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In this paper we study a bonus malus system (bms) with deductibles. A bms is characterized by its premium levels and the transition rules among them. An insured is being moved among premium levels according to his/her claim record. Thus, an insured has to find an optimal strategy of submitting claims. Here optimal is in the sense of minimizing the total expected present value (epv) costs. Such strategies are found both for finite and infinite horizons. Furthermore, premium levels balancing the cost to the insured and the payoff of the insurer are given. The methods used to analyze the problem are from dynamic programming and Markov chains.