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 in- surance. 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 general- ization 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 gener- alized in the same vein. We also present an analogue of our method for the collective model with a severity distribution having mixed support. Contents 1. Introduction 2 2. Extended distributions from the Panjer class 3 2.1. Extended negative binomial distribution 3 2.2. Extended logarithmic distribution 4 3. An example of numerical instability 5 4. A generalization of the Panjer recursion 6 5. Application to numerical stability 10 5.1. Extended negative binomial distribution 10 5.2. Extended logarithmic distribution 12 5.3. Poisson mixed over generalized tempered stable distributions 13 5.4. Convolutions and reciprocal generalized inverse Gaussian distribution 21 5.5. Application and extension of CreditRisk + 22 6. Further distributions for the generalized Panjer recursion 24 7. Study of the recurrence relation 29 7.1. Characterization and moments of distributions 29 Date : February 8, 2013. 2000 Mathematics Subject Classification. 91B30 (primary); 65Q05 and 62P05 (secondary). Key words and phrases. 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. This work was financially supported by the Christian Doppler Research Association (CDG). The authors gratefully acknowledge the fruitful collaboration and support by the Bank Austria and the Austrian Federal Financing Agency ( ¨ OBFA) through CDG and the CD-Laboratory for Portfolio Risk Management (PRisMa Lab). Financial support by WWTF MA13 is gratefully acknowledged. 1