Stochastic Payments per Claim Incurred

We propose a Bayesian model to quantify the uncertainty associated with the payments per claim incurred (PPCI) algorithm. Based on the PPCI algorithm, two submodels are proposed for the number of reported claims run-off triangle and the PPCI run-off triangle, respectively. The model for the claims amount is then derived from the two submodels under the assumption of independence between the number of incurred claims and the PPCI. The joint likelihood of the number of reported claims and claims amount is derived. The posterior distribution of parameters is estimated via the Hamiltonian Monte Carlo (HMC) sampling approach. The Bayesian estimator, the process variance, the estimation variance, and the predictive distribution of unpaid claims are also studied. The proposed model and the HMC inference engine are applied to to an empirical claims dataset of the 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 a PwC report. An alternative model is compared with the proposed model based on various information criteria.

[1]  M. Wüthrich,et al.  Development Pattern and Prediction Error for the Stochastic Bornhuetter-Ferguson Claims Reserving Method , 2011 .

[2]  Michael I. Jordan,et al.  An Introduction to Variational Methods for Graphical Models , 1999, Machine Learning.

[3]  Richard Verrall,et al.  A Stochastic Model Underlying the Chain-Ladder Technique , 1998, British Actuarial Journal.

[4]  Andrew Gelman,et al.  The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo , 2011, J. Mach. Learn. Res..

[5]  G. Taylor,et al.  Loss Reserving: An Actuarial Perspective , 2000 .

[6]  Reversible Jump Markov Chain Monte Carlo Method for Parameter Reduction in Claims Reserving , 2012 .

[7]  Radford M. Neal.  An improved acceptance procedure for the hybrid Monte Carlo algorithm , 1992, hep-lat/9208011.

[8]  A. Gelman,et al.  Weak convergence and optimal scaling of random walk Metropolis algorithms , 1997 .

[9]  Malay Ghosh,et al.  ON THE INVARIANCE OF NONINFORMATIVE PRIORS , 1996 .

[10]  M. Merz,et al.  Stochastic Claims Reserving Methods in Insurance , 2008 .

[11]  Mario V. Wuthrich,et al.  Stochastic Claims Reserving Manual: Advances in Dynamic Modeling , 2015 .

[12]  P. England,et al.  Stochastic Claims Reserving in General Insurance , 2002, British Actuarial Journal.

[13]  Richard Verrall,et al.  A Bayesian Generalized Linear Model for the Bornhuetter-Ferguson Method of Claims Reserving , 2004 .

[14]  S. Duane,et al.  Hybrid Monte Carlo , 1987 .

[15]  James Guszcza,et al.  A Bayesian non‐linear model for forecasting insurance loss payments , 2012 .

[16]  J. Rosenthal,et al.  Optimal scaling of discrete approximations to Langevin diffusions , 1998 .

[17]  Mean Square Error of Prediction in the Bornhuetter–Ferguson Claims Reserving Method , 2009, Annals of Actuarial Science.

[18]  Bayesian over-dispersed Poisson model and the Bornhuetter & Ferguson claims reserving method , 2012, Annals of Actuarial Science.

[19]  Richard Verrall,et al.  An investigation into stochastic claims reserving models and the chain-ladder technique , 2000 .