Effective Interest Rate for any Time Period

The effective annual rate adjusts the nominal rate as if compounding takes place once at the end of a year.

However, the effective annual rate is a special case where the rate is required for a period of one year. In time value of money calculations, particularly when calculating annuities, an effective rate for a period other than one year is often needed.

The formula for the effective interest rate for n compounding periods is as follows:

Effective rate = (1 + r / m )n - 1
Where
r = Annual nominal rate of interest
m = Number of compounding periods in a year
n = Number of compounding periods the rate is required for

* When the effective interest rate is required for a period of one year, the number of compounding periods the rate is required for is the same as the number of compounding periods in a year (n = m), and the formula simplifies to the formula for the effective annual rate.

Calculating Effective Interest Rate

The calculation of the effective rate is best demonstrated using examples. The are three situations which can arise depending on whether the period the rate is required for (the effective period) is equal to, greater than, or less than the period over which compounding takes place (the compounding period).

Effective Period = Compounding Period

If the nominal rate is 8% compounded quarterly, what is the effective interest rate for one quarter shown?

In this example, the rate is required for one quarter (the effective period), which is the same as the compounding period. This is shown in the diagram below.

Effective period the same as the compounding period
Year (Nominal rate 8%)
Compounding 1 Compounding 2 Compounding 3 Compounding 4
Effective period

The effective interest rate for the quarter is calculated using the effective rate formula as follows:

Effective interest rate = (1 + r / m )n - 1
r = annual nominal rate = 8%
m = compounding periods in a year = 4
n = number of compounding periods the rate is required for = 1
Effective interest rate = (1 + 8% / 4 )1 - 1
Effective interest rate = 2.00%

If the nominal rate is 8% compounded quarterly, then the effective interest rate for a quarter is 2%.

Notice that as the compounding period and the effective period are the same, the effective rate and the nominal rate are the same, and therefore the effective interest rate for a quarter is simply the nominal rate divided by four = 8%/4 =2%.

Effective Period > Compounding Period

If the nominal rate is 8% compounded quarterly, what is the effective rate for a 6 month period?

In this example, the effective period (6 months) is greater than the compounding period (quarterly), as shown in the diagram below.

Effective period greater than the compounding period
Year (Nominal rate 8%)
Compounding 1 Compounding 2 Compounding 3 Compounding 4
Effective period

The effective interest rate for the 6 month period is calculated using the effective rate formula as follows:

Effective interest rate = (1 + r / m )n - 1
r = annual nominal rate = 8%
m = compounding periods in a year = 4
n = number of compounding periods the rate is required for = 2
Effective interest rate = (1 + 8% / 4 )2 - 1
Effective interest rate = 4.04%

If the nominal rate is 8% compounded quarterly, then the effective interest rate for the 6 month required period is 4.04%.

To demonstrate this is the case, had 100.00 been deposited into the account at the start of compounding period 1, then by the end of compound period 1 (3 months) the interest in the account using the stated nominal rate of 8% would be

Balance at the end of compounding period 1 = 100.00 + 100.00 x 8%/4 = 102.00

This is now compounded for one more period and the balance at the end of compounding period 2 (the end of the 6 months) is:

Balance at the end of compounding period 2 = 102.00 + 102.00 x 8%/4 = 104.04

The interest earned on the account at the end of the 6 month period (effective period) is 104.04 less the original sum of 100.00 and is equal to 4.04. The effective interest rate on the account for a 6 month period is the interest (4.04) divided by the sum invested (100.00), so is equal to 4.04 / 100.00 = 4.04%, the same answer as the formula gave. This is demonstrated in the diagram below.

Effective period greater than the compounding period
Year (Nominal rate 8%)
Compounding 1 Compounding 2 Compounding 3 Compounding 4
Effective period
Effective interest rate 4.04% ↑

Effective Period < Compounding Period

If the nominal rate is 8% compounded six monthly, what is the effective interest rate for a quarter?

In this example, the effective period (quarter) is less than the compounding period (6 months), as shown in the diagram below.

Effective period less than the compounding period
Year (Nominal rate 8%)
Compounding 1 Compounding 2
Effective period

Using the effective interest rate formula, the rate can be calculated as follows:

Effective interest rate = (1 + r / m )n - 1
r = annual nominal rate = 8%
m = compounding periods in a year = 2
n = number of compounding periods the rate is required for = 1/2
Effective interest rate = (1 + 8% / 2 )1/2 - 1
Effective interest rate = 1.98% per quarter

If the nominal rate is 8% compounded every 6 months, then the effective interest rate for a quarter is 1.98%. Note that this assumes that the calculation for compounding is carried out every quarter (effective period) at this rate.

The calculation of an effective interest rate is a useful tool which can be applied in numerous time value of money calculations. It is particularly relevant when dealing with annuities where the payment period (effective period) does not always coincide withe the compounding of interest (compounding period).

Effective Interest Rate for any Time Period November 6th, 2016Team

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