Compound Interest
A complete guide to how compound interest actually works — the period-by-period mechanism, why the growth curve bends, where the Rule of 72 comes from, and what a steady average return quietly assumes away.
The mechanism: interest earning interest, one period at a time
Every explanation of compound interest eventually says some version of "interest on interest," but the phrase hides how mechanical the process actually is. Picture $1,000 sitting at a flat 12% a year, compounded once annually, with nothing added and nothing withdrawn. Year 1 closes at $1,120. Year 2 doesn't start over from the original $1,000 — it applies the same 12% to $1,120, the whole prior balance, and closes at $1,254. Year 3 does the same to that new number, closing at $1,405, then year 4 at $1,574, then year 5 at $1,762.
Notice what actually grew each year: not the rate — it stayed fixed at 12% the entire time — but the base the rate was applied to. Year 1 added $120 to the balance. Year 5 added $188, on the exact same 12% rate, purely because the base it was applied to had grown for four years first. That's the whole mechanism. There's no second force, no bonus multiplier, no acceleration hidden in the formula — just the same percentage, reapplied every period to a number that keeps getting larger because last period's interest is never withdrawn.
Why the curve bends: time is not a linear input
Run identical inputs — $5,000 to start, $200/month added, 7% a year — for three different lengths of time, and the results don't scale the way most people expect. Ten years produces $44,665. Doubling the horizon to twenty years doesn't double the balance to $89,331 — it reaches $124,379, 2.8× the ten-year figure. Stretch it to thirty years — three times the original horizon, not two — and the balance is $284,577, 6.4× the ten-year figure.
The reason is the same mechanism from the previous section, just accumulated over more periods: a dollar contributed in year one spends thirty years compounding by the time the clock runs out, while a dollar contributed in year twenty-nine spends one year compounding. A dollar's growing power depends entirely on how long it has left to compound — which is why "start early" isn't really advice about discipline, it's a direct description of how this specific curve behaves. Two savers who deposit the identical total on different schedules do not end up in the same place; the one whose dollars arrived earlier wins by construction, not by luck.
Nominal rate vs. APY: the same formula, worked all the way through
A "6% annual rate compounded monthly" doesn't mean 6% is applied once a year. It means 6% ÷ 12 = 0.50% is applied every single month, twelve separate times. Start with $10,000: month one's interest is $10,000 × 0.50% = $50, bringing the balance to $10,050. Month two applies that same 0.50% to $10,050, not back to the original $10,000 — the identical reapplication from the first section, just running on a monthly clock instead of an annual one.
Carry that forward through all twelve months and the account ends the year at $10,617 — not the $10,600 that a single flat 6% would have produced. The extra $17 exists because each of the twelve monthly interest payments started earning its own interest before the year was out. Expressed as one annual rate, that actual result is 6.17% — the APY. A nominal rate answers "what's the stated percentage"; an APY answers "what did the money actually grow by, every compounding effect included." The two are only the same number when compounding happens exactly once a year.
The Rule of 72: where it comes from, and how far you can trust it
Divide 72 by an annual rate and you get an estimate of how many years the money takes to double. Divide the natural log of 2 by the log of (1 + rate) and you get the exact answer — the Rule of 72 is a shortcut for that logarithm, close enough to do in your head because 72 divides cleanly by more small numbers than the mathematically pure constant it's approximating (roughly 69.3) ever would.
The shortcut isn't uniformly accurate — it's tuned to be almost exact around 8%, and drifts in opposite directions on either side of that point. At 3%, the rule estimates 24.00 years against an exact 23.45 years — it overshoots, guessing about 0.55 years too slow. At 8%, it estimates 9.00 years against an exact 9.01 years — a gap of roughly 2 days, effectively exact. At 20%, it estimates 3.60 years against an exact 3.80 years — now it undershoots, guessing about 0.20 years too fast. That sign flip is the tell: 72 was never meant to be uniformly accurate, only accurate exactly where most long-run stock-market assumptions happen to sit. Trust it as a fast sanity check anywhere near typical savings or diversified-portfolio rates, and reach for the exact math whenever the rate itself is unusually low or high.
What a "steady average return" quietly assumes away
Every projection on this site — like every compound interest calculator — holds the rate constant across every period. Real investment returns don't arrive that way; they arrive as a bumpy sequence that only averages out to the assumed rate over the long run, and the order of that sequence matters more than most people expect. Take $10,000 through two different two-year paths that both average 5% a year using a simple average. Path one: a flat 5% both years, compounding to $11,025. Path two: +20% the first year, then −10% the second — the same 5% simple average — but it compounds to $10,800, $225 short of the flat path, because the down year applied its loss to a larger base than the up year applied its gain to.
That's the honest limit of any compound-interest projection, including the calculator this guide funnels to: it's an arithmetic model of a constant rate, not a forecast of a real, variable one. Two accounts can share the identical long-run average return and still finish at different balances, purely because of the order the good and bad years arrived in. Fixed-rate products — savings accounts, CDs — don't carry this risk; a stated APY is what the account actually pays. Variable-return investments do, which is exactly why the rate typed into any projection should be read as a planning assumption, never a promise.