Copula Regression for Deductible--Claim Dependence in Non-Life Insurance: A Stratified Evaluation

Abstract

Deductible choice may be statistically associated with claim outcomes after observed rating variables have been taken into account. The contractual deductible is distinguished from the normalized deductible ratio used as the continuous working margin, while claim count remains discrete and individual severity remains continuous. A Canonical Vine pair-copula construction links the deductible ratio, claim frequency, and individual claim severity. The application uses 1,038 entities from the Wisconsin Local Government Property Insurance Fund observed from 2006 to 2010; models are estimated on 2006-2009 data and evaluated on the 2010 temporal holdout. The estimated dependence is negative for deductible-frequency, positive for deductible observed severity, and negative for frequency-severity conditional on the deductible. These patterns are interpreted as conditional associations that may be consistent with selection and claim-reporting mechanisms, rather than as causal evidence of adverse selection or moral hazard. The dependence model attains a higher observation-level logarithmic score than the exogenous-deductible benchmark for 64.55% of frequency observations and 71.09% of severity observations. Its global Insurance Gini  improvement is 22.54 points, with positive point estimates across all five deductible strata but greater uncertainty in the highest-deductible group. The results show that deductible information can improve out of sample risk discrimination, while also demonstrating the importance of support assumptions, benchmark definitions, and cautious behavioral interpretation.