Joint estimation of insurance loss development factors using Bayesian hidden Markov models

Abstract: Loss development modelling is the actuarial practice of predicting the total 'ultimate' losses incurred on a set of policies once all claims are reported and settled. This poses a challenging prediction task as losses frequently take years to fully emerge from reported claims, and not all claims might yet be reported. Loss development models frequently estimate a set of 'link ratios' from insurance loss triangles, which are multiplicative factors transforming losses at one time point to ultimate. However, link ratios estimated using classical methods typically underestimate ultimate losses and cannot be extrapolated outside the domains of the triangle, requiring extension by 'tail factors' from another model. Although flexible, this two-step process relies on subjective decision points that might bias inference. Methods that jointly estimate 'body' link ratios and smooth tail factors offer an attractive alternative. This paper proposes a novel application of Bayesian hidden Markov models to loss development modelling, where discrete, latent states representing body and tail processes are automatically learned from the data. The hidden Markov development model is found to perform comparably to, and frequently better than, the two-step approach, as well as a latent change-point model, on numerical examples and industry datasets.