A motor insurance company uses a no claim discount (NCD) system with four levels of discount listed as below: Level Discount (%)
0 0
1 25
2 30
3 40
A claims free year results in a transition to the next higher discount level or stays at the maximum discount level in the following year. On the other hand, claiming in a year results in moving down to the next lower discount level or stays at the zero-discount level in the following year. The probability of no claims in a year can be assumed to be equal to p in all years. (a) Write down the transition probability matrix P in terms of p, for the Markov chain with the given discount levels as the state space. (b) Determine the stationary distribution in terms of x, where x= p / 1−p ( 9 marks) (c) If the premium paid by a policyholder at level 0 is RM1,000, write down the average premium paid per policyholder per year in terms of x. (d) Assume further that a policyholder either makes no claims or only one claim in a year, and the average claim size is RM2,500, write down the average profit per policyholder per year that the insurance company makes in the long run. (e) Suppose that p=0.8. (i) Calculate the average premium paid per policyholder per year and the average profit per policyholder per year that the insurance company makes in the long run. (ii) Calculate the probability that a policyholder who starts at level 0 will be at the maximum NCD level after two years. (1 mark)