Investing · Reference

Compound Interest, and the Two Things That Break It

The formula, the Rule of 72, and why the real-world result is always below the projection.

Compounding is earning a return on returns already earned. The formula is short and the consequences are not intuitive, which is why it is worth working through with numbers rather than adjectives.

The formula

Final = Principal × (1 + r)ⁿ

with r the rate per period and n the number of periods. $10,000 at 7% for 30 years is 10,000 × 1.07³⁰ = $76,123. The $66,123 of growth breaks down as $21,000 of simple interest on the original sum and $45,123 earned by the interest itself.

Why time dominates the rate

The exponent is time. That makes the last years of a long horizon worth far more than the first ones, because they compound on the largest balance.

$10,000 at 7%	Balance	Gained that decade
After 10 years	$19,672	$9,672
After 20 years	$38,697	$19,025
After 30 years	$76,123	$37,426

The third decade produces almost four times what the first did, on the same money at the same rate. Nothing changed except the size of the balance doing the work.

The Rule of 72

Divide 72 by the percentage rate to get the approximate doubling time. At 7%, about 10.3 years; at 9%, about 8. Accurate to within a few months for rates between roughly 4% and 12%, which makes it genuinely useful for mental arithmetic.

Run it in reverse and it exposes the cost of fees. A fund charging 1% a year against one charging 0.05% is giving up nearly a full percentage point of compounding — over 30 years at 7% versus 6.05%, the gap is around 22% of the ending balance.

The first thing that breaks it: volatility

Compounding assumes a fixed rate. Real returns vary, and variability does real damage that an average hides. Gain 50% then lose 50% and you have 75% of what you started with, not 100% — the loss applies to the bigger balance.

The result is that the return you actually compound at (the geometric mean) is always below the arithmetic average, and the gap grows with volatility. Two strategies with the same average annual return produce different ending wealth, and the steadier one wins. This is the honest argument for caring about volatility, and it is a better one than "risk feels bad".

The second thing: inflation

The formula compounds nominal money. At 3% inflation, that $76,123 after 30 years buys what about $31,400 buys today. Compounding works against you at exactly the same exponential rate, and any projection quoted without stating whether it is real or nominal is not telling you much.

Reading a projection

Anything showing a smooth exponential curve is showing the arithmetic, not an outcome. Three questions make it honest: is the rate real or nominal, is it net of fees, and what volatility is assumed. A 30-year projection at a fixed 10% net of nothing is a picture of the formula, not of a plausible future.

Reference material, not investment advice. Worked figures use exact compounding, rounded to the nearest dollar.