APY to APR Calculator: Convert Annual Yield to Rate

This calculator converts an Annual Percentage Yield (APY) to an Annual Percentage Rate (APR) using the stated compounding frequency. You enter the APY as a percentage and choose how often interest compounds (daily, monthly, quarterly, etc.), and you get back the equivalent nominal APR. This is useful when comparing loans or savings accounts that quote different rate types.

Formula

The relationship between APR and APY is:

APY = (1 + APR/n)^n – 1

where n is the number of compounding periods per year. To solve for APR given APY:

APR = n × [(1 + APY)^(1/n) – 1]

In this calculator, APY is entered as a percentage (e.g., 5 for 5%), and the formula converts it to a decimal before applying the root. The result is the nominal annual rate that, when compounded n times per year, yields exactly the entered APY.

Worked Example

Using the default values: APY = 5.00%, compounding = Daily (n = 365).

1. Convert APY to decimal: 5.00 / 100 = 0.05
2. Add 1: 1 + 0.05 = 1.05
3. Take the 365th root: 1.05^(1/365) ≈ 1.00013361
4. Subtract 1: 1.00013361 – 1 = 0.00013361
5. Multiply by 365: 0.00013361 × 365 = 0.04877
6. Convert to percentage: 0.04877 × 100 = 4.877%

So a 5% APY with daily compounding corresponds to an APR of about 4.877%. The calculator rounds to three decimal places, giving 4.877%.

Why the Difference Matters

APR (Annual Percentage Rate) is the simple interest rate per year, without compounding. APY (Annual Percentage Yield) includes the effect of compounding within the year. Because of compounding, APY is always higher than APR for the same nominal rate, unless compounding happens only once per year (then they are equal).

When you see a savings account advertising 5% APY, the actual interest credited each day is based on a lower APR. Conversely, a loan with a 5% APR will cost more than 5% per year if interest compounds monthly, so the effective rate (APY) would be about 5.12%.

How to Choose Between APY and APR

• For savings or investments: APY is the better measure because it shows what you actually earn after compounding. Compare APYs when choosing a high-yield savings account or CD.
• For loans or credit cards: APR is the standard quoted rate, but the true cost depends on compounding frequency. A loan with a lower APR but more frequent compounding could cost more than a higher APR with annual compounding.
• When comparing different products: Always compare the same metric—either both APRs or both APYs—never mix them.

Common Mistakes and Misconceptions

One common error is assuming APR and APY are interchangeable. They are not. Another is forgetting that the compounding frequency matters: daily compounding produces a larger difference between APR and APY than annual compounding. For example, a 5% APR compounded daily gives an APY of about 5.13%, while compounded annually it gives exactly 5%.

People also confuse the nominal APR with the effective annual rate. The APR is the quoted rate before compounding; the effective rate (APY) is what you actually pay or earn. When comparing offers, always check whether the rate is nominal or effective.

Limits of This Calculation

This calculator works for simple interest with a fixed compounding schedule. It does not account for fees, changing rates, or continuous compounding (where n approaches infinity). For continuous compounding, the formula would be APR = ln(1 + APY), which gives a slightly lower APR than daily compounding. Most consumer products use daily or monthly compounding, so the calculator covers the common cases accurately.

Also, the calculator assumes the APY is already known and the APR is being derived. In practice, lenders often quote APR and you might want to find the APY. That is the reverse calculation, which can be done with the same formula rearranged.

Practical Examples

• Savings account: A bank offers 4.5% APY compounded monthly. The APR is 4.5 / 12 = 0.375% per month, or about 4.41% nominal. The calculator gives 4.414%.
• Credit card: A card quotes 18% APR compounded daily. The APY is (1 + 0.18/365)^365 – 1 ≈ 19.72%. To convert back, enter 19.72% APY with daily compounding to get 18% APR.
• CD with quarterly compounding: A 3% APY CD compounded quarterly has an APR of about 2.96%.

Frequently Asked Questions

Is APR always lower than APY?

Yes, for any compounding more frequent than annually. If compounding happens once per year, APR equals APY. With more frequent compounding, APY exceeds APR because interest is earned on interest more often.

What is the difference between APR and APY in simple terms?

APR is the base interest rate without compounding. APY is the actual rate you earn or pay after compounding is factored in. Think of APR as the sticker price and APY as the total cost including compounding.

Can I use this calculator for loans too?

Yes, the math is the same. If a loan has an APR of 6% compounded monthly, the APY is about 6.17%. Conversely, if you know the APY, you can find the nominal APR with this calculator.

What if my compounding is continuous?

For continuous compounding, use the formula APR = ln(1 + APY). For example, a 5% APY with continuous compounding corresponds to an APR of about 4.879%, which is very close to the daily compounding result of 4.877%.

Why do banks advertise APY instead of APR for savings?

Because APY is a larger number, it makes the account look more attractive. It also gives a more accurate picture of what you will earn, which is why regulators require it for deposit accounts.

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in apy | APY | % | 5.00 | 0 | 100
sel freq | Compounding | Daily=365, Monthly=12, Quarterly=4, Semi-annually=2, Annually=1
out apr | APR | % | ((1 + apy/100)^(1/freq) - 1) * freq * 100 | 3
out eff | Effective annual rate | % | apy | 2