Savings Interest Calculator

A lump sum left alone grows by repeated multiplication:

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

• $P$ — opening balance
• $r$ — nominal annual rate as a decimal ($4\% = 0.04$)
• $n$ — compounding periods per year: 1 yearly, 4 quarterly, 12 monthly, 365 daily
• $t$ — time in years

Interest earned alone is $A - P$.

Regular deposits need the future value of an annuity. With a deposit $D$ made every period, a periodic rate $i = r/n$ and $N = nt$ deposits:

$FV = D \cdot \frac{(1+i)^N - 1}{i}$

Each deposit compounds for a different length of time, which is exactly what that sum collapses into. Do both at once by adding them:

$A = P(1+i)^N + D \cdot \frac{(1+i)^N - 1}{i}$

This is the standard end-of-period (ordinary annuity) form. If deposits land at the start of each period, multiply the annuity term by $(1+i)$.

Nominal rate versus APY

The advertised rate and the rate you actually earn differ whenever $n > 1$:

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

APY is the only fair way to compare two accounts that compound differently. Continuous compounding is the ceiling: $APY = e^{r} - 1$.

Rearranging for the unknown

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

The last is the present value — what a future balance is worth today.

Doubling time

Exactly, $t = \ln 2 / \ln(1+i)$ periods. The "rule of 72" is a mental approximation of that and drifts at higher rates.

Real growth

Inflation erodes buying power, so the real rate is $(1+r)/(1+\pi) - 1$ rather than $r - \pi$. Rates on savings accounts and inflation both change over time and vary by country; supply your own figures and this page will do the arithmetic on them.

• Leaving the annual rate in the bracket: the rate inside is per period, $r/n$. Pairing an annual rate with a monthly exponent is the classic blow-up.
• Entering a percent instead of a decimal: $r = 0.04$, not $4$. Check the magnitude of the answer as a sanity test.
• Double-counting time: $t$ is in years and $n$ supplies the periods. Do not put 60 months into $t$ and set $n = 12$.
• Using the lump-sum formula with monthly deposits: it cannot handle contributions. Add the annuity term.
• Comparing nominal rates across different frequencies: 3.95% compounded daily can beat 4.00% compounded annually. Only APY settles it.
• Assuming daily compounding is a big win: at 4% nominal, monthly gives an APY of 4.0742% and daily 4.0808% — the rate matters far more than the frequency.
• Ignoring withdrawals, fees and tax on interest: each changes the effective rate, and tax treatment depends on your jurisdiction.