The Compound Interest Formula

Introduction
Compound interest is a concept in finance that refers to the interest earned not only on the initial amount of money invested (known as the principal), but also on the accumulated interest from previous periods. This means that over time, the interest earned becomes a part of the principal and generates additional interest. The formula to calculate compound interest takes into account the principal, the interest rate, the compounding frequency, and the time period.

The Formula
The formula to calculate compound interest is as follows:

A = P(1 + r/n)^(nt)

Where:
- A represents the future value of the investment or loan, including both the principal and accumulated interest.
- P is the principal amount, or the initial sum of money invested or borrowed.
- r is the annual interest rate, expressed as a decimal.
- n denotes the number of times interest is compounded per year.
- t is the time period, typically measured in years.

Explanation of the Formula
The compound interest formula incorporates the concept of exponential growth. Let's break down the formula and understand each component:

- (1 + r/n): This part represents the interest rate per compounding period. By dividing the annual interest rate (r) by the number of compounding periods per year (n), we get the interest rate for each compounding interval.

- ^(nt): This exponentiation represents the compounding effect over time. By multiplying the number of compounding periods (n) by the time period (t), we determine the total number of compounding intervals.

- P(1 + r/n)^(nt): This multiplication takes the principal amount (P) and multiplies it by the factor (1 + r/n)^(nt), which accounts for both the principal and the accumulated interest.

Example
Let's consider an example to illustrate the compound interest formula. Suppose you invest $5,000 at an annual interest rate of 5%, compounded quarterly for a period of 3 years.

Using the formula:
A = 5000(1 + 0.05/4)^(4*3)

Simplifying the equation, we have:
A = 5000(1.0125)^12

Calculating further, we find:
A ≈ 5000 * 1.159274

Therefore, A ≈ $5,796.37

Hence, the future value of the investment after 3 years would be approximately $5,796.37.

Conclusion
The compound interest formula is a powerful tool in finance that allows us to determine the future value of an investment or loan. By understanding and applying this formula, individuals can make informed decisions regarding their financial endeavors and evaluate the potential growth or cost over time.

Principal (1+r/100)-1