Compound Interest Calculator

Compound interest means interest is added to the balance and then itself earns interest. The closing amount is

$A = P\left(1 + \frac{r}{n}\right)^{nt}$

• $P$ — the principal you start with
• $r$ — the nominal annual rate as a decimal (5% is $0.05$)
• $n$ — compounding periods per year: 1 yearly, 4 quarterly, 12 monthly, 365 daily
• $t$ — time in years (a value like $t = 0.5$ is fine)

The interest earned alone is $A - P$.

Each period the balance is multiplied by $\left(1 + \frac{r}{n}\right)$, and doing that $nt$ times gives the exponent. That is the whole derivation — compounding is repeated multiplication, which is why growth is exponential rather than linear.

With simple interest, $A = P(1 + rt)$: interest is charged on the original principal only, so it grows in a straight line and always trails compounding over the same term.

Letting $n$ grow without bound

As the compounding gets more frequent, $\left(1 + \frac{r}{n}\right)^{nt}$ approaches a limit:

$A = Pe^{rt}$

This is continuous compounding — the theoretical ceiling for a given nominal rate. The step from yearly to monthly is noticeable; from daily to continuous it is almost nothing.

The effective annual rate

$EAR = \left(1 + \frac{r}{n}\right)^{n} - 1$

Compare two accounts by EAR, never by nominal rate alone, unless they compound identically.

Solving for what you do not know

Rearranged, the same formula answers three more questions:

$r = n\left[\left(\frac{A}{P}\right)^{\frac{1}{nt}} - 1\right], \qquad t = \frac{\ln(A/P)}{n\ln\!\left(1 + \frac{r}{n}\right)}, \qquad P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}$

The last one is the present value: what a future amount is worth today.

Doubling time

Exactly, $t = \ln 2 / \ln(1 + i)$ where $i$ is the periodic rate. The familiar "rule of 72" is just a mental shortcut to that expression and drifts at high rates.

• Forgetting to divide $r$ by $n$: the rate in the bracket is per period. Using the annual rate with a monthly exponent is the classic blow-up.
• Percent not converted: $r = 0.05$, not $5$. Check the magnitude of your answer as a sanity test.
• $t$ measured in periods: $t$ is in years and $n$ supplies the periods. Do not put 36 months into $t$ and set $n = 12$.
• Comparing nominal rates directly: 5.9% compounded daily can beat 6% compounded annually. Only EAR settles it.
• Using $A = P(1+r/n)^{nt}$ for regular deposits: a lump-sum formula cannot handle contributions — each deposit compounds for a different length of time and needs the annuity formula.
• Assuming daily compounding is a big win: at 6% nominal the EAR is about 6.1678% monthly versus 6.1831% daily. The rate matters, the frequency barely does.
• Ignoring withdrawals, fees and taxes: any of them changes the effective rate that belongs in the formula, and all three depend on your account and jurisdiction.