A Generalization of Panjer's Recursion and Numerically Stable Risk Aggregation

Stefan Gerhold, Uwe Schmock and Richard Warnung

Abstract: Portfolio credit risk models as well as models for operational risk can often be treated analogously to the collective risk model coming from insurance. Applying the classical Panjer recursion in the collective risk model can lead to numerical instabilities, for instance if the claim number distribution is extended negative binomial or extended logarithmic. We present a generalization of Panjer's recursion that leads to numerically stable algorithms. The algorithm can be applied to the collective risk model, where the claim number follows, for example, a Poisson distribution mixed over a generalized tempered stable distribution with exponent in (0,1). De Pril's recursion can be generalized in the same vein. We also present an analogue of our method for the collective model with a severity distribution having mixed support.

Keywords: Portfolio credit risk, CreditRisk+, operational risk, collective risk model, extended negative binomial distribution, extended logarithmic distribution, compound distribution, extended Panjer recursion, numerical stability, De Pril's recursion, Poisson mixture distribution, generalized tempered stable distribution, (generalized) inverse Gaussian distribution, reciprocal generalized inverse Gaussian distribution, inverse gamma distribution, severities with mixed support

2010 Mathematics Subject Classification:

Zentralblatt MATH: Zbl 1224.91060

Reference: Finance and Stochastics 14 (2010), 81-128

The preprint (41 pages, version April 16, 2013, six typos corrected, same numbering as published version) is available in:

The published version (48 pages) is available at:

Slides used at the International Workshop on Credit Risk, Université d'Evry Val d'Essonne, France, June 25-27, 2008

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