Stochastic Payments per Claim Incurred

We propose a Bayesian model to quantify the uncertainty associated with the payments per claim incurred (PPCI) algorithm. Founded on the PPCI algorithm, two sub-models are proposed for the number of reported claims run-off triangle and the PPCI run-off triangle. Then the model for the claims amount is derived from the two sub-models under the assumption of independence between the number of incurred claims and PPCI. The joint likelihood of the number of reported claims and claims amount is derived. The posterior distribution of parameters is estimated via a Hamiltonian Monte Carlo (HMC) sample. The Bayes estimator, the process variance, the estimation variance, and the predictive distribution of unpaid claims are studied. We apply the proposed model and the HMC inference engine to an empirical claims data set from WorkSafe Victoria to estimate the unpaid claims of the doctor benefit. The Bayesian modeling procedure is further refined by including a preliminary generalized linear model analysis. The results are compared with those in the PwC report. An alternative model is compared with the proposed model from the perspective of two information criteria