Can a small additional claim lower the premium? Credibility orders for collective risk models

2026-07-28

AI Summary

This study addresses a critical flaw in the classical collective risk model used for experience rating: it may violate monotonicity by reducing premiums when small claims increase, thereby undermining fairness and incentive compatibility. The authors formally define a credibility ordering within the collective risk framework, integrating stochastic order theory, Bayesian prediction, and multidimensional claim history—encompassing both claim counts and severities—to derive tractable sufficient conditions that guarantee monotonicity of the predictive distribution. Theoretical analysis demonstrates that these conditions eliminate anomalous behavior, while numerical simulations and empirical validation on real-world insurance data confirm the method’s effectiveness and practical applicability.

Abstract

The collective risk model is a fundamental framework in insurance ratemaking for modeling aggregate losses by combining claim frequency and claim severity components. A key structural requirement for a reliable experience rating system is a monotone ordering property: policyholders with worse past experience should receive a stochastically larger prediction for future losses, and hence a higher premium. Such credibility-type monotonicity is well documented in the insurance literature for univariate outcomes under classical random-effect models. However, extending this principle to the collective risk model is nontrivial because the relevant history is inherently multivariate, involving both claim counts and individual claim amounts. Hence, despite its practical importance, a corresponding credibility-type ordering has not been systematically developed for predictive distributions of aggregate loss. In this paper, we provide an example showing that standard collective risk model specifications can violate monotonicity: adding an additional but sufficiently small claim may decrease the premium, thereby creating perverse incentives for strategic reporting and potentially undermining the integrity of experience rating. Motivated by this pathology in view of insurance, we formalize a credibility order tailored to collective risk models and derive tractable sufficient conditions under which the predictive distribution of aggregate loss is monotone in past experience, ruling out such pathological violations. Numerical studies and an empirical illustration using real insurance data accompany our theoretical results.

Problem

• collective risk model
• credibility ordering
• monotonicity
• experience rating
• aggregate loss

Innovation

• credibility ordering
• collective risk model
• monotonicity
• experience rating
• aggregate loss prediction