Compound interest

2020-11-25T12:35:43+02:00

The changes that occur during percentage calculations are always momentary. This means that time does not matter in percentage calculations. In interest calculations, however, time is taken into account. The initial number increases by a certain percentage, but the magnitude of the total increase depends on the time in which the interest accrues.

The amount of interest is calculated based on the rate of interest or the interest percentage and the amount of time in which the interest is accrued. For example, many international banking institutions calculate interests over a period of 360 days. This period is known as the interest year. If no other time specifications are given, use interest periods of 1 year = 12 months = 52 weeks = 360 days.

Example 1

The annual interest rate on a bank account is [[$ 2 \: \% $]]. Calculate how big a deposit of [[$ 1000 $]] QAR will grow in eight years?

This is a percentage increase, so the calculation is performed as learned in the previous chapter.

Deposit after the 1st year: [[$ 1,02 \cdot 1000 \: \text {QAR} = 1020 \: \text {QAR} $]]

Deposit after the 2nd year:[[$ 1,02 \cdot 1020 \: \text {QAR} =1,02 \cdot 1,02 \cdot 1000 \: \text {QAR} = 1,02 ^{2} \cdot 1000 \: \text {QAR} = 1040,40 \: \text {QAR} $]]

Deposit after the 3rd year:
[[$ 1,02 \cdot 1040,40 \: \text {QAR} = 1,02 \cdot 1,02 \cdot 1,02 \cdot 1000 \: \text {QAR} $]] [[$ = 1,02 \cdot 1.02 \cdot 1,02 \cdot 1000 \: \text {QAR} = 1,02 ^{3} \cdot 1000 \: \text {QAR} = 1061,208 \: \text {QAR} $]]

Based on the above, it is seen that the deposit after 8 years is:

[[$ 1,02 ^{8} \cdot 1000 \: \text {QAR} ≈ 1174,66 \: \text {QAR} $]]

Capital after [[$n$]] years

Capital after [[$ n $]] years, when the annual interest rate is p% and the initial capital is [[$ a $]], is calculated as follows:

[[$$ \left ( 1 + \displaystyle\frac {p} {100} \right )^n \cdot a $$]]

Example 2

How much interest will a deposit of [[$ 15 000 \: \text {QAR} $]] accrue over the period of ten years, when the annual interest rate of the savings account is [[$ 1,7 \: \% $]]?

First, calculate the value of the deposit after ten years:

[[$ \left ( 1 + \displaystyle\frac {1,7} {100} \right )^n \cdot 15000 \: \text {QAR} ≈ 17754,19 \: \text {QAR} $]]

The share of interest is obtained by subtracting the original deposit from the new deposit:

[[$ 17754,19 \: \text {QAR} \: – 15000 \: \text {QAR} = 2754,19 \: \text {QAR} $]]

Example 3

Aunt Freda left a deposit of [[$ 118022,23 \: \text {QAR} $]] for Tina. The deposit had accrued interest over a period of ten years with an annual interest rate of [[$ 12 \% $]]. What was the orginal value of the deposit?

[[$ 118022,23 = \left ( 1 + \displaystyle\frac {p} {100} \right )^{10} \cdot a $]]

[[$ 118022,23 = 1,12 ^{10}a $]]

[[$ –1,12 ^{10}a = \: –118022,23 \space ||–1,12 ^{10} $]]

[[$ a = \displaystyle\frac {–118022,23} {–1,12 ^{10}} $]]

[[$ a ≈ 38 000 \: ( \: \text {QAR}) $]]

7. Compound interest

Compound interest

Example 1

Capital after [[$n$]] years

Example 2

Example 3

Exercises