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Understanding the Effective Annual Rate (EAR) Formula

The Effective Annual Rate (EAR) is a crucial concept in finance, particularly when comparing investment options with different compounding frequencies. It represents the true rate of return earned on an investment over a one-year period, taking into account the effect of compounding.

Why EAR Matters

The stated annual interest rate (also known as the nominal interest rate) doesn’t always tell the whole story. If interest is compounded more frequently than annually (e.g., monthly, quarterly, daily), the actual return will be higher than the nominal rate. EAR provides a standardized way to compare investments, ensuring an “apples-to-apples” comparison regardless of compounding frequency.

The EAR Formula

The formula to calculate the Effective Annual Rate is:

EAR = (1 + (i / n))^n – 1

Where:

EAR is the Effective Annual Rate
i is the stated annual interest rate (nominal rate)
n is the number of compounding periods per year

Breaking Down the Formula

i / n: This calculates the interest rate per compounding period. For example, if the annual interest rate is 10% and interest is compounded monthly, the interest rate per month would be 10% / 12 = 0.008333.
1 + (i / n): This represents the growth factor for a single compounding period. Adding 1 to the interest rate per period gives the total value at the end of that period (principal + interest).
(1 + (i / n))^n: This raises the growth factor to the power of ‘n’ (the number of compounding periods per year). This calculates the total growth over the entire year, accounting for the compounding effect. Essentially, it shows how much the initial investment grows by the end of the year.
(1 + (i / n))^n – 1: Finally, subtracting 1 from the result isolates the actual return earned on the investment, expressed as a decimal. This gives you the EAR.

Example

Let’s say you have two investment options:

Option A: 8% annual interest, compounded annually.
Option B: 7.8% annual interest, compounded monthly.

At first glance, Option A might seem better due to the higher stated interest rate. However, let’s calculate the EAR for Option B:

EAR = (1 + (0.078 / 12))^12 – 1

EAR = (1 + 0.0065)^12 – 1

EAR = (1.0065)^12 – 1

EAR = 1.0809 – 1

EAR = 0.0809 or 8.09%

In this case, Option B, with a nominal rate of 7.8% compounded monthly, has an EAR of 8.09%, which is higher than Option A’s 8% EAR. Therefore, Option B is the better investment.

Conclusion

The EAR formula is a powerful tool for comparing investment options with different compounding frequencies. By understanding and using the EAR, investors can make more informed decisions and choose the investments that truly offer the best returns.

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