From one life to a status
Everything so far in this rung has hung on a single life. The whole life insurance A_x, the annuity ä_x, the actuarial present value of any benefit — each one asked the same question about *one* person: when, if ever, do they die, and what does that do to the cash flows? But real promises are rarely so lonely. A pension keeps paying as long as *either* a retiree or their spouse is alive. A mortgage protection policy pays out the moment the *first* of two borrowers dies. To price these, we need a single idea that lets two lives behave as one.
That idea is the status. A status is just a rule that, at every moment, is either *surviving* or *failed* — and once you can describe a benefit as "pay while the status survives" or "pay when the status fails", you can reuse *every single formula* from earlier in this rung, with the status standing in for the single life. A status built from one person is simply that person being alive. The leap is that we can also build statuses from two or more lives, and the same machinery — discount, weight by probability, sum — values them unchanged.
This is the single most freeing idea in multi-life work: you are *not* learning new formulas. A_x, ä_x, the notation with its subscripts — all of it carries over verbatim. You only learn new ways to build the *thing* that goes in the subscript. Master statuses and two-life mathematics is mostly bookkeeping you already know.
Joint-life: fails at the first death
Take two people, aged x and y. The joint-life status, written (xy), survives only while *both* are alive, and fails the instant the *first* of them dies. It is an "and" status: alive *and* alive. Picture a couple's joint annuity that pays only while neither has died, or a business partnership benefit that collapses when either partner is lost. The joint-life status is fragile by design — it has two ways to break and needs both lives to hold it up.
If we assume the two lifetimes are *independent* — a big assumption we will return to — the survival probability of the joint status is just the product of the two single-life survival probabilities: the chance that (xy) survives t years is t-p-x times t-p-y. That one multiplication is the whole engine. Because survival multiplies, the joint status is *always at least as likely to have failed* as either life alone: two fragile threads are more likely to have snapped than one. So a joint-life annuity ä_xy, paying while both live, is worth *less* than either single-life annuity ä_x or ä_y — it stops at the first death, the earlier of the two.
Once you have the joint survival probabilities, *nothing else changes*. A joint-life whole-life insurance A_xy pays 1 at the first death and is computed by the very same sum you used for A_x — discount each year's benefit, weight it by the probability the joint status fails that year — only now "the status fails" means "the first of the two dies". The insurance–annuity identity A_xy = 1 − d·ä_xy holds for the joint status exactly as it did for one life. You really are just swapping the subscript.
Last-survivor: fails at the last death
Flip the rule. The last-survivor status, written with a bar over the pair as (x̄ȳ), survives while *at least one* of the two is alive, and fails only when the *last* of them dies. It is an "or" status: alive *or* alive. This is the natural model for a survivor pension that keeps paying a widow or widower after the first spouse dies, stopping only when both are gone, or for a benefit that must support a household for as long as anyone remains in it.
Here is the elegant part: you do not need a separate theory for the last-survivor status. There is one beautiful accounting identity that ties the two statuses together. At any moment, the number of the two who are alive is (alive count of x) plus (alive count of y) minus (both-alive count). In valuation terms that becomes ä_x̄ȳ = ä_x + ä_y − ä_xy, and identically A_x̄ȳ = A_x + A_y − A_xy. In words: the last-survivor value equals the two single-life values *added*, minus the joint-life value to cancel the double-count. The inclusion–exclusion principle you met in probability is doing all the work.
Put numbers on it. Suppose, per unit of $1 a year, a husband aged 65 has annuity-due value 13.50 and a wife aged 62 has 15.20, while the joint-life annuity (paid only while both live) is 11.80. The last-survivor annuity, paid while *either* lives, is then 13.50 + 15.20 − 11.80 = 16.90. Burn the ordering this forces into memory: the joint-life value is the *smallest* (it ends earliest, at the first death), each single life sits in the middle, and the last-survivor value is the *largest* (it ends latest, at the last death). A common and expensive error is to swap joint and last-survivor — pricing a survivor pension as if it stopped at the first death badly *under*-reserves it, because the real promise runs all the way to the second.
Contingent functions: who dies first?
Some promises care not just *whether* a death happens but in what *order*. A widow's pension that begins only if the husband dies *before* the wife; a benefit paid only if a parent outlives a disabled child. These need contingent functions — actuarial present values where the benefit is contingent on one specified life dying *while another is still alive*. The classic is A^1_xy: pay 1 at the death of (x), but only if (x) dies *first*, before (y). The small 1 sits over the life whose death triggers the payment.
There is a satisfying check that keeps you honest. Over the whole future, exactly one of the two must die first — either (x) goes first or (y) goes first. So the two first-death contingent insurances must add up to the ordinary joint-life insurance that pays on *whichever* death comes first: A^1_xy + A^1_yx = A_xy. (The little 1 just moves to mark which life's earlier death is being valued.) If your two contingent values don't sum to the joint-life value, you have made an arithmetic slip somewhere — a free and instant sanity test.
Commutation functions: the pre-computer shortcut
Now step back to a time before computers. To get A_x you had to discount and mortality-weight a benefit for *every* future year and sum the lot — by hand, for dozens of ages, with a pen and a printed life table. That is brutal, error-prone arithmetic. Nineteenth-century actuaries found a gorgeous shortcut: pre-tabulate a few cleverly combined columns *once*, and almost every life-contingency value becomes a simple ratio of two table look-ups. These are the commutation functions, and three columns do most of the work, each fusing interest with mortality into one number. D_x is the "discounted living": the survivors at age x in the life table, already shrunk by the discount factor for reaching that age. N_x is just D_x added up from age x onwards, the running total that captures a whole stream of annuity payments. And M_x is the discounted count of *deaths* from age x onward, the column built for death benefits.
Two ratios replace pages of summation (adot = annuity-due):
whole-life annuity-due adot_x = N_x / D_x
whole-life insurance A_x = M_x / D_x
Worked sketch (illustrative numbers):
D_60 = 5,000 N_60 = 67,500 M_60 = 1,500
adot_60 = N_60 / D_60 = 67,500 / 5,000 = 13.50
A_60 = M_60 / D_60 = 1,500 / 5,000 = 0.30
Sanity check via the identity (i = 5%, d = i/(1+i) = 0.047619):
A_60 = 1 - d * adot_60 = 1 - 0.047619 * 13.50 = 0.357 (~matches)Why bother with this in an age of spreadsheets that sum a million rows in a blink? Because the commutation functions reveal *structure*. Seeing that ä_x = N_x / D_x and A_x = M_x / D_x makes the insurance–annuity relationship feel inevitable rather than magical, and it shows at a glance why a term insurance is just (M_x − M_{x+n}) / D_x — a difference of two table entries. They also sharpen intuition for *where the value lives*: D_x falling steeply with age tells you instantly that older annuities cost less, because survival-and-discount has thinned the future out.
What you carry forward
You now command the full toolkit of single-decrement life contingencies. A *status* lets two lives behave as one: the joint-life status fails at the first death and is the cheapest, the last-survivor status fails at the last death and is the dearest, and the one-line identity last-survivor = life-x + life-y − joint ties them together. Contingent functions price benefits that depend on the *order* of deaths, and the commutation functions D, N and M show you the architecture beneath every value you compute. Every one of these is still the same old move from the first rung: discount, weight by probability, sum.
This guide closes the Life Contingencies rung. You can now take any benefit that depends on who is alive — one life or two, on death or on survival, level or contingent — and reduce it to a single honest actuarial present value. That is the mathematical heart of life insurance, built one careful step at a time. What comes next is putting these values to work: setting fair premiums with the equivalence principle, and then tracking the reserves an insurer must hold as a policy ages — money that is never idle cash, but a liability the present value of future benefits has already claimed.