Automatic Segmentation of Insurance Rating Classes Under Ordinal Constraints via Group Fused Lasso

This paper proposes a sparse regularization technique for ratemaking under practical constraints. In tariff analysis of general insurance, rating factors with many categories are often grouped into a smaller number of classes to obtain reliable estimate of expected claim cost and make the tariff simple to reference. However, the number of rating-class segmentation combinations is often very large, making it computationally impossible to compare all the possible segmentations. In such cases, an L1 regularization method called the fused lasso is useful for integrating adjacent classes with similar risk levels in its inference process. Particularly, an extension of the fused lasso, known as the group fused lasso, enables consistent segmentation in estimating expected claim frequency and expected claim severity using generalized linear models. In this study, we enhance the group fused lasso by imposing ordinal constraints between the adjacent classes. Such constraints are often required in practice based on bonus–malus systems and actuarial insight on risk factors. We also propose an inference algorithm that uses the alternating direction method of multipliers. We apply the proposed method to motorcycle insurance claim data, and demonstrate how some adjacent categories are grouped into clusters with approximately homogeneous levels of expected claim frequency and severity.