A Method to Estimate Probability Level for Loss Reserves

This paper explores the collective risk model as a vehicle for estimating the probability distribution for reserves. Though this basic model has been suggested in the past and It provides a direct means to estimate process uncertainty, it does not directly address the potentially more significant problem of parameter uncertainty. This paper presents some techniques to estimate parameter uncertainty and, to some extent, also uncertainty regarding projection model selection inherent in reserve estimates. The collective risk model, see for example Beard, Pentiktiinen and Pesonen [l]. provides a conceptually simple framework to model total claims In the insurance process. In its simplest form this model calculates the total loss from an insurance portfolio as the sum of N random claims chosen from a single daim size distribution where the number N is itself a random variable. With some fairly broad assumptions regarding the number and size of claims we can draw conclusions regarding the varlous moments of distribution of total claims. Thus thls model seems to be a reasonable choice as a starting p&t in estimating the dish-ibutlon of reserves for an insurer. The dlstrlbution resulting from this slmp4e collective risk model provides an estimate of the potential variation in total payments assumina all distributions are correct. We often refer to this variation as process variation, that inherent due to the random nature of the process itself. Not directly addressed in this simple collective risk model Is the possibility that the estimates of the parameters for the underlying distributions, are incorrect. Variation due to this latter uncertainty is oflen called parameter variation. Parameter variation is itself an important aspect in assessing the variability inherent in insurance related estimates. Meyers and Schenker [2] discuss this aspecl of collective risk applications. They conclude, not surprisingly, that for a " large " volume of claims, that expected to be experienced by most insurers, parameter uncertainty is a much more significant contributor to overall variability than the random, or process, portlon. 299 As indicated above, the collective risk model does not directly address parameter uncertaintynor does It address the methodology used in obtaining reserve estimates themselves. In practice actuaries often apply several methods, based on different underlying assumptions, to derive different projections of required reserves. The actuary then selects a " best estimate " of required reserves, based on the various projections used, keeping in mind the nature of the data and the assumptions inherent …