Design of Optimal Bonus-Malus Systems With a Frequency and a Severity Component On an Individual Basis in Automobile Insurance

Abstract

The majority of optimal Bonus-Malus Systems (BMS) presented up to now in the actuarial literature assign to each policyholder a premium based on the number of his accidents. In this way a policyholder who had an accident with a small size of loss is penalized unfairly in the same way with a policyholder who had an accident with a big size of loss. Motivated by this, we develop in this paper, the design of optimal BMS with both a frequency and a severity component. The optimal BMS designed are based both on the number of accidents of each policyholder and on the size of loss (severity) for each accident incurred. Optimality is obtained by minimizing the insurer's risk. Furthermore we incorporate in the above design of optimal BMS the important a priori information we have for each policyholder. Thus we propose a generalised BMS that takes into consideration simultaneously the individual's characteristics, the number of his accidents and the exact level of severity for each accident.

Generalized Bonus-Malus Systems with a Frequency and a Severity Component on an Individual Basis in Automobile Insurance

“…This BMS is derived as a function of the years that the policyholder is in the portfolio, of the number of accidents and of the significant -for the number of accidents -individual characteristics. Frangos and Vrontos (2001) extend BMS model by introducing the severity component. They proposed a BMS that integrates a priori and a posteriori information on an individual basis based on the both frequency and the severity component.…”

“…It is assumed that the number of claims of each policyholder is independent from the severity of each claim in order to deal with the frequency and the severity component separately. In this paper, we present a generalized form of those obtained in Frangos and Vrontos (2001).…”

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ABSTRACT Frangos and Vrontos (2001) proposed an optimal bonus-malus systems with a frequency and a severity component on an individual basis in automobile insurance. In this paper, we introduce a generalized form of those obtained previously.

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Vehicle and Fleet Random Effects in a Model of Insurance Rating for Fleets of Vehicles

“…The model is a parametric one which can account directly for both observable and unobservable characteristics of the vehicles, drivers, and owners associated with a particular vehicle fleet. The model proposed is a direct extension of bonus-malus-type automobile insurance models (Lemaire 1985(Lemaire , 1995Vanasse 1989 and1992;Pinquet 1997Pinquet , 1998Frangos and Vrontos 2001;Purcaru and Denuit 2003) to individual premiums (see Pinquet, 2000, for a review of the literature). The extension adds a random fleet effect to the vehicle effect in the model, in order to take into account the unobservable effects of carriers, vehicles, and their drivers on truck accident rates in the Bayesian or a posteriori calculation of premiums.…”

“…The model proposed accounts directly for explicit changes in the various components of the probability of accidents. It represents an extension of bonus-malus-type automobile insurance models for individual premiums (Lemaire, 1985;Vanasse, 1989 and1992;Pinquet, 1997 and1998;Frangos and Vrontos, 2001;Purcaru and Denuit, 2003). The extension adds a fleet effect to the vehicle effect so as to account for the impact that the unobservable characteristics or actions of carriers can have on truck accident rates.…”