IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Markov chains
A no-claims-discount chain has states 0% and 20% discount. From 0% the policyholder moves to 20% with probability 0.8 (no claim), else stays. From 20% the policyholder stays with probability 0.9, else drops to 0%. Starting at 0%, what is the probability of being at 20% after two years?
The probability is 0.88. Sum the two paths: stay at 0% then move up (0.2 times 0.8 = 0.16) and move up then stay (0.8 times 0.9 = 0.72). Total 0.88, confirmed because the complement 0.12 matches the paths ending at 0%.
- A0.72
- B0.80
- C0.88Correct
- D0.90
- 0.96
Explanation
p(0->20, 2) = p00*p0,20 + p0,20*p20,20 = 0.2*0.8 + 0.8*0.9 = 0.16 + 0.72 = 0.88. Check: p(0->0,2)=0.2*0.2+0.8*0.1=0.12, and 0.88+0.12=1. The value 0.72 omits the path that stays at 0% first.
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