To report or not to report: Optimal claim reporting in a bonus-malus system

TL;DR

This paper models an optimal claim reporting decision in a bonus-malus insurance system as a control problem, deriving strategies that maximize the policyholder's expected wealth through a mathematical framework involving Hamilton-Jacobi-Bellman equations.

Contribution

It introduces a novel optimal control formulation for claim reporting decisions in bonus-malus systems, including a viscosity solution approach for strategy computation.

Findings

Optimal barrier strategies can be numerically approximated.

The value function is the unique viscosity solution to the HJB system.

Policyholder's decision impacts long-term wealth maximization.

Abstract

We study an optimal claim reporting problem in a bonus-malus setting. We assume, that the insurance contract consists of two regimes, where reporting a claim leads to a transition to a higher-premium regime, whereas remaining claim-free for a prespecified time period results in a shift to the lower premium regime. The insured can decide whether or not to report an occurred claim. We formulate this as an optimal control problem, where the policyholder follows a barrier-type reporting strategy, with the goal of maximizing the expected value of a function of their terminal wealth. We show that the associated value function is the unique viscosity solution to a system of Hamilton-Jacobi-Bellman equations. This characterization allows us to compute numerical approximations of the optimal barrier strategies.