If you leave money in the bank for a period of time, the interest earned is automatically added to your account. After the interest is added, it also begins to earn interest in the next time period. This process is called compound interest.
Example of compound interest: \(\dollar 1\,000\) is placed in an account that earns \(10\pourcent\) interest per annum (p.a.), and the interest is allowed to compound over three years. This means the account is earning \(10\pourcent\) p.a. in compound interest.
We can illustrate this in a table:
Year | Amount | Interest Earned |
0 | \(\dollar 1\,000\) | \(10\pourcent\) of \(\dollar 1\,000 = \dollar 100\) |
1 | \(\dollar 1\,000 + \dollar 100 = \dollar 1\,100\) | \(10\pourcent\) of \(\dollar 1\,100 = \dollar 110\) |
2 | \(\dollar 1\,100 + \dollar 110 = \dollar 1\,210\) | \(10\pourcent\) of \(\dollar 1\,210 = \dollar 121\) |
3 | \(\dollar 1\,210 + \dollar 121 = \dollar 1\,331\) | --- |
After 3 years, there will be a total of \(\dollar 1\,331\) in the account, meaning we have earned \(\dollar 331\) in compound interest.
We can calculate the final amount using another method as well:
- \(\begin{aligned}[t]\text{Amount after 1 year} &= \text{Initial amount} + \text{Interest on the initial amount}\\&= 1\,000 + 0.1 \times 1\,000 \\&= (1 + 0.1) \times 1\,000 \quad \text{(factoring out 1\,000)}\\&= 1.1 \times 1\,000\end{aligned}\)
- \(\begin{aligned}[t]\text{Amount after 2 years} &= \text{Amount after 1 year} + \text{Interest on the amount after 1 year}\\&= 1.1 \times 1\,000 + 0.1 \times 1.1 \times 1\,000 \\&= (1 + 0.1) \times 1.1 \times 1\,000 \quad \text{(factoring out }1.1 \times 1\,000)\\&= 1.1^2 \times 1\,000\end{aligned}\)
- \(\begin{aligned}[t]\text{Amount after 3 years} &= \text{Amount after 2 years} + \text{Interest on the amount after 2 years}\\&= 1.1^2 \times 1\,000 + 0.1 \times 1.1^2 \times 1\,000 \\&= (1 + 0.1) \times 1.1^2 \times 1\,000 \quad \text{(factoring out }1.1^2 \times 1\,000)\\&= 1.1^3 \times 1\,000\end{aligned}\)
These observations lead to the compound interest formula:$$\text{Final amount} = (1 + \text{Interest rate})^{\text{Number of years}} \times \text{Initial amount}$$