Compound interest explained

Compound interest means earning interest on the interest you have already earned. Each period the interest is added to the balance, and the next period’s interest is calculated on that larger balance. Over short periods the difference from simple interest is small; over decades it dominates.

Simple interest versus compound interest

With simple interest, interest is always calculated on the original amount. $10,000 at 5% earns $500 a year, so after 30 years you have $10,000 + 30 × $500 = $25,000.

With annual compounding, year one earns $500, but year two earns 5% of $10,500 = $525, year three 5% of $11,025 = $551.25, and so on. After 30 years the balance is $43,219.42 — $18,219 more, entirely from interest earned on interest.

The compound interest formula

A

amount at the end, including interest

P

starting principal

r

annual interest rate as a decimal (5% → 0.05)

n

number of times interest is compounded per year

t

time in years

The interest earned is A − P. If interest compounds continuously, the formula becomes:

How much does compounding frequency matter?

Here is $10,000 at 5% for 10 years under different compounding schedules:

Moving from annual to monthly compounding adds about $181 here; going from monthly to continuous adds only about $17 more. The rate and the time matter far more than the schedule. To compare accounts with different schedules, look at the APY (annual percentage yield), which folds compounding into one yearly figure: 5% compounded monthly is an APY of 5.12%.

The Rule of 72

A quick mental estimate: divide 72 by the annual rate to get the approximate number of years for money to double. At 6%, 72 ÷ 6 = 12 years; the exact answer with annual compounding is 11.9 years. At 8% the rule gives 9 years (exact: 9.01), at 12% it gives 6 (exact: 6.12). It is most accurate for rates between about 6% and 10% and less accurate far outside that range.

Regular contributions and starting early

Most people don’t invest one lump sum; they add money regularly. Saving $200 a month for 30 years at 6% a year, compounded monthly, grows to about $200,903, of which $72,000 is money you put in and about $128,903 is growth.

Time is the strongest lever. The same $200 a month for 40 years instead of 30 grows to about $398,298 — the extra ten years of saving add $24,000 in contributions but nearly double the result.

What the formula leaves out

• Investment returns aren’t constant. The formula assumes the same rate every year. Stock returns vary widely from year to year, so a projection shows the order of magnitude, not a promise.
• Inflation. At 3% inflation, $100 buys about $55 worth of today’s goods after 20 years. Compare real (after-inflation) returns when planning long term.
• Taxes and fees reduce the rate that actually compounds. A 1% annual fee has a large effect over decades.
• Debt compounds too. Credit card balances compound against you, usually at far higher rates than savings earn.