Problem 43 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

HomE Eight years ago, Kim secured a bank loan of $$\$ 180,000$$ to help finance the purchase of a house. The mortgage was for a term of \(30 \mathrm{yr}\), with an interest rate of \(9.5 \%\) /year compounded monthly on the unpaid balance to be amortized through monthly payments. What is the outstanding principal on Kim's house now?

Step-by-Step Solution

Answer

The outstanding principal on Kim's house after eight years can be calculated using the amortization formula. First, find the monthly interest rate (\(r_m = \frac{0.095}{12}\)) and the total number of monthly payments (\(n = 30 \times 12\)). Then, calculate the monthly payment using the loan payment formula. Finally, use the amortization formula to find the remaining balance after eight years: \(B = P \times \frac{(1 + \frac{0.095}{12})^{30 \times 12} - (1 + \frac{0.095}{12})^{8 \times 12}}{(1 + \frac{0.095}{12})^{30 \times 12} - 1}\).

Step 1: Find the monthly interest rate

First, we need to find the monthly interest rate by converting the annual interest rate. The annual interest rate is \(9.5\%\), so as a decimal, we have: \(r = 0.095\). Since the interest is compounded monthly, we will divide this annual rate by 12 to get the monthly interest rate: \(r_m = \frac{0.095}{12}\)

Step 2: Find the total number of monthly payments

Next, we need to find the total number of monthly payments required, as the loan term is \(30\) years. To do this, multiply the number of years by the number of months in a year: \(n = 30 \times 12\)

Step 3: Calculate the monthly payment

Now, we can calculate the monthly payment using the loan payment formula: \(P = L \times \frac{r_m(1 + r_m)^n}{(1 + r_m)^n - 1}\) where \(P\) is the monthly payment, \(L\) is the loan amount ($$180,000$$), \(r_m\) is the monthly interest rate, and \(n\) is the total number of monthly payments. Plug in the values we found in Steps 1 and 2 into the formula and solve for \(P\): \(P = 180,000 \times \frac{\frac{0.095}{12}(1 + \frac{0.095}{12})^{30 \times 12}}{(1 + \frac{0.095}{12})^{30 \times 12} - 1}\)

Step 4: Calculate the remaining balance after eight years

To find the remaining balance after eight years, we can use the amortization formula: \(B = P \times \frac{(1 + r_m)^n - (1 + r_m)^m}{(1 + r_m)^n - 1}\) where \(B\) is the remaining balance, \(P\) is the monthly payment, \(n\) is the total number of monthly payments, \(m\) is the number of payments made (eight years' worth), and \(r_m\) is the monthly interest rate. Plug in the values we found in Steps 1, 2, and 3, as well as the number of payments for eight years: \(B = P \times \frac{(1 + \frac{0.095}{12})^{30 \times 12} - (1 + \frac{0.095}{12})^{8 \times 12}}{(1 + \frac{0.095}{12})^{30 \times 12} - 1}\) Calculate the remaining balance, \(B\).

Step 5: State the outstanding principal

The outstanding principal on Kim's house after eight years is the remaining balance, \(B\), calculated in Step 4.

Key Concepts

Compound Interest

In this context, compound interest means that interest is calculated not only on the initial amount borrowed (the principal) but also on any accumulated interest from previous periods. Kim's mortgage rate is compounded monthly. This means each month, the interest applies to the outstanding balance, which includes both the original loan amount and any interest that has accrued up to that point.

This has a significant effect over time. Compared to simple interest, where interest is only calculated on the principal, compound interest grows the debt at a faster rate. This aspect of mortgages allows lenders to earn more over the life of the loan which, for Kim, is set for 30 years. Understanding compound interest is crucial because it affects how quickly the outstanding loan balance changes over time.

When dealing with long-term loans, compound interest plays a big role. For instance, the higher the frequency of compounding, the more interest will accumulate, thus increasing the total amount repaid over the duration of the loan.

Monthly Payments

Monthly payments in a loan amortization context refer to regular payments Kim must make to repay the loan over time. Calculating these payments involves understanding the loan formula, which takes into account the principal amount, interest rate, and loan term.

The loan payment formula used is:
\[ P = L \times \frac{r_m(1 + r_m)^n}{(1 + r_m)^n - 1} \]
This formula helps determine the fixed amount Kim needs to pay each month for 30 years.

Here:

• \( P \) is the monthly payment,
• \( L \) is the loan amount,
• \( r_m \) is the monthly interest rate,
• \( n \) is the total number of monthly payments.

Each payment consists of principal and interest components. Initially, a larger portion goes towards interest, but over time, more of each payment is applied to the principal. This is a key feature of amortization.

Loan Term

The loan term defines the period over which the loan is expected to be paid off. In Kim's case, the loan term is 30 years. This long duration affects how much the total monthly payment will be and how the interest accumulates.

A longer loan term generally results in smaller monthly payments, since the debt is spread over a longer time. However, it also means more interest will be paid over the life of the loan because the outstanding balance reduces more slowly. With a 30-year mortgage, like Kim's, the interest paid can be quite significant when compounded monthly at a 9.5% annual rate.

It's important to consider the loan term when taking out a loan. Shorter terms might have higher monthly payments but reduce the total interest paid, while longer terms lower monthly payments but increase the total interest. Each borrower's decision often weighs their ability to make higher payments against the desire to minimize total interest over time.

Outstanding Principal

Outstanding principal refers to the remaining amount of the loan that is yet to be paid off. After making regular monthly payments, part of which reduces the loan principal, there still exists an unpaid balance. In the exercise, we learned how to calculate this amount after some time has passed, in Kim’s scenario, after 8 years.

The outstanding principal can be calculated using another formula:
\[ B = P \times \frac{(1 + r_m)^n - (1 + r_m)^m}{(1 + r_m)^n - 1} \]
where:

• \( B \) is the outstanding balance,
• \( P \) is the monthly payment,
• \( r_m \) is the monthly interest rate,
• \( n \) is the total number of payments over the whole loan term, and
• \( m \) is the number of payments made so far.

For Kim, calculating the outstanding principal helps determine how much is left to be paid after 8 years. It gives a clear picture of the progress made on the mortgage and helps in any future financial planning.