The promise we have to price
Two rungs ago you learned to move money across time; one rung ago you learned to describe how long a life lasts. The previous guide in this rung fused them into a single discipline — the actuarial present value — by asking: *what is a future payment worth today, when we are not even sure it will happen?* Now we point that machinery at the most important promise an insurer ever makes: life insurance, a sum of money paid out the moment a particular person dies.
What makes this hard, and beautiful, is that *two* things are uncertain at once. The benefit is a fixed amount — say $100,000 — so the size is known. But *when* it will be paid is not: it lands whenever death occurs, which could be next year or in sixty years. That single unknown — the future lifetime of the insured — drives everything. A payment in sixty years, discounted by the time value of money, is worth far less today than the same payment next year. So the value of the policy depends entirely on *which* future the person actually walks into.
So we cannot write down one number and be done. Instead we build the value the way an actuary always does: imagine *every* possible time of death, work out what the benefit would be worth today *in that scenario*, weight each scenario by how likely it is, and add them all up. That weighted average is the heart of insurance pricing — and giving it a name is the first thing we will do.
Whole life insurance and the symbol A_x
Start with the simplest, purest contract: whole life insurance, which pays the benefit whenever death occurs, no matter how far away — the payment is certain to happen *eventually*, only the timing is in doubt. Its actuarial present value gets the single most famous symbol in the subject: A_x, read "A-sub-x", the value today of a benefit of 1 paid on the death of a life now aged exactly *x*. Plug in any benefit you like by multiplying: a $100,000 policy is simply 100,000 × A_x.
Here is how A_x is actually built, in plain words. Slice the future into years. In each year there is a chance the person dies during that year — read straight off the life table and force of mortality you met last rung. If they die in year *k*, the insurer pays 1, but only at the end of that year, so that payment is worth a discount factor *v* raised to the *k*-th power today. Multiply that discounted benefit by the probability of dying in year *k*, do this for every year, and sum. A_x is exactly that sum: a probability-weighted bundle of discount factors.
A_x = sum over k of (chance of dying in year k) x (discount for k years)
Toy 3-year life, aged x, benefit = 1, interest i = 5% (v = 1/1.05)
die in year 1 prob 0.10 pay 1 at t=1 -> 0.10 x v^1 = 0.10 x 0.95238 = 0.09524
die in year 2 prob 0.15 pay 1 at t=2 -> 0.15 x v^2 = 0.15 x 0.90703 = 0.13605
die in year 3 prob 0.75 pay 1 at t=3 -> 0.75 x v^3 = 0.75 x 0.86384 = 0.64788
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A_x (this toy table) = 0.87917
So a 100,000 benefit is worth today: 100,000 x 0.87917 = 87,917Three more shapes: term, endowment, deferred
Whole life pays no matter when death comes. But most policies are choosier about *which* deaths they cover, and each choice gets its own symbol — all built from the same weighted-sum recipe, just with some years switched on or off. Term insurance, written A^1_(x:n), pays the benefit only if death falls inside a fixed window of *n* years; survive past the term and the contract quietly expires worth nothing. It is the cheapest, purest protection — and the limiting case as the term stretches to cover the whole of life is exactly A_x.
Endowment insurance, A_(x:n), is term insurance plus a survival reward: it pays 1 if you die within *n* years *and* pays 1 if you are still alive at the end. That second leg — a benefit paid only for living to a date — is a pure endowment, a savings-like building block in its own right. Because endowment insurance pays out either way, on death or on survival, it is much more expensive than term: you are buying protection *and* a guaranteed maturity payout. It behaves half like insurance and half like a savings plan, which is exactly why it was historically sold as both.
Last, deferred insurance, the m|x notation, flips the term window around: it pays *only* if death comes *after* an initial waiting period of *m* years, and covers nothing during the wait. Think of cover that begins ten years from now. The neat consequence is a clean piece of accounting: whole life splits exactly into term-for-the-first-m-years *plus* deferred-after-m-years. Cover everything, or cover only an early window, or cover only a later window — these three never overlap and always reassemble into A_x. That additivity is your sanity check and your shortcut all at once.
The value is a random variable, not just a number
Here is the idea that separates someone who has memorised A_x from someone who truly understands it. A_x is an *expected* value — an average over all the possible futures. But for any *one* policyholder, the present value of the benefit they actually trigger is a single draw from that distribution: a random variable, usually written Z. If they die early, the benefit is paid soon and discounted little, so Z is large. If they live long, the benefit is paid far off and discounted hard, so Z is small. A_x is just the *mean* of Z; it is not what happens to any individual.
Because Z varies, it has a spread — a variance — and for a single life that spread is enormous. The benefit is all-or-nothing in timing: paid soon and barely discounted, or paid late and deeply discounted, with little in between. A delightful shortcut computes this variance: the variance of Z equals "A_x evaluated at twice the force of interest" minus A_x squared. You do not need to follow the algebra; the point is that the spread is real, large, and computable from the same life table. A single life insurance policy is a genuine gamble for the insurer.
Notation, honest limits, and where this goes next
The symbols look forbidding at first — A_x, A^1_(x:n), the little raised 1 and the subscripts — but they are a precise, compact grammar agreed worldwide: the international actuarial notation. Each piece carries meaning. The capital A always means an insurance (a benefit paid on a status changing); the subscript *x* is the age now; an added *n* marks a time limit; a small 1 over a symbol says "this benefit pays on death," while a 1 placed elsewhere flags the survival leg. Once you can *read* the notation, every formula in this rung becomes a sentence rather than a wall of letters.
Be honest about two simplifications baked into everything above. First, we paid the benefit at the *end* of the year of death, which is convenient but slightly understates the true value, since a real claim is paid within days of death, not at year-end. The fully continuous version pays "the instant death occurs" and uses the force of interest to discount over the exact fractional time — a refinement, not a contradiction. Second, and far more important: every probability here came from a *mortality table*, an estimate of the future built from the past. If people live longer than the table assumes, A_x was too high or too low, and the price was wrong. A model is a careful map; it is never the territory.
You now hold the death-benefit half of life contingencies. But almost no one pays for insurance in one lump at signing — they pay premiums, a stream of payments that itself stops at death. That stream is the *life annuity*, and pricing it is the mirror image of what you just did. The next guide builds it, and then a single elegant bridge — the insurance-annuity relationship — ties A_x and the life annuity together so tightly that knowing one hands you the other for free.