Calculate the effective annual rate (EAR) from any nominal rate and compounding frequency, or find the nominal rate needed to hit a target EAR.
Effective Annual Rate Formula
To find the effective annual rate from a stated (nominal) rate that compounds more than once a year, use:
EAR = (1 + r/n)^n - 1
When interest compounds continuously, use this version instead:
EAR = e^r - 1
To work backward and find the nominal rate you need to reach a target effective rate at a given compounding frequency, rearrange the first formula:
r = n * ((1 + EAR)^(1/n) - 1)
The variables are:
- EAR = effective annual rate, written as a decimal before converting to a percent
- r = nominal (stated) annual rate, also written as a decimal
- n = number of compounding periods per year (12 for monthly, 4 for quarterly, and so on)
- e = Euler’s number, about 2.71828, used only for continuous compounding
The nominal rate is the rate a lender or bank quotes you. Dividing it by n gives the rate charged in each period, and raising the result to the power of n compounds those periods across a full year. Subtracting 1 strips out your original principal so you are left with the true yearly rate. The effective annual rate is the same idea expressed as APY on a savings product. Solving for r tells you what stated rate produces a known effective rate, which is useful when a product advertises its APY and you want the underlying nominal figure.
Effective Rate by Compounding Frequency
The more often interest compounds, the larger the gap between the nominal rate and the effective rate. The table below shows the effective annual rate produced by a 10% nominal rate at common frequencies.
| Compounding | Periods per year (n) | EAR on 10% nominal |
|---|---|---|
| Annually | 1 | 10.000% |
| Semiannually | 2 | 10.250% |
| Quarterly | 4 | 10.381% |
| Monthly | 12 | 10.471% |
| Daily | 365 | 10.516% |
| Continuous | ∞ | 10.517% |
Notice that the effective rate rises quickly from annual to monthly compounding, then barely moves between daily and continuous. Past a certain point, more frequent compounding adds very little.
Example Problems
Example 1. A credit card charges a 24% nominal rate compounded monthly. With r = 0.24 and n = 12, the effective annual rate is (1 + 0.24/12)^12 – 1 = (1.02)^12 – 1 = 0.2682, or about 26.82%. The true yearly cost is almost three points higher than the quoted 24%.
Example 2. A savings account advertises a 5.12% APY and you want to know the nominal rate behind it, assuming daily compounding. With EAR = 0.0512 and n = 365, the nominal rate is 365 * ((1.0512)^(1/365) – 1) = 0.0499, or about 4.99%.
Frequently Asked Questions
What is the difference between the nominal rate and the effective annual rate? The nominal rate is the stated rate before compounding is taken into account. The effective annual rate folds compounding in, so it reflects what you actually earn or pay over a full year. They are equal only when interest compounds once per year.
Is the effective annual rate the same as APY? Yes. Annual percentage yield (APY) and effective annual rate (EAR) describe the same number. Banks tend to use APY for deposit accounts, while EAR is more common in lending and corporate finance, but the calculation is identical.
Why does my calculated EAR differ slightly from a quoted figure? Small differences usually come from rounding or from a different day count, such as using 360 days instead of 365. Confirm the exact compounding frequency and rounding rule the institution uses if you need an exact match.
