Compound Interest - Calculating Compound Interest

Definition of Compound Interest: Compound Interest ($C.I.$) is the interest calculated on the initial principal and also on the accumulated interest of previous periods. It is often described as 'interest on interest,' where the interest earned in one period is added to the principal to calculate interest for the next period.

Principal and Amount: The Principal ($P$) is the initial sum of money borrowed or invested. The total money at the end of the time period is called the Amount ($A$). Visually, the Amount can be seen as the original Principal plus the total Compound Interest earned, expressed by the relationship $A = P + C.I.$

The Conversion Period: This is the fixed time interval after which interest is calculated and added to the principal. It can be annual (once a year), semi-annual (every 6 months), or quarterly (every 3 months). The more frequent the compounding, the higher the final amount will be.

Growth Visualization: Unlike Simple Interest, which grows linearly (forming a straight diagonal line on a graph), Compound Interest grows exponentially. On a coordinate plane where the x-axis represents time and the y-axis represents the value, the $C.I.$ curve starts shallow and bends upward, becoming steeper as time passes.

Comparison with Simple Interest: For the first conversion period, the Simple Interest and Compound Interest are exactly the same. However, from the second period onwards, Compound Interest is always higher because the interest itself begins to earn interest.

Compounding Half-Yearly: When interest is compounded semi-annually, the interest is calculated twice a year. For calculation, we halve the annual rate ($\frac{r}{2}$) and double the number of years ($2n$) to determine the number of conversion periods.

Successive Interest Rates: In some financial scenarios, the interest rate may change every year (e.g., $r_1$ in the first year and $r_2$ in the second). The final Amount is calculated by applying each rate consecutively to the balance from the previous year: $A = P(1 + \frac{r_1}{100})(1 + \frac{r_2}{100})$.