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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Key assurance and annuity contracts

For a whole life assurance of 1 payable at the end of the year of death, E[Z] = 0.40 and E[Z²] = 0.20 at a discount rate d = 0.05. What is the variance of the present value of the corresponding whole life annuity-due, ä_{K+1}?

The annuity present value is (1 - Z)/d, so its variance is Var(Z)/d². Var(Z) = 0.20 - 0.40² = 0.04, and dividing by 0.05² = 0.0025 gives 16.00. Using E[Z²] rather than the variance would give 80.

  1. A0.80
  2. B4.00
  3. C16.00Correct
  4. D80.00
  5. 0.04

Explanation

Since Y = (1 - Z)/d, Var(Y) = Var(Z)/d². Var(Z) = 0.20 - 0.16 = 0.04, and d² = 0.0025, so Var(Y) = 16.00. Dividing by d rather than d² gives 0.80, and using E[Z²] instead of the variance gives 80.

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