A Copula Based Bayesian Approach for Paid-Incurred Claims Models for Non-Life Insurance Reserving

Our article considers the class of recently developed stochastic models that combine claims payments and incurred losses information into a coherent reserving methodology. In particular, we develop a family of Heirarchical Bayesian Paid-Incurred-Claims models, combining the claims reserving models of Hertig et al. (1985) and Gogol et al. (1993). In the process we extend the independent log-normal model of Merz et al. (2010) by incorporating different dependence structures using a Data-Augmented mixture Copula Paid-Incurred claims model. The utility and influence of incorporating both payment and incurred losses into estimating of the full predictive distribution of the outstanding loss liabilities and the resulting reserves is demonstrated in the following cases: (i) an independent payment (P) data model; (ii) the independent Payment-Incurred Claims (PIC) data model of Merz et al. (2010); (iii) a novel dependent lag-year telescoping block diagonal Gaussian Copula PIC data model incorporating conjugacy via transformation; (iv) a novel data-augmented mixture Archimedean copula dependent PIC data model. Inference in such models is developed via a class of adaptive Markov chain Monte Carlo sampling algorithms. These incorporate a data-augmentation framework utilized to efficiently evaluate the likelihood for the copula based PIC model in the loss reserving triangles. The adaptation strategy is based on representing a positive definite covariance matrix by the exponential of a symmetric matrix as proposed by Leonard et al. (1992).