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.
Once the rate is on an effective basis, project growth using this rate basis with monthly or annual contributions.
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.
Once a rate is on the basis you need, you can use a converted rate in future value, or project savings at that rate with yearly increases in your deposits.
Convert equivalent rates between compounding frequencies
Banks, lenders and savings products quote rates with different compounding. A nominal annual rate is a yearly quote before compounding, the periodic rate is that quote divided by the number of periods, and the effective annual rate is what the money really earns in a year. To move between two compounding schedules, turn the quote into its effective annual rate first, then solve for the nominal quote that gives the same yield under the new schedule: k × ((1 + y)1/k − 1) for k periods a year, or ln(1 + y) for continuous compounding. The Equivalent rates task on this page does both steps. Enter a nominal rate, the compounding it is quoted at and the compounding you want, and it returns the target nominal rate, the shared yield and the periodic rates. Because both quotes give the same yield, a one-year balance earns the same interest under either. The converter does not rank products or adjust for fees; it only puts the same growth on a different basis. Because the yield is held fixed, the nominal figure changes from one frequency to the next, and the periodic rate changes with it.
Worked example: 6% compounded monthly
A 6% nominal rate compounded monthly has a periodic rate of 0.5% and an effective annual rate of 6.1678%. The same yield can be quoted as 6.0301% compounded quarterly, 6.0755% semiannually, 6.1678% annually or 5.9850% continuously. On a balance of 10,000 each of those quotes earns 616.78 in one year. To see what that yield does to a balance over time, project the balance over time, or estimate CD earnings when the quote comes from a deposit offer. The monthly periodic rate of 0.5% and the quarterly periodic rate of 1.5075% look different, but each compounds to the same year-end result, which is the only thing the equivalent quotes promise.
Daily 365 and average 365.25 are different conventions
This page treats daily compounding as 365 periods a year. Some tools instead use an average year of 365.25 days. The difference is tiny at ordinary rates but visible at extreme ones: 100% nominal compounded daily gives an effective 171.4567% here, while a 365.25-day convention gives about 171.4570%. Neither is wrong; they answer slightly different questions, so match the convention that your product states before comparing two quotes. At 6% nominal, daily compounding gives 6.1831% on a 365-day basis, and a 365.25-day basis differs only in the seventh decimal place.
What the two frequency tables hold constant
The results can show two tables. The first, in the APR → APY and APY → APR tasks, keeps the number you entered fixed: the same nominal rate for APR → APY, the same APY for APY → APR. It shows what the other figures become at six frequencies. The second, in the Equivalent rates task, holds the effective annual yield fixed and shows the nominal quote needed under each of nine frequencies, so the nominal figures differ from row to row. Read the first to see how compounding changes a given quote, and the second to translate one quote into another. Comparing rows within one table is safe; comparing a row of one table with a row of the other is not.
Rate conversion does not include loan fees
A rate conversion changes only how compounding is described. It does not add points, broker fees or other charges. The Consumer Financial Protection Bureau explains that a mortgage interest rate does not reflect fees, while an APR reflects the rate plus points, broker fees and other charges, which is why the APR is usually higher (page last reviewed 28 August 2026; see its interest rate versus APR explanation). To estimate APR with entered loan fees, use the loan payment calculator, then compare the result with the lender's own disclosure. A fee-inclusive APR is a different measure with its own rules, so never substitute a converted nominal rate for the APR a lender discloses.
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.
For a certificate of deposit, estimate CD maturity from an APY without compounding the rate twice.
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.
Bond yields are also quoted as nominal annual rates. To see how a yield turns into a price, you can price a bond and solve its yield from its coupons and face value.
Frequently asked questions
What is an 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 APR to APY?
Apply (1 + r/n)n − 1 with r as a decimal and n as the number of compounding periods per year, or er − 1 for continuous compounding. This converts a nominal quote; it does not compute a legal fee-inclusive APR.
Can I convert monthly to quarterly compounding?
Yes. Hold the annual yield fixed and solve for the target nominal quote. The Equivalent rates task does this for any pair of the nine supported frequencies, including continuous compounding, and shows the periodic rate for each side.
Are equivalent nominal rates the same number?
Usually not. Their effective annual yields match, but the nominal quotes differ, and they are equal only when compounding is annual or the rate is 0%. For example, 6% compounded monthly and about 6.1364% compounded quarterly are the same yield.
Why is daily conversion slightly different elsewhere?
Some tools use 365.25 or 360 days a year instead of 365. Match the day-count basis that your product states. The gap is small at ordinary rates and grows only at extreme ones.
Does this include loan fees?
No. It converts compounding only. Use the loan payment calculator to estimate an APR with entered fees, and then review the lender's own disclosure.
A simple interest calculation uses a fixed principal and does not compound.