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#### Nominal and Effective Interest Rates

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An interest rate takes two forms: nominal interest rate and effective interest rate. The nominal interest rate does not take into account the compounding period. The effective interest rate does take the compounding period into account and thus is a more accurate measure of interest charges.

A statement that the interest rate is 10% means that interest is 10% per year, compounded annually. In this case, the nominal annual interest rate is 10%, and the effective annual interest rate is also 10%. However, if compounding is more frequent than once per year, then the effective interest rate will be greater than 10%. The more often compounding occurs, the higher the effective interest rate.

The relationship between nominal annual and effective annual interest rates is:

i_{a} = [ 1 + (r / m) ] ^{m} - 1

where i_{a} is the effective annual interest rate, r is the nominal annual interest rate, and m is the number of compounding periods per year.

Example: A credit card company charges 21% interest per year, compounded monthly. What effective annual interest rate does the company charge?

r = 0.21 per year

m = 12 months per year

i_{a} = [ 1 + (.21 / 12) ] ^{12} - 1

= [1 + 0.0175 ] ^{12} - 1

= (1.0175)^{12} - 1 = 1.2314 - 1

= 0.2314 = 23.14%

It may be desired to find the effective interest rate for a period other than annual. In this case, adjust the period for r and m as needed. For example, if the effective interest rate per semi annual period (every 6 months) is desired, then

r = nominal interest rate per 6 months

m = number of compounding periods per 6 months

and the effective interest rate, i_{sa}, per semi-annual period, is:

i_{sa} = [ 1 + (r / m) ] ^{m} - 1

### More Interest Formulas

#### Nominal and Effective Interest Rates

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Question 1.

If a lender charges 12% interest, compounded quarterly, what effective annual interest rate is the lender charging?

Choose an answer by clicking on one of the letters below, or click on Review topic if needed.

A i_{a} = [ 1 + (0.12 / 12) ] ^{12} - 1 = (1.01)^{12} - 1 = 1.1268 - 1 = .1268 = 12.68%

B i_{a} = [ 1 + 0.12 ] ^{12} - 1 = (1.12)^{12} - 1 = 3.8960 - 1 = 2.8960 = 289.6%

C i_{a} = [ 1 + (0.12 / 12) ] ^{4} - 1 = (1.01)^{4} - 1 = 1.0406 - 1 = .0406 = 4.06%

D i_{a} = [ 1 + (0.12 / 4) ] ^{4} - 1 = (1.03)^{4} - 1 = 1.1255 - 1 = .1255 = 12.55%

Question 2.

If a lender charges 12% interest, compounded monthly, what is the effective interest rate __per quarter__?

Hint: m = number of compounding periods per quarter

Let i = effective interest rate per quarter.

Choose an answer by clicking on one of the letters below, or click on Review topic if needed.

A i = [ 1 + (0.12 / 3) ] ^{3} - 1 = (1.04)^{3} - 1 = 0.1249 = 12.49%

B i = [ 1 + 0.03 ] ^{12} - 1 = (1.03)^{12} - 1 = 0.4258 = 42.58%

C i = [ 1 + (0.03 / 3) ] ^{3} - 1 = (1.01)^{3} - 1 = 0.0303 = 3.03%

D i = [ 1 + (0.03 / 12) ] ^{3} - 1 = (1.0025)^{3} - 1 = 0.0075 = 0.75%

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