Bonus-Malus Scale models: creating artificial past claims history

Abstract

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.

Bonus-Malus Scale premiums for Tweedie’s compound Poisson models

“…As shown in 2 and 3, the severity could also be modeled using this approach. More generally, to cover all of these possibilities, Boucher (2023) defined two types of variables to be used in experience rating:…”

“…For the experience rating, the Kappa-N and BMS models are generally proposed (Boucher, 2023), which model the conditional distribution of a target variable according to the scope variables. In this section, the CPG and Tweedie are used as an underlying distribution in each model.…”

“…In order to limit the variation of premiums over time and to allow some form of forgiveness of old claims in the rating, another approach constrains the score to be between two limits for all the past contracts. Boucher (2023) called this approach the BMS models. See Lemaire (1995) and Denuit et al (2007) for a historical review of BMS models.…”

“…See Villacorta Iglesias et al (2021), or Adillon et al (2020) for a review of the BMS models. Research on modeling the total annual claims number across several different databases and insurance products has shown that the BMS models can generate excellent fit of training data and excellent prediction on test data, and often outperform the results of Bayesian random-effects models (Boucher, 2023;Boucher and Inoussa, 2014). In this paper, we generalize this BMS approach by working with data with a slightly more complex structure, close to what is used in practice, at least in North America.…”

Numerical methods for model-free pricing in finance, optimal transport, and cyber risk management

“…A key characteristic of Bonus-Malus systems is the bonus hunger mechanism [Holtan, 2001, Charpentier, David, and Elie, 2017, Tzougas et al, 2018, i.e., under a Bonus-Malus system, an insured is willing to carry small losses themselves in order to avoid premium surcharges in the future. Compared to classical Bonus-Malus systems in the literature in which transitions of Bonus-Malus levels are solely determined by the presence/absence of a claim or claim frequencies (see, e.g., [Boucher, 2023, Gómez-Déniz, 2016), the Bonus-Malus system that we consider in our model…”