MPRA Munich Personal RePEc Archive Simulaion based approach for measuring concenraion risk Kim, Joocheol and Lee, Duyeol UNSPECIFIED February 27 Online a hp://mpra.ub.uni-muenchen.de/2968/ MPRA Paper No. 2968, posed 7. November 27 / 2:5
Simulaion based approach for measuring concenraion risk April, 27 Joocheol Kim School of Economics, Yonsei Universiy, 34 Shinchon-dong, Seodaemun-ku, Seoul, 2-749, Korea Duyeol Lee School of Economics, Yonsei Universiy, 34 Shinchon-dong, Seodaemun-ku, Seoul, 2-749, Korea Absrac Asympoic Single Risk Facor (ASRF) model is used o derive he regulaory capial formula of Inernal Raings-Based approach in he new Basel accord (Basel II). One of he imporan assumpions in ASRF model for credi risk is ha he given porfolio is well diversified so ha one can easily calculae he required capial level by focusing only on sysemaic risk. In real world, however, idiosyncraic risk of a porfolio canno be fully diversified away, causing he so called concenraion risk problem. In his paper we sugges simulaion based approach for measuring concenraion risk using bank capial dynamic model. This approach is especially suiable for a porfolio wih relaively small o medium number of obligors and relaively large sized loans. Keywords: Basel II, ASRF model, credi risk, concenraion risk JEL Codes: G32, G33, G38 Corresponding auhor Email addresses : joocheol@yonsei.ac.kr (Joocheol Kim) i22@yonsei.ac.kr (Duyeol Lee). Tel : +82-2-223-5498 (Joocheol Kim) +82--927-22 (Duyeol Lee)
Simulaion based approach for measuring concenraion risk. Inroducion In recen years, many imporan advances have been made in modeling credi risk of a porfolio. One of hem is Asympoic Single Risk Facor (ASRF) model, which is used o derive he regulaory capial formula of Inernal Raings-Based approach in he new Basel accord (Basel II). Under he ASRF framework here are only wo sources of risk, sysemaic risk and idiosyncraic risk. As he number of obligors in a porfolio increases, idiosyncraic risk is diversified away, so is conribuion o porfolio risk disappears. Thus, one can easily calculae required capial level by focusing only on sysemaic risk under he ASRF assumpions. In real world, however, a bank s porfolio is ofen no sufficienly diversified. The fac ha here are some large exposures in he porfolio implies ha here is a residual of undiversified idiosyncraic risk in he porfolio. Under hese circumsances, IRB formula in he Basel accord underesimaes he required regulaory capial. Some hisorical examples such as insolvency of Enron, Worldcom and Parmala show he dangers of misundersanding concenraion risk. The approaches for measuring concenraion risk suggesed in recen sudies can be caegorized ino wo differen ypes. The firs approach is o adap indices of concenraion such as Gini coefficien or Herfindahl-Hirschman Index (HHI). This approach is simple and easy o perform. While hese indices could be good measures for concenraion iself, hey do no seem o serve well for concenraion risk because hey do no ake disribuion of differen qualiy obligors ino consideraion. The second approach is granulariy adjusmen suggesed by Gordy (23). Is difficulies in implemenaion and huge daa requiremen make i hard o be performed in pracice. Usually praciioners use boh approaches o measure he concenraion risk of heir porfolio. While he concenraion measuremen index such as HHI could no measure he acual risk accuraely, granulariy adjusmen someimes overesimaes he acual concenraion risk of a porfolio. Bundesbank(26): Concenraion risk in credi porfolios, monhly repor, June
In his paper, we inroduce a simulaion based approach o measure concenraion risk. We show ha HHI could no provide enough informaion o measure he acual concenraion risk. Wih he proposed approach, we are able o calculae he amoun of required capial for concenraion risk direcly. We believe ha he approach is especially suiable for banks wih porfolios of relaively small number of obligors wih relaively large size of loans. This paper is organized as follows. In secion 2, we presen deailed descripions of concenraion risk and Herfindahl-Hirshman Index, respecively. Secion 3 explains he frameworks of our simulaion based approach o measure concenraion risk. Secion 4 provides some numerical resuls based on he acual example and explains he implicaion of hose numbers. Secion 5 concludes he paper. 2. Concenraion risk under Basel II framework 2. The IRB model and concenraion risk In his secion, we provide a brief summary on he key assumpions of he Asympoic Single-Risk Facor (ASRF) model ha is used o calculae he regulaory capial requiremen by Basel II. In he risk facor model frameworks ha underpin he Inernal Raings-Based (IRB) risk weighs of Basel II, credi risk of a porfolio is caused by wo main sources, sysemaic and idiosyncraic risks. 2 Sysemaic risk represens he effec of unexpeced changes in macroeconomic and financial marke condiions on he performance of borrowers. Borrowers may differ in heir degree of sensiiviy o sysemaic risk, bu few firms are compleely indifferen o he wider economic condiions in which hey operae. Therefore, he sysemaic componen of porfolio risk is unavoidable and only parly diversifiable. Meanwhile idiosyncraic risk represens he effecs of risks ha are paricular o individual borrowers. As a porfolio becomes more fine-grained, in he sense ha he larges individual exposures accoun for a smaller share of oal porfolio exposure, idiosyncraic risk is diversified away a he porfolio level. This risk is oally eliminaed in an infiniely granular porfolio (one wih a very large number of exposures) as unsysemaic risk vanishes in Capial Asse Pricing Model. The ASRF model framework underlying he IRB approach is based on wo key assumpions. The firs one is ha bank porfolios are perfecly fine-grained and he 2 BCBS(26): Sudies on credi risk concenraion, working paper, Basel 2
second one is ha here is only one source of sysemaic risk. When hese wo assumpions hold, one can easily calculae required capial level depending on only one sysemaic risk. In case of well diversified porfolio, he capial required for a loan does no depend on he porfolio i is added o. This simpliciy makes he new IRB framework applicable o a wider range of counries and insiuions. However, if any of wo assumpions is violaed, here is no guaranee ha he IRB approach and ASRF model will be accurae. The violaion of he assumpion of he fine-grained porfolio leads o concenraion risk problem. Concenraion of exposures in credi porfolios arises from imperfec diversificaion of idiosyncraic risk in he porfolio. The small o medium size of credi porfolio or some large exposures o specific individual obligors can lead o concenraion risk. 2.2 Herfindahl-Hirshman Index The Herfindahl-Hirshman index 3 (HHI), beer known as he Herfindahl index, is a saisical measure of concenraion. The HHI accouns for he number of firms in a marke, as well as concenraion, by incorporaing he relaive size of all firms in a marke. I is calculaed by squaring he marke shares of all firms in a marke and hen summing he squares, as follows: HHI = n i= 2 ( MS i ), () where MS i is marke share of i h firm and n is he number of firms. Well-diversified porfolios wih a very large number of very small firms have an HHI value close o zero whereas heavily concenraed porfolios can have a considerably higher HHI value. In he exreme case of a monopoly, he HHI akes he value of one. In he conex of he measuremen of concenraion risk, he HHI formula is included as a main componen of a number of approaches. Bu HHI iself has some drawbacks o be used for measuring concenraion risk. A firs, i does no consider disribuion of exposures across credi raings, so porfolios wih he same HHI values can have differen sizes of concenraion risks. Secondly, i does no allow concenraion risk o be expressed direcly as economic capial, so i needs addiional funcions o 3 Board of Governors of he Federal Reserve Sysem (993): The Herfindahl-Hirshman Index, Federal Reserve Bullein, March, pp 88-89. 3
calculae economic capial for concenraion risk. 3. Framework for simulaions In his secion, We inroduce he framework for simulaions presened in Peura and Jokivuolle(23) and how o use his bank capial dynamics model o calculae concenraion risk. 3. Bank capial dynamics based on raing ransiions To model bank capial dynamics and required capial buffers and o avoid confusion of noaions, we use hree differen ypes of bank capial, he acual capial, he regulaory capial and he economic capial. The acual capial is bank s acual capial and denoed by A. The regulaory capial is he minimum regulaory capial charge of Basel II and denoed by R. And he economic capial is minimum capial level calculaed by bank wihou considering regulaory capial. Now le here be a bank wih asses consising of illiquid corporae loans. Under Basel II framework, he acual bank capial mus saisfy equaion (2). A R (2) Equaion (2) gives us inuiion how o deermine iniial acual capial of a bank. By calculaing required iniial acual capial subjec o equaion (2), we can have he required capial amoun for credi risk of a porfolio. Now, o model bank s acual capial dynamics, we assume ha he bank s profi occurs before credi losses during period. The bank s credi loss during period is denoed by L and he dividends paid ou of he bank capial a ime by V, he issues of new equiy a ime by S. Now, he bank s capial dynamics can be deermined by A A I L V S. (3) + = + + + + + + The bank deermines he acual capial level preparing for severe macroeconomic downurns. In hose condiions when capial is insufficien, i is naural o assume ha here are no dividends. And also in macroeconomic downurns, i is hard o issue new 4
equiy. So, wih lile loss of generaliy, we can assume ha boh he V+ and he S + erms in equaion (3) equal zero in all scenarios. Now we can express he capial dynamics ha we simulae as A A I L (4) + = + + + By rolling he difference equaion (4) forward, we can ge he capial a ime from A s= = A + I L (5) s s= s The equaion (5) implies ha he bank s capial dynamics are deermined by wo sochasic facors, he cumulaive ne profi and he cumulaive change in he minimum capial requiremen. Now, we need o model bank income, credi losses and regulaory capial in order o simulae he dynamics of a bank s capial. The bank income and credi losses depend on raing ransiions because hey depend on defaul evens of obligors. Obligors defauls can be simulaed based on raing ransiions model. And also regulaory capial can be simulaed by raing ransiions because IRB formula of Basel II needs credi raings of obligors as a componen. In Peura and Jokivuolle (23), hey used a one-facor version of he CrediMerics framework (J.P. Morgan, 997) as raing ransiions, exended wih an underlying condiioning variable which is inerpreed as business cycle sae. The Credimerics model akes he ransiion probabiliy marix of raings as given, which is deermined by he business cycle sae in Peura and Jokivuolle (23). In paricular, hey assume ha he business cycle variable is a wo-sae, ime homogenous, Markov Chain, whose possible saes are expansion and recession. In Bangia e al.(22), hey used models of raings dynamics of his ype. Credi porfolio models are ypically implemened as one-period simulaions wih an annual horizon. However, because banks in mos counries repor heir capial adequacy o heir regulaors quarerly, muli-period simulaions of raing changes should be performed in quarerly ime incremens. Boh he raing ransiion probabiliies and he regime ransiion probabiliies in his simulaion are quarerly probabiliies esimaed based on US daa. The condiional ransiion marices for he expansion and he recession saes are from Bangia e al.(22), which are based on Sandard and Poor s daa on US corporae raings over he period 98-998. The regime swiching probabiliies have been esimaed from quarerly daa on US business cycles over 959-5
998. The saionary disribuion of he business cycle sae implied by his ransiion marix is (79%, 2%). Now we will explain how he evoluion of raings in his model deermines bank income, credi losses. For convenion of noaions, we define an indicaor variable which assigns when i h obligor of he bank s porfolio defauls a ime, oherwise. D i, D i, wih probabiliy PD( k oherwise i, ), (6) where uncondiional defaul probabiliy corresponding o raing k i, by PD( k i, ). Regulaory capial is deermined by he capial charge funcion of Basel II and raing ransiions. Bank income is deermined by ineress of loans and usually ineress are deermined direcly proporional o expeced loss of loans. Credi losses are deermined by defaul evens. Using hese properies, we can express he variables R, I and defined earlier equaions in erms of he following sums over obligors in he bank s porfolio: L n R = f ( PD( k )) EAD ( D i, i i= i, ), (7) n = β i ( i, ) i ( i, ), (8) i= I LGD PD k EAD D n L = LGD EAD ( D Di ), (9) i i i,, i= where f ( ) is he capial charge funcion of Basel II, which akes he defaul probabiliy as an argumen. k i, is he credi raing of obligor i a ime. EADi is he nominal exposure of obligor i. β is a parameer which indicaes he raio of he nominal loan margin o he expeced loss rae (he uncondiional defaul probabiliy imes he loss given defaul percenage) in he porfolio. LGD i is he loss given defaul o nominal exposure raio. n is he number of obligors in he bank s porfolio a ime. Formula (9) implies ha he bank earns income as a fixed muliple of is uncondiional expeced loss rae. We assume ha he underlying asse value correlaions, which ogeher wih he ransiion probabiliies deermine he raing ransiion correlaions, do no depend on he sae of he business cycle. Consisen wih he IRB capial charge formula, we use 6
correlaion formula of Basel II. Now, we have bank s capial dynamics and his resul mus saisfy equaion (2). For convenience of noaion, we define capial buffer, which is he difference beween he bank s acual capial and he regulaory capial. I can be inerpreed as capial buffer o absorb he risk from uncerainy and given by B B = A R () Holding capial buffer means an opporuniy cos for banks. In his poin of view, requiring equaion (2) o hold in all possible saes of he world is no economical o he bank. Therefore banks use value-a-risk ype probabilisic regulaory capial requiremen o calculae he size of capial buffer. Value-a-risk is defined as he α h percenile of he disribuion and he consrain of VaR can be expressed as P min B α, () T where α is a confidence level associaed wih regulaory capial adequacy, such as 99% or 99.9%. The dynamics of depend on he iniial capial buffer B because is increasing in B B B. By subsiuing equaion (5) ino equaion (), and applying he inequaliy (2), we can express he regulaory capial requiremen a ime : B = B + I s Ls R + R (2) s= s= Here he capial buffer a ime is expressed in erms of he iniial capial buffer, he inflows and ouflows of capial beween ime and ime, as well as he change in he regulaory capial charge R beween ime and ime. In paricular, R is he capial charge associaed wih he bank s iniial porfolio evaluaed based on raings of he asses in he porfolio a ime. By simulaion based on capial dynamics explained above, we can calculae a minimum value for B which saisfies equaion (2) and we denoe i wih ˆB. The required iniial capial buffer ˆB is given by { } T B ˆ inf : min = B P B α (3) 7
Now, by assuming ha equaion (3) deermines he capial buffer, we can calculae iniial bank capial as Aˆ = R + B, (4) ˆ where  is required iniial capial for credi risk of a loan porfolio. 3.2 Measuring concenraion risk Simulaions based on bank capial dynamics model inroduced in previous secion provide required capial minimum level direcly from disribuion of bank s iniial acual capial. However, in his paper, we need addiional simulaion and model exensions o calculae concenraion risk. Any given porfolio, here exis a benchmark porfolio which have no concenraion. Now using simulaions, we can calculae required iniial capial levels for hese wo porfolios, real one and benchmark case. The difference beween hese wo values is addiional required capial caused by concenraion in he real porfolio. In Peura and Jokivuolle(23), hey form porfolios according he given qualiy disribuions ha each have equal sized loans o sress es bank capial adequacy. In his framework, here is no concenraion in he porfolio. I is no only unrealisic bu also unsuiable for our main goal which is o calculae concenraion risk. Therefore, we form porfolios which have differenly sized loans. In order o perform he simulaion, we need business cycle scenarios. In Peura and Jokivuolle(23), hey used various assumpions concerning he iniial business cycle sae as well as he duraion of recessions. Bu we used randomly seleced scenarios because he main purpose of our model is jus o calculae he VaR ype crierions from he disribuions. 3.3 Tesing Herfindahl-Hirschman Index In he conex of he measuremen of concenraion risk, he HHI formula is included as a main componen of a number of approaches. Bu here are wo ypes of shorcomings of HHI o be used for measuring concenraion risk. Firsly, HHI doesn ake qualiy of a porfolio ino consideraion. I implies ha he porfolios differenly disribued across he credi raings can have he same HHI. In he nex secion, we form wo porfolios which have differen disribuions wih he same HHI and show hese 8
porfolios have differen sizes of concenraion risk. Secondly, HHI doesn reflec locaion of concenraion in a porfolio. Even hough disribuions of loans in porfolios are same, he locaions of concenraion can differ. If he sizes of loans in porfolios are same, hen hey sill have same HHI regardless he locaions of concenraion. I means he porfolios ha have concenraions in differen credi raings can have same HHI. To show his we form wo porfolios which have same disribuion of loans across he credi raings bu have concenraions in differen grades and also show hese porfolios have differen sizes of concenraion risk in he nex secion. 4. Numerical Resuls Our simulaions resuls are calculaed by following seps. Firs, we form wo ses of porfolios. Second, we deermine business cycle scenarios. Las, we perform Mone- Carlo simulaions based on bank capial dynamics described above. We presen our main resuls subjec o he following base case parameers: porfolio mauriy T equal o 2.5 years, a bank income equal o he uncondiional expeced credi loss( β = ), an loss given defaul of 45% across all obligors( LGD =.45 ), and a confidence levelα of 99%. We use, scenarios seleced randomly. The number of simulaions is, for each scenario. So we have,, samples. We ake represenaive porfolios of banks from a Federal Reserve Board survey as repored by Gordy(2). The porfolios are repored in Table. Table. Average bank porfolios i S&P grade Defaul Probabiliy (%) US average qualiy (%) US high qualiy (%) AAA. 3 4 AA. 5 6 A.4 3 29 BBB.24 29 36 BB. 35 2 B 5.45 2 3 CCC 23.69 3 US porfolios are from Federal Reserve Board survey, as repored in Gordy(2). Defaul probabiliies are annual defaul frequencies from S&P daa 98-998 9
These disribuions of porfolios do no reflec concenraion because hey are calculaed using exposure based daa. In he firs case, we form wo porfolios using US average qualiy porfolio and wo porfolios using US high qualiy porfolio. In each case, he benchmark porfolios have he disribuions ha each obligor has same nominal exposure. Porfolio has concenraion in credi raing of BB of average qualiy porfolio. Porfolio 2 has concenraion in he same grade of high qualiy porfolio. In order o eliminae he effec of locaion of concenraion, we le boh porfolios have concenraions in he same grades. Porfolio Porfolio 2 4 35 3 25 2 5 5 AAA AA A BBB BB B CCC 4 35 3 25 2 5 5 AAA AA A BBB BB B CCC Fig.. The disribuions of porfolio and 2. Each porfolio has concenraion in shaded area. These wo porfolios have wo obligors which have exposures of abou % ou of oal porfolio exposures. And wo porfolios have he same HHI (.272). Table 2 shows he main resul of simulaions. Table 2. Iniial acual capials of porfolios (%) Average qualiy porfolio High qualiy porfolio Benchmark Porfolio Benchmark Porfolio 2 Iniial acual capial 8.7. 5.8 7.9 Addiional capial 2.4 2. The differences beween each porfolio and benchmark porfolio are addiional required capials arise from concenraion. I can be inerpreed as addiional risks from concenraions. In his case, he addiional risk of porfolio is 2.4% and he addiional
risk of porfolio 2 is 2.%. The difference.3% is large enough o conclude ha he concenraion risk from differenly disribued porfolios wih same HHI can be differen. In he second case, we form hree porfolios using US average qualiy porfolio. The benchmark porfolio has he disribuion ha each obligor has same nominal exposure. Porfolio has concenraion in credi raing of BBB. Porfolio 2 has concenraion in BB. Porfolio (%) Porfolio 2 (%) 4 35 3 25 2 5 5 AAA AA A BBB BB B CCC 4 35 3 25 2 5 5 AAA AA A BBB BB B CCC Fig. 2. The disribuions of porfolio and 2. Each porfolio has concenraion in shaded area. These wo porfolios have wo obligors which have exposures of abou % ou of oal porfolio exposures. And wo porfolios have he same HHI (.272). Table 3 shows he main resul of simulaions. Table 3. Iniial acual capials of porfolios (%) Benchmark Porfolio Porfolio 2 Iniial acual capial 8.7.3. Addiional capial.6 2.4 In his case, he addiional risk of porfolio is.6% and he addiional risk of porfolio 2 is 2.4%. The difference.8% is large enough o conclude ha he concenraion risk from porfolios ha have concenraions in differen grade wih same HHI can be differen. In order o show he problems caused by using HHI for concenraion risk measure more clearly, we form randomly seleced porfolios(wih HHI from.2~.5) of average qualiy porfolio. Fig. 3 shows he scaer diagram for HHI and concenraion
2 risk. Using simple linear regression, we found R equal o.43. I implies ha HHI could no provide enough informaion o measure he acual concenraion risk. Fig. 3. Scaer diagram for Herfindahl-Hirschman Index and concenraion risk of average qualiy porfolio 5. Conclusion This paper provides a simulaion based approach o measure concenraion risk. In addiion, i is shown ha Herfindahl-Hirshman Index can no be a good measure for concenraion risk. Given bank capial dynamic model, simulaions direcly provide he amoun of required capial for concenraion risk of a loan porfolio while more simple mehods such as HHI or Gini coefficien need an addiional funcion. And also i provides more precise resul, compared wih approximaion mehods such as granulariy adjusmen. I migh be more ime-consuming han oher mehods, bu i is sill he beer way especially for banks wih porfolios of relaively small number of obligors wih relaively large size of loans. 2
References Bangia, A, Diebold, F, Kronimus, A, Schagen, C, Schuermann, T(22): Raings migraion and he business cycle, wih applicaion o credi porfolio sress esing, Journal of Banking and Finance, vol 26, pp 445-474. Basel Commiee on Banking Supervision(2): The new Basel capial accord, consulaive documen, Basel, January. (23): The new Basel capial accord, consulaive documen, Basel, April. (26): Inernaional convergence of capial measuremen and capial sandards: a revised freamework, comprehensive version, Basel, June. (26): Sudies on credi risk concenraion, working paper, Basel, November. Board of Governors of he Federal Reserve Sysem (993): The Herfindahl-Hirshman Index, Federal Reserve Bullein, March, pp 88-89 Deusche Bundesbank(26): Concenraion risk in credi porfolios, monhly repor, June. Emmer, S and D Tasche(23): Calculaing credi risk capial charges wih he onefacor model, Journal of Risk, vol 7, no 2, pp 85-. Gordy, M(2): A comparaive anaomy of credi risk models, Journal of Banking and Finance, vol 24, pp 9-49. Gordy, M and E Lukebohmer(24): Granulariy adjusmen in porfolio credi risk measuremen, in G Szego(ed), Risk measures for he 2 s Cenury, Wiley. J.P. Morgan(997): CrediMerics - Technical Documen, J.P. Morgan, New York Peura, S and Jokivuolle, E(23): Simulaion based sress ess of banks regulaory capial adequacy, Journal of Banking & Finance, vol 28, pp8-824. Vasicek, O. A(22): Loan Porfolio Value, Risk Magazine, December, pp 6-62. 3