## What Is an Effective Annual Interest Rate?

The effective annual interest rate is the real return on a savings account or any interest-paying investment when the effects of compounding over time are taken into account. It also reveals the real percentage rate owed in interest on a loan, a credit card, or any other debt.

It is also called the effective interest rate, the effective rate, or the annual equivalent rate.

## The Formula for 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?

A bank certificate of deposit, a savings account, or a loan offer may be advertised with its nominal interest rate as well as its effective annual interest rate. The nominal interest rate does not take reflect the effects of compounding interest or even the fees that come with these financial products. The effective annual interest rate is the real return.

### Key Takeaways

- A savings account or a loan may be advertised with both a nominal interest rate and an effective annual interest rate.
- The effective annual interest rate is the real return paid on savings or the real cost of a loan as it takes into account the effects of compounding and any fees charged.
- The more frequent the compounding periods, the greater the return.

That's why the effective annual interest rate is an important financial concept to understand. You can compare various offers accurately only if you know the effective annual interest rates of each.

## Example of Effective Annual Interest Rate

For example, consider these two offers: Investment A pays 10% interest, compounded monthly. Investment B pays 10.1% compounded semi-annually. Which is the better offer?

In both cases, the advertised interest rate is the nominal interest rate. The effective annual interest rate is calculated by adjusting the nominal interest rate for the number of compounding periods the financial product will experience in a period of time. In this case, that period is one year. 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

Investment B has a higher stated nominal interest rate, but the effective annual interest rate is lower than the effective rate for investment A. This is because Investment B compounds fewer times over the course of the year.

If an investor were to put, say, $5,000,000 into one of these investments, the wrong decision would cost more than $5,800 per year.

### More Frequent Compounding Equals Higher Returns

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%

### The Limits to Compounding

There is a ceiling 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. In this example, it would be 2.171828 ^ (0.1) - 1.