Compound Interest Calculation: Yearly vs. Half-Yearly

Understanding the difference between yearly and half-yearly compounding is crucial for financial planning and investment management. This article explains the process and calculations involved in these two types of compounding, illustrated with an example problem.

Understanding Compound Interest

Compound interest is the interest calculated on both the initial principal and the accumulated interest from previous periods. It allows for exponential growth of a sum of money over time, making it a highly effective tool for wealth building.

Scenarios of Compounding: Yearly vs. Half-Yearly

There are two main scenarios to consider when dealing with compound interest: yearly compounding and half-yearly compounding. The key difference lies in the frequency of compounding periods within a year.

Yearly Compounding

When interest is compounded yearly, the interest rate is applied to the principal once a year. The formula for calculating the amount after t years is:

Formula:

A P(1 rt)n

Where:

A Final amount after t years P Principal amount r Annual interest rate (as a decimal) n Number of years

For our example, we have:

P Principal sum (to be determined) r 20% (0.20) t 2 years

Thus, the amount after 2 years is:

A1 P(1 0.20)2 P(1.20)2 P * 1.44

The interest earned in this case is:

I1 A1 - P P * 1.44 - P P * 0.44

The principal sum, or the initial amount, is what remains unknown in this formula.

Half-Yearly Compounding

When the interest is compounded half-yearly, it means the interest rate is applied to the principal twice every year. The formula for half-yearly compounding is:

Formula:

A P(1 frac{r}{2})2t

Where:

A Final amount after 2t periods P Principal amount r Annual interest rate (as a decimal) 2t Number of compounding periods (for 2 years, 2t 4)

For our example:

r 20% (0.20) 2t 4

Thus, the amount after 2 years is:

A2 P(1 frac{0.20}{2})4 P(1.10)4 P * 1.4641

The interest earned in this case is:

I2 A2 - P P * 1.4641 - P P * 0.4641

Setting Up the Equation

According to the problem, the difference in interest earned between half-yearly and yearly compounding is Rs. 4820. We can set up the equation as follows:

I2 - I1 4820

Substituting the expressions for I2 and I1:

P * 0.4641 - P * 0.44 4820

Factoring out P:

P(0.4641 - 0.44) 4820

Calculating 0.4641 - 0.44:

0.4641 - 0.44 0.0241

Thus, the equation becomes:

P * 0.0241 4820

Solving for P:

P frac{4820}{0.0241} approx 200000

Therefore, the sum of money that was put at compound interest is approximately Rs. 200000.

Key Takeaways

The difference in interest earned between yearly and half-yearly compounding can be determined using the formulas for compound interest. The principal sum can be calculated by setting up and solving an equation based on the given differences in interest earned. Understanding these differences is crucial for making informed financial decisions involving compound interest.

Conclusion

This example illustrates the practical application of compound interest calculations and the importance of considering the compounding frequency. By understanding and applying the appropriate formulas, individuals and businesses can effectively manage their finances and grow their wealth over time.