Understanding Effective Annual Rate (EAR)

Nominal Interest Rate vs. Effective Annual Rate

The **Nominal Interest Rate** (also known as the stated rate or Annual Percentage Rate - APR, though APR can sometimes include other fees) is the interest rate quoted by lenders or advertised for investments, typically on an annual basis, *without* considering the effect of intra-year compounding.

The **Effective Annual Rate (EAR)**, also known as the Annual Equivalent Rate (AER) or Effective Annual Yield (EAY), represents the actual annual rate of return an investor earns or a borrower pays once the effect of compounding more frequently than once a year is taken into account. Due to compounding, the EAR will always be equal to or greater than the nominal rate (it's equal only when compounding is annual).

What is Compounding?

Compounding is the process where interest earned during a period is added to the principal amount. In subsequent periods, interest is then earned on this new, larger principal (original principal + accumulated interest). The more frequently interest is compounded within a year (e.g., monthly instead of annually), the greater the impact of compounding, and thus the higher the EAR will be compared to the nominal rate.

Formula for EAR:

The Effective Annual Rate is calculated using the following formula:

EAR = (1 + (i / n))n - 1

Where:

• i = The nominal annual interest rate (as a decimal, e.g., 10% = 0.10).
• n = The number of compounding periods per year (e.g., 1 for annual, 4 for quarterly, 12 for monthly).

Why is EAR Important?

• Accurate Comparison: EAR allows for a true "apples-to-apples" comparison of different financial products (loans, savings accounts, investments) that may have the same nominal interest rate but different compounding frequencies. A loan with a slightly lower nominal rate but more frequent compounding might actually be more expensive than a loan with a slightly higher nominal rate compounded less frequently.
• Understanding True Return/Cost: It reveals the actual annual percentage return you will earn on an investment or the true annual percentage cost you will pay on a loan.

Example:

Suppose you have an investment with a nominal annual rate of 10%:

• If compounded **annually** (n=1): EAR = (1 + (0.10 / 1))¹ - 1 = 0.10 = **10.00%**
• If compounded **semi-annually** (n=2): EAR = (1 + (0.10 / 2))² - 1 = (1.05)² - 1 = 1.1025 - 1 = 0.1025 = **10.25%**
• If compounded **quarterly** (n=4): EAR = (1 + (0.10 / 4))⁴ - 1 = (1.025)⁴ - 1 ≈ 1.10381 - 1 = 0.10381 = **10.38%**
• If compounded **monthly** (n=12): EAR = (1 + (0.10 / 12))¹² - 1 ≈ (1.008333)¹² - 1 ≈ 1.10471 - 1 = 0.10471 = **10.47%**

As you can see, the more frequent the compounding, the higher the Effective Annual Rate for the same nominal rate.

This calculator helps you quickly determine the EAR based on the nominal rate and compounding frequency you provide. It's useful for making more informed financial decisions.