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
A nominal annual interest rate is the stated interest rate before compounding; a regulated loan APR can include charges and is not necessarily interchangeable with it. 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 modeled one-year compound growth rate. The model excludes fees, repayments, rate changes and taxes. Nominal inputs are nonnegative; custom periods must be a whole number. 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 modeled compounded rate is about 2.824 percentage points above the nominal 24%, assuming no repayments or fees; actual credit-card cost depends on account terms and payment timing.
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 when interest compounds once per year, or when the rate is zero.
Is the effective annual rate the same as APY? For a constant-rate deposit with full reinvestment under matching timing assumptions, APY corresponds to effective annual growth. Official APY disclosures can apply additional day-count, tiered-rate, payout and rounding rules; this calculator is a mathematical conversion, not a regulatory disclosure calculation.
Why does my calculated EAR differ slightly from a quoted figure? Differences may reflect rounding, day-count conventions, rate changes, interest payouts or other product terms. Daily means 365 equal periods in this model. Confirm the exact compounding frequency and rounding rule the institution uses if you need an exact match.
