Problem 32 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Joe secured a loan of $$\$ 12,0003$$ yr ago from a bank for use toward his college expenses. The bank charged interest at the rate of \(4 \% /\) year compounded monthly on his loan. Now that he has graduated from college, Joe wishes to repay the loan by amortizing it through monthly payments over \(10 \mathrm{yr}\) at the same interest rate. Find the size of the monthly payments he will be required to make.

Step-by-Step Solution

Answer

Joe will be required to make monthly payments of \$121.32 to repay his loan of $12,000 with an interest rate of 4% per year, compounded monthly, and amortized over 10 years.

Step 1: Identify the Given Information

From the problem, the following information is given: - Principal loan amount, \(P = \$12,000\) - Annual interest rate, \(r_{annual} = 4\% \) - Loan term, \(t = 10\, \text{ years}\) Since the interest is compounded monthly, it's important to convert the annual interest rate and loan term to their respective monthly equivalents.

Step 2: Convert the Interest Rate and Loan Term to Monthly

First, convert the annual interest rate to a monthly interest rate by dividing by 12 (as there are 12 months in a year). This gives the monthly interest rate, \(r\), as follows: \[ r = \frac{r_{annual}}{12} = \frac{4\%}{12} = \frac{0.04}{12} = 0.00333... \] Similarly, convert the loan term to months by multiplying by 12. The total number of payments, or months, \(n\), is: \[ n = t \times 12 = 10 \text{ years} \times 12= 120 \text{ payments (or months) } \]

Step 3: Calculate the Monthly Loan Payment

Now, substitute \(P = \$12000\), \(r = 0.00333\), and \(n = 120\) into the loan amortization formula. Calculate the monthly payment by using the formula: \[ M = P\frac{r(1 + r)^n}{(1 + r)^n - 1} \] Substituting in the variables gives: \[ M = 12000\frac{0.00333(1 + 0.00333)^{120}}{(1 + 0.00333)^{120} - 1} \] This simplifies to: \[ M = 12000\frac{0.00333(1.00333)^{120}}{(1.00333)^{120} - 1} \] After the calculation, it's found that \(M = \$121.32\). Therefore, Joe will be required to make monthly payments of \$121.32.

Key Concepts

Compound Interest

Compound interest is a core concept in finance where interest is calculated on the initial principal and also on the accumulated interest from previous periods. This means, over time, the amount grows faster compared to simple interest calculations, where interest is computed only on the initial principal. In Joe's case, the bank charged compound interest at an annual rate of 4%, but it was compounded monthly, increasing the total amount he owed.

To understand compound interest better, it's crucial to realize how it builds:

• The interest earned each month is added to the principal, so the interest in the following month is calculated on this new principal.
• This compounding effect can significantly increase the total interest paid over the life of a loan or the amount accumulated over time.

Joe's loan demonstrates this as the initial $12,000 accumulated more interest than it would have with simple interest charged annually.

Monthly Payments

Monthly payments are a key component of loan amortization, representing the regular payments made to pay off a loan over a set period. In Joe's scenario, these payments are calculated to ensure that his loan is fully repaid with interest by the end of the 10-year term.

To determine the amount of each payment, the loan amortization formula is used:\[M = P\frac{r(1 + r)^n}{(1 + r)^n - 1}\]

• \(P\) is the principal amount, $12,000.
• \(r\) is the monthly interest rate, derived by dividing the annual rate by 12.
• \(n\) is the total number of payments, which in Joe's case is 120 months.

By accurately calculating these monthly payments, Joe ensures he repays both the principal and the accumulated interest systematically over the loan's term.

Interest Rate Conversion

Interest rate conversion is a vital step in calculating loan payments when interest is compounded more frequently than once a year. For Joe's loan, since the interest is compounded monthly, the annual interest rate must be converted to a monthly rate.

This conversion is necessary because:

• It aligns the interest rate with the frequency of compounding and the payment schedule.
• Provides a precise rate that fits the monthly calculations linked to his amortization schedule.

To convert, simply divide the annual interest rate by the number of compounding periods per year: \[r = \frac{r_{annual}}{12}\]In Joe's situation, this resulted in a monthly interest rate of approximately 0.333%, which accurately reflects the monthly compounding effect. Understanding how to convert interest rates is crucial for properly calculating monthly payments and understanding the total cost of a loan.