Expected premium in bonus-malus system as a non-increasing claims frequency function

Bonus-malus system is most commonly used in automobile insurance to set the premium. In the system the premium depends on the number of claims reported by the driver in the previous period. That's why claims frequency model determines a posteriori classification. Generally there is assumed, that in the automobile insurance, the mixed Poisson distribution describes number of claims. Taking into a consideration above, in the article the system is analyzed for the individual client whose claims model is characterized by the Poisson distribution.To evaluate expected premium the non-realistic systems were eliminated. The bonus-malus system classification presented in the article allows distinguishing those, present on the competitive market, later called 'fair'. In the research it was proved that in the fair systems, the expected premium is non-increasing claim frequency function, in the stationary as well as in the non-stationary periods. The basic condition for a correct a posteriori risk assessment is non-positive change of the expected premium level while claim frequency is increasing. The aim of the research was to describe conditions determining the above theorem and to present formal prove. The fair systems defined in the article are commonly used in practice. Moreover, the actuarial literature assumes that Poisson distribution describes number of claims for the individual insured. Based on that, proved theorem is related to wide category of the system used on the market. .