A generalization of Panjer’s recursion and numerically stable risk aggregation

Abstract

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.

Analysis of an aggregate loss model in a Markov renewal regime

“…In order to approximate (1) a battery of numerical methods, either based on simulations, inversion of characteristic functions, or recursions, have been proposed in the literature. For an exhaustive review of such methods, see [34,56,71,72] , or Ch. 9 in [40] .…”

Actuarial Applications and Estimation of Extended CreditRisk+

“…The algorithm is basically due to [Giese(2003)] for which [Haaf et al(2004)] proved numerical stability. The relation to Panjer's recursion was first pointed out in [Gerhold et al(2010), section 5.5]. [Schmock(2017)] in section 5.1 generalised the algorithm to the multivariate case with dependent risk factors and risk groups, based on the multivariate extension of Panjer's algorithm given by [Sundt(1999)].…”

Calculation of aggregate loss distributions

“…Proof 1 This follows from the expression for characteristic function of the compound distribution (7) and formulas (15,16). The calculus is simple but lengthy.…”

“…Application of the standard Panjer recursion in the case of the generalised frequency distributions such as the extended negative binomial, can lead to numerical instabilities. Generalization of the Panjer recursion that leads to numerically stable algorithms for these cases is presented in Gerhold et al (2009). Discussion on multivariate version of Panjer recursion can be found in Sundt (1999) and bivariate cases are discussed in Vernic (1999) and Hesselager (1996).…”