Effective Interest Rate or Annual Equivalent Rate

Effective Interest Rate Definition

Effective interest Rate also known as the effective annual interest rate is the rate of interest that is actually paid by the person or actually earned by the person on the financial instrument which is calculated by considering the effect of the compounding over the period of the time.

When an investor invests in an instrument, she earns an interest.

However, there’s a twist in the equation.

The investor doesn’t exactly earn what is written as an interest rate on the instrument.

Rather she earns an effective interest rate which is called annual equivalent rate.

But how we would calculate the annual equivalent rate (AER)?

First, let’s look at the AER formula. And then we will see the interpretation and practical examples.

Formula

The Effective Interest Rate formula is very simple.

Annual Equivalent Rate or Effective Interest Rate Formula = (1 + i/n) ⁿ– 1

Here, i = the annual interest rate that has been mentioned in the instrument.

n = It represents the number of compounding periods per year.

Interpretations

Compounding changes the interest rate.

That’s why the interest rate written on the instrument isn’t an effective interest rate for the investor.

For example, if an 11% interest rate is written on the instrument and the interest rate gets compounded four times a year, then the effective interest rate can’t be 11%.

What would it be then?

It would be – (1 + i/n)ⁿ– 1 = (1 + 0.11/4) ⁴– 1 = 1.1123 – 1 = 0.1123 = 11.23%.

That means 11.23% would be the effective interest rate for the investor.

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Even if the change is meager, it’s not the same as the annual interest rate mentioned in the instrument.

Now, we will look at a few practical examples. In these examples, we will go deep and understand the nitty-gritty of the Effective Interest Rate.

Example

In this section, we will take three examples and will see how the Effective Interest Rate works.

The first two examples would be simple examples.

In the third example, we will go deep.

Example#1

Ting bought a particular instrument. The interest rate mentioned on the instrument is 16%. He has invested around $100,000. The instrument compounds annually. What would be the effective interest rate (AER) for this particular instrument? How much he would get every year as an interest?

This example has a twist.

This example illustrates why the mentioned annual interest rate and the effective interest rate aren’t the same.

The effective interest rate and the annual interest rate aren’t always the same because the interest gets compounded a number of times every year. Sometimes, the interest rate gets compounded semi-annually, quarterly, or monthly. And that’s how the effective interest rate (AER) differs from the annual interest rate.

This example shows you that.

Let’s calculate.

Since the interest rate gets compounded yearly, here’s would be the effective interest rate formula –

(1 + i/n)ⁿ– 1 = (1 + 0.16/1)¹– 1 = 1.16 – 1 = 0.16 = 16%.

That means in this particular example, there would be no difference between the annual interest rate and annual equivalent rate (AER).

Every year Ting would get the interest of = ($100,000 * 16%) = $16,000 on the instrument.

Example#2

Tong bought a particular instrument. The interest rate mentioned on the instrument is 16%. He has invested around $100,000. The instrument compounds six times a year. What would be the annual equivalent rate (AER) for this particular instrument? How much he would get every year as an interest?

This is just an extension of the previous example.

But there’s a huge difference.

In the previous example, the instrument got compounded once a year which made the annual interest rate similar to the annual equivalent rate.

However, in this case, the scenario is completely different.

Here we have the interest rate that gets compounded six times a year.

So, here’s the formula of the annual interest rate –

(1 + i/n)ⁿ– 1 = (1 + 0.16/6)⁶– 1 = 1.171 – 1 = 0.171 = 17.1%.

You can now see that if the interest rate gets compounded six times a year, the annual equivalent rate becomes quite different.

Now, as we have an effective interest rate, we can calculate the interest Tong will get at the end of the year.

Tong will get = ($100,000 * 17.1%) = $17,100.

If we compare the interest Ting gets in the previous example with the Tong gets as the interest rates compound differently, we will see that there’s around $1100 of difference in interest.

Example#3

Ping has invested in an instrument. She has invested $10,000. The interest rate mentioned in the instrument is 18%. The interest gets compounded monthly. Find out how in the first year Ping will receive interest every month.

This is a much detailed example of the annual equivalent rate.

In this example, we will show how the calculation actually happens without using the Effective Interest Rate formula.

Let’s have a look.

Since the interest rate gets compounded monthly, the actual break-up of the mentioned interest rate per month is = (18/12) = 1.5%.

In the first month, Ping will receive an interest of = (10,000 * 1.5%) = $150.
In the second month, Ping will receive an interest of = {(10,000 + 150) * 1.5%} = (10,150 * 1.5%) = $152.25.
In the third month, Ping will receive an interest of = {(10,000 + 150 + 152.25) * 1.5%} = (10,302.25 * 1.5%) = $154.53.
In the fourth month, Ping will receive an interest of = {(10,000 + 150 + 152.25 + 154.53) * 1.5%} = (10,456.78 * 1.5%) = $156.85.
In the fifth month, Ping will receive an interest of = {(10,000 + 150 + 152.25 + 154.53 + 156.85) * 1.5%} = (10,613.63 * 1.5%) = $159.20.
In the sixth month, Ping will receive an interest of = {(10,000 + 150 + 152.25 + 154.53 + 156.85 + 159.20) * 1.5%} = (10,772.83 * 1.5%) = $161.59.
In the seventh month, Ping will receive an interest of = {(10,000 + 150 + 152.25 + 154.53 + 156.85 + 159.20 + 161.59) * 1.5%} = (10,934.42 * 1.5%) = $164.02.
In the eighth month, Ping will receive an interest of = {(10,000 + 150 + 152.25 + 154.53 + 156.85 + 159.20 + 161.59 + 164.02) * 1.5%} = (11098.44 * 1.5%) = $166.48.
In the ninth month, Ping will receive an interest of = {(10,000 + 150 + 152.25 + 154.53 + 156.85 + 159.20 + 161.59 + 164.02 + 166.48) * 1.5%} = (11264.92 * 1.5%) = $168.97.
In the tenth month, Ping will receive an interest of = {(10,000 + 150 + 152.25 + 154.53 + 156.85 + 159.20 + 161.59 + 164.02 + 166.48 + 168.97) * 1.5%} = (11433.89 * 1.5%) = $171.51.
In the eleventh month, Ping will receive an interest of = {(10,000 + 150 + 152.25 + 154.53 + 156.85 + 159.20 + 161.59 + 164.02 + 166.48 + 168.97 + 171.51) * 1.5%} = (11605.40 * 1.5%) = $174.09.
In the twelfth month, Ping will receive an interest of = {(10,000 + 150 + 152.25 + 154.53 + 156.85 + 159.20 + 161.59 + 164.02 + 166.48 + 168.97 + 171.51 + 174.09) * 1.5%} = (11779.49 * 1.5%) = $176.69.

The total interest Ping got for the year is –

(150 + 152.25 + 154.53 + 156.85 + 159.20 + 161.59 + 164.02 + 166.48 + 168.97 + 171.51 + 174.09 + 176.69) = $1956.18.

Now, if we use the effective interest rate formula and calculate, we will see –

Effective Interest Rate formula = (1 + i/n)ⁿ– 1 = (1 + 0.18/12) ¹²– 1 = 1.195618 – 1 = 0.195618 = 19.5618%.

So, the interest Ping would receive = ($10,000 ^ 19.5618%) = $1956.18.

Annual Equivalent Rate – Effective Interest Rate in Excel

For finding the Effective Interest Rate or Annual Equivalent Rate in excel, we use the excel Function EFFECT.

nominal_rate is the interest rate
nper is the number of compounding periods per year

Let us see the example below

If you have a nominal interest rate of 10% compounded annually, then the Effective Interest Rate or Annual Equivalent Rate is the same as 10%.
If you have a nominal interest rate of 10% compounded six-monthly, then the Annual Equivalent rate is the same as 10.25%.
If you have a nominal interest rate of 10% compounded quarterly, then the Annual Equivalent rate is the same as 10.38%.
If you have a nominal interest rate of 10% compounded monthly, then the Annual Equivalent rate is the same as 10.47%.
If you have a nominal interest rate of 10% compounded daily, then the effective interest rate is the same as 10.52%.