### Calculating Compound Interest

Compound interest means that the interest will include interest calculated on interest. For example, if an amount of \$5,000 is invested for two years and the interest rate is 10%, compounded yearly:

• At the end of the first year the interest would be (\$5,000 * 0.10) or \$500

• In the second year the interest rate of 10% will applied not only to the \$5,000 but also to the \$500 interest of the first year. Thus, in the second year the interest would be (0.10 * \$5,500) or \$550.

Unless simple interest is stated one assumes interest is compounded.

When compound interest is used we must always know how often the interest rate is calculated each year. Generally the interest rate is quoted annually. e.g. 10% per annum.

Compound interest may involve calculations for more than once a year, each using a new principal (interest + principal). The first term we must understand in dealing with compound interest is conversion period. Conversion period refers to how often the interest is calculated over the term of the loan or investment. It must be determined for each year or fraction of a year.

e.g.: If the interest rate is compounded semiannually, then the number of conversion periods per year would be two. If the loan or deposit was for five years, then the number of conversion periods would be ten.

Compound Interest Formula:
S = P(1+i)^n

Where
S = amount
P = principal
i = Interest rate per conversion period
n = total number of conversion periods

Example:
Alan invested \$10,000 for five years at an interest rate of 7.5% compounded quarterly

P = \$10,000
i = 0.075 / 4, or 0.01875
n = 4 * 5, or 20, conversion periods over the five years

Therefore, the amount, S, is:
S = \$10,000(1 + 0.01875)^20
= \$ 10,000 x 1.449948
= \$14,499.48

So at the end of five years Alan would earn \$ 4,499.48 (\$14,499.48 – \$10,000) as interest.

Note: How to calculate 1.449948,
(1 + 0.01875)^20 = multiply 1.01875 twenty (20) times
1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 x 1.01875 (you will find the number is used 20 times)

If he had invested this amount for five years at the same interest rate offering the simple interest option, then the interest that he would earn is calculated by applying the following formula:

S = P(1 + rt),
P = 10,000
r = 0.075
t = 5

Thus, S = \$10,000[1+0.075(5)]
= \$ 13,750

Here, the interest that he would have earned is \$3,750.
A comparison of the interest amounts calculated under both the method indicates that Alan would have earned \$749.48(\$4,499.48 – \$3,750) more under the compound interest method than under the simple interest method.

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