Problem 19 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A sum of $$\$ 100,000$$ is to be repaid over a 10 -yr period through equal installments made at the end of each year. If an interest rate of \(10 \% /\) year is charged on the unpaid balance and interest calculations are made at the end of each year, determine the size of each installment so that the loan (principal plus interest charges) is amortized at the end of \(10 \mathrm{yr}\).

Step-by-Step Solution

Answer

The size of each equal annual installment to fully repay the loan of $100,000 with a 10% annual interest rate in 10 years is approximately $16,274.54.

Step 1: Understand the problem

We are given a loan amount of $100,000 that has to be paid back over ten years with equal payments at the end of each year. The outstanding balance accrues interest at a rate of 10% per year, so with each payment, we have to cover the interest charged plus a portion of the principal.

Step 2: Use the formula for equal annual payments

The formula to calculate the size of each annual payment is: \[A = P \frac{r(1 + r)^n}{(1 + r)^n - 1}\] Where: - \(A\) is the annual payment, - \(P\) is the loan amount, - \(r\) is the annual interest rate (decimal form), - \(n\) is the number of years. In our case, \(P = 100,000\), \(r = 0.10\) (since 10% = 0.10), and \(n = 10\).

Step 3: Calculate the size of each annual payment

Now, we'll plug the values into the formula: \[A = 100,000 \frac{0.10(1 + 0.10)^{10}}{(1 + 0.10)^{10} - 1}\]

Step 4: Evaluate the expression in the formula

Let's evaluate the formula step by step: 1. Calculate (1 + 0.10) = 1.10. 2. Calculate \(1.10^{10}\) = 2.5937424601. 3. Calculate \(1.10^{10} - 1\) = 2.5937424601 - 1 = 1.5937424601. 4. Calculate \(0.10 \times 1.10^{10}\) = 0.25937424601. 5. Calculate \(\frac{0.25937424601}{1.5937424601}\) = 0.1627453782. Now, we can multiply the result by the loan amount: 6. Calculate \(100,000 \times 0.1627453782\) = $16,274.54 (rounded to the nearest cent).

Step 5: Interpret the result

The size of each installment to fully repay the loan with principal and interest in 10 years is approximately $16,274.54 per year.

Key Concepts

Annual Interest Rate

Understanding the annual interest rate is key when dealing with loans. This rate represents the cost of borrowing money over a period of one year, expressed as a percentage of the principal loan amount. The annual interest rate affects the total amount you will pay over the term of the loan.
It is crucial to convert the percentage into a decimal before using it in calculations. For example, a 10% annual interest rate is converted to 0.10.
This conversion helps in effortlessly applying the interest rate in various financial formulas, like calculating loan installments.

• Annual Interest Rate: Represents the yearly cost of the loan.
• Expressed as a percentage of the principal.
• Conversion to decimals is important for calculations.

Understanding how the interest rate compounds over time is vital, as it influences the total loan repayment significantly.

Loan Installments

Loan installments are the fixed, regular payments made to repay a loan over a set term. They include both principal and interest portions, ensuring the loan is fully amortized by the end of the term.
These payments are typically made monthly or annually, depending on the loan agreement.
For the exercise at hand, it specifically focuses on equal annual installments over a 10-year period. Each installment decreases the principal balance, with subsequent installments having a greater portion going towards principal reduction.

• Consist of principal and interest.
• Are usually fixed for the loan duration.
• Ensure full repayment by the loan's end.

By maintaining consistent payments, you gradually reduce the balance, ultimately paying off the loan by the agreed schedule.

Amortization Formula

The amortization formula is a crucial tool for calculating the size of each installment needed to fully repay a loan. It distributes the loan amount and the accumulated interest across equal payments over the term.
In our example, where a \(100,000 loan is repaid over ten years with a 10% annual interest, the formula is:\[A = P \frac{r(1 + r)^n}{(1 + r)^n - 1}\]
This formula ensures that each payment covers both the interest accrued for the year and a portion of the principal.
Key elements of the formula include:

• \(A\): The annual payment that keeps consistency in equal installments.
• \(P\): Principal loan amount, here it's \)100,000.
• \(r\): Annual interest rate in decimal, which is 0.10.
• \(n\): Total number of payments, which is 10 years in this case.

Using these components, the formula divides the repayment evenly, making sure the loan is settled by the end of the term.