The claims reserving problem is currently one of the most debated in actuarial literature. The high level of interest in this topic is due to the fact that Solvency II rules will come into operation in 2014. Indeed, it is expected that quantile computations will be compulsory in the evaluation of company risk and for this reason we think that the construction of the claims reserve random variable distribution assumes a fundamental relevance.
The aim of this paper is to present a method for constructing the claims reserve distribution which can take into account IBNyR (Issued But Not yet Reported) claims in a natural way. The construction of the distribution function for each time of the observed interval is done by means of a Monte Carlo simulation model applied on a backward time semi-Markov process. It should be pointed out that this is the first time that a simulation model based on semi-Markov with backward recurrence time has been presented. The method is totally different from the models given in the current literature.
The most important features given in the paper are:
1) for the first time the Monte Carlo simulation method is applied to a backward semi-Markov environment;
2) the Monte Carlo simulation permits the construction of the random variable of the claims reserve for each year of the studied horizon in a natural way;
3) as already pointed out, the backward process attached to the semi-Markov process permits taking into account the evaluation of the IBNyR claims in a natural way.
In the last part of the paper an applicative example constructed from tables that summarise 4 years of claims from an important Italian insurance company will be given.