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

TL;DR

This paper addresses optimal claim reporting in a two-regime bonus-malus insurance setting with premium levels $\pi_1$ and $\pi_2$ under a finite horizon $T$. It formulates a continuous-time barrier-control problem driven by a compound Poisson claim process, yielding wealth dynamics $dX_t^b=(c-\pi_{I_{t-}}(S_{t-}^b,m_{I_{t-}}))dt - \big(Y_{N_t}\mathbf{1}_{\{Y_{N_t}\le b_t\}} + r(Y_{N_t},m_{I_{t-}})\mathbf{1}_{\{Y_{N_t}> b_t\}}\big)dN_t$ across two premium classes and an active boundary. The main result proves that the value function $V(i,t,s,x)$ is the unique viscosity solution to a coupled system of Hamilton-Jacobi-Bellman equations, enabling numerical approximation of optimal barrier strategies. The numerical example demonstrates a PDMP-based scheme to compute state- and time-dependent barriers, revealing practical insights into reporting decisions near maturity and the non-reporting behavior that sustains favorable classes. Overall, the work provides a rigorous framework for computing optimal reporting policies in auto-insurance bonus-malus settings and informs premium-setting and risk-management decisions.

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.

Figures (4)

• Figure 1: Approximated value functions $x \mapsto \tilde{V}(1,0,0,x)$ (blue) and $x \mapsto \tilde{V}(2,0,0,x)$ (orange).
• Figure 2: Approximated value functions $t \mapsto \tilde{V}(1,t,0,5)$ (blue) and $t \mapsto \tilde{V}(2,t,0,5)$ (orange).
• Figure 3: Approximated value functions $s \mapsto \tilde{V}(1,\mathcal{S},s,5)$ (blue) and $s \mapsto \tilde{V}(2,\mathcal{S},s,5)$ (orange).
• Figure 4: Approximated barrier strategies.

Theorems & Definitions (8)

• Lemma 3.1
• proof
• Corollary 3.1: Dynamic Programming Principle
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Remark 5.1