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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Stochastic processes

A Markov jump process models a policyholder in states H (healthy), S (sick) and D (dead). Transition rates per year are: H to S 0.10, H to D 0.02, S to H 0.40, S to D 0.08. What is the expected time spent in state H on each visit, and the probability that a visit to H ends in sickness rather than death?

The holding time in H is exponential with total exit rate 0.12, so the mean is 8.33 years. The chance the exit is to sickness is 0.10 divided by 0.12, which is 0.8333.

  1. AExpected 8.33 years; probability 0.8333Correct
  2. BExpected 10 years; probability 0.10
  3. CExpected 8.33 years; probability 0.10
  4. DExpected 12 years; probability 0.8333
  5. Expected 8.33 years; probability 0.1667

Explanation

Total exit rate from H is 0.10 + 0.02 = 0.12, so the holding time is exponential with mean 1/0.12 = 8.33 years. The probability of leaving to S is 0.10/0.12 = 0.8333. Using 0.10 alone as the exit rate gives 10 years, which ignores death.

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