Optimization of transition rules in a Bonus-Malus system

Gyetvai, Márton; Ágoston, Kolos Csaba

doi:10.1016/j.endm.2018.07.002

Cited by 4 publications

(4 citation statements)

References 3 publications

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“…Recently, Tan et al (2015) have incorporated in their model the effect of the transition rule changes. Gyetvai and Ágoston (2018) gave an optimization model for choosing the best transition rules, while the premium scale is fixed. In the present article, we extend this model to optimize the premium scale and transition rules jointly.…”

Section: Literature Overviewmentioning

confidence: 99%

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

Ágoston¹,

Gyetvai²

2020

ASTIN Bull.

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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.

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Section: Literature Overviewmentioning

confidence: 99%

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

Ágoston¹,

Gyetvai²

2020

ASTIN Bull.

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“…Classic textbooks for BMS with aggregate data are Lemaire (2012) or Denuit et al (2007). However, recently, a new scientific interest in finding optimal transition rules for that type of BMS, via integer programming, can be found in the works of Tan et al (2015), Gyetvai & Ágoston (2018) and Ágoston & Gyetvai (2020).…”

Section: Introductionmentioning

confidence: 99%

Bonus-Malus Scale models: creating artificial past claims history

Boucher

2022

Ann. actuar. sci.

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In recent papers, Bonus-Malus Scales (BMS) estimated using data have been considered as an alternative to longitudinal data and hierarchical data approaches to model the dependence between different contracts for the same insured. Those papers, however, did not discuss in detail how to construct and understand BMS models, and many of the BMS’s basic properties were not discussed. The first objective of this paper is to correct this situation by explaining the logic behind BMS models and by describing those properties. More particularly, we will explain how BMS models are linked with simple count regression models that have covariates associated with the past claims experience. This study could help actuaries to understand how and why they should use BMS models for experience rating. The second objective of this paper is to create artificial past claims history for each insured. This is done by combining recent panel data theory with BMS models. We show that this addition significantly improves the prediction capacity of the BMS and provides a temporary solution for insurers who do not have enough historical data. We apply the BMS model to real data from a major Canadian insurance company. Results are analysed deeply to identify specific aspects of the BMS model.

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“…In this dissertation, I present the research on the optimization of the Bonus-Malus System (BMS), which I did with my supervisor, Kolos Csaba Ágoston. Our contribution to the topic was published in Gyetvai and Ágoston (2018); Ágoston and Gyetvai (2020) and Ágoston and Gyetvai (2021),also, in a Hungarian article, Ágoston and . We also had a study with László Kovács that was published in Ágoston et al (2019).…”

Section: Introductionmentioning

confidence: 92%

“…We also introduce a mixed-integer linear programming (MILP) model, where the classifications' rules are in the scope of the optimization. We introduced this model in Gyetvai and Ágoston (2018).…”

Section: Introductionmentioning

confidence: 99%

Optimization of Bonus-Malus Systems

Gyetvai¹

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Bonus-Malus System (BMS) is a risk managing method mostly used in liability insurances. The most general application of the BMS is in the Motor third-party liability insurance. In a BMS, there are finitely many classes, each having a different premium. At the start of the contract, each policyholder is assigned to the "initial class". Subsequently, suppose the policyholder has a claim in the following period. In that case, he/she moves to a worse class, so the policyholder's payment may increase in the subsequent period. If he/she does not have a claim in a particular period, then he/she moves to a better class; therefore, his/her payment may become less in the following period. The classification rule – how many classes the policyholder will move up or down in the system – is called the transition rule. Hence, a transition rule specifies where the policyholder will be reclassified in the subsequent period for each possible claim. Our contribution to the literature of the optimization of the BMS can be summarized as: • We investigate a model that was introduced by Heras et al. (2004) but with a modified objective function. We proved that an optimal premium-scale always exists with this objective function in which all premiums equal one of the risk groups expected claim. • We considered the same model with a profit constraint. In this case, we proved that an optimal premium-scale always exists in which there is only one type of premium that is unequal to any risk group's expected claim. • We introduced a MILP model for the optimization of transition rules with fixed premiums. We considered unified and non-unified transition rules optimization. In the case of unified transition rules, we gave the rule to exclude those transition rules that would lead to a non-irreducible Markov chain. • We introduced a MILP model for the joint optimization of transition rules and premiums. We can determine the exact solution with the investigated objective function when we do not consider the profit constraint. However, we can only approximate it otherwise. • We introduced an extended version of the model, where instead of the stationary probabilities, we use multi-period optimization. • We introduced modeling approaches to consider the BMS premium with other statistical estimations in the final premium. Finally, we compared the methods with numerical experiments on realistic data. • We introduced an optimization model for a BMS where the classification depends on the claim amount.

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Optimization of transition rules in a Bonus-Malus system

Cited by 4 publications

References 3 publications

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

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

Bonus-Malus Scale models: creating artificial past claims history

Optimization of Bonus-Malus Systems

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