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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Transition intensities dependent on age (exact or census)

A three-state health model (Healthy, Sick, Dead) is observed. Total time Healthy is 4,000 years, with 200 H to S transitions and 40 H to D transitions. Total time Sick is 500 years, with 100 S to H and 50 S to D transitions. Using MLEs, what is the estimated probability that a Healthy life, at the moment of leaving Healthy, goes to Sick, and the estimated expected time spent in Healthy per visit?

The intensities are 0.05 to Sick and 0.01 to Dead, so total exit intensity is 0.06. The chance the exit is to Sick is 0.05/0.06 = 5/6, and the expected holding time is 1/0.06 ≈ 16.7 years, which includes both exit routes.

  1. AProbability 5/6, mean time 16.7 yearsCorrect
  2. BProbability 5/6, mean time 20 years
  3. CProbability 0.05, mean time 16.7 years
  4. DProbability 1/6, mean time 16.7 years
  5. Probability 5/6, mean time 0.06 years

Explanation

mu_HS = 200/4000 = 0.05 and mu_HD = 40/4000 = 0.01. Total exit intensity is 0.06. Probability of going to Sick is 0.05/0.06 = 5/6. Mean holding time is 1/0.06 = 16.7 years. Using 1/0.05 gives 20 years, ignoring the death exit.

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