Pricing a promise that ripens in 2060
The previous guide introduced the defined-benefit plan: the employer promises a 30-year-old worker a lifelong income starting at 65, and the *employer* — not the worker — owns the risk that it costs more than expected. That promise is the most extreme cash flow an actuary ever values. The first cheque might not be written for 35 years, the last not for 60, and the worker receiving it may still have grandchildren on the way. Everything you learned about valuing streams now meets its hardest test.
The good news is that you already own every tool. The pension is a *life annuity* — a stream of payments contingent on the retiree being alive — and it starts after a long wait, so it is a deferred annuity too. Valuing it is exactly the actuarial present value move from the life-contingencies rung: for each future year, multiply the payment by the probability the person survives to receive it, then discount it back to today. Sum that over every year the pension could possibly run, and you have one number: today's value of a lifetime of future cheques.
But here is the new twist that makes pensions their own subject. A 30-year-old has not *earned* the whole pension yet — she has earned only the slice corresponding to her years of service so far. By the time she retires she will have earned the rest. So the actuary's job is not just "what is the full pension worth?" but "how much of that promise has already been earned, and how much will be earned each year going forward?" Splitting the promise into the part already owed and the part still accruing is the whole game, and it gives us the two numbers in the title.
Accrued liability: the part already earned
Most defined-benefit plans pay a pension proportional to years worked. A common formula is *1.5% of final salary, per year of service*. Work 40 years and retire on 60% of your final pay; work 10 years and you have earned 15%. The pension you have earned so far is therefore your service-to-date times that accrual rate — this idea of building the benefit up brick by brick is exactly the accrual you met in the previous guide.
The accrued liability is the actuarial present value, today, of the pension *earned so far* — that slice of the benefit attributable to past service. It is the single most important number on a pension plan's books: it is what the plan would owe, in today's money, if everyone stopped earning new benefits this instant but the promises already made still had to be honoured. Notice the three ingredients baked into it: an interest rate to discount with, a mortality assumption to weight each future payment by survival, and a guess about future salaries (since the benefit is tied to *final* pay). Change any one and the liability moves.
Normal cost: the part earned this year
If the accrued liability is the value of the past, the normal cost is the value of *this single year*. Each year a worker stays on the job, she earns one more slice of pension — one more 1.5%-of-final-salary chunk. The normal cost is the actuarial present value, today, of that one extra year's worth of benefit. Think of it as the true *price* of one more year of the promise, computed before a single dollar of it will ever be paid out.
Why does this number matter so much? Because the normal cost is what the plan *ought to set aside this year* to stay on track. The whole logic of pension funding is to pay for each year's promise as it is earned, so the money has decades to grow with interest before the worker retires — the same time-value-of-money tailwind that makes saving early so powerful. Skip a year's normal cost and you have not made the promise cheaper; you have merely pushed an unfunded chunk into the future, where it will cost far more to catch up.
Accrued liability and normal cost are two views of the same promise, split by time. The accrued liability is the present value of all the slices *already* earned; the normal cost is the present value of the *next* slice. How you draw the line between "past slice" and "future slice" is precisely what an actuarial cost method decides — and there are several reasonable conventions. Some methods spread the cost as a level dollar amount over a career; others as a level percentage of pay. They agree on the *total* lifetime cost but disagree on its *timing*, which reshuffles how much shows up as accrued liability versus normal cost in any given year.
Funded ratio: can the plan keep its word?
So far we have only valued the *promise*. The other half of the story is the pile of money the plan has actually saved up: its assets — the contributions made over the years, plus the investment returns they have earned. The funded ratio is the simplest, most-quoted health check on any plan: plan assets divided by accrued liability. Above 100% (or 1.0) the plan holds more than enough today to cover the benefits earned to date; below it, there is a shortfall.
Accrued liability (PV of benefits earned to date) = 1,000 million
Plan assets on hand = 850 million
Funded ratio = 850 / 1,000 = 0.85 (85% funded)
Unfunded liability = 1,000 - 850 = 150 million shortfall
Normal cost this year (PV of benefits earned this year) = 28 million
-> sponsor should put in ~28m just to stand still,
PLUS extra to amortise the 150m shortfall over time.When assets fall short of the accrued liability, the gap has a name: the unfunded liability. It is not a sign of fraud or even of failure — it can open up simply because investments underperformed, people lived longer than the mortality table predicted, or interest rates fell (which, as you know, *raises* the present value of every distant payment). The sponsor's job is to feed the plan the normal cost each year *and* an extra amortisation payment that whittles the unfunded liability down over a set number of years, the way you would pay off a mortgage on top of your ongoing rent.
Why the number is so slippery
Here is the uncomfortable truth a pension actuary lives with: the accrued liability is breathtakingly sensitive to the discount rate. Because the cheques are paid so far in the future, a payment 40 years out is multiplied by a discount factor that shrinks dramatically when the rate moves even a little. A rough rule of thumb: dropping the discount rate by one percentage point can swell the liability of a mature plan by roughly 15–20%. The promise has not changed by one cent — only the rate we used to value it has.
This sensitivity is why the choice of discount rate is so fiercely debated. Should you discount at the return you *hope* your stocks will earn (which flatters the liability and makes the plan look cheap), or at a low, near-risk-free bond yield (which is more honest about a promise that *must* be kept regardless of how markets do)? Reasonable, expert people disagree, and the answer reshapes the funded ratio overnight. The number is real and necessary, but never mistake its three-decimal precision for accuracy.
What you carry forward
You can now read a pension plan's books in three numbers. The accrued liability is the actuarial present value of benefits earned so far — what the plan owes, in today's money, for the past. The normal cost is the value of one more year of benefit — the price of standing still. And the funded ratio, assets over accrued liability, tells you whether the money on hand matches the promise; the gap, when there is one, is the unfunded liability. Every one of these rests on the same actuarial present value engine you have carried up the whole ladder: probability of payment, times amount, discounted back.
One honest closing thought. Every figure in this guide assumed we *know* how long the retiree will live, when in fact we are only guessing from a mortality table. If a whole generation lives a few years longer than the table predicts, every pension runs longer, every liability was understated, and funded ratios quietly slip below where everyone thought they sat. That systematic, can't-pool-it-away danger is longevity risk — the single hardest thing about the entire promise, and exactly where the next guide takes you.