Granularity Adjustment for Regulatory Capital …

1 Granularity Adjustment for RegulatoryCapital Assessment Michael B. Gordyaand Eva L utkebohmertbaFederal Reserve BoardbUniversity of FreiburgThe credit value-at-risk model underpinning the internalratings-based approach of Basel II and III assumes that idio-syncratic risk has been fully diversified in the portfolio, so thateconomic Capital depends only on systematic risk contribu-tions. We propose a simplegranularity Adjustment (GA) forapproximating the effect of undiversified idiosyncratic risk onrequired Capital .

2 To mitigate operational burden in implemen-tation, we derive upper and lower bounds on the GA underincomplete information on the portfolio. We assess the mag-nitude and accuracy of the proposed GA on a set of bankportfolios drawn from the German credit Codes: G32, G28, IntroductionIn the portfolio risk-factor frameworks that underpin both indus-try models of credit value-at-risk (VaR) and the internal ratings-based (IRB) risk weights of Basel II and Basel III, credit risk in aportfolio arises from two sources, systematic risk and idiosyncratic This paper originally circulated under the title Granularity Adjustmentfor Basel II.

3 Much of this work was completed while M. Gordy was a visit-ing scholar at the Indian School of Business and while E. L utkebohmert was atthe Deutsche Bundesbank and University of Bonn. We thank Klaus D ullmann,Chris Finger, Christian Schmieder, Dirk Tasche, and Birgit Uhlenbrock for help-ful comments and input, and Jim Marrone for excellent research assistance. Theopinions expressed here are our own and do not reflect the views of the DeutscheBundesbank or of the Board of Governors of the Federal Reserve System.

4 Authore-mails: and Journal of Central BankingSeptember 2013risk. Systematic risk arises because of unexpected changes in macro-economic conditions and financial market conditions to which mostborrowers are exposed. This risk cannot be eliminated through diver-sification across borrowers. All remaining sources of risk are idiosyn-cratic, , particular to individual borrowers. As a portfolio becomesmore fine grained, in the sense that the largest individual exposuresaccount for a smaller share of total portfolio exposure, idiosyncraticrisk is diversifiedaway at the portfolio the asymptotic framework that underpins the IRBapproach, it is assumed that bank portfolios areperfectlyfinegrained, that is, that idiosyncratic risk has been diversifiedaway,so that economic Capital depends only on systematic risk.

5 It is alsoassumed that there is only a single systematic source of risk, so thatunder mild regularity conditions Capital charges can be calculatedanalytically. Relative to Monte Carlo simulation, simple closed-formcapital rules are preferred in a Regulatory setting for reasons of trans-parency, verifiability, and ease of implementation across institutionsof varying capacity. Real-world portfolios are not, of course, perfectlyfine grained. The asymptotic assumption might be approximatelyvalid for some of the largest bank portfolios, but could be much lesssatisfactory for portfolios of smaller or more specialized there are material name concentrations of exposure.

6 Therewill be a residual of undiversified idiosyncratic risk in the IRB formula omits the contribution of this residual to requiredeconomic impact of undiversified idiosyncratic risk on portfolio capitalrequirements can be assessed via a methodology known asgranular-ity Adjustment . In this paper, we propose and empirically evaluate agranularity Adjustment (GA) suitable for application by banks sub-ject to IRB Capital requirements and by supervisors of such methodology is similar in form and spirit to the GA introducedin a 2001 draft of Basel II, but exploits theoretical advances over thepast practical application, it is the data inputs (and not the formu-lae applied to those inputs) that can pose the most serious obsta-cles to cost-effective implementation.

7 The data inputs to our GAare drawn from quantities already required for the calculation ofIRB Capital charges and loan-loss reserve requirements, but with oneVol. 9 No. 3 Granularity Adjustment35important caveat. When a bank has multiple exposures to the sameunderlying borrower, it is required that these multiple exposures beaggregated into a single exposure for the purpose of calculating GAinputs. For the purpose of calculating IRB Capital requirements,by contrast, the identity of the borrower is immaterial, as capitalcharges depend only oncharacteristicsof the loan and borrower( , type of loan, default probability, maturity) and not on theidentityof the borrower per se.

8 This is a great convenience whendata on different sorts of exposures are held on different computersystems, as the job of calculating Capital may be delegated to thoseindividual systems and reported back as sub-portfolio aggregateswhich can then be added up in a straightforward fashion to arriveat the bank-level Capital and loan-loss reserve requirements. Whenwe measure Granularity , we cannot ignore borrower identity. Fromthe perspective of single name concentration, ten loans to a singleborrower together carry much more idiosyncratic risk than the sameten loans made to ten distinct borrowers of similar risk characteris-tics.

9 For many institutions, the need to aggregate information acrosscomputer systems on multiple exposures to a single borrower is themost significant challenge in implementing a Granularity defense of this aggregation requirement, we note that such aggre-gation would be necessary inanyeffective measure of Granularity ,and so is not a drawback peculiar to the GA we propose in this , one might ask how a bank can effectively manage itsname concentrations without some ability to aggregate exposuresacross the different activities of the reduce the burden associated with exposure aggregation, ourrevised GA provides for the possibility that banks be allowed to cal-culate the GA on the basis of the largest exposures in the portfolio,and thereby be spared the need to aggregate data on each and everyborrower.

10 To permit such an option, regulators must be able to cal-culate the largest possible GA that is consistent with the incompletedata provided by the bank. Our approach, therefore, is based on anupper-bound formula for the GA as a function of complete dataon themlargest exposures measured in Capital contributions outof a portfolio ofnloans (withm n) and summary data on theremainder of the portfolio. Asmgrows towardsn( , as the bankprovides data on a larger share of its portfolio), the upper-bound36 International Journal of Central BankingSeptember 2013formula converges to the whole portfolio GA.