Effective Annual Rate Calculator

A nominal annual rate $i$ quoted with a compounding frequency $n$ is not what a balance actually grows by in a year. The effective annual rate is:

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

• $i$ — the nominal annual rate as a decimal ($6\% = 0.06$); this is the quoted or stated rate
• $n$ — compounding periods per year: 1 annually, 2 semiannually, 4 quarterly, 12 monthly, 365 daily
• $i/n$ — the periodic rate actually applied each period

The gap exists because interest credited partway through the year earns interest for the rest of it. When $n = 1$ the two coincide; the larger $n$ gets, the wider the gap, up to the continuous limit:

$EAR = e^{i} - 1$

This is the only fair way to compare two quoted rates with different compounding. Two accounts both advertising 6% pay different amounts if one compounds monthly and the other annually, and EAR is what exposes the difference.

From EAR back to a nominal rate

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

Use it when you know the yield you need and want the quoted rate that produces it. Note that $i < EAR$ whenever $n > 1$.

Between two compounding frequencies

To restate a rate from $n_1$ to $n_2$ periods, go through the EAR:

$i_2 = n_2\left[\left(1 + \frac{i_1}{n_1}\right)^{n_1/n_2} - 1\right]$

Periodic and continuous rates

$i_{\text{periodic}} = \frac{i}{n}, \qquad i_{\text{continuous}} = \ln(1 + EAR)$

Where the names differ

On deposits the effective figure is usually called APY; on borrowing, EAR or effective interest rate. Meanwhile APR is a separate, legally defined disclosure that folds certain fees into a rate — how it is computed is set by regulation and differs between jurisdictions and product types. This page converts between nominal and effective rates exactly; it does not compute any jurisdiction's regulatory APR and is not a lender or advisor.

• Multiplying the periodic rate by $n$ and calling it effective: $0.5\%$ a month is $6\%$ nominal but $6.1678\%$ effective. Multiplication ignores compounding entirely.
• Percent left unconverted: $i = 0.06$, not $6$. Entering $6$ produces an absurd answer immediately.
• Forgetting the $-1$: $(1 + i/n)^n$ is a growth factor, around $1.06$. The rate is that minus one.
• Comparing nominal rates across frequencies: $5.9\%$ compounded daily and $6.0\%$ compounded semiannually cannot be ranked by their headline numbers. Convert both first.
• Assuming higher frequency always wins: it helps, but only slightly. Frequency is a second-order effect next to the rate itself.
• Treating APR as EAR: APR is a regulatory disclosure that may include fees and may not compound the way EAR does. They answer different questions.
• Mismatching the period when reversing: $i = n[(1+EAR)^{1/n} - 1]$ needs the same $n$ you intend to compound at.