On Approximating the Individual Risk Model

Abstract

Error bounds are developed for Kornya-Presman-type approximations to the individual risk model. In particular, error bounds are given for the compound Poisson approximation, with the Poisson parameter equal to the expected number of claims.The approximations considered are a modification by Hipp of an approximation originally developed by Kornya, as well as Kornya's original approximation. The error bounds are similar in concept to Hipp's original error bounds using concentration-functions, as refined by Č ekanavičius, Roos, and others. Computation of the bounds, however, is considerably simplified. In particular, concentration function type bounds called width-norm bounds are calculated directly from the values of the approximation, avoiding completely the need for calculating a special auxiliary compound Poisson distribution. Two examples are given.Depending on the portfolio parameters, the width-norm error bounds may or may not be sharper than the Hipp-Roos error bounds. In any event, it is recommended that the simpler widthnorm calculation, possibly together with an increase in the order of the approximation, be considered in practical applications if the goal is to achieve a specified accuracy with the least amount of computation.For large portfolios, where the maximum claim on any one policy is relatively small compared to the expected aggregate claims, the width-norm error bounds compare quite favorably with error bounds developed by De Pril and by Dhaene.

Recursions for the Individual Risk Model

“…Since the computational effort is about the same in the three cases, preference should be given to De Pril's approximations. It is important to note that the method with the smallest error bound will not always be the best approximation as is illustrated in an example in Kornya [21] . A comparison of the compound Poisson approximation with De Pril's, Kornya's and Hipp's methods is given in Kuon, Reich, Reimers [22] .…”