How compound interest works

Interest is what money costs over time. Simple interest applies that cost to the original amount and nothing else, so it adds the same figure every year forever. Compound interest applies it to the balance, and the balance already contains the interest from every earlier period, so the figure it adds keeps growing.

A couple of years at 6 percent shows the whole mechanism. In year one the two are identical, because the balance and the original amount are still the same number. In year two the compound version charges 6 percent on 1.06 rather than on 1.00, so each dollar earns 0.0636 instead of 0.0600. That extra 0.0036 is interest paid on interest, and everything else on this page is that single step repeated.

The gap in the last column starts at nothing and widens faster each year. Over five years it is small enough to ignore. Over forty it is most of the balance.

Written as a formula, an amount $P$ compounded $n$ times a year at annual rate $r$ for $t$ years reaches:

$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$

The rate goes in as a decimal, so 6 percent is 0.06. The pair that trips people up is $r/n$ and $nt$: the first is the rate for one period, the second is how many periods there are. Six percent compounded monthly is 0.5 percent applied 12 times, not 6 percent applied once.

A compounding balance does not grow evenly. Each year multiplies the entire balance, including every part of it produced by earlier years, so the amount added rises while the rate stays fixed. Most of the growth in a long run arrives at the end of it rather than spread through the middle.

At 6 percent a year, with nothing paid in after the start:

The fourth decade adds 4.54 per dollar, more than the first two decades put together, at the same rate and with no new money. Nothing changes over that run except the number of periods that have already passed.

This is also why delay is expensive in a way that feels out of proportion. Starting five years later does not cost five years of small early growth. It removes five years of multiplying from the far end of the sequence, so a single amount left alone ends up divided by $1.06^5$, a cut of about 25 percent. A ten year delay cuts it by about 44 percent.

Those two figures are for one amount left untouched. Money paid in monthly against a fixed end date loses more, because a late start drops deposits as well as growth: over a forty year horizon at 6 percent, beginning five years late costs about 28 percent rather than 25. What a delay does not do is put the result out of reach. Paying in about 34 percent more, at the same 6 percent, exactly undoes a five year late start on a single amount. The cost is real and it is quantifiable, which is different from being unrecoverable.

What is unrecoverable is the choice itself. Of the three inputs, the amount paid in, the rate and the number of years, only the years cannot be revisited later. That is a point about what is under your control at what moment, not a claim that years always outweigh the other two. The savings goal calculator shows the same effect from the other end: the monthly amount needed to reach a target falls sharply as the horizon lengthens.

The rule of 72 turns a rate into a doubling time without a calculator. Divide 72 by the annual rate written as a percent, and the answer is roughly how many years a balance takes to double. At 6 percent that is 12 years. At 9 percent it is 8 years.

It works because the true doubling time is $\ln 2 / \ln(1 + r)$, and for small rates $\ln(1 + r)$ is close to $r$ itself, which leaves roughly $0.693 / r$. In percentage points that is 69.3 divided by the rate. The number 72 is used instead because it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, and because the error that swap introduces happens to cancel the approximation's own error at around 8 percent.

The rule is exact at about 8 percent and drifts either side of it: it runs about a year long at 2 percent, six weeks short at 12 percent, and two and a half months short at 20 percent. For anything in the range a saver actually meets it is close enough to do in your head, which is the only reason to use it at all.

Two ways to run it backwards. Divide 72 by the number of years you have, and you get the rate you would need to double in that time. And point it at prices rather than balances: at 3 percent inflation, the price level doubles in about 24 years, which is the same arithmetic aimed at what your money buys.

Two accounts can quote the same 6 percent and pay different amounts, because the rate says nothing about how often interest is added. Every time interest is credited it starts earning, so more frequent crediting means a slightly higher yearly return. The effect is real and it is smaller than most people expect.

Going from yearly to monthly adds 0.168 percentage points. Going from monthly all the way to continuous adds 0.016, about a tenth as much. There is a hard ceiling on this: as $n$ rises, $(1 + r/n)^n$ approaches $e^r$, so nothing at a 6 percent nominal rate can ever pay more than 6.1837 percent a year, however the compounding is arranged.

Measured against that ceiling, a quarter of a percentage point on the rate itself is worth more than every possible improvement in frequency combined. A plain 6.25 percent credited once a year beats 6 percent credited continuously. Compare rates first, then read the schedule.

Better still, compare yields. A yearly yield already contains the compounding, which is what makes two products comparable in a single number, and that translation is the whole job of the APR against APY calculator. In the United States, deposit accounts advertise an annual percentage yield, so a quoted APY should be treated as an annual figure rather than compounded a second time. The same idea carried on a nominal rate is called the effective annual rate.

Here is the comparison the whole idea rests on, worked out in full below.

Ella pays $200 at the end of every month from 25 to 35, then stops and never adds another cent. Liam pays the same $200 a month from 35 all the way to 65. Both earn 6 percent compounded monthly, and both are counted at 65.

Ella pays in $24,000. Liam pays in $72,000, three times as much over three times as long. At 65 Ella has $197,395.14 and Liam has $200,903.01. Tripling the money and tripling the years buys Liam a lead of under 2 percent.

Ella's ten years of deposits ended at $32,775.87. She then did nothing at all for 30 years while that balance multiplied by just over six, and $164,619.27 of her final total arrived after her last deposit. Across the whole run she paid in $24,000, so everything above that figure is interest.

Be honest about what is driving that result: it turns on the rate. These two schedules cross at about 6.1 percent. Below that, Liam's larger deposits win. Above it, Ella's extra 30 years of compounding wins, and at 7 percent she finishes roughly 15 percent ahead of him. The claim that starting early always beats saving more is not pure arithmetic, it is arithmetic plus an assumption about returns.

Two further things the arithmetic assumes. First, that 6 percent arrives as a fixed, certain rate. A deposit rate can be fixed for a term; an investment return cannot. Returns arrive as a sequence, and a sequence whose yearly figures average 6 percent compounds to less than 6 percent whenever those figures move around, so an average return is not the same object as a compound one. Second, every figure here is nominal. At 3 percent inflation, what $197,395.14 buys in 40 years is around a third of what the same sum buys today.

What survives all of that is still worth the space. Ella got within 2 percent of Liam on a third of the money and made no decisions at all after 35, her position improves with every extra year of horizon and every extra point of return, and nobody can add years to the front of their own run later on. Time in the account is the one input that is only available now.

Compounding is not a savings feature. It is what happens whenever a rate is applied to a balance that already includes what the rate produced last time, so it runs on the other side of a balance sheet just as reliably.

On debt, unpaid interest is added to what you owe and starts being charged interest immediately. A card quoting 18 percent as a nominal annual rate compounded daily costs about 19.7 percent over a year if nothing is repaid, and one quoting 22 percent costs about 24.6 percent. Whether a headline figure works that way depends on local disclosure rules rather than on arithmetic: a United States card APR is a nominal rate, so the effective cost sits above the number advertised, while a United Kingdom or European Union card APR is already an effective annual figure and should not be compounded a second time. Read which one you are holding before applying either sum. This is why a minimum payment can leave a balance looking almost unchanged: the payment covers that period's interest and little else, so the principal generating next month's interest barely moves.

An amortising loan is the same mechanism run deliberately, with a payment set high enough to shrink the balance every period. The loan payment calculator shows how the split between interest and principal shifts as that balance falls, which is what switches the compounding off one period at a time.

Costs compound too. A fee of 1 percent a year does not cost 1 percent of the final balance. It comes off the rate being multiplied, so a 6 percent gross return becomes about 5 percent net, and over 40 years the balance ends close to a third below where it would otherwise have been. Charging that 1 percent against the balance each year rather than against the return gives 33 percent over the same span, so the answer is near a third either way. That is a statement about what a repeating percentage costs once it is compounded, not a claim that a paid service is never worth its fee. What it changes is the comparison: a fee has to be weighed against what it delivers after costs, over the same horizon, rather than against nothing.

Tax behaves the same way wherever it applies. If interest is taxed in the year it is credited, the balance compounds on the after tax amount, which lowers the rate being multiplied in exactly the way a fee does. How large that effect is, and whether it applies at all, depends on the account type and the country, so it belongs in the calculation as a rate adjustment made against local rules rather than as a fixed haircut.

Inflation is the same shape aimed at what money buys. Prices compound, so a return only means something after they are subtracted, which is what the real return calculator works out.

In calculus this is not a special financial rule at all. A balance growing at a fixed proportional rate is the standard exponential growth model, the same one used for populations and radioactive decay: exponential growth.