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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Assurance and annuity functions involving two lives

Two lives (x) and (y) have independent future lifetimes. Which relationship between the joint-life annuity a_xy, the last-survivor annuity a_x̄ȳ (both payable annually in advance) and the single-life annuities is correct?

The last-survivor annuity equals the sum of the two single-life annuities minus the joint-life annuity. Adding both single-life annuities double counts payments while both lives survive, so the joint-life annuity must be deducted once to correct this.

  1. Aa_x̄ȳ = a_x + a_y − a_xyCorrect
  2. Ba_x̄ȳ = a_x + a_y + a_xy
  3. Ca_x̄ȳ = a_x × a_y / a_xy
  4. Da_x̄ȳ = a_x + a_y − 2·a_xy
  5. a_x̄ȳ = a_xy − a_x − a_y

Explanation

Payments under the last-survivor status are made while at least one life survives. Summing the two single-life annuities counts the period when both are alive twice, so the joint-life annuity is subtracted once. The other options either add it or subtract it twice, which gives wrong payment counts.

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