Compound Interest Formula
Future value when interest is compounded n times per year over t years.
A = P × (1 + r/n)^(n·t)
How it works
Compound interest differs from simple interest because each period earns interest on the interest already accrued, not just on the original principal. Over ten years at 5%, $1,000 grows to $1,500 under simple interest and $1,628.89 compounded annually — and the gap widens the longer you leave it. How often the interest compounds matters much less than people expect, and there is a hard ceiling on it. That same $1,000 at 5% over ten years reaches $1,628.89 compounded annually, $1,643.62 quarterly, $1,647.01 monthly and $1,648.66 daily. Compounding continuously, the theoretical limit, gives $1,648.72. So the whole distance from annual to infinitely frequent compounding is under $20, and almost all of it is captured by the time you reach monthly. Two practical notes. The Rule of 72 is a good mental shortcut for doubling time: 72 divided by the percentage rate gives the years needed, so 8% suggests 9.0 years against a true value of 9.006. And the same mathematics runs in reverse on debt. A credit card at 22% compounding monthly grows a balance on exactly the curve that makes a retirement account attractive, which is why paying down high-interest debt reliably beats investing at typical market returns.
Variables
- AFinal amount
- PPrincipal (starting amount)
- rAnnual interest rate (decimal)
- nCompounding periods per year
- tYears
Worked examples
| Input | Result |
|---|---|
| $1,000 at 5%, monthly, 10 yr | ≈ $1,647.01 |
| $10,000 at 7%, daily, 30 yr | ≈ $81,649 |
Practical applications
- Projecting a retirement or index-fund balance decades ahead
- Comparing savings accounts and certificates quoted at different compounding frequencies
- Forecasting how a credit-card or loan balance grows if only the minimum is paid
- Sanity-checking a quoted annual percentage yield against the stated rate and frequency
Common mistakes
- Entering the rate as a percentage rather than a decimal. 5% is 0.05; using 5 produces a nonsensical answer.
- Mismatching r and n. If r is the annual rate then n is the number of compounds per year, and t is in years.
- Confusing the nominal rate with the effective yield. 5% compounded monthly is an effective 5.12% a year.
- Assuming more frequent compounding matters a great deal. Beyond monthly the difference is a rounding error over a decade.
- Comparing two products on headline rate alone when they compound at different frequencies.
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FAQs
What is continuous compounding?+
As n → ∞ the formula collapses to A = P × e^(r·t). At everyday rates the difference vs daily compounding is negligible.
How does this differ from simple interest?+
Simple interest = P × r × t — no exponentiation, so a $1,000 / 5% / 10-year deposit earns only $500 vs $629 compounded annually.