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Insurance companies have to build a reserve for their future payments which is usually done by deterministic methods giving only a point estimate. In this paper two semi-stochastic methods are presented along with a more sophisticated hierarchical Bayesian model containing MCMC technique. These models allow us to determine quantiles and confidence intervals of the reserve which can be more reliable as just a point estimate. A sort of cross-validation technique is also used to test the models.

Our aim was to examine the territorial dependence of risk for household insurances. Besides the classical risk factors such as type of wall, type of building, etc., we consider the location associated to each contract. A Markov random field model seems to be appropriate to describe the spatial effect. Basically there are two ways of fitting the model; we fit a GLM to the counts of claims with the classical risk factors and regarding their effects as fixed we fit the spatial model. Alternatively we can estimate the effects of all covariates (including location) jointly. Although this latter approach may seem to be more accurate, its high complexity and computational demands makes it unfeasible in our case. To overcome the disadvantages of the distinct estimation of the classical and the spatial risk factors proceed as follows: use first a GLM for the non-spatial covariates, and then fit the spatial model by MCMC. Refit next the GLM with keeping the obtained spatial effect fixed and afterwards refit the spatial model, too. Iterate this procedure several times. We achieve much better fit by performing eight iterations.

Latent, that is Incurred But Not Reported (IBNR) claims influence heavily the calculation of the reserves of an insurer, necessitating an accurate estimation of such claims. The highly diverse estimations of the latent claim amount produced by the traditional estimation methods (chain-ladder, etc.) underline the need for more sophisticated modelling. We are aimed at predicting the number of latent claims, not yet reported. This means the continuation the so called run-off triangle by filling in the lower triangle of the delayed claims matrix. In order to do this the dynamics of claims occurrence and reporting tendency is specified in a hierarchical Bayesian model. The complexity of the model building requires an algorithmic estimation method, that we carry out along the lines of the Bayesian paradigm using the MCMC technique. The predictive strength of the model against the future disclosed claims is analysed by cross validation. Simulations serve to check model stability. Bootstrap methods are also available as we have full record of the individual claims at our disposal. Those methods are used for assessing the variability of the estimated structural parameters.