Life Insurance and Annuity Contracts
This domain tests the present value random variables for life insurance and annuity contracts, and the relationships that let you switch between them.
Core contracts
- A whole-life assurance pays on death whenever it occurs. With a unit sum assured payable at the end of the year of death, A_x = sum_{k>=0} v^{k+1} k p_x q_{x+k}.
- A term assurance pays only if death occurs within the term; an endowment assurance pays on death within the term or on survival to the end of the term.
- A whole-life annuity-due of 1 per year pays at the start of each year while alive: a-due_x = sum_{k>=0} v^k k p_x.
- The annuity-immediate a_x = a-due_x - 1 for annual payments. For continuous contracts, a-bar_x = integral_0^infinity v^t t p_x dt.
- The key identity is A_x = 1 - d a-due_x, where d = i/(1+i) = 1 - v. Hence a-due_x = (1 - A_x)/d.
Worked example
A whole-life assurance pays 100,000 at the end of the year of death. At i = 4%, A_x = 0.25. Then d = 0.04/1.04 = 0.0384615. The annual annuity-due is a-due_x = (1 - 0.25)/0.0384615 = 19.5. The expected present value of the assurance is 100,000 x 0.25 = 25,000. If the same life buys a whole-life annuity-due of 1,000 per year, its EPV is 1,000 x 19.5 = 19,500. These figures are consistent because the annuity and assurance are linked by the identity above.
Traps that cost marks
Check whether death benefits are payable immediately on death or at the end of the year. Check whether annuities are due, immediate or continuous. Endowment assurance includes a survival benefit; term assurance does not. In the identity A_x = 1 - d a-due_x, d is not i and not v.