo Gupton et al (997), Gordy (2000), and Bluhm et al (2003) for more detailed information on thi type of model. The origin of thee model can be found in the eminal work by Merton (974). 4

6 Dependencie between borrower arie only from their affiliation with the indutry ector and from the correlation between the ytematic ector factor. The intra-ector aet correlation for each pair of borrower i imply the ector weight 2 r quared. If a firm default, the amount of lo i determined by the tochatic lo everity ψ, i. We aume that the lo everity i known at default and that before thi event it i ubject only to fully diverified idioyncratic rik. 7 Credit loe of the whole portfolio are then given by (3) S M =. = i= L w, iψ, i { X, i N ( p, i )} In ummary, the model need the following input parameter: the relative expoure ize w,i of borrower i in ector the default probability p,i of borrower i in ector the expected lo everity µ, i = E ψ, i of borrower i in ector the factor correlation matrix and the ector weight r. In the following we aume that the factor correlation matrix can be proxied by a correlation matrix of equity index return. In theory, aet correlation could be directly etimated from the time erie of aet value. However, aet value are uually not obervable. Since equity can be viewed a a call option on a firm aet, it i frequently argued that correlation etimate from equity return provide a viable approximation of factor correlation The CreditMetric default-mode model To obtain the lo ditribution, CreditMetric applie Monte Carlo imulation by generating aet return and counting the default event. In each imulation run the portfolio lo i determined from equation (3). For each expoure, the aet return are determined from equation (a/b) and compared with the default threhold. If the value of the aet return fall below the threhold γ = N ( p ), the borrower i in default. The portfolio lo of a, i, i imulation run i calculated by adding up the incurred loe from defaulted borrower. The number of imulation run in our analye i typically 500,000. Portfolio loe obtained in each imulation run are orted to form the ditribution of portfolio loe from which can 7 8 The model analyzed in thi paper can alo be extended to incorporate idioyncratic rik in lo everitie, if required. Previou analye (See, for example, Zeng and Zhang (200)) have hown that equity correlation may not be the bet proxie for aet correlation, given that equity correlation may be ubject to noie which may not necearily be related to firm fundamental. However, the main advantage of uing equity data i that there i an abundance of data. Therefore, it i no urprie to ee that it ha become market practice to ue equity correlation a a proxy for aet correlation. 5

7 be calculated a the difference between the q-quantile of thi lo ditribution and the expected lo. Since it i obtained by imulation, we refer to it in the following a im..3. Analytic approximation In thi ection, we decribe an analytical approximation to the VaR in the framework of a multi-factor model, which i a implified verion of the model developed by Pykhtin (2004). The main advantage of thi model i it tractability, ince it doe not require Monte Carlo imulation. Furthermore, we have implified the model in uch a way that it only require expoure ize, PD, and expected lo everitie aggregated on a ector level intead of on an expoure level. The factor correlation matrix and the factor loading are till needed a in the CreditMetric model. On the bai of the work by Gouriéroux et al (2000) and Martin and Wilde (2002), we can approximate the true lo L of a portfolio by a perturbed lo variable L = L + ε U, where U can be interpreted a L L and ε decribe the caling parameter in the perturbation. L denote the lo in the aymptotic ingle rik factor (ASRF) model with infinitely granular ector. It depend on the default probability pˆ( Y ) conditional on the ytematic rik factor Y S = with = (4) L w ˆ µ p ( Y ) pˆ ( Y ) = N ε N ( p ) cy c The parameter c can be interpreted a the correlation between the ytematic rik factor Y and the aet return X, i. The q-quantile of the true lo ditribution, t q (L) can be approximated by ( ) q t L ε a the um of the VaR in the ASRF model, t ( L ) and a multi-factor adjutment. Thi multi-factor adjutment can be determined from a econd-order Taylor erie expanion of t q (L). The firt-order effect vanihe becaue we require L = E L Y. By keeping term up to quadratic and neglecting higher-order term, we can approximate t ( L) a follow: 9 (5) 2 d tq ( Lε ) tq ( L) tq ( L) dε The firt ummand in (5) denote the VaR in the ingle rik factor model and can be calculated by replacing Y in (4) by tq ( Y ). If Y follow a tandard normal ditribution, tq ( ) ε = 0 q 2. q Y i the q-quantile of thi ditribution. Note that thi ingle rik factor model differ from the well- 9 See Pykhtin (2004) for proof. 6

8 known ASRF model in that ˆ ( q ( )) p t Y depend on a ector-dependent aet correlation. To avoid confuion, we will call thi model the ASRF* model, reerving the term ASRF model for the model with uniform aet correlation. The econd ummand in (5) denote the multi-factor adjutment, according to Pykhtin (2004) by tq, which can be calculated (6) l ( y) tq = v ( y) v( y) + y 2 l ( y) l ( y) y= N ( q) where l ( y) and l ( y) denote, repectively, the firt and econd derivative of equation (4). v( y ) give the conditional variance of L L (conditional on y = Y ) which equal the conditional variance of L. It firt derivative i v ( y). The detail and the input of thee equation are preented in Appendix B. The remaining open iue i how to etablih the link between L and L. Thi i achieved by retricting Y to the pace of linear mapping of the rik factor Z,,Z S : Y S = bz. = The correlation between the indutry rik factor Y and the ytematic rik factor Y are denoted by ρ. Thee are ued to calculate the (alo ector-dependent) correlation in the ASRF* model uing the following mapping function, for {,..., S} : c = r ρ where ρ α, b S = j= j j There i no unique olution for determining the coefficient b,..., b S. In the following, we will ue the approach in Pykhtin (2004), which i briefly ummarized in Appendix C. 3. Portfolio Compoition 3.. Data et and definition of ector Our analye are baed on loan portfolio which reflect characteritic of real bank portfolio obtained from European credit regiter data. Our benchmark portfolio repreent the overall ector concentration of the German banking ytem which wa contructed by aggregating the expoure value of loan portfolio of 2224 German bank in September The portfolio include branche of foreign bank located in Germany. Credit expoure to foreign borrower, however, are excluded. We deem thi to be a reaonable approximation of a well- 7

9 diverified portfolio on the intuitive aumption that a portfolio cannot be more diverified than in the cae in which it repreent the average relative ector expoure of the national banking ytem. In principle, we could alo have created a more diverified portfolio in the ene of having a lower VaR. However, uch a portfolio would be pecific to the credit rik model ued and would not be obtainable for all bank. All credit intitution in Germany are required by the German Banking Act (Kreditweengeetz) to report quarterly expoure amount of thoe borrower whoe indebtedne to them amount to.5 million or more at any time during the three calendar month preceding the reporting date. In addition, bank report national code that are compatible with the NACE claification cheme and indicate the economic activity of the borrower and hi country of reidence. Individual borrower are ummarized in borrower unit which are linked, for example, by invetment and contitute an entity haring roughly the ame rik. The aggregation of expoure on a buine ector level wa carried out on the bai of borrower unit. If borrower in that unit belong to different ector, the dominating expoure amount determine the final ector allocation. Therefore, the credit regiter include not only expoure above.5 million, but alo maller expoure to individual borrower belonging to a borrower unit that exceed thi expoure limit. Thi characteritic ubtantially increae it coverage of the credit market. The indutry claification choen by CreditMetric i the Global Indutry Claification Standard (GICS), which wa jointly launched by Standard & Poor' and Morgan Stanley Capital International (MSCI) in 999. The claification cheme wa developed to etablih a global tandard for categorizing firm into ector and indutrie according to their principal buine activitie. It comprie 0 broad ector which are divided into 24 indutry group. 0 GICS further divide thee group into indutrie and ub-indutrie. However, the latter detailed cheme are not ued by vendor model. In the following, we ue the broad ector claification cheme. Becaue ome of the indutry group that form the broad Indutrial ector are very heterogeneou, we decided to plit thi ector into three indutry group: Capital Good (including Contruction), Commercial Service and Supplie, and Tranportation. Credit regiter dataet, however, ue the NACE indutry claification ytem, which i quite different from the GICS ytem. In order to ue the information from the credit regiter, we mapped 2 the NACE code onto the GICS code. Similar mapping i ued by other vendor model, uch a S&P Portfolio Rik Tracker developed by S&P. We have excluded expoure to the financial ector (ector G) which comprie expoure to Bank (G), 0 2 See Table 2 in Appendix A, which how the broad ector and the more detailed indutry group. Unreported imulation have hown that reult are not affected by uing the more detailed claification cheme. See Table 3 in Appendix A for the mapping. 8

10 Diverified Financial (G2), Inurance Companie (G3) and Real Etate (G4). We have excluded expoure to the financial ector becaue of the pecificitie of thi ector. Expoure to the real etate ector are heavily biaed a it comprie a large number of expoure to borrower that are related to the public ector. Since we could not differentiate between private and public enterprie in the real etate ector, we have excluded thi ector from the following analye. We have alo diregarded expoure to houehold ince there i no repreentative tock index for them. Thi i a typical limitation of model relying on equity data for the etimation of aet correlation. In um, we ditinguih between ector, which can be conidered a broadly repreenting the Bael II aet clae Corporate and SME..2. Comparion with French, Belgian and Spanih banking ytem A rough comparion of the relative hare of the ector decompoition between the aggregated German, French, Belgian and Spanih banking ytem how that the number are imilar. 3 The only noticeable difference i the greater hare of the Capital Good ector (33%) in Spain compared to Germany and Belgium and the maller hare of the Commercial Service and Supplie ector in Spain compared to Germany and Belgium. In general, however, the average ector concentration are very imilar acro the four countrie, which ugget that our reult are to a large extent tranferable. Figure : Comparion of average ector concentration for Germany, Spain, Belgium, and France (*) 40,00% 35,00% 30,00% 25,00% 20,00% 5,00% Germany Spain Belgium France 0,00% 5,00% 0,00% Energy Material Capital good Commercial ervice and upplie Tranportation Conumer dicretionary Conumer taple Health care Information technology Telecommunication ervice Utilitie (*) A breakdown of Indutrial ector C into the three categorie Capital Good, Commercial Service and Supplie, and Tranportation i not available for France. The ector hare of the aggregated ector C, however, are quite imilar for all four countrie. 3 The exact figure are provided by Table 4 in Appendix A. 9

11 .3. Decription of the benchmark portfolio The ectoral ditribution of expoure in the benchmark portfolio, which i hown in Table, aume that the total portfolio ha a volume of 6 million. A mentioned above, thi portfolio repreent the ectoral ditribution of aggregate expoure in the German banking ytem. The degree of concentration in our reference portfolio i purely national and driven by the firm' ector compoition becaue we do not conider the impact of regional or country factor in our analyi. It i not uncommon for bank to ue a more detailed ector claification cheme. We conider it more conervative to ue a relatively broad ector claification cheme rather than a very detailed one. In a broad ector claification cheme, a larger proportion of expoure i attached to one ector. Therefore, correlation between expoure of the ame ector, which are typically greater than the correlation between expoure of a different ector, will play a larger role. In order to focu on the impact of ector concentration we aume an otherwie homogeneou portfolio by requiring that all other characteritic of the portfolio are uniform acro ector. We aume a total portfolio volume of 6 million that conit of 6,000 expoure of equal ize and a uniform probability of default (PD) of 2%. Every expoure i to a different borrower, thu circumventing the need to conider multiple expoure default. We et a uniform LGD of 45%, which i the correponding uperviory value for a enior unecured loan in the Foundation IRB approach of the Bael II framework. 4 In the CreditMetric approach, indutry weight can be aigned to each borrower according to it participation. Here, we aume that every firm i expoed to only one ingle ector a it main activity. Furthermore, we aume bank do not reduce expoure to certain ector by purchaing credit protection. Table : Compoition of the benchmark portfolio (uing the GICS ector claification cheme) Total expoure Number of expoure % expoure A: Energy, % B: Material 36, % C: Capital Good 692, % C2: Commercial Service and Supplie 2,020,000 2, % C3: Tranportation 429, % D: Conumer Dicretionary 898, % E: Conumer Staple 389, % F: Health Care 545, % H: Information Technology 92, % I: Telecommunication Service 63, % J: Utilitie 400, % Total 6,000,000 6,000 4 See BCBS (2004b). 0

12 .4. Sequence of portfolio with increaing ector concentration In order to meaure the impact on of more concentrated portfolio than the benchmark portfolio, we contruct a equence of ix portfolio, each with increaed ector concentration relative to the previou one. To thi end, we gradually increae ector concentration in our benchmark portfolio by uing the following algorithm. In each tep we remove x expoure from all ector and add them to a previouly elected ector. Thi procedure i repeated until a ingle-ector portfolio which i the portfolio with the highet poible concentration i obtained. The ector which receive x expoure at every tep and alo the amount x that i tranferred to thi ector are determined in uch a way that ome of the generated portfolio reflect a degree of ector concentration that i actually obervable in real bank. 5 Table 2 how a equence of even portfolio in the order of increaing ector concentration. The increae in ector concentration i alo reflected in the Herfindahl-Hirchmann Index (HHI), 6 given in the lat row which i calculated at ector level. Portfolio ha been contructed from the benchmark portfolio by re-allocating one third of each ector expoure to the ector Capital Good. The even more concentrated portfolio 2, 3, 4 and 5 have been created by repeated application of thi rule. Portfolio 2 and 5 are imilar to portfolio of exiting bank 7 inofar a the ector with the larget expoure ize ha a imilar hare of the total portfolio. Furthermore, the HHI i imilar to what i oberved in real-world portfolio. Finally, we created portfolio 6 with the highet degree of concentration a a one-ector portfolio by hifting all expoure to the Capital Good ector. Table 2: Sequence of portfolio with increaing ector concentration Benchmark portfolio Portfolio Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 Portfolio 6 A: Energy 0% 0% 0% 0% 0% 0% 0% B: Material 6% 4% 3% 2% 2% % 0% C: Capital Good 2% 4% 56% 7% 78% 82% 00% C2: Commercial Service & Supplie 34% 22% 7% % 8% 7% 0% C3: Tranportation 7% 5% 4% 2% 2% % 0% D: Conumer Dicretionary 5% 0% 7% 5% 4% 3% 0% E: Conumer Staple 6% 4% 3% 2% 2% % 0% F: Health Care 9% 6% 5% 3% 2% 2% 0% H: Information Technology 3% 2% 2% % % % 0% I: Telecommunication Service % % % 0% 0% 0% 0% J: Utilitie 7% 4% 3% 2% 2% % 0% HHI Due to confidentiality requirement, we are unable to reveal more detailed information. See Hirchmann (964). Confidentiality require thoe bank with a high ector concentration remain anonymou.

13 .5. Intra and inter-ectoral correlation The ector factor correlation are etimated from hitorical equity index correlation. Table 2 how the equity correlation matrix of the relevant MSCI EMU indutry indice. 8 The ector factor correlation are baed on weekly return data covering the period from November 2003 to November Sector that are highly correlated with other ector (i.e. ector that have an average inter-ector equity correlation of greater than 65%) are Material (B), Capital Good (C), Tranportation (C3) and Conumer Dicretionary (D). Sector that are moderately correlated with other ector, i.e. ector that have an average inter-ector equity correlation of between 45% and 65%, are Commercial Service and Supplie (C2), Conumer Staple (E) and Telecommunication (I). Sector that are the leat correlated with other ector, i.e. ector that have an average inter-ector equity correlation of le than 45%, are Energy (A) and Health Care (F). The relative order of thee ector i broadly in line with reult reported in other empirical paper. 9 The heterogeneity between Capital Good, Commercial Service and Supplie, and Tranportation are confirmed by noticeable difference in correlation. The intra-ector correlation and/or inter-ector correlation between expoure are obtained by multiplying thee ector correlation of Table 3 with the ector weight. Table 3: Correlation matrix baed on MSCI EMU indutry indice (baed on weekly log return data covering the Nov Nov 2004 period; in percent) A B C C2 C3 D E F H I J A: Energy B: Material C: Capital Good C2: Commercial Service & Supplie C3: Tranportation D: Conumer dicretionary E: Conumer taple F: Health Care H: Information Technology I: Telecommunication Service J: Utilitie 00 More difficult than the etimation of ector correlation i the determination of the ector weight, which depend alo on the intra-ector aet correlation. We do not ue the formula provided in CreditMetric to determine the ector weight a recent reearch ha uggeted 8 9 The correlation matrix baed on MSCI US data i imilar. See, for example, De Servigny and Renault (200), FitchRating (2004) and Moody' (2004). It i difficult to compare the abolute inter-ector correlation value a different paper report different type of correlation. De Servigny and Renault (200) report inter-ector default correlation value, FitchRating (2004) report interector equity correlation while Moody' (2004) provide correlation etimate inferred from co-movement in rating and aet correlation etimate. Furthermore, the different paper ditinguih between a different number of ector. 2

14 that thi formula doe not fit the German data very well. 20 Intead, we aume a unique ector weight for all expoure and calibrate the value of the factor loading to match the correponding IRB regulatory capital charge. More preciely, we determine a factor loading r =0.50 for all ector {,..., S} uch that the economic capital im equal the IRB capital charge for corporate expoure, auming a default probability of 2%, a lo given default (LGD) of 45% and a maturity of one year. Setting the ector factor weight to 0.5 i lightly more conervative than empirical reult for German companie ugget. The average of all the correlation entrie in the factor correlation matrix i 0.59, which implie by evoking equation (2) an average aet correlation of 0.4 between expoure. Empirical evidence 2 ha hown that German SME typically have an average aet correlation of 0.09, which ugget r = Large firm, however, are typically more expoed to ytematic rik than SME and therefore uually have higher aet correlation value. 22 Equation (2) implie that intra-ector aet correlation are thu fixed at 25%. Inter-ector aet correlation can be calculated by multiplying the factor weight of both ector by the interector equity correlation. The lowet equity correlation between the Energy ector index and the Information Technology ector index of 0% tranlate into inter-ector aet correlation of 2.5%. The highet equity index correlation occur between the Commercial Service and Supplie and the Conumer Dicretionary ector index. At 92%, it tranlate into an interector aet correlation of 23%. 2. Impact of ector concentration on economic capital In thi ection we analyze the impact of increaing ector concentration on economic capital, which i defined a the difference between the 99.9% percentile of the lo ditribution and the expected lo. The reult are given in Table 4. We oberve for the corporate portfolio that economic capital increae from the benchmark portfolio to portfolio 2 by 20%. Economic capital for the concentrated portfolio 5 increae by a ubtantial 37% relative to the benchmark portfolio. Thee reult demontrate the importance of taking ector concentration into account when calculating. Typically, the corporate portfolio comprie only a fraction of the total loan portfolio (which alo contain loan to overeign, other bank and private retail client). Although the increae in ector concentration may have a ignificant impact on the economic capital for the corporate credit portfolio, it may have a much maller impact in term of a bank total credit See Hahnentein (2004) for a detailed analyi. See Hahnentein (2004). See, for example, Lopez (2004) for empirical evidence of thi relation for the US. 3

15 portfolio. For a meaningful comparion, we aume that the corporate credit portfolio comprie 30% of the total credit portfolio and that the bank need to hold capital amounting to 8% of their total portfolio. By auming that there are no diverification benefit between corporate expoure and the bank' other aet, the of the total portfolio can be determined a the um of the for the corporate expoure and the for the remaining expoure. Table 4: Impact of ector concentration on economic capital ( im ) for the equence of corporate portfolio and for the equence of total portfolio of a bank (in percent) Benchmark portfolio Portfolio Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 Portfolio 6 Corporate portfolio Total portfolio The reult for the total portfolio of the bank are alo hown in Table 4. A expected, the impact of an increae in ector concentration i much le evere when looking at the for the total portfolio. Economic capital for portfolio 5, for example, increae by about 6% relative to the benchmark portfolio intead of 37% if only the corporate portfolio i taken into account. In order to verify how robut our reult are to the input parameter, we carried out the following four robutne check (RC - RC4): a lower uniform PD of 0.5% intead of 2% for all ector (RC), heterogeneou PD etimated from hitorical default rate of the individual ector (RC2) and given in Table 5, a different factor correlation matrix (See Table 5, Appendix A) repreenting the correlation matrix with the highet average annual correlation over the period between 997 and 2005 (RC3) and a uniform intra-ector aet correlation of 5% and a uniform inter-ector aet correlation of 6% (RC4), which are value ued by Moody for the rik analyi of ynthetic CDO See Fu et al (2004). 4

16 Table 5: Average hitorical default rate ( ; before and after caling to an expoureweighted expected average default rate of 2% for the benchmark portfolio; in percent) Sector Uncaled default rate Scaled default rate A: Energy.5.0 B: Material C: Capital Good C2: Commercial Service and Supplie C3: Tranportation D: Conumer Dicretionary E: Conumer Staple F: Health Care.6. H: Information Technology I: Telecommunication Service J: Utilitie Source: own calculation, baed on S&P (2004) The hitorical default rate in Table 5 are, on average, higher than the value of 2% which i ued for the PD in the cae of homogeneou PD for all ector. In order to iolate the effect of PD heterogeneity between ector, we cale the hitorical default rate, ector a follow, (7) p = p caled hit S = 0.02 w p hit hit p, for every In thi way we enure that the weighted average PD of the benchmark portfolio tay at 2% even in the cae of PD heterogeneity acro ector. The reult of the four robutne check are ummarized in Table 6. Although the abolute level of varie between thee robutne check, the relative increae in compared with the benchmark portfolio i imilar to previou reult in thi ection. For Moody correlation aumption in RC4, the increae in i tronger than for the other robutne check. Thi can be explained by the larger difference between intra-ector and inter-ector correlation, which i jutified by the higher number of ector they ue, and which lead to a tronger increae when the portfolio become more and more concentrated in a ingle ector. We conclude that the oberved ubtantial relative increae in due to the introduction of ector concentration i robut againt realitic variation of the input parameter. Furthermore, thi increae in may be even greater, depending on the underlying dependence tructure.. 5

17 Table 6: for the benchmark portfolio and it relative increae for the more concentrated portfolio - 6 (in percent of total expoure) Portfolio Benchmark portfolio Uing "Realrule" RC: PD=0.5% RC2: Heterogeneou (caled) PD RC3: Higher correlation RC4: Moody' Proportional change of in % Portfolio Portfolio Portfolio Portfolio Portfolio Portfolio Evaluation of the Approximation for Sector-dependent PD and High Granularity The purpoe of thi ection i to analyze the performance of the approximation, given homogeneity within each ector and auming a highly granular expoure ditribution in each ector. Since thee are two model aumption, the reult can be undertood a an upper bound in term of approximation quality. The analyi i performed by varying one-by-one the ector ditribution, the factor correlation, the factor weight, the number of factor and the ector PD. Cae of le granular portfolio and heterogeneou PD on an expoure level are tudied in Section 6 and 7. We again aume a confidence level q of 99.9% and employ the following three rik meaure S M = i i i ): = i= (where EL w, ψ, p, = t L EL * economic capital in the ASRF* model, which i defined a 99.9% ( ) economic capital baed on the multi-factor adjutment, ( ) = t L + t EL 99.9% 99.9% economic capital baed on Monte Carlo (MC) imulation, im Firtly, we preent reult for the benchmark portfolio and for the more concentrated portfolio - 6 in Table 7. The model parameter are the ame a in Section 4. 6

18 Table 7: Comparion of, and im for different expoure ditribution acro ector with increaing ector concentration given a default probability of 2% (in percent of total expoure) Portfolio im Relative error of Benchmark portfolio % Portfolio % Portfolio % Portfolio % Portfolio % Portfolio % Portfolio % The figure for the benchmark portfolio in Table 7 how that and provide extremely accurate proxie for im. Thi reult ugget that in the given example the calculation of may, in practice, be ufficiently accurate for certain rik-management purpoe. The four etimate for the more highly concentrated portfolio - 6 indicate that economic capital increae a expected, but that our reult for the approximation performance of and till hold. According to Table 7, relative error of are in a relatively mall range between 0.0% and.3%. Secondly, we check whether our reult differ when we vary the underlying correlation tructure. To thi end we calculate in Table 8 the three rik meaure for different factor correlation matrice. More pecifically, we aume homogeneou factor correlation matrice in which the entrie (outide the main diagonal) vary between 0 and in increment of 0.2. The lat cae, in which all factor correlation are equal to one, correpond to the cae of a inglefactor model. Table 8: Comparion of, and im for different factor correlation ρ, given a default probability of 2% (in percent of total expoure) Factor correlation ρ im Relative error of % % % % % % Table 8 how im and it proxie and for increaing factor correlation. A expected, economic capital increae with increaing factor correlation, ince a higher factor 7

19 correlation reduce the diverification potential by hifting probability ma to the tail of the lo ditribution. The highet relative error of of all factor correlation conidered i 2.5% which till reveal a good approximation performance. With increaing factor correlation the multi-factor model approache the tructure of a one-factor model for which and coincide. In all cae i relatively cloe to. Therefore, our earlier reult concerning the good approximation performance of and alo hold under different factor correlation aumption. Thirdly, we relax the aumption that the factor weight r i fixed by increaing the factor weight r from 0.2 to 0.8. There i a trong increae in with the factor weight but thi doe not affect the approximation quality, neither of nor of. Fourthly, we explore how the reult depend on the number of factor. For thi purpoe we vary the number of factor from 2 to 6. Figure 2 how how, and im depend on the number of ector and the factor correlation. i only plotted for 2 ector becaue it value are inditinguihable from Sim for 6 and for 6 ector. Figure 2: Economic capital (, and Sim ) for different factor correlation value for 2, 6 and 6 ector (in percent of total expoure) 2% 0% Economic capital (%) 8% 6% 4% 2% 0% *_2 Sec _im_2 Sec *_6 Sec _im_6 Sec *_6 Sec _im_6 Sec _2 Sec Factor correlation For a given number of ector, increae in Figure 2 with factor correlation a expected. If the factor correlation approache one, then value coincide, irrepective of the number of ector. The reaon i that in the limiting cae of a factor correlation equal to one, the model collape to a ingle-factor model. For a factor correlation of 0.6, which i alo the average of the entrie in the correlation matrix in Table 3, and alo for higher factor correlation, the relative approximation error i below 8

20 % for and below 2% for. Therefore, the previou reult howing a good approximation performance of and an even better one for are found to be robut with repect to the number of ector, at leat for realitic factor correlation. Figure 2 alo how that and im generally decreae when the number of ector increae for given aet correlation value. Thi reult can be explained by rik reduction through diverification acro ector. Fifthly, we teted whether our reult for the approximation performance of and are enitive to PD heterogeneity on a ector level. For thi purpoe we employ the caled default rate for ector from Table 5. The reult are given in Table 9. Table 9: Comparion of, and im, baed on ector-dependent default probabilitie, etimated from hitorical default rate (in percent of total expoure) Portfolio im Relative error of Benchmark portfolio % Portfolio % Portfolio % Portfolio % Portfolio % Portfolio % Portfolio % For all rik meaure the reult in Table 9 are relatively cloe to thoe in Table 7. The more concentrated the expoure are in one ector, however, the maller the difference to Table 7 i. Thi i explained by the fact that the ector PD are calibrated to an average value of 2% which i alo the PD ued for Table 7. The approximation quality of and i imilar to Table 7. We conclude that, in qualitative term, the reult obtained for a uniform PD alo hold for heterogeneou ector-dependent PD. 4. Evaluation of the Approximation for Sector-dependent PD and Low Granularity Simulation reult in the previou ection, which reveal a reaonably good approximation quality for and, were obtained conditional to a uniform PD in every ector and highly granular portfolio becaue each individual expoure ha a relatively mall hare of 0.07% (=/6000) of the total portfolio volume. However, portfolio of mall bank, in particular, are le granular. In the following we explore the impact of lower granularity. From the et of even portfolio, only the benchmark portfolio and portfolio 6 are conidered ince 9

21 they have the lowet and the highet ector concentration. The impact of granularity i conidered in two cae. In the firt cae, characterized by a portfolio of repreentative granularity, the ditribution of expoure ize wa elected from a ample of typical mall, regional German bank to reflect an average granularity in term of the HHI. The purpoe i to meaure the impact of granularity for an expoure ditribution that i repreentative for real bank. However, ince the expoure ditribution i baed on central credit regiter data, only larger expoure are captured 24 in the underlying data et with the conequence that thi expoure ditribution i le granular than what we can expect for real bank portfolio. The HHI of the portfolio i and the decriptive tatitic on expoure ize are hown in Table 6 in Appendix D. The aignment of expoure to ector wa achieved by randomly drawing expoure from the data et uch that expoure ize follow the ame ditribution in every ector. In the econd cae, characterized by low granularity, we conider the highet individual expoure hare that are admiible under the EU large expoure rule. 25 In thi way we obtain an upper limit for the potential impact of granularity. According to the EU rule, an expoure i conidered large if it amount require 0% or more of regulatory capital. Bank are generally not allowed to have an expoure that require at leat 25% of regulatory capital. Furthermore,, the um of all large expoure mut not require more than 8 time the regulatory capital. 26 We aume that a bank regulatory capital i 8% of it total loan volume. For a total portfolio value of 6,000 currency unit, bank are required to hold 480 currency unit in capital. Each large expoure require a minimum amount of capital of 48 currency unit and a maximum amount of 20 currency unit. The total um of all large expoure mut not exceed 3,840 currency unit. With thee retriction, the leat granular admiible expoure ditribution of our portfolio conit of 3840/20=32 loan of 20 currency unit 260/47= 45 loan expoure of 47 currency unit (jut below the large expoure limit of 48) and a remaining ingle expoure of 45 currency unit While keeping the average ector concentration of the portfolio contant, we increae the granularity of the portfolio to reflect the expoure ize ditribution of thi leat granular portfolio. More detail of thi portfolio can be found in Table 7, Appendix D. Simulated economic capital, im, and the analytic proxie * and are given in Table See ection 3. for more information on the characteritic of expoure included in the German central credit regiter. See Directive 93/6/E of 5 March 993 on the capital adequacy of invetment firm and credit intitution. The lat two retriction may be breached with permiion of the German Federal Financial Superviory Authority (BaFin), in which cae the exce mut be fully backed by capital. 20

22 Table 0: Comparion of, and im for portfolio with repreentative and low granularity (in percent of total expoure ) Portfolio Granularity im Relative error of Benchmark portfolio repreentative % low % Single ector portfolio low % In Table 0 and for the repreentative granular portfolio are lightly higher than for the low granular portfolio. Thi difference i caued by minor difference in the expoure ditribution acro ector which arie when the repreentative dicrete expoure ditribution i mapped to the ector allocation of the benchmark portfolio. The im value of 9.3% for the low granular benchmark portfolio i.3 percentage point (or 4% in relative term) higher than for the highly granular benchmark portfolio in Table 9. Thi difference appear to be ubtantial, but we have to conider that the granularity of the portfolio in Table 0 i very low ince it reflect the lowet granularity permiible under European bank regulation. The im meaure for the ingle ector portfolio 6 in Table 0 i higher than for the benchmark portfolio, which i conitent with earlier reported reult. The im value of 8.8% for the benchmark portfolio with typical granularity i relatively cloe to the value of 9.3% for the portfolio with low granularity, at leat if compared with im of 8.0% for the infinitely granular benchmark portfolio in Table 9. One reaon i that ome expoure in the portfolio with typical granularity technically violate the large expoure rule. 27 Therefore, a mentioned before, the portfolio of repreentative granularity hould till be regarded a conervative in term of granularity. For the purpoe of thi analyi, the approximation error of the proxie, * and, are more important than the level of. Both proxie are baed on the aumption of infinite granularity in each ector, while the heterogeneity acro ector and granularity. We find that im calculation take into account PD * and underetimate by up to 4%, in particular for portfolio with low granularity. can ubtantially 27 Thi can be explained either by pecial BaFin approval or, mot likely, by data limitation given that our credit regiter data do not contain loan below.5 million. The latter implie that their um i lower than the total portfolio expoure of the data, providing real bank and, therefore, our relative expoure weight are biaed upward. In other word, it i well poible that the large expoure limit i breached becaue the total expoure a reference i downward biaed, although the limit i till met by the data-providing bank. 2

23 5. Evaluation of Approximation for Heterogeneou Sector So far we have only conidered ector-dependent PD, which mean PD variation on a ector level, but not on the expoure level. In the following we explore the impact of heterogeneou PD inide a ector together with the impact of granularity. For the benchmark portfolio of repreentative granularity analyzed in the previou ection, expoure-dependent PD were computed from a logit model baed on firm accounting data. In order to apply the logit model, borrower information from the central credit regiter on expoure ize had to be matched with a balance heet databae, alo maintained by the Deutche Bundebank. 28 Uing empirical data on expoure ize and PD automatically capture a potential dependence between thee two characteritic. In order to enure comparability with previou reult, we apply the ame caling procedure a in Section 6 to achieve the ame average ector PD. Information on thi PD ditribution i given in Table 8, Appendix D. The portfolio with the lowet granularity admiible under the EU large expoure rule i an artificially generated portfolio, o that we have no PD information for ingle expoure. Therefore, we randomly aign PD from an empirical aggregate PD ditribution baed on the ame balance heet databae, but thi time aggregated over a ample of bank. The empirical PD ditribution i given in Table 20 and information on the PD ditribution of the low granular portfolio i provided in Table 9, Appendix D. 29 The reult for PD heterogeneity in every ector are given in Table. The reduction of im compared to Table 0, which occur for both portfolio, i due to the PD heterogeneity on the expoure level. Thi impact of PD heterogeneity ha alo been noted by Hanon et al (2005) and can be explained by the concavity of the dependence of on PD. Table : Comparion of, (in percent of total expoure) and im for portfolio with heterogeneou ector Portfolio Granularity im Relative error of Benchmark portfolio repreentative % low % Single ector portfolio low % Since and do not account for PD heterogeneity on the expoure level, thee value tay unchanged from Table 0 while im decreae. A a conequence the More detail on the databae and the logit model that wa ued to determine the PD can be found in Krüger et al. (2005). Since a negative correlation between expoure ize and PD emerged a a tylized fact in recent empirical literature (See, for example, Dietch and Petey (2002) or Lopez (2004)), we alo conidered the cae that the PD are perfectly ordered in term of decreaing expoure ize. We found that our reult are robut in thi cae. 22

24 underetimation by uing and intead of im i reduced relative to Table 0. Thi i confirmed by the approximation error in the lat column of Table, which i lower when uing heterogeneou PD compared to the cae of ector-dependent PD in Table 0. For the portfolio of repreentative granularity, and approximate extremely well a the relative error of i only %. In the ingle-ector portfolio, the approximation error of the proxie are poitive, implying that the effect of PD heterogeneity i tronger than the granularity effect, meaured relative to the highly granular portfolio with homogeneou ector PD. A a conequence, the ASRF* model actually overetimate. In ummary, the approximation error for all portfolio conidered vary between -6% and +8%. For the benchmark portfolio they are alway lower than the correponding value in Table 0 and for the ingle-ector portfolio the etimate are more conervative. The reult of Table 0 and Table taken together demontrate that the effect of PD heterogeneity counterbalance the effect of granularity. In general it i not poible to determine which of the two oppoing effect dominate. For the portfolio with a repreentative granularity in Table, both effect nearly cancel each other out, which i a rather encouraging reult. Granularity in thi portfolio i till conervatively meaured ince the underlying ample include only the larger expoure of a bank. Therefore, the effect of granularity for thi portfolio i arguably weaker than for portfolio of average granularity in real bank. Thi ugget that for uch portfolio, PD heterogeneity would tend to overcompenate the granularity effect and and would provide conervative etimate. Further empirical work i warranted to confirm thi indicative reult. Our analyi ha hown that PD heterogeneity on the expoure level improve the performance of the analytic approximation relative to the ituation of a granular portfolio with (only) ector-dependent PD. The reaon i that PD heterogeneity reduce the underetimation of that i caued by the granularity of the portfolio. Thi effect i even tronger if larger expoure or firm have lower PD than maller one. Furthermore, PD heterogeneity appear not to affect the relative difference between and *. 6. Summary and Concluion The minimum capital requirement for credit rik in the IRB approach of Bael II implicitly aume that bank portfolio are well diverified acro buine ector. Potential concentration rik in certain buine ector i covered by Pillar 2 of the Bael II Framework 23

25 which comprie the uperviory review proce. 30 To what extent the regulatory minimum capital requirement can undertate economic capital i an empirical quetion. In thi paper we provide a tentative anwer by uing data from the German central credit regiter. Credit rik i meaured by economic capital in a multi-factor aet value model by Monte Carlo imulation. In order to meaure the impact of concentration rik on economic capital, we tart in the empirical part with a benchmark portfolio that reflect average ector expoure of the German banking ytem. Since the expoure ditribution acro buine ector are imilar in Belgium, France and Spain, we expect that our main reult alo hold for other European countrie. Starting with the benchmark portfolio, we have ucceively increaed ector concentration, conidering degree of ector concentration which are obervable in real bank. The mot concentrated portfolio contained expoure only to a ingle ector. Compared with the benchmark portfolio, economic capital for the concentrated portfolio can increae by almot 37% and be even higher in the cae of a one-ector portfolio. Thi reult clearly underline the neceity to take inter-ector dependency into account for the meaurement of credit rik. We ubjected our reult to variou robutne check. We found that the increae in economic capital may even be greater, contingent to the dependence tructure. Since concentration in buine ector can ubtantially increae economic capital, a tractable and robut calculation method for economic capital which avoid the ue of computationally burdenome Monte Carlo imulation i deirable. For thi purpoe the theoretical part evaluate the accuracy of a model developed by Pykhtin (2004) which provide an analytical approximation of economic capital in a multi-factor framework. We have applied a implified, more tractable verion of the model which require only ector-aggregate of expoure ize, PD and expected lo everity. The dependence tructure i captured by the correlation matrix of the original multi-factor model. Furthermore, we have evaluated the extent to which, a the firt of two component in the analytic approximation of economic capital, already provide a reaonable proxy of. refer to the economic capital for a ingle-factor model in which the ector-dependent aet correlation are defined by mapping the richer correlation tructure of the multi-factor model. The benchmark for the approximation quality i alway the of the original multi-factor model which i obtained from MC imulation. We have hown that the analytic approximation formulae perform very well for portfolio with relatively granular and homogeneou ector. Thi reult hold for portfolio with different ector concentration and for variou factor weight and correlation aumption. Furthermore, we have found that i relatively cloe to the imulation-baed economic capital for mot of the realitic input parameter tupel conidered. 30 See BCBS (2004b), paragraph

26 Finally, we explore the robutne of our reult againt the violation of two critical model aumption, namely infinite granularity in every ector and ector-dependent. We find that lower granularity and PD heterogeneity (on the ingle expoure level) have two counterbalancing effect on the performance of the analytic approximation for economic capital. The reduction of granularity induce the analytic approximation formulae to have a light downward bia. In extreme cae of portfolio with the lowet granularity permiible by EU large expoure rule, the downward bia increae to 4%, depending on the ector tructure of the portfolio. Introducing heterogeneou PD on the individual expoure level reduce economic capital, but doe not affect the analytic approximation. A a conequence, the downward bia decreae. The relative error of the analytic approximation, meaured relative to the imulation-baed figure, lie in a range between 6% and +8%. In ummary, we find that heterogeneity in individual PD and low granularity partly balance each other in their impact on the performance of the analytic approximation. Which effect prevail depend on the pecific input parameter. Indicative reult ugget that in repreentative credit portfolio, PD heterogeneity will at leat compenate for the granularity effect which ugget that the analytic formulae approximate reaonably well or err on the conervative ide. In the cae tudied, it i poible to ue the implified verion of the model which provide analytic approximation of without acrificing much accuracy. Thi i an important reult a it ugget that upervior and bank can reaonably well approximate their economic capital for their credit portfolio by a relatively imple formula and without running computationally burdenome Monte Carlo imulation. Further reearch eem to be warranted, particularly in further advancing Pykhtin methodology in a direction which improve it approximation accuracy without extending data requirement. Thi could be achieved, for example, by exploring alternative way to map the correlation matrix of the multi-factor model into ector-dependent aet correlation. 25

27 Reference BCBS (2004a), Bael Committee on Banking Superviion, Bank Failure in Mature Economie, BCBS (2004b), Bael Committee on Banking Superviion, International Convergence of Capital Meaurement and Capital Standard: A revied Framework, BLUHM, C., L. OVERBK, AND C. WAGNER (2003), An Introduction to Credit Rik Modeling, Chapman&Hall/CRC. BURTON, S., S. CHOMSISENGPHET AND E. HEITFIELD (2005), The Effect of Name and Sector Concentration on the Ditribution of Loe for Portfolio of Large Wholeale Credit Expoure, forthcoming in the Journal of Credit Rik. CESPEDES, J.C.G., J.A DE JUAN HERRERO, A. KREININ, AND D. ROSEN (2005), A Simple Multi-Factor Adjutment, for the Treatment of Diverification in Credit Capital Rule, forthcoming in the Journal of Credit Rik. COURIEROUX, C., J.-P. LAURENT, AND O. SCAILLET (2000), Senitivity Analyi of Value at Rik, Journal of Empirical Finance 7, DE SERVIGNY A. AND O. RENAULT (2002), Default correlation: Empirical evidence, Standard and Poor Working Paper. DIETSCH, M. AND J. PETEY (2002), The Credit Rik in SME Loan Portfolio: Modeling Iue, Pricing and Capital Requirement, Journal of Banking and Finance 26, DÜLLMANN, K. (2006), Meauring Buine Sector Concentration by an Infection Model, Deutche Bundebank Dicuion Paper Serie 2, No 3. FITCHRATINGS (2004), Default Correlation and it Effect on Portfolio of Credit Rik", Credit Product Special Report. GORDY, M. (2000), A Comparative Anatomy of Credit Rik Model, Journal of Banking and Finance 24 (-2), GORDY, M. (2003), A Rik-Factor Model Foundation for Rating-Baed Bank Capital Rule, Journal of Financial Intermediation 2, GUPTON, G., C. FINGER AND M. BHATIA (997), CreditMetric - Technical Document. HAHNENSTEIN, L. (2004), Calibrating the CreditMetric Correlation Concept - Empirical Evidence from Germany, Financial Market and Portfolio Management 8 (4), HANSON, S., M.H. PESARAN, AND T. SCHUERMANN (2005), Firm Heterogeneity and Credit Rik Diverification, Working Paper ( HESTON, L. and G. ROUWENHORST (995), Indutry and Country Effect in International Stock Return, Journal of Portfolio Management 2 (3), HIRSCHMANN, A. O. (964), The Paternity of an Index, American Economic Review 54, JOINT Forum (999), Rik Concentration Principle, Bael. 26

28 KRÜGER, U., M. STÖTZEL AND S. TRÜCK (2005), Time Serie Propertie of a Rating Sytem Baed on Financial Ratio, Deutche Bundebank Dicuion Paper Serie 2, No 4. LOPEZ, J., (2004), The Empirical Relationhip between Average Aet Correlation, Firm Probability of Default and Aet Size, Journal of Financial Intermediation 3, MARTIN, R. and T. WILDE (2002), Unytematic Credit Rik, Rik Magazine, November, MERTON, R. (974), On the Pricing of Corporate Debt: The Rik Structure of Interet Rate, Journal of Finance 34, MOODY'S (2004), Moody' Reviit it Aumption regarding Corporate Default (and Aet) Correlation for CDO, November. PESARAN, M.H., T. SCHUERMANN AND B. TREUTLER (2005), The Role of Indutry, Geography and Firm Heterogeneity in Credit Rik Diverification, NBER Working Paper. PYKHTIN, M. (2004), Multi-Factor Adjutment, Rik Magazine, March, S&P (2004), Rating Performance 2003, S&P Special Report. ZENG, B. and J. ZHANG (200), Modeling Credit Correlation: Equity Correlation i not Enough, Moody' KMV Working Paper. 27

29 Appendix A Table 2: GICS Claification Scheme: Broad Sector and Indutry Group A: Energy A: Energy B: Material B: Material C: Indutrial C: Capital good C2: Commercial Service and Supplie C3: Tranportation D: Conumer Dicretionary D: Automobile and Component D2: Conumer Durable and Apparel D3: Hotel, Retaurant and Leiure D4: Media D5: Retailing E: Conumer Staple E: Food and Drug Retailing E2: Food, Beverage and Tobacco E3: Houehold and Peronal Product F: Health Care F: Health Care Equipment and Service F2: Pharmaceutical and Biotechnology G: Financial G: Bank G2: Diverified Financial G3: Inurance G4: Real etate H: Information Technology H: Software and Service H2: Technology Hardware & Equipment H3: Semiconductor & Semiconductor Equipment I: Telecommunication Service I: Telecommunication Service J: Utilitie J: Utilitie Table 3: Mapping NACE code to GICS code 2 (or more) -digit code Decription Mapped to GICS Agriculture and hunting E 2 Foretry B 5 Fihing E 0 Coal mining B Crude petroleum and natural ga extraction A 2 Mining of uranium and thorium B 3 Mining of metal ore B 4 Other mining and quarrying B 5 Food and beverage manufacturing E 6 Tobacco manufacturing E 7 Textile manufacturing D 8 Textile product manufacturing D 9 Leather and leather product manufacturing D 20 Wood product D 2 Pulp, paper and paper product B 22 Publihing and printing C2 23 Manufacture of coke, refined petroleum product and nuclear fuel A 28

30 24 (excl 244) Chemical and chemical product manufacturing B 244 Pharmaceutical F 25 Rubber and platic manufacturing D 26 Other non-metallic mineral product B 27 Baic metal manufacturing B 28 Fabricated metal manufacturing B 29 Machinery and equipment manufacturing C 30 Office machinery and computer manufacturing H 3 Electrical machinery manufacturing H 32 TV and communication equipment manufacturing H 33 Medical and optical intrument manufacturing F 34 Car manufacturing D 35 Other tranport equipment manufacturing D 36 Furniture manufacturing D 37 Recycling J 40 Ga and electricity upply J 4 Water upply J 45 Contruction C 50 Car ale, maintenance and repair D 5 Wholeale trade C2 52 (excl 52, 522,523) Retail trade D 522, 523 Conumer taple E 55 Hotel and retaurant D 60 Land tranport C3 6 Water tranport C3 62 Air tranport C3 63 Tranport upporting activitie and travel agencie C3 64 Pot and telecommunication I 65 Financial intitution G 66 Inurance G3 67 Support to financial intitution G 70 Real etate G4 7 Machinery and equipment leaing manufacturing C 72 Computer and related activitie H 85 Health care and ocial work F 90 Sewage and refue dipoal J 96 Reidential property management G4 29

31 Table 4: Comparion of ector concentration, aggregated expoure value over bank in Germany, France, Belgium and Spain Sector Germany France Belgium Spain A: Energy 0.8% 0.88% 0.05%,05% B: Material 6.0% 3.97% 7.45% 9,34% C: Indutrial % 63.82% 54.77% 48,53% C: Capital Good.53% 9.89% 32,90% C2: Commercial Service and Supplie 33.69% 37.74% 0,20% C3: Tranportation 7.4% 7.4% 5,43% D: Conumer Dicretionary 4.97%.9% 5.77% 8,60% E: Conumer Staple 6.48% 7.2% 7.05% 0,20% F: Health Care 9.09% 5.00% 5.64%,85% H: Software and Service 3.20%.47%.86%,99% I: Telecommunication Service.04%.9% 0.54% 2,67% J: Utilitie 6.67% 3.82% 6.87% 5,77% Table 5: Correlation matrix baed on MSCI EMU indutry indice (baed on weekly log return data covering the Nov Nov 2003 period; in percentage). A B C C2 C3 D E F H I J A: Energy B: Material C: Capital Good C2: Comm. v & upplie C3: Tranportation D: Conumer dicretionary E: Conumer taple F: Health Care H: Information technology I: Telecommunication ervice J: Utilitie 00 3 Aggregate of C, C2 and C3 only ued for comparion with French data. Not ued in the analyi. 30

32 Appendix B The multi-factor adjutment tq can be calculated according to Pykhtin (2004) a follow: (A) l ( y) tq = v ( y) v( y) + y 2 l ( y) l ( y) y= N ( q) where y denote the ingle ytematic rik factor. The firt and econd derivative of the lo ditribution function in a one-factor model are (A2) N l ( y) = w µ pˆ ( y) = N l ( y) = w µ pˆ ( y) = ˆ ' pˆ ' where p ( y) and ' ( y) are, repectively, the firt and the econd derivative of the conditional probability of default. (A3) c N ( p ) c y pˆ ( y) = N 2 2 c c. c N ( p ) c y N ( p ) c y pˆ ( y) = N c c c The factor loading in the ASRF* model i denoted by c which can be written a c = r ρ whereρ denote the correlation between the compoite ector factor factor Y in the ASRF* model. The conditional variance v( y) var( L Y = y) S = t= = S Y ( ( ) ( ) ) Y ( ( ) ( ) ) Y and the ytematic v ( y) = w ˆ ˆ ˆ ˆ wt µ µ t N2 N p ( y), N pt ( y), ρt p ( y) pt ( y) S + w µ pˆ ( y) N N pˆ ( y), N pˆ ( y), ρ S S Y N ( pˆ ( )) ˆ t y ρt p ( y) v ( y) = w ˆ ( ) ˆ wt µ µ t p y N p ( ) 2 t y = t= Y ( ρt ) S r + w ˆ ( ) 2 2 ( ˆ µ p y N N p ( y) ) = + r 3

33 where N 2 ( ) denote the cumulative ditribution function of the bivariate-normal ditribution and Y ρ ha the meaning of a conditional aet correlation for two expoure in t ector t and, conditional on Y. Thi conditional aet correlation can be written a r r ρ c c Y t t t ρ =. t 2 2 c c t 32

34 Appendix C In Pykhtin (2004) the coefficient b,..., bs are obtained by maximizing the correlation between Y and the rik factor Y,, Y S which lead to the following optimization problem: max S b,..., bs, j j = j= S γ α b. ubject to S 2 b =. The olution of thi problem i given by = b j γ α = S, j = λ. λ i the Lagrange multiplier choen to atify the contraint. Again there i no unique olution for γ. We follow Pykhtin who reported good reult when defining ( ) ( ) N p r N q γ = wµ N 2 r. 33

35 Appendix D Table 6: Decriptive tatitic of expoure ditribution of a portfolio of ector, repreentative in term of granularity Sector Expoure No. Minimum 25% percentile Median 75% percentile Maximum NA.2 NA Table 7: Decriptive tatitic of expoure ditribution of a low granular portfolio of ector Sector Expoure No. Minimum 25% percentile Median 75% percentile Maximum NA NA

36 Table 8: Scaled PD ditribution of a portfolio of ector, repreentative in term of granularity Sector Expoure No. Minimum 25% percentile Median 75% percentile Maximum.0% NA.0% NA.0% % 0.5%.0% 2.0% 8.8% % 0.7%.4% 2.3% 8.9% % 0.9%.6% 3.0% 4.4% % 0.6%.4% 2.6% 5.7% % 0.7%.2% 2.5% 2.7% % 0.8%.9% 3.4% 5.% % 0.3% 0.6%.3% 8.4% % 0.5%.6% 2.5% 5.5% % 2.4% 4.4% 5.7% 7.8% % 0.2% 3.4% 0.6% 2.4% Table 9: Scaled PD ditribution of a low granular portfolio of ector Sector Expoure No. Minimum 25% percentile Median 75% percentile Maximum.0% NA.0% NA.0% %.3%.3%.3% 4.2% %.5%.5%.5% 5.% %.8%.8%.8% 5.9% %.5%.5%.5% 4.9% %.4%.4%.4% 4.7% %.7%.7%.7% 5.8% % 0.7% 0.7% 0.7% 2.4% %.2%.2%.2% 3.9% % 2.4% 2.4% 2.4% 2.4% 9 0% 0.3% 0.3% 0.3% % Table 20: Quality ditribution of German firm in the Bundebank databae Rating grade AAA AA A BBB BB B Share in percent PD in percent