Effective Annual Rate (EAR) Calculator

Effective Annual Rate Formula

The standard formula for EAR is:

i = (1 + r/m)^m − 1

Where:

• i = effective annual rate (decimal form)
• r = nominal interest rate (decimal form) → r = R / 100
• R = nominal interest rate (percent form)
• m = number of compounding periods per year

Example with Continuous Compounding

If compounding is continuous, the formula changes to:

i = e^r − 1

Where e is Euler’s number (≈ 2.71828).

After finding your effective annual return, explore these related calculators to deepen your analysis:

• Compare with stated rates using the Nominal Interest Rate Calculator.
• Translate rates to other periods using the Periodic Interest Rate Calculator.
• Examine compounding growth in the Compound Interest Calculator.
• For quick reference, consult the Interest Rate Table.

EAR Calculator Variables Explained

• Annual Interest Rate (R): The nominal or stated rate in percent (e.g., 6%).
• Compounding Periods (m): How many times per year the interest is compounded (12 for monthly, 4 for quarterly, 52 for weekly, etc.).
• Continuous Compounding: Enter “Continuous” (or equivalent) to calculate using the continuous formula.
• Effective Annual Rate (I): The result, expressed as a percentage, that shows the real yearly rate.

Example: Comparing Loan Offers

Suppose two banks offer the following:

1. Bank A: 7.24% interest, compounded quarterly.
2. Bank B: 7.18% interest, compounded weekly.

Which one is cheaper?

Bank A Calculation:

Formula: i = (1 + r/m)^m − 1

• r = 0.0724
• m = 4

i = (1 + 0.0724/4)^4 − 1
i = 0.074389 → 7.439%

Bank B Calculation:

Formula: i = (1 + r/m)^m − 1

• r = 0.0718
• m = 52

i = (1 + 0.0718/52)^52 − 1
i = 0.074387 → 7.439%

Result: The EAR is practically identical. Despite different nominal rates and compounding frequencies, both loans cost nearly the same annually.

EAR Comparison

Nominal Rate	Compounding Frequency	EAR (%)
6.00%	Annual	6.000%
6.00%	Semi-Annual (2x)	6.090%
6.00%	Quarterly (4x)	6.136%
6.00%	Monthly (12x)	6.168%
6.00%	Weekly (52x)	6.183%
6.00%	Daily (365x)	6.185%
6.00%	Continuous	6.186%

This table illustrates how compounding frequency increases the EAR, even when the nominal rate stays constant.

Why EAR Matters

The Effective Annual Rate is crucial because it levels the playing field:

• Loan Comparisons: Credit cards, personal loans, and mortgages often advertise nominal rates that hide the real cost. EAR exposes the true rate.
• Investment Returns: Bonds, savings accounts, and certificates of deposit (CDs) may look similar, but EAR shows which earns you more over time.
• Credit Cards: Many credit cards compound interest daily. EAR helps you understand how quickly debt grows.

Step-by-Step Guide: How to Calculate EAR Manually

1. Convert the nominal rate into decimal form (R ÷ 100).
2. Divide by the compounding periods (r ÷ m).
3. Add 1 to the result (1 + r/m).
4. Raise it to the power of m ((1 + r/m)^m).
5. Subtract 1 to get EAR in decimal form.
6. Multiply by 100 to convert to a percentage.

Example:

Nominal rate = 8%, compounded monthly.

• r = 0.08
• m = 12

EAR = (1 + 0.08/12)^12 − 1
EAR = (1 + 0.006666)^12 − 1
EAR = 0.0830 → 8.30%

EAR vs APR: Key Differences

Many confuse EAR with APR (Annual Percentage Rate). While both measure yearly interest, they differ in application:

• APR: Includes interest plus some fees, but ignores compounding.
• EAR: Excludes fees, but includes compounding.

For borrowers: APR shows the legal disclosure cost, but EAR shows the true compounding effect.

Practical Applications of EAR

• Comparing Mortgages: A 6.9% mortgage compounded semi-annually may actually be more expensive than a 7% mortgage compounded annually.
• Evaluating Savings Accounts: Banks often use EAR (sometimes called APY—Annual Percentage Yield) to show actual returns.
• Corporate Finance: Businesses use EAR to evaluate the real cost of short-term loans or credit lines.

EAR Calculator: Key Takeaways

• EAR reveals the true annual interest rate after compounding.
• Higher compounding frequency = higher EAR, even if the nominal rate stays the same.
• Use the EAR calculator to compare loans, investments, and savings accounts on equal terms.
• Always check both APR and EAR to fully understand borrowing costs.

The Effective Annual Rate (EAR) Calculator is more than just a formula—it’s a reality check. It exposes the hidden effect of compounding and ensures you never compare financial products unfairly. Whether you’re a borrower, investor, or simply trying to grow your savings, understanding EAR empowers you to make smarter money decisions.