Compound interest, mechanically: what "interest on interest" really does

Simple interest pays you on what you put in. Compound interest pays you on what you put in plus everything it’s already earned, and that one change to the rule turns a straight line into a curve that eventually goes nearly vertical.

Both of the usual intuition failures about compounding, underestimating it over decades and overestimating what frequency does, come from never looking at the formula. So here it is.

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 a year interest is credited, t the years. The heart of it is the exponent, because growth multiplies instead of adding. €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’re identical. All the divergence comes from repetition. By year 40 compound has produced more than four times the interest that simple did.

The lesson in that table is that compounding’s famous power is really time’s power. The curve is unimpressive early and relentless late, which is why every chart of it spans decades.

Compounding frequency matters less than you’d 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 entire gap from annual to the theoretical maximum is about 1.2% of the final amount, and frequency converges fast: going from monthly to daily gains you almost nothing.

So when you’re comparing two rates with different compounding, 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 type in.

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 holds up well between about 2% and 15%, and its real use is sanity-checking. Somebody promising to double your money in 5 years is implicitly claiming around 14.4% a year, a number you can now weigh for plausibility on the spot.

Contributions change the shape

Most real saving isn’t a lump sum but a monthly amount, which makes the maths a sum of many small compoundings, each contribution growing for however long it’s been in. Two mechanical consequences:

• Early money dominates. In a 30-year monthly plan the first years’ contributions matter disproportionately to the final total, because they ride the longest part of the curve.
• Late on, growth outweighs deposits. There’s a crossover point after which the balance grows more per year from returns than from contributions. That’s the curve taking over from the saver.

The savings goal calculator runs this backwards: give it a target, a horizon and a rate, and it solves for the monthly amount.

The same curve, pointed at you

Everything above is sign-agnostic. The mechanics don’t care whether the compounding works for you or against you.

Debt compounds identically. An unpaid balance at 20% annual grows on exactly the curve a 20% investment would. That’s why unattended debt outruns what people expect of it.

Recurring costs compound in reverse. A charge of 1% a year on a growing balance doesn’t cost you “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 brings the outcome down 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 thing to take away. 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.