Introduction to Annual Payments and Simple Interest

When dealing with financial obligations such as a debt, understanding how to calculate the annual payments required to discharge a debt over a specific period at a given interest rate is crucial. This article will delve into the detailed calculations and methods to achieve this, with a focus on the simple interest formula and the future value of an ordinary annuity.

Understanding the Problem

Suppose you have a debt of $6,450 outstanding for 4 years at an annual simple interest rate of 5%. The goal is to determine the annual payment required to settle the debt by the end of the 4-year period.

Step 1: Calculate the Total Amount Due with Interest

The total amount due with simple interest can be calculated using the formula:

[ A P(1 rt) ]

Where:

A is the total amount due P is the principal amount (debt) r is the annual interest rate as a decimal t is the time in years

Given the data:

P 6450 r 0.05 t 4

Substituting these values into the formula:

[ A 6450(1 0.05 times 4) 6450 times 1.20 7740 ]

This means that the total amount due, including interest, is $7,740.

Step 2: Calculate the Annual Payment

To find the annual payment (PMT) that will discharge this amount over 4 years, we use the future value of an ordinary annuity formula:

[ FV PMT times frac{1 - r^t}{r} ]

To find PMT, we rearrange the formula:

[ PMT frac{FV times r}{1 - r^t} ]

Given:

FV 7740 r 0.05 t 4

Substituting these values into the rearranged formula:

[ PMT frac{7740 times 0.05}{1 - 0.05^4} approx frac{387}{0.21550625} approx 179.47 ]

Therefore, the annual payment that will discharge the debt is approximately $179.47.

Alternative Method for Simplification

An alternative, simpler method is to calculate the total amount of the installments over the four years directly. Let's denote the installments as:

1st year: 1x 2nd year: 1.5x 3rd year: 1.1 4th year: 1.15x

The total amount of installments is:

[ 1x 1.5x 1.1 1.15x 6450 ]

Solving for x:

[ x frac{6450}{4.30} 645000/430 64500/43 approx 1500 ]

This simplifies the calculation to an annual payment of approximately $1,800, with monthly payments of $150.

Using an Amortization Table or Calculator

For a more precise and straightforward solution, one can use an amortization table or a financial calculator. Alternatively, in Excel, the PMT function can be used as follows:

PMT(0.05, 4, -6450, 0)

This formula calculates the payment amount for a loan of $6,450, with a 5% annual interest rate, over a 4-year period, with a future value of 0.

Conclusion

By using the simple interest formula and the future value of an ordinary annuity, you can determine the annual payment required to discharge a debt. This method provides a clear, step-by-step approach to calculate the payment, ensuring that the debt is fully paid off within the specified time frame. For more complex scenarios or for detailed breakdowns, tools such as an amortization table or a financial calculator can be very helpful.