How to Calculate Effective Annual Rate (EAR) from APR

Understanding how interest rates apply to loans and investments is important for financial planning. The Annual Percentage Rate (APR) and the Effective Annual Rate (EAR) are two primary ways interest or returns are expressed. While APR is frequently advertised, EAR offers a more accurate representation of the actual cost of borrowing or the true return on an investment, particularly because it accounts for the impact of compounding. Knowing how to convert APR to EAR is fundamental for making informed financial decisions.

Understanding APR and Compounding

The Annual Percentage Rate (APR) represents the stated annual interest rate for a loan or investment. For instance, a loan might advertise a 12% APR. However, this rate does not always convey the total cost if interest is applied more frequently than once a year.

Compounding refers to the process where interest is calculated not only on the initial principal but also on the accumulated interest from previous periods. This “interest on interest” effect can significantly increase the total amount owed on a loan or earned on an investment over time. The frequency of compounding, such as monthly, quarterly, or semi-annually, directly impacts how quickly interest accrues. For example, monthly compounding means interest is calculated and added to the principal 12 times a year, leading to a higher overall cost or return compared to annual compounding, even with the same APR. This accelerated growth makes understanding compounding essential for grasping the true financial implications beyond the stated APR.

The EAR Formula Explained

The Effective Annual Rate (EAR) provides a precise measure of the annual interest rate by incorporating the effect of compounding. It offers a standardized way to compare financial products, regardless of their stated APR or compounding frequency. The EAR formula is expressed as: EAR = (1 + i/n)^n – 1.

In this formula, ‘i’ represents the nominal annual interest rate, which is the Annual Percentage Rate (APR) expressed as a decimal. For example, a 6% APR would be 0.06. The variable ‘n’ denotes the number of compounding periods within one year. This means ‘n’ would be 12 for monthly compounding, 4 for quarterly, 2 for semi-annual, and 365 for daily compounding.

Step-by-Step Calculation Examples

These examples illustrate how different compounding frequencies can result in varying EARs, even when the APR remains constant.

Consider a credit card with an APR of 18% that compounds monthly. To find the EAR, first convert the APR to a decimal, so 18% becomes 0.18. Since interest compounds monthly, there are 12 compounding periods per year (n=12). Using the EAR formula, EAR = (1 + 0.18/12)^12 – 1, the calculation is: (1 + 0.015)^12 – 1 = (1.015)^12 – 1 ≈ 1.1956 – 1 = 0.1956. The EAR is approximately 19.56%. This shows that an 18% APR compounded monthly effectively costs nearly 20% annually.

Now, consider an investment account offering an APR of 5% compounded quarterly. Here, the APR as a decimal is 0.05. With quarterly compounding, there are 4 compounding periods per year (n=4). Applying the formula, EAR = (1 + 0.05/4)^4 – 1, the calculation is: (1 + 0.0125)^4 – 1 = (1.0125)^4 – 1 ≈ 1.0509 – 1 = 0.0509. The EAR is approximately 5.09%. This demonstrates how compounding more frequently than annually increases the actual return beyond the stated APR. Comparing these examples highlights the importance of EAR in revealing the true cost or return, allowing for a more accurate assessment of financial products.