joint-life and last-survivor statuses

So far every benefit depended on one person. But many real promises hinge on two (or more) lives at once — a couple's pension, a survivor benefit, a family policy. There are two opposite ways two lives can define a 'status' that is alive or dead. A joint-life status survives only while both people are still alive; it fails the moment the first of them dies. A last-survivor status survives as long as at least one of them is alive; it fails only when the last of them dies. A joint annuity that stops when the first partner dies uses the joint-life status; a survivor annuity that keeps paying until both are gone uses the last-survivor status.

Once you can compute the probability that a status is still 'alive', all the earlier machinery — insurances, annuities, pure endowments — carries straight over, just with the status replacing the single life. The two statuses are linked by a simple and very useful identity: the value (or probability) for the joint life plus the value for the last survivor equals the sum of the two single-life values. In words, joint plus last-survivor equals individual one plus individual two. So if you know each person's annuity and the joint-life annuity, the last-survivor annuity is just ä_x + ä_y - ä_{xy}. For example, with two single-life annuities worth 12 and 14 and a joint-life annuity worth 10, the last-survivor annuity is 12 + 14 - 10 = 16 — it pays longest, so it costs most.

These statuses are essential for pensions and couple products, where survivor benefits are the norm. The big honest caveat is independence: the clean formulas assume the two lives die independently of each other, but in reality couples' lifetimes are correlated — shared environment, lifestyle, and the well-documented 'broken-heart' effect where one partner's death raises the other's mortality. Treating correlated lives as independent biases the valuation, usually making joint annuities look cheaper and last-survivor benefits look dearer than they truly are.