Compound interest explained, with worked examples

Compound interest is interest earning interest. That is the whole idea, and it sounds too small to matter until you watch it run for thirty years.

What actually happens, year by year

Take $10,000 at 7% a year. Simple interest would pay $700 every year forever. Compound interest pays 7% of whatever the balance has become:

In year one the two are identical. By year thirty compound interest has produced $76,123 against $31,000 — nearly two and a half times as much, from the same rate on the same money. Look at the interest column: it grows from $700 to $4,980, because it is a percentage of a balance that keeps rising.

The formula

A = P(1 + r/n)^(nt)

The interest alone is A − P. The exponent is what makes the curve bend — every extra year multiplies rather than adds.

Time beats rate, and it is not close

Because time sits in the exponent and rate does not, an extra decade usually outperforms an extra percentage point. $10,000 invested at 7%:

Ten more years nearly doubles what an extra percentage point achieves. This is the practical argument for starting early with a modest amount rather than waiting until you can invest a serious one.

Put the other way: an investor who contributes $200 a month from 25 to 35 and then stops entirely ends up ahead, at 65, of one who starts at 35 and contributes for thirty years. The first put in $24,000; the second put in $72,000. Ten years of head start beat three times the money.

The rule of 72

years to double ≈ 72 ÷ interest rate as a percentage

It is accurate to within a few months between about 4% and 12%, which covers most real decisions. It works in reverse too: at 3% inflation, prices double in 24 years, which is why a pension projected in today's money needs adjusting.

Compounding frequency matters less than people think

The entire span from annual to continuous compounding is $313 — about 1.7% — and it converges. Daily and continuous differ by one dollar. Choose products on the rate and on fees, not on the compounding frequency.

Regular contributions change everything

Most people are not investing a lump sum; they are adding monthly. That needs the future value of an annuity added on top:

A = P(1 + i)^N + PMT × ((1 + i)^N − 1) ÷ i

where i is the rate per period and N the number of periods. $10,000 at 7% with $300 a month for 30 years:

Contributions dominate. For most people the amount saved each month matters far more than the return achieved on it, which is an unglamorous conclusion that the financial industry rarely emphasises.

Where the idea stops working

• Inflation. A 7% return with 3% inflation is a real return of about 3.9% — subtract, do not ignore. $76,123 in thirty years buys what roughly $31,000 buys today.
• Fees compound too. A 1% annual fee on a 7% return does not cost you 1%; over 30 years it removes roughly a quarter of the final balance, because the fee is charged on the compounding balance every year.
• Tax. Interest taxed annually cannot compound. The same rate inside a tax-sheltered account substantially outperforms the same rate outside one.
• Returns are not a fixed rate. Markets deliver an average, not an annuity. Sequence matters: a bad decade at the start of retirement withdrawals is far worse than the same decade at the end.

It runs backwards too

Compounding is direction-neutral. A credit card at 22% APR compounded daily turns an untouched $3,000 balance into about $3,738 in a year, and $5,568 in three. Applying the rule of 72 to it: at 22%, an ignored balance doubles in about three years and three months.

This is why paying down high-interest debt usually beats investing the same money. A guaranteed 22% you stop paying is larger, and far more certain, than any return you can reasonably expect to earn.