Compound interest explained: formula, examples and the rule of 72

With simple interest you earn a return only on the money you put in. With compound interest each period's return is added to the balance, and the next period's return is calculated on that larger amount. Interest starts earning interest — and because the base keeps growing, the curve bends upward over time.

The same mechanism applies to reinvested dividends, to an index fund whose gains are left invested and to any savings product that rolls over its returns. Time is the most powerful ingredient: the longer money compounds, the more of the final balance comes from returns on returns rather than from what you paid in.

For a lump sum P growing at an annual rate r, compounded n times a year for t years:

A = P × (1 + r ÷ n)^(n × t)
Continuous compounding: A = P × e^(r × t)
Rule of 72: years to double ≈ 72 ÷ rate (%)

More frequent compounding raises the effective annual rate: 12% a year paid monthly is (1 + 0.12 ÷ 12)^12 − 1 ≈ 12.68% effective. The rule of 72 is a mental shortcut: at 6% money doubles in about 12 years, at 9% in about 8.

Invest $10,000 at 8% a year, compounded annually. After 10 years: 10,000 × 1.08^10 ≈ $21,589. With simple interest you would have $18,000 — the extra $3,589 is interest on interest. After 30 years the compounded balance reaches about $100,600, more than ten times the original sum, while simple interest would have produced only $34,000.

Now subtract a 1% annual fee, so the investment compounds at 7%: after 30 years it is worth about $76,100 instead of $100,600 — roughly a quarter less, from a fee that sounds small. Try your own numbers in the compound interest calculator.

A compound return in nominal terms can hide a much smaller gain in purchasing power. The real return is (1 + nominal return) ÷ (1 + inflation) − 1: a 10% return with 7% inflation is only about 2.8% in real terms.

Common mistakes: assuming a high past return will keep compounding for decades; ignoring taxes and fees, which compound too; and confusing a stated annual rate with the effective rate after compounding. Compounding also works in reverse on debt — unpaid credit-card interest grows the same way.