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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Projecting and valuing cashflows contingent on multiple decrement events

For a scheme member, the dependent probabilities of leaving service by any cause are 0.08 in year 1 and 0.10 in year 2. In year 3, the dependent probability of withdrawal is 0.06 for those active at the start of year 3. What is the probability that a new member remains active for two years and then withdraws during year 3?

The probability is 4.97%. A new member stays active through year 1 with probability 0.92 and through year 2 with probability 0.90, giving 0.828, then withdraws in year 3 with probability 0.06, so 0.828 × 0.06 = 0.04968.

  1. A4.97%Correct
  2. B5.52%
  3. C5.30%
  4. D6.00%
  5. 82.80%

Explanation

The probability of being active after two years is (1 − 0.08)(1 − 0.10) = 0.92 × 0.90 = 0.828. Multiply by the year 3 withdrawal rate: 0.828 × 0.06 = 0.04968, or 4.97%. Using only 0.92 × 0.06 = 5.52% ignores the year 2 exits.

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