7.5-Deducing a Formula for Compound Interest

Compound interest is the interest calculated on the initial principal as well as the accumulated interest from previous periods. Unlike simple interest, where the interest is always calculated on the principal, compound interest takes into account the interest that accumulates over time. This section focuses on deriving the formula for compound interest.

Let us start by considering the following parameters:

• P = Principal amount (initial investment)
• r = Rate of interest per period (in decimal form)
• n = Number of periods (years, months, etc.)
• A = Amount after n periods
• CI = Compound Interest (the interest earned after n periods)

For compound interest, the amount after 1 period is calculated by:

$A = P \times (1 + r)$

In the next period, the interest will be calculated not just on the principal P, but also on the interest accumulated in the previous period. So, after 2 periods, the amount becomes:

$A = P \times (1 + r) \times (1 + r) = P \times (1 + r)^2$

After 3 periods, it will be:

$A = P \times (1 + r) \times (1 + r) \times (1 + r) = P \times (1 + r)^3$

Thus, after n periods, the amount is:

$A = P \times (1 + r)^n$

Now, the compound interest CI is the difference between the final amount A and the initial principal P. Therefore, the formula for compound interest is:

$CI = A - P$

Substituting the value of A from the previous formula:

$CI = P \times (1 + r)^n - P$

Factoring out P from the right-hand side:

$CI = P \left( (1 + r)^n - 1 \right)$

This is the formula for compound interest. It gives the amount of interest earned on the initial principal as well as on the interest accumulated in each period.

In cases where the interest is compounded more than once a year, we use the formula:

$A = P \times \left( 1 + \frac{r}{m} \right)^{m \times n}$

Where:

• m = Number of times the interest is compounded per year
• r = Annual rate of interest (in decimal form)
• n = Number of years

Thus, the compound interest when compounded m times a year is:

$CI = P \times \left( 1 + \frac{r}{m} \right)^{m \times n} - P$

This formula allows us to calculate the compound interest when the interest is compounded more frequently, such as monthly, quarterly, or half-yearly.

Through this approach, we can deduce the formula for compound interest both for annual compounding and for more frequent compounding periods.