A conventional practice in standard risk theory considerations has been to assume that claims are paid immediately as they have incurred (see BPP, item 3.1c, BPP is used as an abbreviation for the book “Risk Theory”, 1984 edition, by Beard, Pentikäinen, Pesonen). The delay of the claims settlement has been, of course, a central aspect in reserve calculation theory and practices, and numerous valuable works have been published on this topic in recent years. However, its regard in general model building and in risk theory considerations has gained little attention until recent years. The purpose of this paper is to contribute to this research work by discussing how the “run-off” risk, i.e., the variability due to the delay of the claims payment, could be incorporated into the standard risk theory models as a separate entry (see BPP, item 10.2e) and to find some evaluation of the order of magnitude of the “extra” (if any) fluctuation so rendered. We expect that the proposed technique can also be utilized in testing different reserve calculation methods and in comparing their effectiveness. The main ideas follow very much along the lines given by Rantala in his doctoral thesis (1984).
One should appreciate the fact that any risk theory model can never be more than an idealization of real-life processes. An intricate problem for practitioners is to evaluate the uncertainties ensuing from the fact that the model, more or less, ignores or only approximates the factors affecting the real events, and that the practical applications are often based on and their necessary parameters estimated from observed data that are subject to random fluctuations and to many other kinds of uncertainties. The problem complex of the run-off risk, when understood in a broad sense, is so wide that it requires a series of studies, and this paper should be regarded as a first step only. The posing of the problem follows the conventional risk theory approach by using the mixed compound Poisson process further allowing for long-term variations of risk exposure (“cycles”), and now extending the model to cope with the delayed settlement of the claims. At this stage of the on-going researchwork the impact of the parameter estimation is excluded from consideration. Therefore, our results and the numerical examples, as given in what follows, do not describe the total uncertainty of the claims or the reserves.