Annuities

A pension pays a retired teacher £18\,000 every year for the rest of her life. A car loan is cleared by the same payment every month for four years. A lottery winner is offered "£10 million" — but paid in thirty yearly slices. Each of these is an annuity: a stream of equal payments spread over time.

The one question that unlocks all of them is: what is a stream of future payments worth today? Answer it and you can compare a lump sum against instalments, work out the payment that clears a loan, or price a pension. This single idea — equal payments over time — is the maths underneath mortgages, pensions, bonds and that lottery choice, and it all rests on one stubborn fact: money now is worth more than money later.

The time value of money

Would you rather have £100 today or £100 in a year? Today, obviously — you could bank it and, with compound interest at, say, 5\%, it would grow to £105. Turned around, a promise of £105 next year is worth only £100 today. To bring a future payment back to today we discount it — divide by (1 + i) for each period we wait:

\text{value today of } x \text{ paid after } k \text{ periods} = \frac{x}{(1 + i)^{k}} = x\,(1 + i)^{-k}.

An annuity's present value is just every future payment discounted back to today and added up. Because each term is the one before it multiplied by the same factor (1 + i)^{-1}, the sum is a geometric series, and it collapses to a tidy formula.

Suppose you pay x at the end of each period and the interest rate per period is i. For saving up, the first payment compounds for n - 1 periods, the next for n - 2, and so on. Adding the grown-up payments is a geometric series with common ratio 1 + i, and its sum is the future value F:

F = x\,\frac{(1 + i)^n - 1}{i}.

For paying off, we discount instead of compound, and get the present value P — the lump sum today that a loan of P can be repaid by n equal instalments of x:

P = x\,\frac{1 - (1 + i)^{-n}}{i}. For n equal payments of x at interest rate i per period:

Worked example 1 — the present value, term by term

You are promised £1000 at the end of each year for 3 years, with interest at i = 10\% = 0.10. What is that stream worth today?

The slow, honest way — discount each payment and add:

\frac{1000}{1.10} + \frac{1000}{1.10^{2}} + \frac{1000}{1.10^{3}} = 909.09 + 826.45 + 751.31 = 2486.85.

The fast way — the formula gives the identical answer:

P = 1000\,\frac{1 - 1.10^{-3}}{0.10} = 1000 \times 2.48685 = £2486.85.

Look hard at that result. Three payments of £1000 are worth £2486.85 today — not £3000. The missing £513.15 is the time value of money: the later pounds are worth less because you have to wait for them.

Worked example 2 — the payment that clears a loan

You borrow £10\,000 and repay it in 5 equal year-end payments at i = 8\% = 0.08. What is each payment?

Now the loan is the present value, and we solve the present-value formula for x:

x = \frac{P\,i}{1 - (1 + i)^{-n}} = \frac{10\,000 \times 0.08}{1 - 1.08^{-5}} = \frac{800}{0.319417} = £2504.56.

So five payments of £2504.56 clear the debt. Add them up: 5 \times 2504.56 = £12\,522.82. You borrowed £10\,000 and paid back £12\,522.82 — the extra £2522.82 is the interest, the price of borrowing. This is exactly how a mortgage payment is worked out.

Worked example 3 — lump sum or instalments?

A prize is advertised as "£10 million", paid as £333\,333 a year for 30 years. A bank offers you a single lump sum now instead. With interest at i = 5\%, how much is the instalment stream really worth today?

P = 333\,333\,\frac{1 - 1.05^{-30}}{0.05} = 333\,333 \times 15.3725 \approx £5.12\text{ million}.

The "£10 million" prize is worth only about £5.12 million today — roughly half the headline. So if the bank offered you a lump sum of, say, £6 million now, you should take it. Never compare a lump sum with instalments until you have discounted the instalments to today.

Saving, payment by payment

Below, a payment of 100 is made each period. The curve is the future value F after n payments; the straight line is the total cash actually paid in (100n). The gap between them is the interest the annuity has earned — and it widens fast as the rate climbs. Drag the rate slider and watch the curve bend away from the line.

The number-one annuity blunder is to just add up the future payments. £1000 a year for 10 years is not £10\,000 today — because future pounds are worth less than present ones, so each payment must be discounted by the interest rate before you sum. At 8\% that £10\,000 of payments is really worth only about £6710 today.

This is the exact error that wrecks "lump sum vs instalments" decisions. Ignoring the time value of money makes distant payments look far more valuable than they are — which is precisely why lotteries love to quote the undiscounted total. And one more trap: keep the rate and the period matched. Monthly payments need a monthly rate — an annual 12\% becomes i = 0.01 per month, with n counted in months.

A pension is a giant annuity: a promise to pay you x every year for decades. Its present value is P = x\,\frac{1 - (1 + i)^{-n}}{i}, and that divide-by-i makes it hugely sensitive to the interest rate. When rates fall, the discounting is gentler, so the same future payments are worth much more today — the fund suddenly owes far more than it thought, and headlines scream about "pension deficits". When rates rise, the debt shrinks. Nothing about the promised payments changed; only i did.

The same geometric-series maths values a bond (a stream of coupon payments plus a final lump), a mortgage, and an insurance payout. The humble idea of "equal payments over time" quietly underpins nearly the whole financial world — which is why the formula on this page is worth more than it looks.