On the mean total reserve of an insurance company with a random number of clients

Recieved: 12/01/2025

Accepted: 12/10/2025

Keywords: insurance mathematics, random number of clients, asymptotic deficiency, Delaporte distribution, generalized Poisson distribution, truncated distributions, insurance reserves

Available in the on-line version with: 24.07.2026

To cite this article

Arifullin A.I., Bening V.E. On the mean total reserve of an insurance company with a random number of clients. // Moscow University Journal. Series 15. Computational Mathematics and Cybernetics. 2026. N 2, p.11-22 https://doi.org/10.55959/MSU/0137–0782–15–2026–50–2–11–22.

This work is licensed under a Сreative Commons Atribiution - NonCommercial 4.0 International (CC BY-NC 4.0)

Arifullin A.I. (Lomonosov Moscow State University (MSU))
Bening V.E. (Lomonosov Moscow State University (MSU))

Abstract

The paper studies the asymptotic behaviour of the mean total reserve of an insurance company in the case of a random number of clients. We use results on asymptotic deficiency and relative efficiency of procedures based on samples of fixed and random sizes. Zero-truncated Delaporte and generalized Poisson distributions are considered as models for the random number of clients. Asymptotic expansions of inverse moments are obtained for these distributions and the coefficients in the asymptotic formulas are computed. It is shown that the asymptotic deficiency of the scheme with a random number of clients can be expressed in terms of distribution parameters and in particular cases equals 1 and (1−θ)⁻². An empirical illustration is given based on data from the U.S. National Flood Insurance Program.