APR and APY calculator
Conversion between the nominal and the effective annual rate.
Fill in the fields and the result will appear here automatically.
This conversion links a nominal annual rate to the annual effective rate produced by compounding alone. More frequent compounding changes the annual multiplier even if the nominal rate stays fixed. APR and APY in the title do not turn this calculation into a statutory credit-cost disclosure: fees, dated cash flows and jurisdiction-specific rules are not inputs.
How it works
Formula and logic
For nominal annual share j and m periods, effective annual share e = (1+j/m)^m−1. In reverse, j = m[(1+e)^(1/m)−1]. The periodic rate is j/m and yearly multiplier is 1+e. Zero stays zero at every frequency; at m = 1 both annual rates match. Stable logarithms retain very small rates.
Example
Nominal 18% with 12 periods gives 1.50% per month, annual effective 19.56% and multiplier 1.1956. With one period, 18% remains 18%. At 0% with 365 periods, both annual rates are 0% and the multiplier is 1.
Fields and units
- Direction of conversion — list option
- Rate, % — unitless
- Compounding periods per year — unitless
How to use
- — Choose a direction and enter the corresponding nonnegative annual rate.
- — Enter a positive whole number of equal compounding periods per year.
- — Use 12 for monthly compounding or 1 for annual compounding.
- — Compare rates on a consistent fee and compounding basis; review the full product cost separately.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- Microsoft EFFECT: effective rate from nominal rate and compounding frequency CFPB, United States: mortgage APR can include costs beyond interest
- Limitation
- Constant-rate compounding conversion only. It is not a product quote, full borrowing cost or statutory APR disclosure.
FAQ
When is the compounded annual rate strictly above nominal?
With a positive rate and more than one period in this model. At zero interest or one period, the rates are equal.
Is daily compounding identical to continuous compounding?
No. At nominal 18%, both round to 19.72%, but their exact values differ. The continuous limit is exp(0.18)−1.
Does this APR conversion calculate a regulated loan APR?
No. Credit disclosures may include fees and prescribed cash-flow rules. The CFPB mortgage APR explanation is specific to the United States, rather than a rule for every country.
Which costs are omitted from the APR/APY conversion?
Upfront and recurring fees, required insurance, taxes and actual payment dates. Equal compounding rates alone do not make two products equivalent.