How Compound Interest Works

Simple interest pays only on what you originally deposited. Put $10,000 into an account paying 10% simple interest and you receive $1,000 a year, for as long as the money is there. After ten years you have $20,000.

Compound interest pays on the deposit and on the interest already credited. The same $10,000 at 10% compounded annually becomes $25,937 after ten years. The extra $5,937 is entirely interest earned on interest — and it is more than half of what simple interest paid in total.

For a lump sum: A = P(1 + r/n)^(nt), where P is the starting amount, r the annual rate as a decimal, n the number of compounding periods per year and t the number of years.

Regular contributions add a second term — the future value of an annuity: PMT × ((1 + i)ⁿ − 1) ÷ i, where i is the rate per contribution period. Adding both together gives the projection any compound interest calculator produces.

Divide 72 by the annual return to estimate how many years money takes to double. At 6%, about twelve years. At 9%, eight. At 12%, six.

It is a mental shortcut, accurate enough between roughly 4% and 12%, and its real value is making the cost of lower returns visible. Two percentage points does not sound like much until you notice it changes doubling time from twelve years to eighteen.

Daily compounding sounds meaningfully better than annual, and it barely is. At a 6% nominal rate, annual compounding yields exactly 6% and daily compounding yields 6.183% — about $18 per $10,000 in the first year.

This is the distinction between a nominal rate and an effective rate, or APY. Banks quote APY precisely so accounts with different compounding schedules can be compared on one number. Compare APYs, and spend your attention on the rate and the time horizon instead.

Two savers, both earning 7%. The first invests $200 a month from age 25 to 35, then stops entirely — $24,000 contributed. The second starts at 35 and invests $200 a month until 65 — $72,000 contributed.

At 65 the first has roughly $300,000 and the second roughly $244,000. Three times the contributions produced less money, because the first saver’s deposits had thirty extra years to compound. This single comparison is the strongest argument for starting early that personal finance has.

Fees compound against you on exactly the same curve. $500 a month for thirty years at 7% reaches about $566,000; at 6% — the same portfolio carrying a 1% fee — it reaches about $475,000. One percentage point cost 16% of the outcome.

Inflation does something similar to purchasing power. A projected balance of $1 million in thirty years buys what roughly $477,000 buys today at 2.5% inflation. Neither fact is an argument against compounding. Both are arguments for modeling it in real, net-of-fee terms rather than headline ones.