On The Approximation of the Total Amount of Claims

Abstract

Several “short cut” methods exist to approximate the total amount of claims ( = χ) of an insurance collective. The classical one is the normal approximationwhere and σx are the mean value and standard deviation of x. Φ is the normal distribution function.It is well-known that the normal approximation gives acceptable accuracy only when the volume of risk business is fairly large and the distribution of the amounts of the individual claims is not “too dangerous”, i.e. not too heterogeneous (cf. fig. 2).One way to improve the normal approximation is the so called NP-method, which provides for the standardized variable a correction Δzwhereis the skewness of the distribution F(χ). Another variant (NP3) of the NP-method also makes use of the moment μ4, but, in the following, we limit our discussion mainly to the variant (2) (= NP2).If Δz is small, a simpler formulais available (cf. fig. 2).Another approximation was introduced by Bohman and Esscher (1963). It is based on the incomplete gamma functionwhere Experiments have been made with both formulae (2) and (4); they have been applied to various F functions, from which the exact (or at least controlled) values are otherwise known. It has been proved that the accuracy is satisfactory provided that the distribution F is not very “dangerous”.

Comparison of Approximations for Compound Poisson Processes

“…This is known in Statistics as the first-order "normalizing" Cornish-Fisher expansion (see, e.g., Hill and Davis, 1968). It seems to us that the first appearance in actuarial sciences is due to Pentikäinen (1977), as a simpler version of the NP2 approximation. We get:…”

“…With respect to related uniform bounds similar to the famous Berry-Esséen one, the present approach has three definite advantages. First, it holds also when the accuracy of the method is not uniform throughout the support (as happens, e.g., for what we call, following Pentikäinen, 1977, the NP2a approximation). Second, it provides a different measure of the quality of the approximation in different points of the real line.…”

Approximative Evaluation of the Distribution Function of Aggregate Claims

“…Then the Haldane-B fails for x = 3 (note the different scale for the different lines of Figure A.3!). By the way, a similar observation was also made when the long version of the NP formula was investigated (PENTIKAINEN, 1977). These approaches, which are based on four characteristics, kurtosis included, instead of three characteristics (mean, standard deviation and skewness), proved to have superior accuracy for slightly skewed distributions but do not tolerate markedly skewed cases.…”

A solvency testing model building approach for business planning

“…If Yl is small «0.05) the normal function can give a fairly satisfactory approximation (Pentikainen, 1977).…”