The Compound Interest Formula, Explained
The compound interest formula, A = P(1 + r/n)nt, explained piece by piece with worked examples and a calculator you can try.
- 5 min read
- Last reviewed October 30, 2026
- Includes a live calculator
The formula, piece by piece
Five symbols carry the whole idea. A is the future value, the number you end with. P is the principal, what you start with. r is the annual interest rate written as a decimal, so 5% is 0.05. n is how many times per year interest is added to the balance: 12 for monthly, 365 for daily, 1 for annually. t is the time in years.
The heart of it is (1 + r/n): each compounding period, the balance is multiplied by one-plus-a-small-slice of the annual rate. Doing that nt times in a row is what the exponent means. Because each multiplication acts on a balance that already includes past interest, growth accelerates instead of staying flat. That is the entire difference from simple interest.
A worked example
Take $10,000 at 5% compounded monthly for 10 years. Filling in: P = 10,000, r = 0.05, n = 12, t = 10.
The monthly rate is 0.05 divided by 12, about 0.41667% per month. There are 120 monthly steps in ten years. Each step multiplies the balance by 1.0041667, and 120 of those multiplications compound into a 64.7% total gain, not the 50% that ten flat years of 5% simple interest would pay. Set the rail calculator to these exact inputs and it lands on the same figure.
How frequency changes the answer
Same $10,000, same 5%, same 10 years, changing only n:
| Compounding | n | Future value |
|---|
Two things worth noticing. Going from annual to monthly buys real money; going from monthly to daily buys pocket change. And the values in this table are computed by this page from the formula at render time, not typed in, so they cannot drift out of date or carry a typo. Our daily calculator makes the same point interactively.
Continuous compounding
Push n toward infinity and the formula collapses into this cleaner one, built on the constant e (about 2.71828). It is the mathematical ceiling on compounding: $10,000 at 5% for 10 years reaches $16,487.21 continuously, barely a dollar above daily compounding's result. Textbooks and finance courses use it constantly; bank accounts do not, because daily already gets within pennies of the limit.
Five compounding schedules, same $10,000 at 5% for 10 years, each computed live from the formulas above. Hover or tap the line to see any year's value.
Simple vs compound, in brief
Simple interest is I = Prt, paid only on the original principal: the same dollar amount every year, forever. Compound interest pays on principal plus accumulated interest, so each year's payment grows. On short timelines the two are close, which is why simple interest persists in auto and short-term lending. Over decades, compounding wins by a widening margin; the worked example above already shows a 14.7-point gap at just ten years.
Common mistakes
Nearly every wrong answer from this formula traces back to one of three setup mistakes. Each one below is a setup someone actually tried; open it to check your instinct against what really happens.
Checked 0 of 3
Setup: r = 5, not 0.05. Does the formula still work?
No. r must be the decimal form: 0.05, not 5. Feeding 5 into the formula produces numbers in the millions, since 1 + 5/12 compounds far faster than 1 + 0.05/12 ever would. This is the single most common wrong-answer cause.
Setup: n = 12, but the exponent uses t instead of 12t. Right or wrong?
Wrong. If n is 12, the exponent must be 12t, not t. Half-converting, changing n but leaving the exponent alone, is the classic spreadsheet bug, and it understates the real growth.
Setup: a bank quotes 5.12% APY, and it gets run through the formula again at n = 12. Any issue?
Yes. APY already includes compounding, so running it through the periodic-rate formula a second time at n = 12 double-counts it slightly. When a bank quotes APY, the honest comparison is APY to APY, not APY through this formula.
Written by Cedrick Reese. The formula, its notation, and the worked examples on this page are checked against the U.S. Securities and Exchange Commission's compound interest resources at investor.gov, and every figure in the frequency table is computed live from the formula by this page's own code rather than transcribed. Reviewed on a maintenance cycle. Last reviewed: October 30, 2026.
Frequently asked questions
How do I solve the formula for the starting amount, rate, or time?
Rearrange it. Principal: P = A divided by (1 + r/n) to the nt. Rate: r = n times ((A/P) to the 1/(nt) power, minus 1). Time: t = ln(A/P) divided by n times ln(1 + r/n). The reverse compound interest calculator does this for you, or try the mini version below.
Time
Assumes the ending amount is greater than the starting amount.
What is the rule of 72?
A mental shortcut for doubling time: divide 72 by the annual rate to estimate the years for money to double. At 8%, about 9 years. It is an approximation that is most accurate for rates of roughly 6% to 10%; the exact answer is ln(2) divided by ln(1 + r). Try your own rate below.
Rule of 72 estimate
Exact answer: 9.01 years.
Why does my bank's number differ slightly from the formula?
Banks round to the cent as they go, credit interest on their own posting schedule, and sometimes count days on a 360-day basis. The formula is the clean mathematical model; a statement is the model plus rounding and calendar quirks. The differences are pennies, not errors.