Discussiones Mathematicae Probability and Statistics 24(2) (2004) 197-213


Gracinda Rita Guerreiro and João Tiago Mexia

Department of Mathematics
FCT - New University of Lisbon
Quinta da Torre, 2829-516 Caparica, Portugal
e-mail: grg@fct.unl.pt


Under the assumptions of an open portfolio, i.e., considering that a policyholder can transfer his policy to another insurance company and the continuous arrival of new policyholders into a portfolio which can be placed into any of the bonus classes and not only in the ßtarting class", we developed a model (Stochastic Vortices Model) to estimate the Long Run Distribution for a Bonus Malus System. These hypothesis render the model quite representative of the reality.

With the obtained Long Run Distribution, a few optimal bonus scales were calculated, such as Norberg's (1979), Borgan, Hoem's and Norberg's (1981), Gilde and Sundt's (1989) and Andrade e Silva's (1991).

To compare our results, since this was the first application of the model, we used the Classic Model for Bonus Malus and the Open Model developed by Centeno and Andrade e Silva (2001).

The results of the Stochastic Vortices and the Open Modelare highly similar and quite different from those of the Classic Model. Besides this the distribution of policyholders in the various bonus classes was derived assuming that the entrances followed adequatestochastic models.

Keywords: bonus malus, stochastic vortices, long run distribution, optimal bonus scales.

2000 Mathematics Subject Classification: 60J05, 60J20.


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Received 7 January 2004
Revised 18 October 2004