**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

**2000
Mathematics Subject Classification:**

- 91B30 Risk theory, insurance
- 65Q05 Difference and functional equations, recurrence relations
- 62P05 Applications to actuarial sciences and financial mathematics

**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:

- Portable document format (834 kB)

The published version (48 pages) is available at:

- DOI: 10.1007/s00780-009-0104-1 (979 kB)

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

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