DRAFT October 2005 DRAFT

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1 DRAFT October 2005 DRAFT The Effect of Name and Sector Concentration on the Ditribution of Loe for Portfolio of Large Wholeale Credit Expoure * Erik Heitfield Federal Reerve Board erik.heitfield@frb.gov Steve Burton Federal Depoit Inurance Corporation burton@fdic.gov Souphala Chomiengphet Office of the Comptroller of the Currency ouphala.chomiengphet@occ.trea.gov Abtract: Thi paper examine the influence of ytematic and idioyncratic rik on credit loe for portfolio of large wholeale bank loan. Information on bank larget credit expoure from US bank regulator Syndicated National Credit (SNC) examination program and a hierarchical factor model calibrated from KMV data are ued to imulate the ditribution of portfolio credit loe for 30 real-world loan portfolio. We find that for very large SNC portfolio idioyncratic rik i of limited importance, but it can meaningfully increae Value-at-Rik (VaR) for maller portfolio. The average contribution of ytematic rik to Value-at-Rik i imilar in group of relatively large and relatively mall SNC portfolio. Simple indexe of name and ector concentration are poitively correlated with portfolio VaR even after controlling for difference in average credit quality and portfolio ize. We ue Monte Carlo imulation to etimate the marginal contribution to portfolio VaR of credit expoure to individual ector and find that expoure to different economic ector have dramatically different influence on VaR. Thee difference reult not only from variation in the average credit quality of obligor acro ector, but alo from feature of the dependence tructure of credit loe. The relative importance of expected lo, ytematic rik, and idioyncratic rik varie coniderably from ector-to-ector and i enitive to the ditribution of expoure within a given portfolio. * The view expreed here are olely thoe of the author. They do not reflect the opinion of the Federal Reerve Board of Governor, the Federal Depoit Inurance Corporation, or the Office of the Comptroller of the Currency.

2 1. Introduction While bank upervior and rik manager have long recognized the importance of managing concentration rik in credit portfolio, recent reviion to the Bael Capital Accord have focued attention on thi iue. According to the Bael Committee on Banking Superviion, [r]ik concentration are arguably the ingle mot important caue of major problem at bank. 2 Under the Bael II Framework Document iued in June of 2004, minimum regulatory capital requirement (o called Pillar I requirement) are calibrated uing model that aume that bank portfolio are well diverified. However, the Framework Document require that bank have in place effective internal policie, ytem and control to identify, meaure, monitor, and control their credit rik concentration, and hould explicitly conider the extent of their credit rik concentration in their aement of capital adequacy. 3 Effort to manage portfolio concentration are intended to mitigate the effect of ytematic rik reulting from dependence in loe acro credit and idioyncratic rik aociated with large expoure to individual obligor. Thi paper examine the influence of ytematic and idioyncratic rik on credit loe for portfolio of large wholeale bank loan. By combining information on bank larget credit expoure from US regulator Syndicated National Credit (SNC) examination program with KMV data we imulate the ditribution of portfolio credit loe for 30 real-world bank portfolio. Uing thee imulation we invetigate the relationhip between portfolio loe and imple indexe of name and ector concentration and explore how expoure to individual economic ector contribute to portfolio Value-at-Rik. The cro-expoure dependence tructure of credit loe which give rie to ytematic rik i decribed uing a hierarchical factor model etimated from KMV data. Under thi model a firm future default tatu depend on an idioyncratic factor and an indutry ector rik factor. The ector factor, in turn, depend on an array of common rik factor that influence all ector a well a factor that are unique to individual ector. Correlation among firm within an indutry ector are driven by each firm enitivity to the ector factor. Correlation among firm in different ector depend on the firm enitivitie to their correponding ector factor and the ector factor enitivitie to the common rik factor. Thi model impoe ufficient tructure on the correlation among firm to allow u to etimate relevant parameter uing available KMV data but it i nonethele reaonably general. The enitivitie of firm to their ector rik factor are allowed to vary acro ector and very little arbitrary tructure i impoed on correlation among indutry ector factor. 2 BCBS (2004), paragraph BCBS (2004), paragraph

3 We ue the hierarchical factor model to invetigate the effect of ytematic and idioyncratic rik on the ditribution of loe for bank yndicated loan portfolio. Among the different type of credit expoure held by bank, yndicated loan are epecially ueful for tudying the implication of ector and name concentration. 4 Thee credit expoure, a repreented by SNC examination data, account for an etimated onethird or more of large U.S. agent bank total corporate loan expoure, and conit primarily of expoure to large dometic and multinational companie. A uch, the performance of yndicated loan often provide a window into broader corporate credit trend and credit condition. 5 For intance, the credit deterioration experienced by variou large U.S. banking organization during the receionary period of 2000 to 2002 correponded to a ignificant rie in problem yndicated loan a identified through the SNC program. Becaue they repreent loan expoure of the larget firm within any given ector, yndicated loan expoure alo erve a a meaningful barometer of indutry condition and trend. By imulating the ditribution of loe for yndicated credit portfolio, we are able to decompoe portfolio Value-at-Rik into expected, ytematic, and idioyncratic rik component. We find that both idioyncratic and ytematic rik can have ignificant effect on the ditribution of loe for bank large wholeale credit portfolio. For very large SNC portfolio idioyncratic rik i of limited importance, but it meaningfully increae Value-at-Rik for maller portfolio. Bank with relatively mall SNC portfolio may therefore need to more actively manage concentration to individual obligor than bank with larger portfolio, or they may need to hold additional capital to offet the effect of idioyncratic rik. The average contribution of ytematic rik to Value-at-Rik i imilar in group of relatively large and relatively mall SNC portfolio. Although larger portfolio tend to be better diverified acro ector than maller one, portfolio ize alone doe not appear to ubtantially mitigate the effect of ytematic rik. Simple indexe of name and ector concentration are poitively correlated with portfolio Value-at-Rik, even after controlling for difference in average obligor credit quality and portfolio ize. However, though correlated with VaR, thee indexe cannot fully explain oberved variation in VaR among the 30 SNC portfolio we tudy. Concentration indexe baed only on portfolio expoure weight can provide ueful metric for aeing the extent to which bank expoure are diverified acro ector or name, but becaue they are not enitive to cro-ector difference in expoure characteritic they are of limited utility for managing expoure to individual ector or obligor. 4 Syndicated loan are large corporate loan expoure held by a group (or yndicate) of lender. 5 See 2

4 In order to effectively manage name and ector concentration bank need to compare the relative impact of expoure to different ector on portfolio lo ditribution. We ue Monte Carlo imulation to etimate the marginal contribution of credit expoure to individual ector to portfolio Value-at-Rik and find that different economic ector have dramatically different influence on VaR. Thee difference reult not only from variation in the average credit quality of obligor in different ector, but alo from feature of the dependence tructure of credit loe acro ector. The relative importance of expected lo, ytematic rik, and idioyncratic rik varie from ector-toector and i enitive to the ditribution of expoure within a particular SNC portfolio. Thu, effort to manage name concentration can be expected to have a greater impact on VaR in ome ector and portfolio than other. The benefit of reducing aggregate expoure to particular ector will imilarly vary acro ector and will depend on the weighting of ector expoure within a given portfolio. The paper i organized a follow. Section 2 decribe the SNC examination data ued in thi analyi. Section 3 introduce everal imple concentration meaure that can be calculated with a minimum of information on the ector weighting and number of expoure within a portfolio. Section 4 and 5 dicu calibration of the hierarchical factor model and how how it can be ued to imulate the ditribution of realized credit loe for a portfolio of loan. In Section 6 and 7 we examine the relationhip between obervable portfolio characteritic and the ditribution of imulated portfolio loe. Section 6 focue on three bank portfolio in detail while Section 7 invetigate pattern in the full ample of 30 bank portfolio. Section 8 how how Monte Carlo imulation can be ued to etimate the marginal contribution of expoure in different ector to portfolio VaR and applie thi approach to elected SNC portfolio. Section 9 ummarize concluion and how how they relate to recent empirical and theoretical reearch on the management of credit rik in bank loan portfolio. 2. Portfolio of Syndicated National Credit The SNC examination program provide U.S. banking upervior with an important ource of information about corporate credit market trend and condition. Thi program wa developed in 1977 for the primary purpoe of enuring the conitency and accuracy of uperviory rik rating for commonly-hared yndicated loan expoure held by regulated banking organization. The program i upported by a databae that i updated annually with SNC expoure information. Reporting requirement are triggered when a yndicated loan commitment exceed $20 million and when that expoure i held by three or more regulated entitie. 6 Baed on the mot recently available 2005 SNC data, the SNC databae capture nearly $1.7 trillion in commercial credit expoure to roughly 4,700 borrower. Thee expoure repreent an etimated one-third of the total commercial loan expoure (on and off-balance heet) of U.S. regulated banking organization. With an average credit expoure of $350 million per borrower, the 6 That i, regulated by one of the three U.S. federal bank regulator: the Federal Reerve Sytem, the Office of the Comptroller of the Currency, or the Federal Depoit Inurance Corporation. 3

5 databae predominantly reflect credit expoure of large dometic and multinational corporation. We analyze the SNC portfolio for the 30 larget SNC lender a meaured by the total dollar value of SNC commitment in Taken together, thee bank hold roughly $760 billion in SNC commitment, which repreent about 46 percent of the total value of commitment included in the SNC examination databae. Among thee bank are the larget US commercial bank, a well a maller regional, monoline, and proceing bank. For each bank SNC expoure are grouped into 50 broadly-defined indutry ector. Thee indutry ector are compried of companie in economically related financial and non-financial buinee a determined by each firm primary indutry claification code uing either North American Indutry Claification Sytem (NAICS) or Standard Indutrial Claification (SIC) code. The compoition of each of thee 50 broad ector, baed on SIC code, i detailed in Table 1. The SNC databae include detailed information on each SNC expoure, but for thi analyi we ue ector-level data. For each of the bank in our ample we have compiled data on the total dollar value of SNC commitment in each indutry ector and the number of SNC expoure to each ector. Importantly, our data do not included information on the ize ditribution of expoure within a ector or the identitie of the obligor aociated with individual expoure. For thi reaon, we aume that all of a bank commitment to a particular ector are the ame ize. Thu, for example, if the total loan commitment by a bank to the Aeropace and Defene ector i fifty-million dollar and the bank ha extended 10 loan to thi ector, we aume that the commitment amount for each facility i five-million dollar. Throughout thi paper, we will ue w to denote the portfolio weight on ector a meaured by the ector hare of total commitment. We will ue n and n to denote the number of expoure in ector and the number of expoure in the portfolio repectively. Table 2 report information on the ditribution and the number of SNC expoure by indutry ector for the aggregate portfolio containing all yndicated national credit including thoe held by bank not repreented in our ample. A we hall ee preently, thi All SNC portfolio provide a ueful benchmark for aeing the degree of concentration of individual bank portfolio. To provide a ene of the relative credit quality of expoure in each ector, Table 2 alo report ector-wide average KMV EDF. 7 7 Some intitution in our ample have a mall number of SNC expoure to real-etate invetment trut (REIT) which are not included in thi analyi. A dicued in Section 3, our model of credit loe i calibrated uing imputed aet value and EDF from KMV. While KMV publihe EDF for ome REIT, it recommend that thee parameter be treated with particular caution. An obligor EDF and imputed aet value i enitive to it liability tructure which, in the cae of a REIT, can change rapidly. Becaue data on REIT liability tructure are updated relatively infrequently, a REIT EDF may not reflect it current liability tructure, and converely, it EDF may change dramatically a new liability information become available. 4

6 Becaue of the confidential nature of SNC examination data, we cannot report detailed information on the ector ditribution of yndicated credit expoure for individual intitution. To preerve the anonymity of bank-level information and to aid in ummarizing the available SNC data, we have divided the 30-bank ample into bank with mall, medium, and large SNC portfolio. Small portfolio are defined a thoe with total commitment of le than $10 billion; medium portfolio are thoe with total commitment between $10 and $20 billion; and large portfolio are thoe with total commitment of greater than $20 billion. The row labeled Portfolio Characteritic in Table 3 report ummary information on the portfolio in each of the three ize categorie a well a the All SNC portfolio. 3. Simple indexe of name and ector concentration The credit rik for a portfolio of expoure at ome future horizon can be decompoed into three component: expected lo (EL), ytematic rik, and idioyncratic rik. Expected lo refer to that component of future loe that can be forecat from currently-available information on the characteritic of portfolio expoure. Becaue it can be predicted, EL can be managed relatively eaily by appropriately pricing newlyoriginated loan and by etting aide reerve for eaoned loan that have declined in credit quality. Sytematic rik i that component of portfolio loe attributable to dependence in loe acro individual credit expoure. Typically, ytematic rik i modeled a ariing from common hock that affect many obligor at once. Thi rik component can be managed but it cannot be eliminated. Bank can leen the influence of ytematic rik by hifting lending toward expoure whoe loe tend to be le highly correlated with one another. For example, if loe on loan to obligor within an economic ector tend to be more highly correlated than thoe of loan to obligor in different ector, then preading expoure acro ector may reduce the volatility of portfolio loe. Idioyncratic rik refer to that component of future loe that can potentially be diverified away. Typically, idioyncratic rik i een a ariing from independent hock to individual obligor. In principle, by ditributing expoure in a portfolio acro a large number of obligor, idioyncratic rik can be largely eliminated. Equity capital erve a a buffer to cover unexpected portfolio loe ariing from ytematic and idioyncratic rik. Effort by bank to limit expoure to particular economic ector or to individual obligor are intended to reduce unexpected loe, thereby leening the need for cotly capital. Indexe deigned to meaure name concentration and ector concentration are often ued to ummarize the ditribution of expoure within a portfolio acro obligor or ector. In thi ection we decribe typical example of uch indexe and ue them to compare SNC portfolio. In Section 6 and 7 we will examine how thee indexe are related expected lo and ytematic and idioyncratic rik in thee portfolio. Broadly peaking, name concentration refer to any granularity in expoure to individual obligor within a portfolio. Gordy (2003) ugget a tandard for meauring name concentration. He define an infinitely-fine-grained portfolio a one in which no expoure to a ingle obligor i large enough relative to the total portfolio to meaningfully 5

7 affect the realized portfolio lo rate. In practice, of coure, no portfolio can achieve thi level of diverification acro name, but the infinitely-fine-grained portfolio erve a a ueful benchmark for meauring name concentration. The name-concentration Herfindahl index h N w n 2 provide a imple meaure of the deviation of an actual portfolio from the infinitely-finegrained portfolio benchmark. Thi index reflect information about both the number and the relative ize of individual obligor expoure in a portfolio. A portfolio coniting of only one expoure would have a Herfindahl index equal to one and a portfolio coniting of an infinite number of very mall expoure Gordy infinitely-fine-grained portfolio would have a Herfindahl index of zero. An alternative to meauring name concentration relative to a theoretical fully diverified portfolio i to meaure uch concentration relative to a well diverified market portfolio. In thi analyi we will ue the All SNC portfolio a a benchmark for aeing name concentration. Let w * and n * denote the ector expoure weight and the number of obligor in ector for the All SNC portfolio. We define the name-concentration entropy index a e * * ( ln ( ) ln ( )) * * ln min ( w n) w w n w n N. ( ) Thi index i equal to zero if all obligor expoure in the portfolio receive the ame weight a thoe in the All SNC portfolio and it approache one in the limiting cae where the portfolio weight on the ingle obligor with the mallet weight in the All SNC portfolio grow to dominate all other. If the ector expoure weight and the relative number of expoure in each ector are held contant, then e N i decreaing in the number of expoure in the portfolio. 8 Sector concentration i a bit harder to define than name concentration. Since the number of indutry ector i finite and fixed, it i not meaningful to conider a benchmark diverified portfolio in which expoure to any given ector i o mall that no ector ha a ignificant effect on portfolio loe. It alo would not be particularly meaningful to aume that a diverified portfolio i one in which each ector receive equal weight becaue ome ector clearly play a much larger role in the economy than other. For 8 Note that both h N and e N are derived under the aumption that all expoure within a ector are the ame ize o that the portfolio weight on obligor i in ector i w /n. Thee meaure are only approximation to thoe indexe that would be calculated from actual obligor expoure weight. h N will undertate the true name concentration Herfindahl index. 6

8 thee reaon, it i particularly deirable to ae ector concentration relative to a realworld benchmark portfolio. The ector-concentration entropy index e S ( ln ( ) ln ( * )) * ( w) w w w ( ) ln min provide a meaure of the difference between a portfolio ector expoure weight and thoe of the All SNC portfolio. Thi entropy index ha the following appealing propertie: (i) when all weight in a bank portfolio i concentrated in that ector with the mallet weight in the All SNC portfolio e S = 1, (ii) when the bank ector weight are equal to thoe of the All SNC portfolio e S = 0, and (iii) in all other cae e S lie between zero and one. The ector-concentration Herfindahl index i defined a h S w. 2 While h S i commonly ued to ummarize ector concentration, it ha ome unappealing feature. It i very enitive to ector definition. For example, if two large ector are aggregated together, h S may increae ubtantially. Moreover, in contrat to the entropy meaure, the Herfindahl index i inenitive to difference in the overall ize of different ector. Thi meaure attain a minimum when a portfolio i equally weighted acro all ector, even though ome ector are preumably much larger than other. The row labeled Concentration Indexe in Table 3 report SNC portfolio group average, minimum, and maximum of the four meaure of portfolio concentration dicued above. A expected, maller portfolio appear by our meaure to be more concentrated than larger portfolio. However, notice that for all four concentration meaure there are ubtantial overlap in the range of index value acro group. Thu, there i not a trict decreaing relationhip between portfolio ize and either name or ector concentration. Table 4 report Kendall correlation coefficient among the four concentration indexe calculated for the 30 SNC portfolio in our ample. 9 The entropy and Herfindahl indexe of name concentration are very highly correlated, uggeting that both thee meaure convey imilar information. The entropy and Herfindahl indexe of ector concentration are omewhat le highly correlated, o there may be ome advantage in uing one of thee indexe over the other. The name- and ector- concentration indexe are poitively correlated with one another becaue, in general, larger portfolio tend to be le concentrated acro both name and ector. 9 Unlike the more common Pearon correlation coefficient, Kendall tau i invariant to monotone tranformation of the variable of interet. Hence it provide a better indication of whether two indexe convey imilar information. 7

9 None of the concentration indexe dicued here are directly linked to tandard portfolio rik metric uch a Value-at-Rik. While it i intuitive to think that a portfolio that i more evenly ditributed acro name or ector may be le ubject to the effect of idioyncratic and ytematic rik, the relationhip between portfolio expoure weight and the ditribution of portfolio credit loe i quite complex and except under very tylized aumption it cannot be accurately decribed uing imple portfolio-wide concentration indexe. 10 To fully undertand the role of expected, ytematic, and idioyncratic rik in credit portfolio a model of the dependence of credit loe acro expoure i needed. 4. Modeling dependence in expoure loe To model dependence in credit loe acro expoure we ue a framework baed on Merton (1974) imilar to that underlying indutry-tandard credit rik model uch a thoe developed by KMV (Crobie and Bohn, 2005) and the Rik Metric Group (Gupton, Finger, and Bhatia, 1997). Under thi framework the default tatu of firm i at a one-year-ahead aement horizon depend on the realization of a continuou index of the firm credit quality denoted Y i. Thi variable i commonly interpreted a a meaure of the firm return on aet over the aement horizon. If the realized value of Y i lie below a critical default threhold γ i then firm i default and expoure to that firm accrue a credit lo; if the realized value of Y i exceed the default threhold no lo arie from expoure to the firm. The probability of default of firm i i imply the probability that Y i lie below γ i. For implicity we aume that the lo-given-default (LGD) per dollar expoure i exogenou and i the ame for all firm. Given thi framework, the lo rate per dollar expoure to firm i i (1) L i 0 if = LGD if Y i Y i > γ i γ i We aume that Y i can be expreed a the um of two rik factor: an idioyncratic rik factor that i unique to firm i and a ector rik factor that affect all firm in firm i indutry ector. If firm i i part of ector then (2) Y = Z λ + E 1 λ 2 i i where Z i the ector rik factor, E i i the idioyncratic rik factor, and λ i a ectorpecific parameter that lie between zero and one. Z and E i are tandard normal random variable that are independent of one-another o that, by contruction, the marginal ditribution of Y i i alo tandard normal. λ, the ector factor loading, decribe the 10 One can conclude that a the name-concentration Herfindahl index approache zero the contribution of idioyncratic rik to portfolio loe approache zero a well. However, there i no direct relationhip between value of thi index that are different from zero and the magnitude of the contribution from idioyncratic rik at the portfolio level. 8

10 enitivity of firm in ector to the ector rik factor Z. Notice that a higher value for λ implie that firm i i more enitive to the indutry factor and le enitive to the idioyncratic factor. We next aume that each ector rik factor can be expreed a a linear combination of K common factor and one ector-pecific factor. The ector rik factor for ector i defined a (3) Z = X ' ω + U 1 ω 'ω where X i a K-dimenional tandard normal random vector of common rik factor, U i a tandard normal random variable that i independent of X, and ω i a K-dimenional parameter vector that i normalized o that ω' ω 1. ω, the common factor loading, decribe the enitivity of the ector rik factor Z to the vector X of common factor. Becaue all ector rik factor depend on ome or all of the common factor, the ector rik factor will be correlated with one-another. The magnitude and direction of uch correlation depend on each the common factor loading for each ector. Correlation in lo rate acro firm are driven by correlation in the firm credit quality indexe. Conider firm i in ector and firm j in ector t. Equation (2) and (3) imply that (4) Cor ( Yi, Yj) 2 λ if = = λλ ω ω ( ' ) t t if t. t Oberve that when λ i high, credit loe for two firm in ector will be highly correlated with one another. The correlation in lo rate among expoure to two firm in different indutry ector depend on each firm enitivity to it ector rik factor and the two ector rik factor enitivitie to the common rik factor. All ele equal, loe for expoure to two firm in the ame ector are more likely to occur together than loe for two firm in different ector. 11 Under thi imple model etimate of the ector factor loading λ and etimate of each ector common factor loading ω are all that are needed to decribe dependencie in lo rate acro expoure. To etimate the factor loading, we ue monthly KMV data on the aet value for 4,516 large corporate firm with average hitorical aet value exceeding $500 million between 1993 and For each firm it rate, KMV impute an aet value from data on the firm leverage and it equity price and volatility. 12 We ue thee imputed aet 11 A ingle-rik-factor model conitent with the aumption ued to derive minimum regulatory capital requirement under Bael II arie a a pecial cae of the hierarchical model in which for every ector, ω ω = 1. In thi pecial cae all ector factor are perfectly correlated, and the correlation between Y i and Y j i imply the product of the ector factor loading for firm i and j. 12 The detail of thi procedure are proprietary to KMV Corporation, but the baic methodology i decribed by Crobie and Bohn (2005). Treating equity a a call option on the aet of a firm, KMV 9

11 value to contruct annual rate of return on aet for each firm which can tand in a proxie for the credit quality indexe, Y i. Equation (4) implie that the ector factor loading λ i equal to the quare-root of the pair-wie correlation between Y i and Y j for any two firm i and j in ector. Uing thi fact, we etimate each ector ector factor loading a the quare-root of the average empirical correlation between all pair of obligor in the ector. Table 5 report etimated ector factor loading. A can be een from thee reult, the correlation among the aet return of the firm in a ector varie coniderably acro ector. Return for firm in the Oil Refining and Delivery ector, for example, appear to move together rather cloely while thoe of firm in the Commercial Banking ector appear to be much le correlated with one-another. In interpreting thee figure it i important to keep in mind that although the aet return for firm in a ector may not be highly correlated, the aggregate return for the ector a a whole may nonethele be cloely linked with thoe of other ector. Within a ector the fortune of individual firm may differ for a variety of largely idioyncratic reaon including management quality, market power, and even imple luck. In aggregate the idioyncraie aociated with individual firm hould average out o that ector-wide return may tend to more directly reflect fundamental economic driver of profitability. To invetigate the relationhip among return in different ector it i therefore ueful to examine correlation among ector-wide mean return captured in our model by indutry ector rik factor. Normalized annual average rate of return for each indutry ector provide proxie for the ector factor and the empirical correlation among thee proxy ector factor provide an etimate of the correlation matrix of ector factor. Deriving etimated common factor loading from an empirical correlation matrix i a tandard factor analyi problem. Let Z be the vector of ector factor (Z ) for each of S ector and let Ω be a K-by-S matrix whoe column are the K-dimenional common factor loading (ω ). Equation (3) implie that (5) Cor [ ] = ' + diag ( ' ) Z ΩΩ I ΩΩ. S Given the empirical proxy ector factor correlation matrix, a tandard iterative principle component method i ued to derive parameter for Ω that minimize dicrepancie between the left- and right-hand-ide of equation (5). The etimate preented here aume ix common factor (K = 6). Several other pecification of K were conidered. The majority of the correlation among ector can be explained by only one common factor. Adding common factor beyond the firt ix doe little to improve the fit of the model. Etimated common factor loading for each ector are reported in Table 5. Shaded cell identify large loading of particular interet. If two ector have loading on a given invert an option pricing formula to impute a firm aet price and volatility from obervable data on firm leverage and the value and volatility of equity. 10

12 factor that are large in magnitude and hare the ame ign then aggregate return for thee ector tend to be more highly correlated with one another. Interpreting loading from a multifactor model i a notoriouly ambiguou exercie that invariably involve a degree of judgment. Nonethele an examination of the etimated common factor loading provide inight into the relationhip among ector return. Loading on the firt common factor are all the ame ign and are relatively large in magnitude. Thi factor would eem to capture the influence of a general macroeconomic cycle on all ector. Loading on the econd factor ugget high correlation in return among high-tech ector. Large loading on the third and fourth factor point to high correlation among return in foil energy and other reource extraction ector. Loading on the fifth factor ugget that the fortune of firm in Healthcare and Medical Equipment ector tend to move together. 5. Simulating portfolio loe To examine the role of expected lo, ytematic rik, and idioyncratic rik in SNC portfolio we ue Monte Carlo imulation to etimate portfolio lo ditribution. We aume that all credit loe occur at the end of a one-year horizon and arie only from obligor default. Thi default-mode approach abtract from expoure revaluation effect that arie in a mark-to-market etting when expoure have maturitie longer than one year. A mentioned earlier, we aume that the lo-given-default for all expoure i non-tochatic. 13 Finally, we aume that the amount of an expoure at default i 100% of the current expoure commitment amount. Taken together, thee aumption imply that the only ource of uncertainty urrounding a portfolio one-year-ahead lo rate i the default tatu of obligor. For practical reaon we all loan within a ector a if the were interchangeable, ex ante. The dataet ued in thi analyi provide information on the total dollar value of loan commitment by in each ector and the number of obligor in each ector. It doe not include the credit amount committed to individual obligor, o we aume that all loan within a given ector receive equal weight in the portfolio. Becaue we do not know the counterparty aociated with individual expoure, we aume that each obligor ha a PD equal to the average KMV EDF for all SNC expoure in that obligor ector. 14 Given thee tylized aumption, the lo per dollar expoure to ector i 13 For all expoure we aume an LGD of 45 percent. Thi i conitent with the regulatory treatment for unecured wholeale expoure under Bael II foundation internal-rating-baed approach. If LGD were tochatic but independent acro expoure, it would contribute to idioyncratic rik. If it were tochatic and wa not independent acro expoure, it would contribute to both idioyncratic and ytematic rik. 14 If PD i the average KMV EDF for ector then for each obligor i in ector we et γ = Φ -1 i (PD ). It i eay to verify that given thi calibration equation (1) implie that the one-year-ahead unconditional default probability for each obligor in ector i equal to PD. 11

13 D (6) L = LGD n where D in an integer-valued random variable that decribe the number of default in the ector. D depend on idioyncratic and ector rik factor. Since we aume that all obligor in a ector hare the ame default probability, the expected lo per dollar expoure to ector i imply (7) [ L ] E = PD LGD. Conditional on the ector rik factor Z, D i a binomial random variable compoed of n independent Bernoulli trial. Equation (1) and (2) imply that the expected lo rate for ector given the ector rik factor Z i (8) E [ ] ( ) where p L Z = p Z LGD ( Z ) ( PD ) 1 Φ λz =Φ 2 1 λ The Law of Large Number implie that a n grow large for a given realization of Z equation (6) converge in probability to equation (8). 15 The lo rate per dollar expoure in a portfolio of ector expoure i (9) L = wl. Equation (9) can be decompoed in a way which allow u to pare out the incremental effect of expected lo, ytematic rik, and idioyncratic rik on portfolio loe. Oberve that (9 ) L= we[ L] + w( E [ L Z] E[ L] ) + w( L E[ L Z] ) A B C Term A in thi expreion i the portfolio expected lo rate. It i non-tochatic and can be readily calculated uing equation (7). Term B i that component of the portfolio lo rate that can be attributed to common hock within and acro ector. Term C i that component of the portfolio lo that can be attributed to lack of diverification acro name within ector. A the number of expoure in all ector grow large, Term C approache zero for any realization of the ector rik factor. 15 See Vaicek (1991) for a derivation of (8). 12

14 The ditribution of L for a given bank portfolio i etimated uing Monte Carlo imulation. We firt draw a realization of each ector rik factor Z uing equation (3) and the etimated common factor loading. For each ector we then imulate the lo rate per dollar expoure conditional on the realized ector rik factor by drawing a value of D from a binomial ditribution with n trial and ucce probability p (Z ). Finally, we ue equation (6) and (9) to calculate a realized portfolio lo rate which embed the effect of both ytematic and idioyncratic rik. For each imulated draw of L, equation (9 ) can be ued to decompoe the lo rate into expected, ytematic, and idioyncratic component. By imply dropping term C from the imulated lo calculation we are able to derive the portfolio lo ditribution conitent with the aumption of perfect diverification acro name. 6. The role of ytematic rik and idioyncratic rik in elected portfolio In thi ection we compare imulated lo ditribution for three bank portfolio choen from the portfolio-ize group decribed in Section 2. We have deliberately elected three portfolio with imilar expected loe to allow u to focu on the effect of ytematic and idioyncratic rik on portfolio lo ditribution. Table 6 report concentration indexe for the three portfolio. A expected, maller portfolio have greater name concentration a meaured by the entropy and Herfindahl concentration indexe. Note, however, that the entropy and Herfindahl indexe give conflicting information about the relative importance of ector concentration in the Small Bank portfolio. The Herfindahl index ugget that thi portfolio i le concentrated among ector than the Medium Bank or Large Bank portfolio, while the entropy index ugget that it i more concentrated. The Lorenz curve in Figure 1 provide a more detailed look at the ditribution of portfolio expoure acro ector. The difference between a portfolio Lorenz curve and the 45-degree line indicate a difference between the portfolio ector weight and the All SNC portfolio weight. In the figure cumulative ector weight are calculated baed on a orting of ector from highet to lowet average EDF, o a Lorenz curve above the 45- degree line indicate a portfolio that i weighted toward ector with higher average default probabilitie relative to the All SNC portfolio. A can be een from thi figure, the Large Bank portfolio i lightly more weighted toward particularly low- and particularly high-edf ector than the All SNC portfolio. The difference between the Medium Bank and Small Bank portfolio and the All SNC portfolio are more ignificant. Both portfolio are ubtantially le weighted toward ector with particularly high EDF than the All SNC portfolio and contain a larger hare of expoure in ector with mid-range EDF.. For each portfolio 400,000 Monte Carlo trial were ued to imulate the ditribution of portfolio loe at a one-year horizon. Table 7 report characteritic of the imulated lo ditribution with and without incorporating the effect idioyncratic rik. Denity plot of the imulated lo ditribution are hown in Figure 2. Incorporating 13

15 idioyncratic rik in the lo imulation ha no effect on expected lo, but hift ma to the tail of the lo ditribution. Table 8 decompoe the Value-at-Rik for each portfolio into expected, ytematic, and idioyncratic rik component baed on equation (9 ). VaR i defined a the 99.9 th percentile of the lo ditribution. Unexpected lo (UL) i defined a the difference between VaR and EL and reflect the combined effect of ytematic and idioyncratic rik. 16 In all four portfolio the majority of Value-at-Rik i attributable to unexpected lo, and mot of UL arie from ytematic rik. While the Large Bank portfolio ha lower EL than the All SNC portfolio, it ha lightly higher UL. The effect of ytematic rik are greater in the Medium Bank portfolio than in either the All SNC or the Large Bank portfolio. The UL per dollar expoure attributed to ytematic rik for the Medium Bank portfolio i about even percent larger than that of the All SNC portfolio. The ytematic rik component of UL for the Small Bank portfolio i only lightly larger than that of the Medium Bank portfolio. For the All SNC and the Large Bank portfolio, idioyncratic rik play very little role. It increae UL for the All SNC portfolio by le than one percent and it increae UL for the Large Bank portfolio by about 1.5 percent. Name concentration ha a omewhat larger effect in the Medium Bank portfolio; it increae UL by about 3.3 percent. The effect of name concentration i very ignificant in the Small Bank portfolio. Idioyncratic rik increae UL for the Small Bank portfolio by about 10 percent. Thee comparion ugget that idioyncratic rik can meaningfully increae Value-at- Rik and UL for maller SNC portfolio that are le well diverified acro name. Sytematic rik i the mot important contributor to VaR and UL for all portfolio, but larger portfolio that are better diverified acro ector do appear to be omewhat le affected by ytematic rik. In the next ection we examine the relationhip between portfolio ize, indexe of name and ector concentration, and unexpected lo for a broader et of SNC portfolio. 7. The relationhip between SNC portfolio characteritic and Value-at-Rik The comparion of imulated lo ditribution for three bank SNC portfolio in the previou ection ugget that difference in portfolio ize and name and ector concentration meaure are cloely related to the ditribution of realized credit loe. Thi ection examine the relationhip between portfolio characteritic and the ditribution of portfolio loe for our full ample of 30 SNC portfolio. The row labeled Component of Value-at-Rik in Table 3 report information on the ditribution of etimated expected and unexpected lo rate for the large, medium, and mall SNC 16 Thi definition of unexpected lo i conitent with that ued to calculate minimum regulatory capital requirement under Bael II internal-rating-baed approache. 14

16 portfolio egment of our ample. A can be een from the tatitic on portfolio expected lo there i no clear relationhip between portfolio ize and EL. On average, the medium- and mall-portfolio group have lower EL than either the large portfolio group or the All SNC portfolio. The range of expected loe among the mall portfolio group i coniderably larger than that among the other two group. Becaue maller portfolio are le diverified acro ector, expoure to particular high- or low-creditquality ector tend to have larger influence on the EL of thee portfolio, leading to greater diperion of EL among member of the mall-portfolio group. Table 3 alo ugget no clear relationhip between portfolio ize and unexpected loe ariing from ytematic and idioyncratic rik. A with EL, the diperion in UL acro portfolio i greatet for the mall-portfolio group. Portfolio in the medium ize group have the lowet UL on average, whether or not imulated UL etimate reflect the effect of name concentration. Not urpriingly, idioyncratic rik affect UL more for maller portfolio than for larger one. On average, accounting for the effect of name concentration increae UL by 3.3 percent, 5.5 percent, and 12.9 percent for the large-, medium-, and mall-portfolio group repectively. Figure 3 how the component of portfolio VaR attributed to expected lo, ytematic rik, and idioyncratic rik for each of the SNC portfolio in our ample. A can be een from thi figure, in every cae ytematic rik i by far the larget contributor to portfolio VaR. 17 While expected lo i correlated with VaR, there i not a one-to-one relationhip between expected and unexpected loe. The portfolio with the lowet EL in our ample ha the lowet VaR and the portfolio with the highet EL ha the highet VaR, but for other portfolio there i no clear relationhip between EL and either the ytematic- or the idioyncratic-rik component of UL. Table 9 report parameter etimate for regreion of portfolio characteritic on imulation of UL (including both ytematic and idioyncratic rik) for the full ample of 30 SNC portfolio. 18 Specification (a) confirm that portfolio ize (a meaured by total expoure) and EL are cloely related to portfolio UL. However, a comparion of pecification (a), (b), and (d) ugget that entropy indexe of name and ector concentration are much more informative about portfolio UL than i portfolio ize. Including the entropy indexe and omitting portfolio ize (pecification (b)) provide a much better fit than including ize but omitting the entropy indexe (pecification (a)).. Adding ize to a model that include the entropy indexe (pecification (d)) doe little to improve the fit of the model. Comparing pecification that ue entropy indexe 17 The relative importance of ytematic rik and expected lo a contributor to portfolio VaR depend on the lo percentile ued to define VaR. The ytematic rik contribution i increaing in the lo percentile while the expected lo contribution i fixed. Our concluion reflect a 99.9 th percentile VaR meaure which i conitent with the olvency tandard ued in Bael II and may be omewhat le conervative than that ued internally by bank. 18 A two-tep generalized-leat-quare procedure wa ued to correct for heterocadaticity aociated with portfolio ize and EL. 15

17 (pecification (b) and (d)) with thoe that ue Herfindahl indexe (pecification (c) and (e)) ugget that the two entropy indexe are more informative about portfolio UL than are the two Herfindahl indexe. The ector concentration Herfindahl index appear to be of little value for aeing UL. In both pecification that involve the Herfindahl indexe, the ector concentration Herfindahl index i not tatitically ignificant, and in pecification (c) it ha the wrong ign. While we find a poitive relationhip between imple portfolio-wide concentration indexe and unexpected lo, it i important to note that a ignificant hare of oberved variation in UL acro SNC portfolio i not explained by a combination of expected lo, portfolio ize, and concentration indexe. Under pecification (e), which provide the bet fit among the pecification conidered, thee factor explain jut over three-quarter of the cro-portfolio variation in UL. A dicued in Section 3, portfolio-wide concentration indexe can provide ueful ummary information on the overall ditribution of expoure within a portfolio, but they are too implitic to fully capture the effect of ytematic and idioyncratic rik on portfolio loe. By their nature, concentration indexe treat expoure to all ector or obligor in a ymmetric fahion. A we hall ee in the next ection, expoure to different ector can have very different effect on the ditribution of portfolio loe. 8. Managing expoure to individual ector The rik aociated with a particular expoure can be managed by evaluating that expoure marginal contribution to portfolio-wide rik meaure uch a Value-at-Rik, expected lo, or unexpected lo. For example, it i common to allocate economic capital to a credit expoure commenurate with it marginal contribution to portfolio UL. Similarly, loan-lo proviion may be allocated baed on an expoure expected lo. Not all expoure contribute equally to portfolio lo meaure. Obviouly, lower quality credit (in our model, expoure to higher-pd obligor) tend to contribute more to both expected and unexpected portfolio credit loe. What i perhap le well appreciated i that difference in expoure portfolio weighting and the dependence tructure of loe acro expoure can have ignificant influence on an expoure marginal contribution to UL, VaR, and other portfolio rik meaure. In thi ection we ue Monte Carlo imulation baed on the hierarchical factor model developed in Section 4 to examine difference acro ector marginal contribution to portfolio VaR. Thee marginal effect are decompoed into UL and EL component, and the marginal UL contribution are further decompoed into component ariing from ytematic and idioyncratic rik. Let l α be the α th percentile of the portfolio lo rate L. For a pecified value of α (99.9 percent in thi analyi) l α i the portfolio Value-at-Rik. L depend on the portfolio ector weight o l α depend on thee weight a well. To how how one can meaure the effect on l α of a mall change in ector weight, we need to introduce ome additional notation. Let E be the vector of obligor-pecific idioyncratic rik factor for expoure in ector, and let F = {Z 1 Z S,E 1 E S } be the vector of all ector rik factor and idioyncratic factor. Given a realization f of F, the portfolio lo rate i fully determined. The default rate for ector i 16

18 1 2 (10) d ( z, e ) = 1{ zλ + ei 1 λ < γ} n i C where C i the et of expoure in ector. The portfolio lo rate i ( ) ( ) (11) f = wd z, e LGD. L i dicrete, wherea F i continuou. Thu, a given realization of L i conitent with a continuum of realization of F. The et of realization of F conitent with the α th percentile of L i 0 (12) I α = { f ( f) = lα }. Uing a reult from Gourieroux, Laurent, and Scaillet (2000, Lemma 1) it can be hown that l w = E d, E F I Z LGD α 0 (13) ( ) α. Equation (13) give a unique expreion for the marginal effect on portfolio VaR of an increae in invetment in ector. In principle, thi expreion can be ued to calculate the marginal contribution of an expoure to ector. In practice, however, evaluating the conditional expectation in (14) may not be tractable. When a portfolio contain more than a trivially mall number of name expoure, it i impractical to write down a et of 0 contrain on F conitent with memberhip in I α. In thi cae, Monte Carlo imulation can be ued to approximate (14). The et ε (14) f ( f) { l } I = < α α ε decribe realization of F that produce portfolio lo rate that lie in a neighborhood of the α th percentile of L. Oberve that lim E d Z, E F I α = E d Z, E F I α. ε 0 (15) ( ) ( ) ε 0 Equation (13) and (15) ugget the approximation l w α (16) ( E ) ε E d, F I Z LGD α. The approximation can be made a precie a deired by chooing a ufficiently mall value for ε. For a given value of ε the conditional expectation in (16) can be etimated a a by-product of the Monte Carlo imulation procedure decribed in Section 5. Recall that 17

19 to imulate the ditribution of L we firt generated peudo-random draw from the marginal ditribution of F. Given a ufficiently large number of uch draw, a conditional expectation of any function of F can be approximated uing Monte Carlo integration and acceptance ampling. The conditional expectation in equation (16) i d z, e evaluated over all thoe approximated by computing the average value of ( ) peudo-random draw of F that are member of ε I α. When Monte Carlo integration i ued to etimate the conditional expectation in equation (16), chooing ε involve a tradeoff. A maller value of ε improve the accuracy of the approximation decribed in the equation, but it mean that fewer peudo-random draw are available for etimating the conditional expectation. For thi analyi, we et ε = Thu, to etimate marginal contribution for the th percentile of L we average over peudo-random draw of the rik factor that are conitent with realized lo rate between the th and th percentile of L. Our Monte Carlo imulation conit of 400,000 draw from the marginal ditribution of F which give u 160 draw with which to evaluate the conditional expectation. 19 Uing equation (7) and (8), we can rewrite equation (16) a δl δ w α PD LGD A (16 ) ( ) ε + E p F I Z PD α LGD B ( E ) ( ) ε + E d, F I Z p Z α LGD C Equation (16 ) provide a decompoition of the marginal contribution for ector. Term A i the component that can be attributed to the expected lo of expoure in the ector; term B i that component ariing from dependence in default acro expoure in the ector (ytematic rik); and term C i that component ariing from name concentration within the ector (idioyncratic rik). Figure 4 how the marginal contribution per dollar expoure to VaR by ector for the All SNC portfolio For elected ector, Table 10 and Figure 5 compare the marginal contribution to VaR for the All SNC portfolio and the three bank portfolio decribed in Section 6. For each ector the marginal contribution are eparated into component attributable to expected lo, ytematic rik, and idioyncratic rik. 19 With very minor modification, the method decribed here can be ued to etimate marginal contribution to other meaure of portfolio rik uch a expected tail lo (ETL). In fact, for a given number of Monte Carlo imulation and a given threhold lo percentile, marginal contribution to ETL can probably be etimated with greater accuracy becaue more peudo-random draw are available for approximating the conditional expectation (i.e. the acceptance region i larger). 18