Stable-1/2 Bridges and Insurance: a Bayesian approach to non-life reserving

We develop a non-life reserving model using a stable-1/2 random bridge to simulate the accumulation of paid claims, allowing for an arbitrary choice of a priori distribution for the ultimate loss. Taking a Bayesian approach to the reserving problem, we derive the process of the conditional distribution of the ultimate loss. The ‘best-estimate ultimate loss process’ is given by the conditional expectation of the ultimate loss. We derive explicit expressions for the best-estimate ultimate loss process, and for expected recoveries arising from aggregate excess-of-loss reinsurance treaties. Use of a deterministic time change allows for the matching of any initial (increasing) development pattern for the paid claims. We show that these methods are well-suited to the modelling of claims where there is a non-trivial probability of catastrophic loss. The generalized inverse-Gaussian (GIG) distribution is shown to be a natural choice for the a priori ultimate loss distribution. For particular GIG parameter choices, the best-estimate ultimate loss process can be written as a rational function of the paid-claims process. We extend the model to include a second paid-claims process, and allow the two processes to be dependent. The results obtained can be applied to the modelling of multiple lines of business or multiple origin years. The multidimensional model has the attractive property that the dimensionality of calculations remains low, regardless of the number of paid-claims processes. An algorithm is provided for the simulation of the paid-claims processes. Email: [email protected] Department of Mathematics, Imperial College London, London SW7 2AZ, UK Department of Mathematics, King’s College London, London WC2R 2LS, UK Institute of Economic Research, Kyoto University, Kyoto 606-8501, Japan