whole life annuity-due function (ä_x)

An insurance pays once, at death; an annuity does the opposite — it pays a stream of money for as long as a person stays alive, and stops when they die. A whole life annuity-due pays 1 at the start of every year that the person is alive: 1 now, then 1 a year from now if still alive, then 1 the year after if still alive, and so on for the rest of their life. This is exactly the structure of a lifetime pension: regular income that you cannot outlive, but which also ceases the moment you die.

Its actuarial present value, written ä_x (the two dots mean the payments come at the start of each year — 'annuity-due'), is built by valuing each yearly payment separately and adding them up. The payment due now is worth its full 1; each later payment is discounted for interest and weighted by the probability of still being alive to receive it. In words, ä_x is the sum over each future year k of (discount factor to year k) times (probability of surviving to year k). Because a long life means many payments, ä_x is usually a sizeable number — for a 40-year-old at moderate interest it might be around 18, meaning the lifetime income stream is worth about 18 years' worth of payments in today's money.

The whole life annuity is the engine of pensions and retirement income, and it is inseparable from insurance through the identity A_x = 1 - d times ä_x (equivalently ä_x = (1 - A_x) / d). That link means every annuity value implies an insurance value and vice versa. One honest subtlety: ä_x assumes payments at the start of each year, while an annuity-immediate (written a_x, no dots) pays at the end of each year; they differ by exactly one payment-due-now, so ä_x = 1 + a_x. Mixing the two up is one of the most common errors in the subject.