Penalized Copula Mixed Models for Intercompany Loss Reserving and Risk Capital

Abstract

Intercompany loss reserving provides an opportunity to improve reserve estimation by borrowing information across insurers while accounting for company-level heterogeneity. We propose a penalized generalized copula mixed model for multivariate loss reserving and risk capital analysis using multiple companies' loss triangles. The framework combines mixed-effects marginal models with company-specific copula dependence parameters, allowing residual dependence between lines of business to vary across insurers. Penalization is introduced through an $L_1$ penalty on the fixed effects to stabilize estimation in the tail of the loss triangles, where observations are limited. Estimation is carried out using an iterative two-stage procedure that combines likelihood-based estimation of the marginal mixed models with rank-based copula estimation using residual pseudo-observations. To obtain predictive reserve distributions, we develop a modified bootstrap procedure that accommodates penalized estimation while preserving the dependence structure. Using Schedule P data from the National Association of Insurance Commissioners, we show that the proposed model provides a more stable decomposition of unpaid losses across lines of business, reduces predictive variability relative to silo and fixed-effect copula benchmarks, and leads to lower risk capital after accounting for diversification. A simulation study further evaluates parameter recovery, sparsity selection, reserve accuracy, and robustness to random-effect misspecification. Overall, the proposed model offers an interpretable and flexible framework for intercompany multivariate reserving and capital assessment.