Commercial Mathematics - Compound Interest (using formula)

Compound Interest (CI) is the interest calculated on the initial principal and also on the accumulated interest of previous periods. Unlike Simple Interest, where the principal remains constant, in CI, the interest earned in one period is added to the principal to form the new principal for the next period. This creates an exponential growth curve when plotted on a graph, where the total amount rises more steeply over time compared to the linear growth of Simple Interest.

The Conversion Period refers to the fixed interval of time after which the interest is calculated and added to the principal. This can be yearly (annually), half-yearly (semi-annually), or quarterly. On a timeline, yearly compounding divides time into one-year segments, while half-yearly compounding divides each year into two equal six-month segments, increasing the frequency of interest calculation.

The Principal ($P$) is the initial sum of money borrowed or invested. In Compound Interest, the principal increases at the end of each conversion period because the interest for that period is added to it. Visualize this as a 'snowball effect' where a small ball of snow (the initial principal) picks up more snow (interest) as it rolls, becoming larger at every turn.

The Rate of Interest ($r$) is usually expressed as a percentage per annum ($\%$ p.a.). If the interest is compounded semi-annually, the rate is halved ($r/2$) for each six-month period. If compounded quarterly, the rate is divided by four ($r/4$) for each three-month period.

The Amount ($A$) is the total sum of money (Principal + Compound Interest) at the end of the specified time period ($n$). The difference between the Final Amount and the Original Principal gives the Compound Interest ($CI = A - P$).

Successive Interest Rates occur when the rate of interest changes every year (e.g., $r_1\%$ for the first year, $r_2\%$ for the second year). In this case, the amount is calculated by multiplying the principal by the growth factors of each individual year sequentially.

Depreciation and Growth: The Compound Interest formula is also used to calculate the reduction in value of assets (depreciation) or the increase in population (growth). For depreciation, the rate is taken as negative, resulting in a formula $A = P(1 - \frac{r}{100})^n$. Graphically, this represents a curve that slopes downwards toward the x-axis, showing the value decreasing over time.