# Equivalent Rate Calculator | AER

Created by Tibor Pal, PhD candidate
Reviewed by Bogna Szyk and Jack Bowater
Based on research by
Cipra T. Financial and Insurance Formulas (2006)
Last updated: Jul 10, 2022

The equivalent rate calculator converts an interest rate from one compound frequency to another while keeping the effective interest rate constant. You can use it on a savings account or investment product where interest compounds more than once a year.

Many financial calculators employ the equivalent interest rate for their computations where the compounding and payment frequencies differ. You may check the calculators below for some examples:

Below we show you how to use the calculator, and also, you can study the equivalent interest rate formula and learn what the annual equivalent rate (AER) is.

## What is the annual equivalent rate (AER)?

The annual equivalent rate (AER), or Effective Annual Interest Rate, is the interest rate on a loan or financial product** that is adjusted for compounding over a year. In other words, the effective annual interest rate is the rate of interest that you can earn in a year after considering the compounding effect.

r = (1 + i/m)m - 1

For example, the annual equivalent rate of an investment having a nominal interest rate of 5% compounded monthly is equivalent to 5.116%. 5% compounded monthly has a periodic rate of 5/12 ≈ 0.4166%. After one year, the initial capital is increased by a factor of (1 + 0.004166)12 ≈ 1.05116, which means AER = 5.116%.

When the frequency of compounding is infinity, the calculation will be:

r = ei - 1 where e stands for the exponent.

The effective interest rate is a special case of the internal rate of return.

If the monthly periodic interest rate p is known and remains constant throughout the year, the effective annual rate can be estimated in the following way:

r = (1 + p)12 -1

## What is the equivalent interest rate formula?

The formula, what we also applied in the equivalent rate calculator, takes the following form:

$i = q * \bigg [ \bigg (1 + \frac{r}{m} \bigg ) ^ \frac{m}{q} - 1 \bigg]$

Where:

• $r$ - nominal annual interest rate;
• $m$ - initial compounding frequency;
• $q$ - desired compounding frequency; and
• $i$ - equivalent interest rate.

For example, you have a loan at an annual rate of 5% that compounds monthly ($m=12$) but you make payments quarterly ($q=4$) so your interest will be calculated quarterly. What is the equivalent annual rate that coincides with quarterly compounding?

$i = 5 * \bigg [ \bigg (1 + \frac{0.05}{12} \bigg ) ^ \frac{12}{4} - 1 \bigg]$
$i = 5.02086 \%$

## How to use the equivalent interest rate calculator?

You need to set four out of the five variables in the equivalent rate calculator to obtain the fifth variable as a result.

• Nominal interest rate: the nominal annual interest rate;
• Compounding frequency: the number of times compounding occurs annually. It is the frequency you would like to transform into another frequency;
• New compounding frequency;
• Equivalent interest rate; and
• Effective rate (annual equivalent rate - AER).

## Annual equivalent rates in different compound frequencies

The following table shows you some nominal annual rates with their corresponding effective rate at different compounding frequencies.

CSV To HTML using codebeautify.org
Nominal annual rate Semi-annual Quarterly Monthly Daily Continuous
1% 1.003% 1.004% 1.005% 1.005% 1.005%
5% 5.063% 5.095% 5.116% 5.127% 5.127%
10% 10.250% 10.381% 10.471% 10.516% 10.517%
15% 15.563% 15.865% 16.075% 16.180% 16.183%
20% 21.000% 21.551% 21.939% 22.134% 22.140%
30% 32.250% 33.547% 34.489% 34.969% 34.986%
40% 44.000% 46.410% 48.213% 49.150% 49.182%
50% 56.250% 60.181% 63.209% 64.816% 64.872%

If you would like to learn more about the annual equivalent rate, you may check our APY calculator, which is based on the same concept that we've applied in this equivalent rate calculator.

Tibor Pal, PhD candidate
Nominal interest rate
%
Compounding frequency
New compounding frequency
Equivalent interest rate
%
Effective Annual Interest Rate (AER)
%
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