About

Misreading the difference between APY and APR on a loan term sheet or deposit disclosure can cost hundreds or thousands of dollars over the life of the instrument. APY (Annual Percentage Yield) reflects the total interest earned or paid after compounding is applied, while APR (Annual Percentage Rate) is the nominal rate before compounding effects. Banks quote deposits in APY to make yields look larger, and loans in APR to make costs look smaller. This calculator performs the exact algebraic inversion of the compounding formula to recover the true APR from any stated APY.

The conversion depends on the compounding frequency n. A 5% APY compounded monthly yields a different APR than the same APY compounded quarterly. This tool assumes standard discrete compounding periods as well as the continuous compounding limit using the natural logarithm. Results approximate real-world rates assuming no fees, points, or day-count conventions are embedded in the quoted APY.

Formulas

The standard relationship between APY and APR for discrete compounding is:

APY = (1 + APRn)ⁿ − 1

Solving for APR by algebraic inversion:

APR = n × [(1 + APY)¹ⁿ − 1]

For continuous compounding (n → ∞), the formula reduces to:

APR = ln(1 + APY)

Where: APY = Annual Percentage Yield (as a decimal, e.g. 0.05 for 5%). APR = Annual Percentage Rate (nominal rate, as a decimal). n = number of compounding periods per year. ln = natural logarithm (base e).

Reference Data

Compounding	Periods (n)	APR for 5% APY	APR for 10% APY	APR for 15% APY	APR for 20% APY
Annually	1	5.0000%	10.0000%	15.0000%	20.0000%
Semi-Annually	2	4.9390%	9.7618%	14.4761%	19.0890%
Quarterly	4	4.9089%	9.6455%	14.2211%	18.6540%
Monthly	12	4.8889%	9.5690%	14.0579%	18.3714%
Semi-Monthly	24	4.8790%	9.5311%	13.9780%	18.2333%
Bi-Weekly	26	4.8771%	9.5239%	13.9628%	18.2071%
Weekly	52	4.8691%	9.4936%	13.8993%	18.0976%
Daily	365	4.8790%	9.5310%	13.9762%	18.2363%
Continuous	∞	4.8790%	9.5310%	13.9762%	18.2322%
Values computed using exact inversion: APR = n × [(1 + APY)^1/n − 1]

Frequently Asked Questions

Because compounding adds interest-on-interest within the year. The APY captures this accumulated effect, so for any compounding frequency n > 1, the nominal APR must be lower than the effective APY to produce the same total yield. When n = 1 (annual compounding), APR and APY are identical because no intra-year compounding occurs.

Higher compounding frequency means interest is reinvested more often, so a smaller nominal rate (APR) can produce the same effective yield (APY). For example, a 5.116% APY requires an APR of approximately 5.00% with monthly compounding (n = 12), but roughly 5.04% with semi-annual compounding (n = 2). The difference grows as APY increases.

Continuous compounding (APR = ln(1 + APY)) is primarily used in theoretical finance, derivatives pricing (Black-Scholes model), and academic contexts. In practice, most banks compound daily (n = 365). The numerical difference between daily and continuous is negligible for typical consumer rates below 30%.

No. The Truth in Lending Act (TILA) definition of APR for loans includes certain fees and closing costs, which makes the regulatory APR higher than the purely mathematical nominal rate. This calculator performs a pure mathematical inversion of the compounding formula. For loan APR that includes fees, you need the lender's specific fee schedule.

An APY of 0% returns an APR of 0% for any compounding frequency. Negative APY values (representing a net loss, such as fees exceeding interest) are mathematically valid and the formula handles them correctly, producing a negative APR. The constraint is that APY must be greater than −100%, since a total loss of principal cannot be expressed as a periodic rate.

This calculator assumes a standard year of 365 days for daily compounding and 12 equal months for monthly. In commercial lending, a 360-day year convention (common in money markets) effectively increases the true cost because interest accrues on a shorter base. If your institution uses ACT/360, the actual effective rate will be approximately 365360 ≈ 1.389% higher than the quoted APR.