3. 4. 3. In most of Europe and Asia annual automobile insurance is determined by the bonus malus system. Each policyholder is givena number (state) and the policy premium is determined as a function of this state. A policy holder's state changes from year to year in response to the number of claims made by the policy holder. Suppose the number of daims made by a particular policyholder in any given year, is a Poisson random variable with parameter A Let. s(k) be the next state of a policyholder who was in state i in the previous year and who made k claims in that year. The following table is a simplified Bonus-Malus system having 4 states. Next state if Annual State O claims 1 claim 2 claims 3 or more claims Premium 1 200 1 250 1 4. 4. 400 2 4. 4. 600 3. 4. 4 For A=2 A. Find the mean time to return to state 1 (if you start in state 1): (round to 2 decimals) (round to 2 B. Find the expected annual premium for a random person after it has been with the insurance company for a while: decimals)

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