11: Seemingly Unrelated Regression (SUR) Copula Mixed Models for Multivariate Loss Reserving

Music City Center 

In property and casualty (P&C) insurance, estimating unpaid claims is a critical task that directly impacts an insurer's reserve levels and risk capital requirements. Insurance companies often underwrite multiple, interrelated lines of business (LOBs), and appropriately modelling dependence across these LOBS is essential for accurate loss prediction and capital allocation.

The Seemingly Unrelated Regression (SUR) copula regression framework has been proposed to model such dependence using loss triangle data from a single company. However, this model can suffer from high bias due to limited data and its inability to fully capture heterogeneity across LOBs and firms.

To address these challenges, we propose a SUR copula mixed model that incorporates data from multiple companies and explicitly models heterogeneity via random effects and flexible distributional assumptions for each LOB. Furthermore, we introduce a shrinkage component to stabilize estimation in high-dimensional settings and improve generalization across heterogeneous company data.

Using multiple pairs of loss triangles from the National Association of Insurance Commissioners (NAIC) database, we demonstrate that our model reduces the bias between predicted and actual reserves when compared to the classical SUR copula regression. We also show that it delivers improved diversification benefits, as reflected in higher estimated risk capital gains. These results are validated through both empirical analysis and a targeted simulation study.