Problem 20 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

What monthly payment is required to amortize a loan of $$\$ 30,000$$ over \(10 \mathrm{yr}\) if interest at the rate of \(12 \% /\) year is charged on the unpaid balance and interest calculations are made at the end of each month?

Step-by-Step Solution

Answer

The monthly payment required to amortize a loan of $30,000 over 10 years at an interest rate of 12% per year is approximately $430.50.

Step 1: Identify the Variables

Let's identify the variables we will use in the formula: - P: Principal (loan amount) = $30,000 - r: monthly interest rate (annual interest rate/12) = (12% / 12) = 1% = 0.01 - n: number of payments (loan term in months) = (10 years * 12 months/year) = 120 We'll use these values in the amortization formula to find the monthly payment (M).

Step 2: Formula for Monthly Payment

Now, we'll use the formula for a monthly payment on an amortized loan expressed as: \( M = P * \frac{r(1+r)^n}{(1+r)^n-1} \) Plugging in the values we found in step 1: \( M = 30000 * \frac{0.01(1+0.01)^{120}}{(1+0.01)^{120}-1} \)

Step 3: Calculate the Monthly Payment

Next, we'll calculate the values inside the parenthesis and exponent, then apply the formula. \( M = 30000 * \frac{0.01(1.01)^{120}}{(1.01)^{120}-1} \) Calculate the exponent \( (1.01)^{120} \approx 3.298127044 \) \( M = 30000 * \frac{0.01*3.298127044}{3.298127044-1} \) Now, calculate the numerator and denominator: Numerator: \( 0.01*3.298127044 \approx 0.03298127044 \) Denominator: \( 3.298127044-1 \approx 2.298127044 \) Now we can plug in these values: \( M = 30000 * \frac{0.03298127044}{2.298127044} \)

Step 4: Find the Monthly Payment

Finally, we divide the numbers to find the value of monthly payment M: \( M \approx 30000 * 0.01434995537 \) \( M \approx 430.498661 \) The monthly payment required to amortize the loan of \(30,000 over 10 years at an interest rate of 12% per year is approximately \)430.50.

Key Concepts

Amortization Formula

Understanding the amortization formula is essential when figuring out the payment schedule for a loan.
Amortization is the process of paying off a debt over time through regular payments. A part of each payment goes towards the loan principal, and another part goes towards interest. This formula provides a way to calculate the fixed monthly payments that a borrower needs to make in order to pay off a loan.

• Principal (P): The initial amount borrowed.
• Monthly interest rate (r): The annual interest rate divided by 12 (for 12 months a year).
• Number of payments (n): The total number of monthly payments over the life of the loan.

Using the formula \[\begin{equation}M = P * \frac{r(1+r)^n}{(1+r)^n-1}\end{equation}\] we can determine the monthly payment (M). The numerator represents the growth due to interest, and the denominator is the total number of payments reduced by the influence of that interest.

Calculating Monthly Payments

The step-by-step solution highlights how to calculate the monthly payment on an amortized loan—and this concept is crucial for prospective borrowers who want to understand their repayment obligations.
Calculating these payments requires plugging the principal, monthly interest rate, and number of payments into the amortization formula. It's important to pay close attention to converting the annual interest rate to a monthly rate and adjusting the loan term into months.
The formula simplifies the complex process of accounting for both principal and interest accrued over the life of the loan. The monthly payment calculation ensures that by the end of the loan term, the borrower has fully paid off both the original loan amount and the interest charges.

Interest Rate Calculations

Interest rate calculations are integral to understanding the cost of borrowing money. These calculations determine how much extra you'll need to pay back on top of the principal loan amount.
For the given exercise, the annual interest rate is 12%, but since interest calculations are done monthly, we divide this rate by 12, arriving at 1% per month, or a decimal rate of 0.01.

Financial Implications

The higher the interest rate and the longer the loan term, the more interest you pay. When you calculate the monthly payment, you're not just dividing the principal over the payment periods; you are also incorporating the cost of borrowing which includes this interest calculation. It's a critical skill to be able to convert annual rates to monthly since it affects the total cost of the loan significantly.