APY Interest Calculator

A quoted nominal rate says nothing about how often interest is added. APY — the annual percentage yield, also called the effective annual rate — folds the compounding in so two accounts can be compared with one number:

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

• $r$ — the nominal annual rate as a decimal ($4.5\%$ is $0.045$)
• $n$ — compounding periods per year: 12 monthly, 365 daily, 4 quarterly

Once you have the APY, a balance held for $t$ years grows as

$A = P(1 + APY)^{t}$

with $t$ allowed to be fractional: 18 months is $t = 1.5$.

Going the other way, the nominal rate that produces a given APY at frequency $n$ is

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

APY is always at least the nominal rate, and equal to it only when $n = 1$. The gap grows with the rate but shrinks fast with frequency: the jump from annual to monthly is visible, from daily to continuous almost nothing.

Interest on an account whose balance moves is not computed on the closing figure. Under the average daily balance method the statement cycle is weighted by how long each balance was held:

$\overline{B} = \frac{\sum_i B_i \cdot d_i}{\sum_i d_i}, \qquad I = \overline{B}\cdot r \cdot \frac{D}{365}$

with $B_i$ the balance held for $d_i$ days and $D$ the number of days in the cycle. Some issuers instead post interest daily on each day's closing balance and compound it, which gives a slightly larger figure; the day-count basis may be 365 or 360.

The limit of frequent compounding

$\lim_{n\to\infty}\left(1 + \frac{r}{n}\right)^{n} = e^{r}$

so continuous compounding gives $APY = e^{r} - 1$ — the ceiling any nominal rate can reach.

CDs and early withdrawal

A CD compounds at its stated APY for the term. Withdraw early and the penalty is a contract term subtracted from the interest, not part of the formula.

Rates, day-count conventions and penalty terms differ by institution and change constantly. Enter the figures from your own account disclosure; this page does the arithmetic on them.

• Comparing nominal rates directly: $4.5\%$ compounded daily beats $4.55\%$ compounded annually. Only APY settles it.
• Compounding an APY again: APY is already an annual figure. Using $(1 + APY/12)^{12}$ double-counts the compounding.
• Using the closing balance for a cycle with deposits: a \5{,}700$balance held 12 days out of 31 does not earn a month of interest at$$5{,}700$.
• Assuming 365 days: some products use a 360-day basis, which raises the daily rate by about $1.4\%$ of itself.
• Forgetting fractional years: an 18-month term is $t = 1.5$, not $t = 2$ prorated.
• Ignoring tax and fees: both reduce the yield you actually keep, and both depend on your jurisdiction and account — neither is inside the APY figure.