Joint Optimization of Transition Rules and the Premium Scale in a Bonus-Malus System

Abstract

Bonus-malus systems (BMSs) are widely used in actuarial sciences. These systems are applied by insurance companies to distinguish the policyholders by their risks. The most known application of BMS is in automobile third-party liability insurance. In BMS, there are several classes, and the premium of a policyholder depends on the class he/she is assigned to. The classification of policyholders over the periods of the insurance depends on the transition rules. In general, optimization of these systems involves the calculation of an appropriate premium scale considering the number of classes and transition rules as external parameters. Usually, the stationary distribution is used in the optimization process. In this article, we present a mixed integer linear programming (MILP) formulation for determining the premium scale and the transition rules. We present two versions of the model, one with the calculation of stationary probabilities and another with the consideration of multiple periods of the insurance. Furthermore, numerical examples will also be given to demonstrate that the MILP technique is suitable for handling existing BMSs.

On monotone likelihood ratio of stationary probabilities in bonus-malus systems

“…In this section, we give a possible application of the monotone likelihood property. In actuarial sciences, a well-studied problem to set optimal premium scale in a BM system, see Lemaire (1995), Heras et al (2004), Ágoston and Gyetvai (2020) For a small example, we assume that there are two equal-sized risk groups (A and B) in a risk community. To the risk community, the premium is set based on a common BM system.…”

Discrete-Time Risk Models with Claim Correlated Premiums in a Markovian Environment

“…Dionne and Ghali (2005) conducted an empirical evaluation of the 1992 bonus-malus system in Tunisia. More recently, Denuit et al (2019) considered multivariate credibility modelling for usage-based motor insurance pricing with behavioural data, Vilar-Zanon et al (2020) discussed an average model approach to experience based premium rates discounts using the Spanish agricultural insurance data, and Ágoston and Gyetvai (2020) studied a joint optimisation problem of transition rules and the premium scale in a bonus-malus system. A lot more research work can be found in the references therein.…”

Boosting Poisson Regression Models with Telematics Car Driving Data

“…In car insurance, a posteriori information is often encoded in a bonus-malus system (BMS) which scores past claims history and directly affects the insurance prices of policy renewals by a multiplicative factor. One stream of literature on BMS studies optimal design, efficiency and economic questions related to BMS, see Loimaranta (1972), De Pril (1978, Lemaire (1995), Denuit et al (2007), Brouhns et al (2003) and Ágoston and Gyetvai (2020). A second stream of literature rather addresses the question of how an existing BMS can be used to improve the predictive power for forecasting future claims since a BMS reveals past policyholder behavior, see e.g., Boucher and Inoussa (2014), Boucher and Pigeon (2018) and Verschuren (2021).…”