A distorted copula-based evolution model: risks’ aggregation in a Bonus–Malus migration system

In this paper, we put forward a new model to compute the loss distribution of an automobile insurance company’s portfolio evolving by a bonus–malus system. We allow for a continuous evolution of the demographic-economic system based on a migration’s rule which is refreshed in discrete time, i.e., at the monitoring times. Therefore, the migration’s probabilities are discretely updated through a technique based on the combinatorial distributions of claims’ arrival in the rating classes. This technique is hierarchical copula-based, a natural tool permitting us to represent the co-movement between claims’ arrivals and distorted due to the formalization of an arrival policy of claims, that restricts the set of combinatorial distributions to those representing the most probable scenarios, therefore distorting the loss function. At every monitoring date, the copula-based model computes the migration’s probabilities and the loss function which accommodates for a discrete-time dynamic of the claims’ reserving and the capital requirements. An empirical application, the evaluation of the claims’ reserving and the capital requirements for different kinds of hierarchies are analyzed, with real data originating with the General Insurance Association of Singapore.