Understanding Compound Interest with Half-Yearly Compounding: A Practical Example

In the world of finance, understanding compound interest is crucial, especially when it comes to half-yearly compounding. This article will walk you through a practical example to help you grasp the concept better. We will use a real-world scenario to calculate the total interest gained over a year with given deposits and a specified interest rate.

The Problem at Hand

Consider a scenario where a bank offers a compound interest rate of 5% per half-year. A customer deposits Rs 9600 each on January 1 and July 1 of the same year. We will calculate the total amount gained by way of interest at the end of the year.

Breakdown of the Deposits and Computation

The total amount deposited is Rs 19200 (Rs 9600 Rs 9600). The interest rate is 5% per half-year, which means:

Deposit on January 1: The principal amount is Rs 9600 and it will earn interest for two half-year periods. Deposit on July 1: The principal amount is also Rs 9600, but it will only earn interest for one half-year period.

Step-by-Step Calculation

Let’s proceed with the calculations step by step.

January 1 Deposit Calculation

Using the compound interest formula: A P left(1 frac{r}{100}right)^n

Where:

P (Principal): Rs 9600 r (Interest rate): 5% (half-yearly) n (Number of compounding periods): 2 A_{Jan} 9600 left(1 frac{5}{100}right)^2 9600 left(1 0.05right)^2 9600 times 1.05^2 9600 times 1.1025 10584 July 1 Deposit Calculation

Similarly, for the July 1 deposit:

Using the same formula:

A_{Jul} 9600 left(1 frac{5}{100}right)^1 9600 left(1 0.05right)^1 9600 times 1.05 10080 Total Amount at the End of the Year

Adding the amounts from both deposits:

A_{total} A_{Jan} A_{Jul} 10584 10080 20664 Total Interest Earned

The total deposit amount is Rs 19200 (Rs 9600 Rs 9600). The interest earned is:

Interest A_{total} - Total Deposits 20664 - 19200 1464

Therefore, the total amount gained by way of interest at the end of the year is Rs 1464.

Understanding the Concept with a Simplified Example

To further clarify, let’s break down the process:

When interest is compounded half-yearly, the rate of interest becomes half of the annual rate. For the first six months, the interest will be calculated on the first deposit of Rs 9600. After six months, the total amount will be Rs 10500 (10000 5% interest on 10000 another 5% on 10500).

On July 1, the customer deposits another Rs 9600, and the total amount becomes Rs 20500 (10500 9600).

For the second half of the year, this total amount will earn 5% interest. Thus, the final amount at the end of the year is Rs 20664, and the interest gained is Rs 1464.

Conclusion

This example demonstrates how to calculate the total interest earned with half-yearly compounding. Understanding and applying these concepts correctly can help you optimize your financial decisions and understand your bank's interest rates better.

Further Reading

Understanding Compound Interest Half-Yearly Compounding Formula