Computing compound distributions faster!

The use of Panjerʹs algorithm has meanwhile become a widespread standard technique for actuaries (Kuon et al., 1955). Panjerʹs recursion formula is used for the evaluation of compound distributions and can be applied to life and general insurance problems. The discrete version of Panjerʹs recursion formula is often applied to continuous distributions by discretizing the underlying distribution at n equidistant points, covering a large enough interval, say [0, nΔ]. Panjerʹs recursion returns a discrete function as an approximation of the compound distribution. It is claimed that this procedure is fast, (O(n2)), accurate, (O(Δ2)) and easy to understand, cf. Bühlmann (1984), Dickson (1995) and Xie (1989). In this article we propose a method based on cubic splines. The accuracy of the method is better, namely O(Δ5). The computation time is O(n2) and hence for the same accuracy much faster and furthermore, the method returns a twice continuously differentiable function.