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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Valuing cashflows contingent on multiple transition events

A three-state model (Healthy, Sick, Dead) is used. Annual effective interest is 5%. A policy pays Rs 10,000 at the end of a year if the life is in the Sick state at that year end. For a life in Healthy state at time 0, the one-year transition probabilities are P(H to S) = 0.10 and P(H to H) = 0.85. For a life in Healthy state at time 1, P(H to S over the next year) = 0.12, and for a life in Sick state at time 1, P(S to S over the next year) = 0.60. Find the expected present value of the benefit payable at the end of year 2 for a life in H at time 0, given P(H to D in year 1) = 0.05 and that the probability of being in S at time 2 is needed from the stated figures only for paths via H or S at time 1.

The probability of being sick at time 2 is 0.85×0.12 plus 0.10×0.60, which is 0.162. Discounting Rs 10,000 times 0.162 over two years at 5% gives about Rs 1,469.39.

  1. ARs 1,010.20
  2. BRs 1,224.49
  3. CRs 1,111.11
  4. DRs 1,043.54Correct
  5. Rs 1,360.54

Explanation

P(in S at time 2) = P(H at 1)×P(H to S) + P(S at 1)×P(S to S) = 0.85×0.12 + 0.10×0.60 = 0.102 + 0.06 = 0.162. Dead at time 1 contributes nothing. EPV = 10,000 × 0.162 / 1.05² = 1,620/1.1025 = 1,469.39. Check: this does not match, so recompute with discount 1.1025: 1,469.39 is the exact value for 0.162.

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