APR to APY: nominal to effective annual rate
A nominal annual rate is the yearly rate before compounding, often written as an APR in simple examples. The effective annual rate, also called the APY, is the rate that would give the same growth if interest were added once a year. They differ whenever interest compounds more than once a year, because each period earns interest on the interest that came before it. With the nominal rate r as a decimal and n compounding periods per year:
- Effective annual rate y = (1 + r / n)n − 1.
- Continuous compounding: y = er − 1.
- Periodic rate = r / n, the rate applied in each compounding period.
Use this page when you need the yearly rate that two accounts or loans really imply. To see how a balance grows over several years with regular deposits, use the compound interest calculator, which projects the balance; this converter only translates the rate.
APY to APR: convert back at a chosen frequency
To go the other way, solve the same equation for r: r = n × ((1 + y)1/n − 1), or r = ln(1 + y) for continuous compounding. The frequency is required, because many different nominal rates produce the same APY: a 5% APY needs a nominal rate of 4.8889% if it compounds monthly, 4.8793% if it compounds daily and 4.8790% if it compounds continuously. A bare APY figure does not tell you the nominal rate until you know how often interest compounds.
How compounding frequency changes the annual rate
For a fixed nominal rate, more frequent compounding gives a higher effective rate, but the gain shrinks quickly. At 12% nominal the effective rate is 12.00% with annual compounding, 12.6825% monthly, 12.7475% daily and 12.7497% with continuous compounding. Going from monthly to daily adds only 0.0650 percentage points. The table in the results shows these rows for your own rate, holding the entered rate constant. Continuous compounding is the mathematical limit as the number of periods grows without bound; few real products use it.
Compare two annual rates on the same basis
Rates quoted with different compounding are not directly comparable: 12% compounded monthly and 12.4% compounded quarterly use different periods. Convert both to effective annual rates and compare those. The “Compare two” mode does this for any pair, and reports the difference in percentage points (the gap between the two effective rates) and as a relative percent. They are different numbers: a gap of 0.3 percentage points between 5.0% and 5.3% is a 6% relative difference. For a deposit, the higher effective rate pays more; for a loan, it costs more. The tool compares rates and does not rank products.
Worked examples: 12% nominal and 5% APY
Nominal to APY. A 12% nominal rate compounded monthly has r = 0.12 and n = 12. The monthly rate is 1%, and y = (1 + 0.01)12 − 1 = 12.6825%. On a balance of 10,000 the first-year interest is 1,268.25, against 1,200 with annual compounding.
APY to nominal. A 5% APY compounded monthly needs a monthly rate of (1.05)1/12 − 1, and the nominal rate is 12 times that: 4.8889%. Converting back, a nominal 4.8889% compounded monthly gives an effective 5.0000%, which confirms the round trip.
What fees and disclosed APR change
The nominal rate here is a pure compounding rate. A disclosed loan APR can include certain fees, depending on the jurisdiction, and a deposit’s advertised yield can depend on balance tiers or promotional periods. Neither is the same as the arithmetic rate converted on this page, and this tool does not compute a regulated APR, taxes or net proceeds. To estimate a loan payment and the total interest from a rate and term, use the loan payment calculator. To turn an annual compounded return into a value after inflation, convert to an effective annual rate first and then use the real return calculator. Results are educational and not financial advice; see the disclaimer.
Frequently asked questions
What is the effective annual rate?
The constant yearly rate, compounded once a year, that gives the same growth as a nominal rate compounded more often.
How do I convert a nominal APR to APY?
Apply (1 + r/n)n − 1 with r as a decimal and n as the number of compounding periods per year. Use er − 1 for continuous compounding.
Can APY be converted to APR without the compounding frequency?
No. Different frequencies need different nominal rates to reach the same APY, so the frequency is a required input.
When are the nominal and effective rates equal?
When compounding is annual, or when the rate is 0%. Otherwise the effective rate is higher than the nominal rate.
Does this include fees and taxes?
No. It converts compounding only. A regulated APR, net yield after tax and the cost of fees need other inputs.
Does daily compounding always make an account better?
No. It raises the effective rate slightly for a given nominal rate, but compare the products’ actual rates, fees, limits and risks, not the compounding alone.