Two engines, finally bolted together
You arrive at this guide carrying two tools you have already sharpened. From the interest rung you can take any fixed stream of guaranteed payments and collapse it into one number — that was the annuity-certain, pure interest, no dice. From the survival rung you can read off the chance that a person now aged x is still alive at any future age — that was the survival function read out of a life table. A life annuity is what you get when you bolt these two engines together: a stream of payments that continues *only for as long as the annuitant is alive*, and stops the instant they die.
This is a genuinely new kind of promise, and it is worth pausing on why. An annuity-certain promises, say, twenty payments — and you will receive all twenty, dead or alive, because they fall to your estate either way. A life annuity promises a payment *this year only if you are here to collect it*. Nobody knows in advance how many payments will actually be made; that count is itself a random variable, governed by the future-lifetime distribution. So we cannot simply add up payments. We have to weight each future payment by the probability that it ever gets made.
The atom: a pure endowment is one survival-weighted payment
Here is the single idea the whole guide hangs on, and it is small. Picture just *one* future payment: $1 paid at the end of year k, but only if our person — aged x today — is still alive then. What is that worth today? Two things must happen for the dollar to arrive, so we multiply two factors. First, discount the dollar back k years for the time value of money: that is the same v-to-the-k discount factor you have used since the interest rung. Second, weight it by the probability the person actually survives those k years to collect it, which you read straight off the survival curve. Multiply discount by survival probability and you have the value of this single contingent dollar. That object has a name: a pure endowment.
Now the punchline, and it is the heart of valuing every life annuity: a life annuity is *nothing but a sum of pure endowments*, one for each future payment date. A whole life annuity is a $1 pure endowment due in 1 year, plus a $1 pure endowment due in 2 years, plus one due in 3 years, and so on for every year the person could conceivably live. Each term is its own little discount-times-survival product; the annuity's APV is simply their grand total. You already know how to value one pure endowment — so you already know how to value a life annuity. You just do it many times and add.
ONE pure endowment, $1 at the end of year k, life aged x:
value = (discount k years) x (prob. survive k years)
= v^k x k_p_x
WHOLE LIFE annuity = sum of pure endowments, one per future year.
Annuity-DUE (pay $1 at the START of each year you are alive):
adot_x = 1 + v*p_x + v^2*(2_p_x) + v^3*(3_p_x) + ...
^the first $1 is paid NOW, while certainly alive (prob 1)
Annuity-IMMEDIATE (pay $1 at the END of each year survived):
a_x = v*p_x + v^2*(2_p_x) + v^3*(3_p_x) + ...
The ONLY difference: adot_x has the leading 1, a_x does not:
adot_x = 1 + a_xDue vs immediate, with a life attached
The end-of-period versus start-of-period distinction you met for annuities-certain returns here, but with a twist worth savouring. The whole life annuity-due, written a-double-dot-x and read "a-double-dot sub x", pays $1 at the *start* of each year the annuitant is alive — beginning right now, at age x. That very first payment is special: it happens today, when the person is *certainly* alive, so it is not discounted and not weighted at all. It simply counts as 1. Every later payment is a proper pure endowment, discounted and survival-weighted. This is why life-insurance mathematics overwhelmingly prefers the annuity-due form: pensions, premiums, and benefits are nearly always reckoned at the start of each year.
The whole life annuity-immediate, written a-x, pays $1 at the *end* of each year survived, so it has no leading payment-today term. The relationship between the two is now even tidier than in the certain case. For annuities-certain we multiplied by (1 + i) to shift the whole stream one period. Here the two streams are identical *except* that the due version carries one extra payment — the certain $1 today. So a-double-dot-x simply equals 1 plus a-x. Get one and you have the other for the cost of adding or subtracting a single dollar.
Three flavours: whole life, temporary, deferred
With the atom in hand, the three standard life annuities are just three different choices of *which* pure endowments to include in the sum. The whole life annuity (a-double-dot-x) includes every year from now until the oldest age the table allows — it pays for life, however long that turns out to be. The temporary, or term, life annuity (written a-double-dot-x for n years) stops after at most n payments: you collect each year only if you are both alive *and* still within the first n years. It is the natural model for an income that runs, say, until age 65 and then ceases.
The third flavour is the deferred life annuity: it pays nothing for the first m years, then becomes an ordinary life annuity from age x+m onward. This is exactly the retirement product a 35-year-old buys — pay in now, draw a lifelong income starting at 65, but only if you live to reach it. Valuing it needs no new machinery: it is simply the whole-life annuity minus the temporary annuity for the deferral years, or equivalently a pure endowment to survive the deferral period multiplied by an annuity that starts then. And notice the three are not independent — a temporary annuity plus the matching deferred annuity reassembles the whole-life annuity, because together they cover every future year exactly once.
- Pick the payment dates the contract actually covers — every future year (whole life), only the first n (temporary), or only after year m (deferred).
- For each such date, write down one pure endowment: discount factor times the probability of surviving from age x to that date.
- Add them all up. That single total is the actuarial present value — the fair expected cost of the annuity today.
- If the payments are at the start of each year, you have the annuity-due; if at the end, drop the leading $1 to get the annuity-immediate.
A tiny worked feel for the numbers
Let's make it concrete with a deliberately tiny example so the mechanism shows through. Imagine an annuitant for whom only three more years are even possible, with survival probabilities (read from a life table) of 0.95 to the end of year 1, 0.88 to year 2, and 0.70 to year 3, and an interest rate of 5%. A whole life annuity-due pays $1 now (certain), then $1 at the start of each later year *if alive*. The first $1 is undiscounted and unweighted. The next is worth one year of discounting times the chance of living one year, and so on. Add them and you get roughly $1 + 0.905 + 0.798 + 0.605 — about $3.31, not the naive $4 a careless eye might expect from "four possible payments."
Two forces pull every term below $1, and the example shows both at once. Discounting alone would shrink a future dollar (the time value of money you already know); survival weighting shrinks it again, because the dollar might never be paid. Together they explain why a lifetime of $1 payments is worth only a few dollars today, and why that number falls as the annuitant gets older (less survival ahead) or interest rises (harder discounting). To turn this into a real product, an insurer applies the equivalence principle: the single premium it charges is set equal to the APV, so that on average, across the whole pool, premiums collected just cover annuity payments made.
Honest limits, and what comes next
Three honest caveats keep you from over-trusting these clean numbers. First, the survival probabilities are not facts — they come from a mortality table that is itself an estimate, built from past data and bound to drift as people live longer. An annuity insurer's nightmare is exactly that drift: if annuitants live longer than the table assumed, every APV was too low and the annuities were underpriced. Second, the single-rate assumption from the interest rung is still here; serious work discounts with a yield curve, not one frozen rate. Third, the APV is an *average* — it tells you the fair price for a large pool, never what any one person will actually receive.
Practitioners rarely re-sum hundreds of pure endowments by hand. Two shortcuts you will meet soon make the bookkeeping vanish. Recursion lets you build this year's annuity value from next year's with a single step, walking backward through the table. And actuaries precompute clever running totals called commutation functions so that a whole-life or temporary annuity becomes a quick ratio of two table look-ups. Both are pure efficiency — they change nothing about the meaning. The idea you must actually own is the one from the middle of this guide: a life annuity is a stack of pure endowments, each a future dollar shrunk twice, once for time and once for the chance of being alive to claim it.