# Effective Annual Rate (EAR)

There are two types of discount rates (interest rates) to consider when carrying out time value of money calculations, nominal rates and effective rates.

The basic distinction is that nominal rates do not allow for the effects of compounding, whereas effective rates do allow for the effects of compounding. An effective rate is a nominal rate adjusted to allow for the effects of compounding.

## How to Calculate Effective Annual Interest Rate

Both nominal and effective rates can be quoted for any period but are usually quoted for a year.

To convert a nominal rate into an effective annual rate we need to allow for the effect of compounding over a year, this will change depending on how many compounding periods there are in year.

For example, if an amount is deposited into an account for a year earning compound interest, it will grow from its present value (PV) to a future value (FV), the difference between the two values is the amount of compound interest earned on the deposit for the year. This is discussed more fully in our How to calculate compound interest article.

The effective rate of interest is then calculated by dividing the interest by the original deposit (PV)

```Compound interest = FV - PV
Effective rate of interest = (FV / PV) - 1```

Using the we can find a value for FV / PV as follows:

```FV = PV x (1 + i)n
FV / PV = (1 + i)n
i = nominal rate per compounding period
n = number of compounding periods per year
If r is the annual nominal rate, then we know that
i = r / n, and
FV / PV = (1 + r / n)n```

Using this value for FV / PV in our interest rate equation, we have the effective annual rate formula which shows the EAR in terms of the nominal rate of interest as follows:

```EAR = (1 + r / n )n - 1
```
Where
r = Annual nominal rate of interest
n = Number of compounding periods in a year

## Effective Annual Rate Example

Suppose a savings account has a stated annual nominal rate of 8%, and compounding takes place every 6 months or 2 times per year.

 Year (Nominal rate 8%) Compounding period 1 Compounding period 2 Compound 8%/2 ↑ Compound 8%/2 ↑

The effective annual rate (EAR) can now be calculated using the formula as follows:

```Effective rate of interest = (1 + r / n )n - 1
r = annual nominal rate = 8%
n = compounding periods in a year = 2
Effective annual rate = (1 + 8% / 2 )2 - 1
Effective annual rate = 8.16%
```

The effective annual interest rate is higher than the annual nominal rate due to the effect of compounding during the year.

To see this in action, had 100.00 been deposited into the account at the start of compounding period 1, then by the end of compound period 1 (6 months) the interest in the account using the stated nominal rate of 8% would be

```Balance at the end of period 1 = 100.00 + 100.00 x 8%/2 = 104.00
```

This is now compounded for one more period and the balance at the end of compounding period 2 (the end of the year) is:

```Balance at the end of period 2 = 104.00 + 104.00 x 8%/2 = 108.16
```

The interest earned on the account is 108.16 less the original sum of 100.00 and is equal to 8.16. The effective rate of interest on the account for a year is the interest (8.16) divided by the sum invested (100.00), so is equal to 8.16 / 100.00 = 8.16%, the same answer as the formula gave.

 Year (Nominal rate 8%) Compounding period 1 Compounding period 2 Compound 8.16% ↑

## Effective Annual Rate Example 2

If the compounding had been daily, then the number of compounding periods in the year would be 365 and the interest compounded daily formula would give:

```Effective annual rate = (1 + r / n )n - 1
r = Annual nominal rate = 8%
n = Compounding periods in a year = 365
Effective annual rate = (1 + 8% / 365 )365 - 1
Effective annual rate = 8.33%
```

As the number of compounding periods increases the effective annual rate increases. Notice that in each case the nominal rate has remained the same.

A nominal rate of 8% compounded every 6 months gives an effective annual rate (EAR) of 8.16%, and
A nominal rate of 8% compounded every day gives an effective annual rate (EAR) of 8.33%.

The nominal rates cannot be compared as the compounding periods are not the same, the effective rate has allowed for this compounding and so can be compared.

## Effective Annual Rate Example 3

Consider another (unusual) example where compounding takes place once every two years.

 Year (Nominal rate 8%) Compounding period 1 Compound ↑

In this example there is only 1/2 a compounding period (2 years) in the year, and the effective annual interest rate is given as:

```Effective annual rate = (1 + r / n )n - 1
r = annual nominal rate = 8%
n = compounding periods in a year = 1/2
Effective annual rate = (1 + 8% / (1/2) )1/2 - 1
Effective annual rate = 7.70%
```

The effective annual rate is now less then the nominal rate as compounding is only taking place once every two years.

## Compounding period is one year

The situation where the compounding takes place once per year at the end of the year is shown below.

 Year (Nominal rate 8%) Compounding period 1 Compound ↑

The effective annual rate is calculated as follows:

```Effective annual rate = (1 + r / n )n - 1
r = Annual nominal rate = 8%
n = Compounding periods in a year = 1
Effective annual rate = (1 + 8% / 1 )1 / 1 - 1
Effective annual rate = 8.0%
```

When compound takes place once in a year the effective annual rate is the same as the nominal rate.

The effective annual rate adjusts the stated nominal rate to allow for the effects of compounding. Different compounding periods earn different amounts of interest and the effective annual rate allows comparisons to be made by converting these to an equivalent rate as if compounding took place once at the end of the year, hence the alternative name of annual equivalent rate (AER).

The effective annual rate is a special case where the number of compounding periods the rate is required for (1 year) is the same as the number of compounding periods in a year. Effective interest rates can be calculated for periods other than one year, and this is discussed in more detail in our effective interest rate for any time period tutorial.

Effective Annual Rate (EAR) November 6th, 2016

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