The subject of the paper are basic properties of bonus-malus system fair by the transition rules between classes (BMSF(TR)), of which definition excludes unrealistic bonus-malus systems. The paper presents an ergodic Markov chain which is a BMSF(TR) model and which allows to analyze the properties of expected value of insurance premium according to the features characterizing an insured and a system i.e. claims frequency, class in the initial year, insurance duration and maximum number of claims acknowledged in the system.
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