Compound Interest

Interest exists because lending money isn't free for the lender. If you lend a friend $20 for a month, you can't spend that $20 yourself while it's out on loan — and there's some chance they never pay you back at all. Interest is how a lender gets compensated for both of those things: giving up the use of their money for a while, and taking on the risk it might not come back. The same idea scales up from a friend lending $20 to a bank lending someone money for a car or a house — whoever lends money is owed something extra in return, and that something extra is interest. See What an Interest Rate Fundamentally Is for more on how that compensation gets priced.

Putting money in a savings account, or investing it, is really just lending your money out — to a bank, or to a company — and collecting interest in return. That's also why a dollar today is worth more than a dollar a year from now: a dollar in hand right now can start earning that “something extra” immediately, while a dollar arriving later can't. See Time Value of Money for more on that idea. Compound interest is what happens when the interest you've already earned gets added to the amount you're lending out, so it starts earning its own interest too.

If you only earned interest on your original principal (simple interest), your balance would grow by the same fixed amount every period — a straight line. But once earned interest is left in the account, it starts earning its own interest. Each period's gain builds on a larger base than the last, so growth curves upward instead of running flat. This is why long time horizons matter more than most people intuitively expect: the effect is small early on and large later, because each period's growth multiplies the balance (scales it up by a percentage) rather than just adding the same fixed amount on top each time.

The balance after compounding is given by:

A = P × (1 + r/n)^(n×t)

• P — principal, the starting balance
• r — the annual nominal interest rate (the rate you were quoted or advertised, before taking into account how often it compounds), as a decimal
• n — number of times interest compounds per year (1 = annual, 12 = monthly, 365 = daily)
• t — number of years the money is invested
• A — the resulting balance after t years

Total interest earned is simply A − P. As n increases, the effective annual rate (the actual percentage growth over one year) rises slightly above the nominal rate r, because interest gets folded back into the balance more often — the quoted rate stays the same, but you actually earn a bit more than that number suggests.

Suppose you invest $10,000 at a 7% annual rate, compounded monthly, for 20 years.

n = 12, r = 0.07, t = 20
A = 10,000 × (1 + 0.07/12)^(12×20)
A = 10,000 × (1.005833...)^240
A ≈ $40,387

Of that $40,387 balance, about $30,387 is interest — over triple the original principal — even though the nominal rate never changed. Try the compound interest calculator to see how changing the frequency or time horizon shifts the result.