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

In a continuous-time multiple state model for a policyholder, the states are Healthy (H), Sick (S) and Dead (D). The policy pays a sickness benefit at a continuous rate while the life is in state S. Which expression gives the expected present value, at time 0, of the benefit of 1 per annum for a life currently in state H, using force of interest δ?

The expected present value is the integral of v^t times the probability that the life, starting in H, occupies state S at time t. Payments depend on occupancy of S, so the occupancy probability is needed, not transition intensities or probabilities for other states.

  1. AThe integral over t of v^t times the probability of being in state S at time t, given the life starts in HCorrect
  2. BThe integral over t of v^t times the probability of staying continuously in state H until time t
  3. CThe integral over t of v^t times the transition intensity from H to S
  4. DThe integral over t of v^t times the probability of being in state D at time t
  5. The sum over integer t of v^t times the probability of being in state S at time t, divided by δ

Explanation

A benefit paid continuously while in state S has an expected present value equal to the integral of v^t multiplied by the probability of occupying S at time t from the starting state H. Using the intensity instead gives a rate of transition, not occupancy. Using the staying-in-H probability values the wrong state.

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