[Paper Review] A Claim Score for Dynamic Claim Counts Modeling

Jean‐Philippe Boucher, Mathieu Pigeon|arXiv (Cornell University)|Dec 14, 2018

Data-Driven Disease Surveillance19 references4 citations

TL;DR

This paper introduces a claim score based on a bonus-malus system (BMS) to model dynamic claim counts in auto insurance, offering improved fit and predictive performance over existing panel count models. The BMS-panel model simplifies parameter estimation, ensures decreasing dependence over time, and avoids the counterintuitive penalty inflation seen in Harvey-Fernandès models, making it more practical and interpretable for actuarial use.

ABSTRACT

We develop a claim score based on the Bonus-Malus approach proposed by [7]. We compare the fit and predictive ability of this new model with various models for of panel count data. In particular, we study in more details a new dynamic model based on the Harvey-Fernandès (HF) approach, which gives different weight to the claims according to their date of occurrence. We show that the HF model has serious shortcomings that limit its use in practice. In contrast, the Bonus-Malus model does not have these defects. Instead, it has several interesting properties: interpretability, computational advantages and ease of use in practice. We believe that the flexibility of this new model means that it could be used in many other actuarial contexts. Based on a real database, we show that the proposed model generates the best fit and one of the best predictive capabilities among the other models tested.

Motivation & Objective

• To develop a more practical and interpretable model for dynamic claim count data in auto insurance by replacing complex random effects or copula structures.
• To address the shortcomings of the Harvey-Fernandès (HF) model, which assigns increasing penalties to older claims and becomes impractical for long-term insureds.
• To propose a simplified parameter estimation procedure for the BMS-panel model, reducing computational complexity while preserving statistical flexibility.
• To evaluate the BMS-panel model’s fit and predictive performance against multiple standard and dynamic panel count models using real insurance data.
• To demonstrate the claim score’s utility as a summary statistic of claim history that captures temporal decay in claim impact, enhancing model interpretability and usability.

Proposed method

• Proposes a claim score derived from a bonus-malus system (BMS), where past claims are summarized into a single numerical level reflecting claim history and time decay.
• Introduces a simplified estimation procedure for the BMS-panel model, drastically reducing calibration time and complexity compared to prior methods.
• Employs a Markovian structure where the BMS level evolves stochastically based on claim occurrence, with claim penalties decreasing over time.
• Uses a negative binomial distribution with a mean parameter dependent on the BMS level and exposure, allowing overdispersion and flexibility.
• Applies a dynamic weighting scheme where claims are weighted by their age, with older claims contributing less to future premium increases.
• Compares the BMS-panel model against standard models (Poisson, NBBeta) and dynamic models (HF-NBBeta), using real portfolio data to assess fit and prediction.

Experimental results

Research questions

• RQ1 Can a BMS-based claim score provide a more interpretable and computationally efficient alternative to complex random effects or copula models for panel count data?
• RQ2 How does the dynamic weighting of claims by age in the BMS-panel model affect model fit and predictive performance compared to static or uniformly weighted models?
• RQ3 Why does the Harvey-Fernandès (HF) model fail in practice despite its theoretical appeal, and what are the implications of its increasing penalty structure?
• RQ4 To what extent does the BMS-panel model’s Markovian structure and time-decaying claim impact improve the modeling of claim dependence across policy periods?
• RQ5 Can the claim score be generalized to multivariate claim types or other actuarial contexts beyond frequency modeling?

Key findings

• The BMS-panel model achieves the best statistical fit among all tested models, including Poisson, NBBeta, and HF-NBBeta, based on real insurance data.
• The BMS-panel model exhibits decreasing covariance between claim counts over time, reflecting the natural decay of claim impact, which enhances predictive accuracy.
• Unlike the HF model, which assigns increasing penalties to claims based on policyholder tenure (e.g., a 78.9% premium increase for a claim at time 15), the BMS-panel model avoids such counterintuitive behavior.
• With parameters s=11, ℓ*=1, and Ψ=6, the BMS model assigns a 12% premium increase per claim level, with a 64% increase for a claim at level 2, and an 11% reduction for a claim-free year.
• The model’s covariance structure depends on the initial BMS level ℓ₁, with higher initial levels increasing short-term dependence, but the effect diminishes with lag, confirming dynamic dependence decay.
• The BMS-panel model’s claim score is both interpretable and practical, summarizing full claim history into a single, actionable metric that supports real-world ratemaking.