IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains
A No Claims Discount scheme has states 0, 1 and 2 (increasing discount). The probability of a claim-free year is 0.8 each year. After a claim-free year the policyholder moves up one state (staying at 2 if already there); after a claim the policyholder moves down one state (staying at 0 if already there). What is the long-run proportion of policyholders in state 2?
The long-run proportion in the top state is 16/21, about 0.762. Balance between adjacent states gives stationary weights in the ratio 1 : 4 : 16 for states 0, 1 and 2, since moving up is four times as likely as moving down, and these sum to 21.
- A0.640
- B0.800
- C0.941
- D0.762Correct
- 0.500
Explanation
The chain is a birth-death chain with up probability 0.8 and down probability 0.2. The stationary distribution satisfies pi1 = 4*pi0 and pi2 = 4*pi1 = 16*pi0. The sum is 21*pi0 = 1, so pi0 = 1/21 and pi2 = 16/21 = 0.762. The value 0.941 would result from wrongly using ratio 4 with only two states in the denominator.
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