[Paper Review] Estimation of Claim Numbers in Automobile Insurance

Miklós Arató, László Martinek|arXiv (Cornell University)|Mar 5, 2012

Probability and Risk Models10 references3 citations

TL;DR

This paper proposes and compares three methods for estimating claim frequency in automobile insurance: Bayesian estimation using bonus-malus class, average claim frequency per class, and individual claim history. Using Brier and logarithmic scores in Monte Carlo simulations, it finds that claim history-based estimation becomes superior earlier in systems with more classes, with the Belgian system requiring 16–17 years to outperform class-based methods, while the Brazilian system achieves this in just 2 years.

ABSTRACT

The use of bonus-malus systems in compulsory liability automobile insurance is a worldwide applied method for premium pricing. If certain assumptions hold, like the conditional Poisson distribution of the policyholders claim number, then an interesting task is to evaluate the so called claims frequency of the individuals. Here we introduce 3 techniques, two is based on the bonus-malus class, and the third based on claims history. The article is devoted to choose the method, which fits to the frequency parameters the best for certain input parameters. For measuring the goodness-of-fit we will use scores, similar to better known divergence measures. The detailed method is also suitable to compare bonus-malus systems in the sense that how much information they contain about drivers.

Motivation & Objective

• To evaluate and compare three methods for estimating individual claim frequency in bonus-malus automobile insurance systems.
• To determine which method provides the most accurate estimation of the true claim frequency (λ) under varying system structures and claim history lengths.
• To use statistical scores (Brier and logarithmic) as a metric for assessing the goodness-of-fit of frequency estimates.
• To assess the information content of different bonus-malus systems by measuring how quickly each method converges to accurate estimates.
• To provide a practical, simulation-based decision framework for insurers to select the optimal estimation method based on available data and system design.

Proposed method

• Employs a Bayesian approach to estimate λ by updating a prior distribution using the policyholder’s bonus-malus class, assuming conditional Poisson claim counts.
• Uses the average claim frequency observed in each bonus-malus class as a frequentist benchmark method for estimating λ.
• Applies individual claim history (number of claims over time) to estimate λ via empirical frequency, treating it as a third estimation method.
• Utilizes Brier score and logarithmic score as proper scoring rules to evaluate the accuracy of predicted λ distributions against true values.
• Conducts Monte Carlo simulations with synthetic portfolios (N=80,000, M=20,000) over 15 years to compare method performance across different systems.
• Models the bonus-malus system as a homogeneous Markov chain with transition probabilities dependent on λ, using a stochastic matrix M(λ) derived from Poisson claim probabilities.

Experimental results

Research questions

• RQ1 Which of the three frequency estimation methods—Bayesian class-based, class-average, or history-based—provides the most accurate estimate of true claim frequency λ?
• RQ2 How does the performance of each method vary across different bonus-malus systems (Belgian, Brazilian, Hungarian) with differing numbers of classes?
• RQ3 At what point in time does the claim history-based method surpass the class-average method in accuracy, and how does this vary by system?
• RQ4 To what extent do Brier and logarithmic scores yield consistent rankings of method performance?
• RQ5 How does the number of bonus-malus classes influence the information content and estimation efficiency of the system?

Key findings

• The Bayesian method based on bonus-malus class consistently performed the worst across all systems and time points.
• The claim history-based method became the most accurate estimator in the Brazilian system after just 2 years of data.
• In the Hungarian system, the history-based method surpassed the class-average method after 7–8 years of claim history.
• In the Belgian system, which has 23 classes, the history-based method required 16–17 years to outperform the class-average method.
• The Brier and logarithmic scores produced nearly identical rankings of method performance, indicating robustness of the evaluation framework.
• The number of classes in a bonus-malus system directly correlates with the time required for history-based estimation to become optimal, indicating higher information content in larger systems.