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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Means and variances of assurance and annuity payments

For a whole life assurance payable at the end of the year of death and a whole life annuity-due, both on a life aged x with annual effective interest rate i and discount rate d, which relationship holds?

The correct relationship is A_x = 1 − d·ä_x. The whole life annuity-due and the end-of-year-of-death assurance are linked through the discount rate d, because the annuity-due payments are made at the start of each year and d converts them to an equivalent assurance value.

  1. AA_x = 1 − d·ä_xCorrect
  2. BA_x = 1 − i·ä_x
  3. CA_x = v·ä_x − 1
  4. DA_x = 1 + d·ä_x
  5. A_x = d − ä_x

Explanation

The annuity-due pays 1 at the start of each year while alive. Equating cash flows gives 1 = d·ä_x + A_x, since each annual payment is worth 1 − v of interest-adjusted cost. So A_x = 1 − d·ä_x. Using i instead of d is the usual error and would apply to an immediate annuity.

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