Effective Annual Rate Calculator

What is the Effective Annual Rate (EAR)?

The Effective Annual Rate (EAR), also known as the effective annual interest rate, effective interest rate (EIR), or annual equivalent rate (AER), represents the actual annual interest earned on an investment or paid on a liability after accounting for intra-year compounding.

When financial institutions advertise interest rates, they commonly quote the nominal annual rate or stated Annual Percentage Rate (APR). However, because interest is typically added back to principal multiple times per year (such as monthly, daily, or quarterly), each subsequent compounding cycle accrues interest on prior interest. The EAR quantifies the true economic yield or total borrowing cost over one full calendar year. For comparing consumer savings instruments, evaluate yields with our APY calculator, or convert loan figures directly using our APR to APY calculator. To model multi-year balance growth with periodic contributions, explore our compound interest calculator, or analyze fixed-term deposit yields with our CD calculator.

The Effective Annual Rate mathematical formula

For interest that compounds discretely $m$ times per year at a nominal stated annual rate $i$, the Effective Annual Rate is determined by compounding the periodic rate:

Where:

• $i$ is the stated nominal annual interest rate as a decimal (for instance, 0.06 for 6.00%).
• $m$ is the compounding frequency per year (1 for annual, 2 for semi-annual, 4 for quarterly, 12 for monthly, 52 for weekly, or 365 for daily).
• $i / m$ represents the periodic interest rate charged or earned in each single interval.

When compounding occurs continuously across infinitesimal intervals, the formula relies on Euler's mathematical constant $e \approx 2.71828$:

Inverting the formula: Calculating nominal stated rate from EAR

When you know the target effective yield and need to determine the required nominal stated rate, the formula is inverted by solving for $i$:

For continuous compounding, inverting the formula employs the natural logarithm:

If you need to convert a nominal rate directly from one compounding frequency to another while maintaining the exact same effective return, use our equivalent interest rate calculator. To evaluate how anticipated inflation erodes your effective return into real purchasing power, test your rate with our Fisher effect calculator.

Published worked example

Consider an investor evaluating a corporate debt security or bank certificate offering a nominal annual rate of 6.00% ($i = 0.06$) on a $10,000 principal deposit. Let us calculate how compounding frequencies change the true annual yield:

• Semi-Annual Compounding ($m = 2$):
  $\mathrm{EAR} = (1 + 0.06/2)^2 - 1 = (1.03)^2 - 1 = 6.0900\%$
  Annual interest earned: $609.00 (a compounding boost of +$9.00 over the simple $600.00 base).
• Quarterly Compounding ($m = 4$):
  $\mathrm{EAR} = (1 + 0.06/4)^4 - 1 = (1.015)^4 - 1 = 6.1364\%$
  Annual interest earned: $613.64 (a compounding boost of +$13.64).
• Monthly Compounding ($m = 12$):
  $\mathrm{EAR} = (1 + 0.06/12)^{12} - 1 = (1.005)^{12} - 1 = 6.1678\%$
  Annual interest earned: $616.78 (a compounding boost of +$16.78).
• Daily Compounding ($m = 365$):
  $\mathrm{EAR} = (1 + 0.06/365)^{365} - 1 = (1.00016438)^{365} - 1 = 6.1831\%$
  Annual interest earned: $618.31 (a compounding boost of +$18.31).
• Continuous Compounding:
  $\mathrm{EAR} = e^{0.06} - 1 \approx 6.1837\%$
  Annual interest earned: $618.37 (representing the theoretical upper bound of intra-year compounding).

Why EAR is essential for financial comparisons

Nominal rates can create misleading comparisons because two products quoting the identical nominal interest rate of 6.00% will deliver different financial results if one compounds annually while the other compounds daily.

In lending, financial institutions frequently highlight the nominal APR because it appears lower than the actual effective cost of borrowing. Conversely, for deposit accounts and savings vehicles, institutions often emphasize the APY or EAR because intra-year compounding yields a higher published percentage. Standardizing all financing options to their Effective Annual Rate enables a direct, objective comparison across consumer loans, credit facilities, bonds, and high-yield savings accounts. If you are specifically analyzing fixed-income securities and periodic coupon reinvestment, use the dedicated effective annual yield calculator.

Frequently asked questions