APY to APR Calculator

Understanding APY to APR conversion

When financial institutions advertise high-yield savings accounts, certificates of deposit (CDs), and money market funds, they headline the Annual Percentage Yield (APY). APY captures the total interest you earn over a full year, incorporating compound interest. However, when modeling cash flows, programming loan amortizations, or calculating periodic monthly interest payouts, you need the underlying nominal interest rate: the Annual Percentage Rate (APR).

While APR reflects the simple annualized base rate before compounding, APY reflects the realized annual return after compounding. If you already have a stated nominal rate and want to find the forward yield, use our APR to APY converter or project multi-year deposit growth with the APY calculator. To evaluate loan terms with upfront origination fees and closing points, explore our dedicated APR calculator, or analyze fine rate movements with our basis point calculator.

The mathematical formula: Converting APY to APR

To isolate the nominal annual rate from an effective annual yield, we invert the standard compound interest formula. When interest compounds discretely $m$ times per year, the relationship is defined as:

Where:

• $\text{APY}$ is the stated annual percentage yield expressed as a decimal (for example, 0.05 for 5.00%).
• $m$ is the number of compounding cycles per year (365 for daily, 52 for weekly, 12 for monthly, 4 for quarterly, 2 for semi-annually, 1 for annually).
• $\text{APR}$ is the resulting nominal annual percentage rate in decimal form.

For continuous compounding, where interest accrues constantly at every infinitesimal moment, the formula uses the natural logarithm:

Published worked example

Suppose a high-yield certificate of deposit offers a 5.00% APY ($0.05$). Let us determine the required nominal APR under different compounding frequencies:

• Monthly Compounding ($m = 12$):
  $\text{APR} = 12 \left[(1 + 0.05)^{1/12} - 1\right] = 12 \times [1.0040741 - 1] = 4.889\%$
  The periodic monthly rate is $4.8889\% / 12 = 0.4074\%$.
• Daily Compounding ($m = 365$):
  $\text{APR} = 365 \left[(1 + 0.05)^{1/365} - 1\right] = 365 \times [1.0001337 - 1] = 4.879\%$
  The periodic daily rate is $4.8790\% / 365 = 0.013367\%$.
• Quarterly Compounding ($m = 4$):
  $\text{APR} = 4 \left[(1 + 0.05)^{1/4} - 1\right] = 4 \times [1.012272 - 1] = 4.909\%$
• Continuous Compounding ($m = \infty$):
  $\text{APR} = \ln(1 + 0.05) = \ln(1.05) = 4.879\%$

Notice that more frequent compounding allows a smaller nominal APR to achieve the exact same 5.00% annual yield. On an initial deposit of $10,000, all these compounding frequencies produce exactly $500.00 in interest over one full year.

Why convert APY back to APR?

There are several practical financial scenarios where finding the nominal APR is necessary:

• Calculating periodic payouts: If an annuity or bond pays interest quarterly or monthly, the periodic distribution check equals the principal multiplied by the periodic interest rate ($\text{APR} / m$), not the APY divided by $m$.
• Comparing debt against investments: Lenders quote debt in APR under the Truth in Lending Act (TILA), whereas deposit accounts quote APY under the Truth in Savings Act (TISA). Converting APY to APR provides a direct apple-to-apple comparison.
• Amortization schedules: Standard loan calculators require the nominal rate to compute fixed monthly payments. To calculate loan schedules with principal and interest breakdowns, use our EMI calculator or advanced loan calculator.

Frequently asked questions