## What Is an Effective Annual Interest Rate?

The effective annual interest rate is the interest rate that is actually earned or paid on an investment, loan or other financial product due to the result of compounding over a given time period. It is also called the effective interest rate, the effective rate or the annual equivalent rate.

## The Formula for the Effective Annual Interest Rate Is

$\begin{aligned} &Effective\ Annual\ Interest\ Rate=\left ( 1+\frac{i}{n} \right )^n-1\\ &\textbf{where:}\\ &i=\text{Nominal interest rate}\\ &n=\text{Number of periods}\\ \end{aligned}$

#### The Effective Annual Interest Rate

## What Does the Effective Annual Interest Rate Tell You?

The effective annual interest rate is an important concept in finance because it is used to compare different products—including loans, lines of credits, or investment products like deposit certificates—that calculate compounded interest differently.

For example, if investment A pays 10 percent, compounded monthly, and investment B pays 10.1 percent compounded semi-annually, the effective annual interest rate can be used to determine which investment will actually pay more over the course of the year.

## Example of How to Use the Effective Annual Interest Rate

The nominal interest rate is the stated rate on the financial product. In the example above, the nominal rate for investment A is 10 percent and 10.1 percent for investment B. The effective annual interest rate is calculated by taking the nominal interest rate and adjusting it for the number of compounding periods the financial product will experience in the given period of time. The formula and calculations are as follows:

**Effective annual interest rate**= (1 + (nominal rate / number of compounding periods)) ^ (number of compounding periods) - 1- For investment A, this would be: 10.47% = (1 + (10% / 12)) ^ 12 - 1
- And for investment B, it would be: 10.36% = (1 + (10.1% / 2)) ^ 2 - 1

As can be seen, even though investment B has a higher stated nominal interest rate, because it compounds fewer times over the year, the effective annual interest rate is lower than the effective rate for investment A. It is important to calculate the effective rate because if an investor were to invest, for example, $5,000,000 into one of these investments, the wrong decision would cost over $5,800 per year.

As the number of compounding periods increases, so does the effective annual interest rate. Quarterly compounding produces higher returns than semi-annual compounding, monthly compounding more than quarterly, and daily compounding more than monthly. Below is a breakdown of the results of these different compound periods with a 10% nominal interest rate:

- Semi-annual = 10.250%
- Quarterly = 10.381%
- Monthly = 10.471%
- Daily = 10.516%

There is a limit to the compounding phenomenon. Even if compounding occurs an infinite amount of times—not just every second or microsecond but continuously—the limit of compounding is reached. With 10%, the continuously compounded effective annual interest rate is 10.517%. The continuous rate is calculated by raising the number "e" (approximately equal to 2.71828) to the power of the interest rate and subtracting one. It this example, it would be 2.171828 ^ (0.1) - 1.