Compound interest, mechanically: what "interest on interest" really does
The compound interest formula, why compounding frequency matters less than people think, the rule of 72, and how the same mechanics work against you in fees.
Published
Simple interest pays you on what you put in. Compound interest pays you on what you put in plus everything it has already earned — and that one change in the rule turns a straight line into a curve that eventually goes nearly vertical. The mechanics are worth seeing precisely, because both intuition-failures about compounding (underestimating it over decades, overestimating what frequency does) come from not looking at the formula.
As elsewhere on this site: this is the arithmetic, not advice about what to do with it.
The formula and what each part does
A = P × (1 + r/n)^(n·t)P is the starting amount, r the annual rate, n how many times per year interest is credited, t the years. The heart of it is the exponent: growth multiplies, it does not add. €10,000 at 5% annually:
| Year | Simple (5% of P each year) | Compound |
|---|---|---|
| 1 | 10,500 | 10,500 |
| 10 | 15,000 | 16,289 |
| 25 | 22,500 | 33,864 |
| 40 | 30,000 | 70,400 |
For one year they are identical; the divergence is all in the repetition. By year 40 compound has produced more than four times the interest of simple. The lesson in the table is that compounding’s famous power is really time’s power — the curve is unimpressive early and relentless late, which is why every illustration of it involves decades.
Compounding frequency matters less than you would think
Monthly compounding beats annual, but by far less than intuition suggests. €10,000 at 5% for 10 years:
| Compounded | Result |
|---|---|
| Annually | €16,289 |
| Monthly | €16,470 |
| Daily | €16,487 |
| Continuously (the limit) | €16,487 |
The whole gap from annual to the theoretical maximum is about 1.2% of the final amount. Frequency converges fast: going from monthly to daily gains almost nothing. So when comparing two rates with different compounding, the honest method is to convert both to their effective annual rate — (1 + r/n)ⁿ − 1 — and compare those. 5% monthly-compounded is effectively 5.116% annual. This is exactly the trick lenders’ “nominal” rates exploit in reverse, and the compound interest calculator reports the effective rate for whatever combination you enter.
The rule of 72
A useful approximation: money doubles in roughly 72 ÷ rate years. At 6%, about 12 years; at 3%, about 24. (The mathematically exact constant is ln 2 ≈ 69.3, but 72 divides so conveniently by 2, 3, 4, 6, 8, 9 and 12 that the slightly-off version won.) It works well between about 2% and 15%, and its real use is sanity-checking: someone promising to double your money in 5 years is implicitly claiming ~14.4% a year, a number you can now evaluate for plausibility instantly.
Contributions change the shape
Most real saving is not a lump sum but a monthly amount, which makes the maths a sum of many small compoundings — each contribution grows for however long it has been in. Two mechanical consequences:
- Early money dominates. In a 30-year monthly plan, the first years’ contributions contribute disproportionately to the final total because they ride the longest part of the curve.
- Late in the game, growth outweighs deposits. There is a crossover point after which the balance grows more per year from returns than from contributions — the curve taking over from the saver.
The savings goal calculator runs this in reverse: given a target, a horizon and a rate, it solves for the monthly amount.
The same curve, pointed at you
Everything above is sign-agnostic — the mechanics do not care whether the compounding works for you or against you:
- Debt compounds identically. An unpaid balance at 20% annual grows on the same curve a 20% investment would, which is why interest-bearing debt left unattended grows faster than intuition expects.
- Recurring costs compound in reverse. A charge of 1% a year on a growing balance does not cost “1%” over 30 years — it removes money that would itself have compounded. On the €10,000/40-year/5% example above, a 1% annual drag reduces the outcome from €70,400 to about €47,900: nearly a third of the result, from a fee that sounds like a rounding error. The mechanics are neutral; the compounding always belongs to whoever owns the growing quantity.
That symmetry is the practical takeaway: whenever a percentage recurs on a balance — interest earned, interest owed, growth, fees, inflation — the exponent is in charge, and over long horizons the exponent beats the coefficient every time.