Effective Annual Interest Rate Calculator

The effective annual rate (EAR) tells the true value of an interest rate when compounding is taken into account. Even if two accounts show the same nominal rate, a higher frequency of compounding will result in more money paid or received each year.

This calculator allows you to convert any nominal rate (the rate shown) to an effective annual rate with different frequencies of compounding. It also supports continuous compounding; when the nominal rate field is left blank, it calculates the rate that would be shown for a desired effective annual rate.

What is an effective annual interest rate?

The effective annual rate (EAR) is also known as the Annual Equivalent Rate (AER) or Annual Percentage Yield (APY), and it's the actual yearly interest rate taking into account compound interest. The nominal interest rate only gives you the value of the interest rate, without telling you how often the interest is calculated.

Because each time the interest is calculated and compounded, it's applied to a slightly higher balance, the amount actually received or paid per year will be slightly more than the stated figure. The effective annual rate expresses this difference in one number, which makes it easier to compare different products. It doesn't depend on the deposit, but only on the interest rate and how often the interest is compounded.

Here's how to use this calculator.

Enter the nominal annual interest rate and select the compounding frequency: annually, semi-annually, quarterly, monthly, weekly or daily. The effective interest rate will be immediately displayed, along with the amount of the compound interest (the difference between the effective and nominal rates) and the interest rates for each compounding period. If the continuous compounding option is enabled, you can display the maximum that would be achieved by infinitely frequent compounding. Conversely, leave the field for the nominal rate blank and enter the desired annual effective rate to calculate the corresponding nominal rate. There are also two additional toggle functions: one to compound interest over multiple periods, and another to apply the effective rate to an initial balance and display the result in currency units.

Formula for calculating effective interest rate

For an interest rate that is compounded a certain number of times per year, the effective annual interest rate will be as follows.

In this case, i is the nominal annual interest rate as a decimal, and n is the number of compounding periods per year. For example, a bank account with an annual interest rate of 5 percent that pays interest monthly has i equal to 0.05 and n exactly 12.

The published interest rate of 5 percent is actually about 5.116 percent per year. The same value can be calculated in a spreadsheet using the EFFECT function: EFFECT(nominal_rate, npery) where npery is the number of compounding periods per year. The following table shows each input parameter and examples.

Continuous interest rate

The more frequently interest is compounded, the shorter each period becomes and the number of periods approaches infinity. The effective rate does not increase infinitely but rather approaches a limit which is given by the exponential constant e. This upper limit is continuous interest.

If the interest rate is 5%, then continuous compounding gives a value of e raised to the 0.05th power minus 1, which is approximately 5.127 percent. Since daily compounding is already very close to this value, converting from monthly to daily and then continuous compounding will result in ever smaller changes in the outcome.

Nominal and effective interest rates

The nominal interest rate is the value quoted by banks while the effective interest rate is what actually applies. Lending institutions often quote lower nominal rates while savings accounts often show higher effective or annual percentage yields (APY). Comparing only the stated nominal rates can be misleading because of differences in compounding frequency. When both are converted to effective rates, they can be compared on an equal basis. The same is true for the relationship between APR (annual percentage rate) and APY. The APR is the nominal interest rate while the APY is the effective interest rate, and this calculator allows you to convert from one to the other.

The effect of compounding frequency on outcome

The more frequently interest is compounded, the higher the effective rate of interest will be compared to the nominal rate, but the increase becomes smaller with each additional compounding. The following table shows the effect for some common nominal rates.

The comparison and charting tool allows you to check the situation for an entered interest rate taking into account all compounding frequencies including continuous compounding.

Applications of the effective interest rate.

When comparing two interest rates that use different compounding periods, you should consider the effective annual rate. This will allow you to see which savings account is actually offering a higher rate of return and will give you an idea of what your actual cost on loans or credit cards will be if they compound monthly or daily. It can also be helpful when comparing bonds or certificates of deposit with accounts that have their interest rates listed as annual percentage yields (APYs). The effective annual rate is a fair way to compare returns because it takes into account the differences in compounding periods.

Important points for an accurate comparison:

Match the compounding frequency to the product in question. Most credit cards and many savings accounts compound daily, while mortgages and other loans usually use monthly compounding. Make sure you're comparing the same type of interest rate - either nominal rates with nominal rates or effective annual rates with effective annual rates - and avoid confusion. If a savings account gives an APY figure, that's already the effective annual rate and can be directly compared to the value produced by this calculator.