Introduction to Calculating Annual Payment to Discharge Debt with Compound Interest

Debts and loans often come with varying interest rates and time periods, which can complicate the repayment process. Effective financial planning requires understanding how different factors, such as compound interest, can affect the total amount to be paid. In this article, we will delve into the specifics of how to calculate the annual payment needed to discharge a debt using compound interest.

Understanding Compound Interest and the Problem

Compound interest is the addition of interest to the principal sum of a loan or deposit, or in this case, the debt. It is calculated on the initial principal and also on the accumulated interest of previous periods. The interest grows exponentially, making it a crucial factor to consider when dealing with financial obligations.

Problem Statement

The problem at hand is to find the annual payment to discharge a debt of Rs 1025 due in 2 years at a compound interest rate of 5%.

Mathematical Formulation and Calculation

To solve this problem, we can use the formula for the present value of an annuity:

PV PMT times left(frac{1 - 1}{r^{-n}}right)

Where:

PV: Present value or the debt amount (Rs 1025) PMT: Annual payment (to be determined) r: Interest rate per period (5% or 0.05) n: Number of periods (2 years)

Rearranging the formula to solve for PMT:

PMT PV times left(frac{r}{1 - 1}{r^{-n}}right)

Step-by-Step Calculation

PV 1025

r 0.05

n 2 years

First, calculate 1 - 1}{r^{-n}}:

1}{r^{-n}} 1}{0.05^{-2}} approx 1.05^{-2} approx 0.9070

1 - 0.9070 approx 0.0930

Substituting back into the formula for PMT:

PMT 1025 times left(frac{0.05}{0.0930}right) approx 1025 times 0.5376 approx 550.41

Therefore, the annual payment required to discharge the debt is approximately Rs. 550.41.

Alternative Approach: Using Excel

The alternative approach involves using Microsoft Excel's PMT function for a more straightforward calculation. Here’s how you can do it:

Excel Calculation Using PMT Function

Open an Excel sheet.

Use the PMT function with the following inputs:

Rate: 5% (or 0.05) Nper: 2 years PV: -1025 (negative sign for loan/debt) FV: 0 (the debt is fully discharged at the end of 2 years) Type: 0 (end of period payments)

The PMT function will then calculate the payment amount as Rs. 551.25.

It's important to note that when interest is expressed as a rate (5%), it is assumed to be compounded annually unless otherwise specified. If the compounding period were different, the calculation would need to be adjusted accordingly.

Conclusion

In conclusion, we explored how to calculate the annual payment necessary to discharge a debt with compound interest. Using the present value of an annuity formula and Excel's PMT function, we derived that the annual payment of approximately Rs. 550.41 or Rs. 551.25 (depending on the context) would suffice. Ensuring clarity on the compounding periods is crucial to accurately determine the payment amount.

Frequently Asked Questions (FAQs)

What is compound interest?

Compound interest is the interest calculated on the initial principal and the accumulated interest of previous periods. It causes the total amount of interest to grow exponentially over time.

Why is it important to mention the compounding periods?

It’s important to specify the compounding periods (e.g., annually, semi-annually, quarterly) because it affects the actual interest rate charged and, consequently, the total payment amount.

How can Excel be used to calculate the payment for a debt with compound interest?

Excel's PMT function can be used by inputting the interest rate, number of periods, and present value to calculate the payment amount accurately.