The so-called bonus-malus premium calculation principle is widely spread in car insurance practice. There is a...

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The so-called bonus-malus premium calculation principle is widely spread in car insurance practice. There is a finite number of classes (tariff groups), and the premium a policy holder pays depends on what class he/she belongs to. Each year, a policy holder's class improves depending upon there were no claims during past year (a bonus: transfer to a lower class) or it deteriorates in the event of claims (a malus: transfer to a higher class). The table below shows the premium scale (as a percentage of the premium paid by class 13 drivers) for the German bonus-malus system: Class 18 17 16 15 14 13. 12 11 10 ܚ ܩ ܣ ܙ ܟ ܗ ܟ ܚ ܢ ܝ 7 2 Premium % ClassAfter One Year (# claims) 24 0 1 2 3 13 18 18 18 18 13 18 18 18 18 13 17 18 18 18 13 16 17 18 18 13 15 17 18 18 14 16 17 18 13 14 16 18 10 13 14 16 18 9 12 13 14 18 8 11 13 14 18 7 11 13. 14 18. 6 11 13 14 18 11 13 14 18. 10 12 13. 18 9 11 13 18 8 200 200 175 175 125 100 85 70 65 60 55 50 45 40 40 40 40 40 12 11 10 5 +3 ~ - 4 2 1 7 1 7 11 13 18 11 13 18 11 13 18 1 Assuming that for any policy holder the collection of random variables "number of claims per year" is id, with Poisson distribution and parameter λ = 1.6, construct the transition probability matrix for the Markov chain associated with a policy holder's class. Classify the states of the chain. If the initial condition for a certain individual is that of being in Class 11, find the probability distribution of class to which he/she belongs after 10 years. Assuming the premium paid by class 13 drivers is $400, find the the long-run average premium charged to German drivers. The so-called bonus-malus premium calculation principle is widely spread in car insurance practice. There is a finite number of classes (tariff groups), and the premium a policy holder pays depends on what class he/she belongs to. Each year, a policy holder's class improves depending upon there were no claims during past year (a bonus: transfer to a lower class) or it deteriorates in the event of claims (a malus: transfer to a higher class). The table below shows the premium scale (as a percentage of the premium paid by class 13 drivers) for the German bonus-malus system: Class 18 17 16 15 14 13. 12 11 10 ܚ ܩ ܣ ܙ ܟ ܗ ܟ ܚ ܢ ܝ 7 2 Premium % ClassAfter One Year (# claims) 24 0 1 2 3 13 18 18 18 18 13 18 18 18 18 13 17 18 18 18 13 16 17 18 18 13 15 17 18 18 14 16 17 18 13 14 16 18 10 13 14 16 18 9 12 13 14 18 8 11 13 14 18 7 11 13. 14 18. 6 11 13 14 18 11 13 14 18. 10 12 13. 18 9 11 13 18 8 200 200 175 175 125 100 85 70 65 60 55 50 45 40 40 40 40 40 12 11 10 5 +3 ~ - 4 2 1 7 1 7 11 13 18 11 13 18 11 13 18 1 Assuming that for any policy holder the collection of random variables "number of claims per year" is id, with Poisson distribution and parameter λ = 1.6, construct the transition probability matrix for the Markov chain associated with a policy holder's class. Classify the states of the chain. If the initial condition for a certain individual is that of being in Class 11, find the probability distribution of class to which he/she belongs after 10 years. Assuming the premium paid by class 13 drivers is $400, find the the long-run average premium charged to German drivers.

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