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Risk Contributions in an Asymptotic Multi-Factor Framework

by Dirk Tasche of Deutsche Bundesbank

May 20, 2005

Abstract: So far, regulatory capital requirements for credit risk portfolios are calculated in a bottom-up approach by determining the requirements at asset level and then adding up them. In contrast, economic capital for a credit risk portfolio is calculated for the portfolio as a whole and then decomposed into risk contributions of assets or sub-portfolios for, e.g., diagnostic purposes like identifying risk concentrations. In the "Asymptotic Single Risk Factor" model that underlies the most important part of the "Basel II Accord", bottom-up and top-down approach yield identical results. However, the model fails in detecting exposure concentrations and recognizing diversification effects. We investigate multi-factor extensions of the ASRF model and derive exact formulae for the risk contributions to Value-at-Risk and Expected Shortfall. As an application of the risk contribution formulae we introduce a new concept for a diversification index. The use of this new index is illustrated with an example calculated with a two-factor model. The results with this model indicate that there can be a substantial reduction of risk contributions by diversification effects.

Keywords: credit portfolio model, capital requirement, quantile derivative.

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