Efficient computation of Value-at-Risk and Expected Shortfall in large and heterogeneous credit portfolios: application to Default Risk Charge

Abstract

Within the new Basel regulatory and capital framework for market risks (FRTB, 2016), the Basel Committee on Banking Supervision’s (BCBS’s) sets out significant revisions to the market risk capital requirements framework. Key areas include moving from Value-at-Risk (VaR) to Expected Shortfall (ES). Non securitized credit positions in the trading book are also subject to a separate default risk charge (DRC) and banks using the internal model approach are required to use a two-factor model and a 99.9% VaR over a one year capital horizon to calculate the DRC capital charge. In this framework, the multi-factor Merton-type models of credit risk are topical since banks are required to measure the capital charge using Value-at-Risk (VaR) and Expected Shortfall (ES). But with this new framework, the implementation of these indicators using the semi-analytical formulas is no longer straightforward and the practice of financial institutions seems to head towards Monte Carlo simulations which are usually more time consuming.

In this paper, we present a new approach for simultaneously calculating the VaR and ES of large and heterogeneous credit portfolios. VaR in the multi-factor Merton framework where its numerical implementation is carried out using a semi-analytical formula. This is of particular relevance in the Fundamental Review of Trading book framework (FRTB) since the measure of the two indicators VaR and ES are in general required: the risk measure to be employed in the market risk capital charge computation is changed from the VaR to the ES but the VaR calculation is also needed to for the validation and back-testing process. For this purpose, we use a result of Rockafellar and Uryasev [20] where they characterize the Value-at-Risk (VaR) and the Expected Shortfall (ES) as a solution of a convex optimization problem of a target function chosen in an appropriate way. The target function is built across the portfolio loss and its cumulative distribution. This has been applied by the authors to portfolio optimization.

The set-up of our approach requires the use of the Central Limit Theorem (CLT). Although the assumption of large portfolios ensures the classical conditions of the CLT, the heterogeneous nature of the credit portfolio can sometimes prevent the direct use of this theorem. To overcome this difficulty, the portfolio loss can be decomposed in two components with CLT assumptions verified for the first one, whereas the second is treated by a recursive method.

Even if our new approach can be used in a general framework of the FRTB, we first apply it to a large and heterogeneous credit portfolio to compute the default risk charge and we explore its ability to deal with the exposure concentration. Results of several numerical tests support our quantitative analysis and confirm the theoretical aspects of our approach.