A financially Balanced Bonus/Malus System

Abstract

The premiums for a bonus-malus system which stays in financial equilibrium over the years are calculated. This is done by minimizing a quadratic function of the difference between the premium for an optimal BMS with an infinite number of classes and the premium for a BMS with a finite number of classes, weighted by the stationary probability of being in a certain class, and by imposing various constraints on the system.

Confidence intervals of the premiums of optimal bonus malus systems

“…In this Section, using the NPML estimator given by (5) and based on the framework developed by Karlis and Patilea (2008), we are going to build Wald Type confidence intervals and Efron percentile bootstrap confidence intervals for λ t+1 .…”

“…Consider a bootstrap procedure where the bootstrap samples N * j,1 , ..., N * j,n are the number of claims of a policyholder i, i = 1, .., n, during the jth year of their presence in the portfolio generated according to the finite Poisson mixture given by (5). This is a parametric bootstrap procedure where the unknown parameter is the mixing distribution and the parameter space is the set of all probability measures on [0, M ] , that is, the parameter space is of infinite dimension.…”

“…An excellent account of BMS can be found in Lemaire (1995). Also, references for BMS include, among others, Vanasse (1989, 1992), Coene and Doray (1996), Walhin and Paris (1999), Pinquet (1998) However, even though the construction of optimal BMS has been a basic interest of actuarial literature for over four decades, scientific attention has only now focused on deriving credibility updates of the claim frequency based on the employment of an abundance of alternative parametric distributions, nonparametric distributions and advanced regression models. In this respect, a major drawback in the design of such systems was neglected: namely the fact that they do not give a measure of uncertainty of the resulting premium estimates by providing a confidence interval that contains plausible values.…”

Optimal relativities and transition rules of a bonus–malus system

“…A financial equilibrium constraint is added to the optimization such that the set of optimal relativities has an average of 100%. This constraint was first considered by Coene and Doray (1996) who devised a financially-balanced BMS. The solution can then be derived analytically by using the Lagrangian method.…”

Measuring sensitivity in a bonus–malus system

“…A model often used for experience rating in a BMS assumes that each individual risk has its own Poisson distribution for a number of claims, λ, assuming that the mean number of claims is distributed across individual policyholders according to a Gamma distribution (Coene and Doray, 1996;Corlier et al, 1979;Lemaire, 1979Lemaire, , 1988Lemaire, , 1998.…”