What is compound interest? How money grows on itself

Compound interest is interest calculated on the original amount and on all the interest added to it so far, so the balance grows on itself. Simple interest is paid on the original amount only. At 5% a year, $1,000 earns $50 in the first year either way, but with compounding the second year's interest is paid on $1,050. Over 30 years, simple interest turns $1,000 into $2,500; compound interest turns it into about $4,322.

In short

• Compounding pays interest on interest, so the yearly gain gets bigger every year instead of staying the same.
• The formula is A = P(1 + r/n)^(nt); the rule of 72 says money doubles in about 72 ÷ rate years.
• The same math works against borrowers, and inflation shrinks what the final number will buy.

How does compound interest work, step by step?

Interest is the price of using money over time. A lender gives up spending it now, its opportunity cost, and because money is scarce, borrowers pay for the use of it. The question compound interest answers is what happens to interest once it is paid: if it is left in the account, it starts earning interest of its own. Here is $1,000 at 5% a year, year by year:

Year	Balance at start	Interest at 5%	Balance at end	Simple interest balance
1	$1,000.00	$50.00	$1,050.00	$1,050
2	$1,050.00	$52.50	$1,102.50	$1,100
3	$1,102.50	$55.13	$1,157.63	$1,150
10	—	—	$1,628.89	$1,500
20	—	—	$2,653.30	$2,000
30	—	—	$4,321.94	$2,500

In year 2 the extra $2.50 is interest on the first year's $50. In year 3, interest is paid on that $2.50 too. The gap looks trivial at first and then opens fast. By year 30, the account earns about $206 a year, four times the $50 it earned in year 1, from the same 5% rate. Of the $3,322 gained in 30 years, $1,500 is interest on the original $1,000 and $1,822 is interest on interest.

That shape, growth in proportion to what is already there, is exponential growth. Simple interest grows in a straight line, by the same $50 every year. Compound interest curves upward, and the longer it runs, the steeper it gets.

What is the compound interest formula?

OpenStax's Principles of Economics writes the yearly version as principal × (1 + rate)^years. The general formula, for interest added more than once a year, is:

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

• A is the amount at the end.
• P is the principal, the amount you start with.
• r is the yearly interest rate, as a decimal: 5% is 0.05.
• n is how many times a year interest is added: 1 for yearly, 12 for monthly, 365 for daily.
• t is the number of years.

For the table above, P = 1,000, r = 0.05, n = 1 and t = 30, so A = 1,000 × 1.05^30 = 1,000 × 4.3219 = $4,321.94.

Adding interest more often helps, but less than people expect:

Compounding	Formula	$1,000 at 5% after 30 years
Yearly	1,000 × (1 + 0.05)^30	$4,321.94
Quarterly	1,000 × (1 + 0.05/4)^120	$4,440.21
Monthly	1,000 × (1 + 0.05/12)^360	$4,467.74
Daily	1,000 × (1 + 0.05/365)^10,950	$4,481.23
Continuously	1,000 × e^(0.05 × 30)	$4,481.69

Going from yearly to monthly adds about $146; going from daily to continuous adds 46 cents. The total approaches a limit, and that limit involves the number e, about 2.718. Jacob Bernoulli found the constant in 1683 by asking exactly this question about compound interest, as Wikipedia's history of compound interest notes; the note on e follows it from there. The yearly rate that includes compounding is what US banks quote as APY, which how interest rates work explains alongside APR.

What is the rule of 72?

The rule of 72 is a mental shortcut: divide 72 by the yearly interest rate in percent, and the answer is roughly how many years money takes to double. At 5%, 72 ÷ 5 = 14.4 years. The exact answer is 14.2. The rule is old: it appears in Luca Pacioli's arithmetic textbook of 1494, according to Wikipedia's article on the rule, without any explanation, which suggests it was older still.

Rate	Rule of 72	Exact doubling time
2%	36 years	35.0 years
4%	18 years	17.7 years
6%	12 years	11.9 years
8%	9 years	9.0 years
12%	6 years	6.1 years
24%	3 years	3.2 years

It works best between about 6% and 10%. The same rule runs backward for prices: at 3% inflation, prices double in about 24 years.

Why does starting early matter so much?

Because time is the exponent. Compare two savers who each put $2,000 a year into an account earning 5% a year, compounded yearly, until they turn 65. Ana saves from 25 to 34, then stops and never adds another dollar: $20,000 in all. Ben starts at 35 and saves every year until 64: $60,000 in all. At 65, Ana has about $114,000 and Ben about $140,000. Ben put in three times as much money and ends up only about a fifth ahead, because Ana's early deposits had 30 to 40 years to compound. The numbers are illustrative; real returns vary from year to year and are never guaranteed.

Not advice

This note explains how the arithmetic works. It isn't financial advice, and it doesn't recommend any account, product or rate.

How does compound interest work against borrowers?

The same formula runs the other way on debt. Interest that isn't paid is added to the balance, and then interest is charged on it. Many credit cards add interest daily. A card with an annual percentage rate of 24%, compounded daily, has an effective yearly rate of (1 + 0.24/365)^365 − 1, or about 27.1%. Leave $1,000 unpaid for a year with no payments and it becomes about $1,271. By the rule of 72, an unpaid balance at that rate doubles in under three years. The interest rates note works through paying down a card balance month by month.

Governments face the same arithmetic on what they owe. Interest on a national debt that is paid by borrowing more compounds just as a card balance does.

Does inflation change the result?

Yes. A balance of $4,321.94 in 30 years won't buy what $4,321.94 buys today if prices rise in the meantime. What matters is the real return: the interest rate after inflation. The exact formula is (1 + nominal rate) ÷ (1 + inflation rate) − 1; the quick version is the rate minus inflation.

At 5% interest and 3% inflation, the real return is 1.05 ÷ 1.03 − 1, or 1.94% a year. Compounded for 30 years, $1,000 grows to $4,321.94 in future dollars but only about $1,781 in today's buying power. Compounding is still working, but inflation compounds too, against the saver. If inflation ran above the interest rate, the real balance would shrink every year even as the number on the statement grew.

Did Einstein call compound interest the most powerful force in the universe?

There is no good evidence he did. The line, in many versions, "the eighth wonder of the world," "man's greatest invention," "the most powerful force in the universe," is widely credited to Albert Einstein. Quote Investigator, which traced the "greatest invention" version, found no significant evidence that he said it; the earliest link to Einstein it located was "Letter From a Pensioner," a piece in the Wall Street Journal of 25 August 1976, 21 years after his death. The arithmetic needs no famous name: a balance that grows in proportion to itself will, given enough time, outgrow any balance that grows by a fixed amount.

Check yourself

• Why does $1,000 at 5% earn $52.50 in year 2 instead of $50?
• Using the rule of 72, roughly how long does money take to double at 6%, and why is it only an approximation?
• Why can a savings balance rise every year while what it can buy falls?

Sources

• 17.3 How to Accumulate Personal Wealth — OpenStax, Principles of Economics 3e, 2022
• Compound interest — Wikipedia
• Rule of 72 — Wikipedia
• Compound Interest Is Man's Greatest Invention — Quote Investigator, 2011