Within the Internal Ratings-Based (IRB) approach of Basel II it is assumed that idiosyncratic risk has been fully diversified away. The impact of undiversified idiosyncratic risk on portfolio Value-at-Risk can be quantified via a granularity adjustment (GA). We provide an analytic formula for the GA in an extended single-factor Credit Risk setting incorporating double default effects. It accounts for guarantees and their effect of reducing credit risk in the portfolio. Our general GA very well suits for application under Pillar 2 of Basel II as the data inputs are drawn from quantities already required for the calculation of IRB capital charges.
In the portfolio risk-factor frameworks that underpin both industry models of credit Value-at-Risk (VaR) and the Internal Ratings-Based (IRB) risk weights of Basel II, credit risk in a portfolio arises from two sources, systematic and idiosyn cratic. Idiosyncratic risk represents the effects of risks that are particular to individual borrowers. Under the Asymptotic Single Risk Factor (ASRF) framework on which the IRB approach is based, it is assumed that bank portfolios are perfectly fine-grained in the sense that the largest individual exposures account for an infinitely small share of total portfolio exposure. In such a portfolio idiosyncratic risk is fully diversified away, so that economic capital depends only on systematic risk. Real-world portfolios are not, of course, perfectly fine-grained. The asymptotic assumption might be approximately valid for some of the largest bank portfolios, but clearly would be much less satisfactory for portfolios of smaller or more specialized institutions. When there are material name concentrations, there will be a residual of undiversified idiosyncratic risk in the portfolio. The IRB formula omits the contribution of this residual to the required economic capital.
The impact of undiversified idiosyncratic risk on portfolio VaR can be assessed via a methodology known as granularity adjustment (GA). It is derived as a first-order asymptotic approximation for the effect of diversification in large portfolios. The basic concepts and approximate form for the granularity adjustment were first introduced by Michael Gordy in 2000 for application in Basel II (see Gordy [2003]). It was then substantially refined and put on a more rigorous foundation by Wilde [2001] and Martin and Wilde [2003] using theoretical results from Gouriéroux et al. [2000]. Recently, Gordy and Lütkebohmert [2007] proposed and evaluated a granularity adjustment suitable for application under Pillar 2 of Basel II.
All these methods, however, do not account for guarantees and general hedging instruments within a credit portfolio. This paper aims at filling this gap as the exclusion of hedging instruments represents, of course, a rather severe limitation since it is not at all rare that credit exposures are hedged in some way. For example, granting loans and transferring the risk afterwards is a typical business for a bank. The relevance of hedging instruments is also acknowledged by the Basel Committee as Basel II (Basel Committee on Banking Supervision [2006]) discusses extensively credit risk mitigation (CRM) techniques. These include, for example,ordinary guarantees, collateral securitization and credit derivatives such as credit default swaps. Today, credit derivatives might be the most common guarantee instrument. At least their market has grown rapidly over the last years. In the Mid-Year 2007 Market Survey Report of the International Swaps and Derivatives Association (ISDA), the notional amount of outstanding credit derivatives was estimated to be $45.46 trillion.
It is reasonable that a financial institution should be able to decrease its capital requirements if it buys protection for its exposures. This is also important from a regulatory point of view, because it sets the incentive for banks to hedge their credit risk. Therefore, in 2005 the Basel Committee made an amendment to the 2003 New Basel Accord concerning the treatment of guarantees in the IRB approach (see Basel Committee on Banking Supervision [2005]).3 In the New Basel Accord of 2003, banks were allowed to adopt a so-called substitution approach to hedged exposures. Roughly speaking, under this approach a bank can compute the risk-weighted assets for a hedged position as if the credit exposure was a direct exposure to the obligor’s guarantor. Therefore, the bank may have only a small or even no benefit in terms of capital requirements from obtaining the protection. Since the 2005 amendment, for each hedged exposure the bank can choose between the substitution approach and the so-called double default treatment. The latter, inspired by Heitfield and Barger [2003], takes into account that the default of a hedged exposure only occurs if both the obligor and the guarantor default (“double default”). There are rather strict requirements on the obligor and the guarantor for application of the double default treatment. Moreover, the parameters chosen in calculating the double default probability are quite conservative. We refer toGrundke [2008] for a meta-study on this issue. It has been shown in Heitfield and Barger [2003]) that this double default treatment can lead to a significant decrease in capital requirements under the Advanced IRB approach.
Since the double default treatment in the IRB approach is also based on the assumption of an infinitely granular portfolio, it seems natural to investigate the impact of guarantees on possible adjustments for undiversified idiosyncratic risk as represented e.g. by the GA. In this paper we address this issue and derive aGA that takes into account double default effects. The GA is derived as a first order asymptotic approximation for the effect of diversification in large portfolios within an extended version of the CreditRisk model that allows for idiosyncratic recovery risk.4 Note, however, that our methodology could in principle be applied to any model of portfolio credit risk that is based on a conditional independence framework. We derive an analytic solution for the granularity adjustment in a very general setting with several partially hedged positions where the guarantors can also act as obligors in the portfolio themselves. Moreover, we present some results on the performance of our new formula. In particular, we study the impact of guarantees and double default effects on the risk weighted assets of Basel II. Similar to the revised GA of Gordy and Lutkebohmert [2007] our generalization only requires data inputs which are already available when calculating IRB capital charges and reserve requirements. The fact that the GA is analytical allows for a fast computation and avoids the simulation of the rare double default events. Thus it very well suits for application under Pillar 2 of Basel II.
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Treatment of Double Default Effects within the Granularity Adjustment for Basel II
