On the ordering of credibility factors

TL;DR

This paper investigates how the ordering of credibility factors in insurance risk models relates to the covariance structure, demonstrating that certain state-space models ensure a logical, monotonic weighting of recent claims over older ones.

Contribution

It clarifies the relationship between covariance structures and credibility factor ordering, proposing that AR(1)-type models guarantee a proper monotonic weighting of claims.

Findings

Covariance structure does not guarantee credibility factor ordering.

AR(1)-type models ensure monotonic credibility factor ordering.

Simulation and case study validate the theoretical results.

Abstract

Traditional credibility analysis of risks in insurance is based on the random effects model, where the heterogeneity across the policyholders is assumed to be time-invariant. One popular extension is the dynamic random effects (or state-space) model. However, while the latter allows for time-varying heterogeneity, its application to the credibility analysis should be conducted with care due to the possibility of negative credibilities per period [see Pinquet (2020a)]. Another important but under-explored topic is the ordering of the credibility factors in a monotonous manner -- recent claims ought to have larger weights than the old ones. This paper shows that the ordering of the covariance structure of the random effects in the dynamic random effects model does not necessarily imply that of the credibility factors. Subsequently, we show that the state-space model, with AR(1)-type autocorrelation function, guarantees the ordering of the credibility factors. Simulation experiments and a case study with a real dataset are conducted to show the relevance in insurance applications.