CreditRisk⁺ Model with Dependent Risk Factors

Abstract

The CreditRisk⁺ model is widely used in industry for computing the loss of a credit portfolio. The standard CreditRisk⁺ model assumes independence among a set of common risk factors, a simplified assumption that leads to computational ease. In this article, we propose to model the common risk factors by a class of multivariate extreme copulas as a generalization of bivariate Fréchet copulas. Further we present a conditional compound Poisson model to approximate the credit portfolio and provide a cost-efficient recursive algorithm to calculate the loss distribution. The new model is more flexible than the standard model, with computational advantages compared to other dependence models of risk factors.

Keywords:

• CreditRisk⁺ model
• conditional independence
• dependent risk factors
• Panjer’s recursion
• multivariate copulas

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APPENDIX

Proposition 3.1 follows directly from the following lemmas.

Lemma 6.1.

(Credit Suisse First Boston 1997) A power series expansion H(z) = ∑^∞_{n = 0}A_nzⁿ has a recurrence relation if where In other words, the logarithmic derivative of H(z) is a rational function.

Lemma 6.2.

The logarithmic derivative of is a rational function, and

Proof.

From the risk theory, the corresponding random variable of is an independent sum of compound negative binomial risks, that is, Put Then Since P_k(z) are polynomials with finite terms, the logarithmic derivative of is a rational function.