A Multivariate Bayesian Claim Count Development Model With Closed Form Posterior and Prdictive Distributions

We present a rich, yet tractable, multivariate Bayesian model of claim count development. The model combines two conjugate families: the gamma-Poisson distribution for ultimate claim counts and the Dirichlet-multinomial distribution for emergence. We compute closed form expressions for all distributions of actuarial interest, including the posterior distribution of parameters and the predictive multivariate distribution of future counts given observed counts to date and for each of these distributions give a closed form expression for the moments. A new feature of the model is its explicit sensitivity to ultimate claim count variability and the uncertainty surrounding claim count emergence. Depending on the value of these parameters, the posterior mean can equal the Bornhuetter-Ferguson or chain-ladder reserve. Thus the model provides a continuum of models interpolating between these common methods. We give an example to illustrate use of the model.