Problem 24 - Precalculus Mathematics for Calculus

Question

A couple needs a mortgage of \(\$ 300,000\). Their mortgage broker presents them with two options: a 30-year mortgage at \(6 \frac{1}{2} \%\) interest or a 15-year mortgage at \(5 \frac{3}{4} \%\) interest. (a) Find the monthly payment on the 30 -year mortgage and on the 15 -year mortgage. Which mortgage has the larger monthly payment? (b) Find the total amount to be paid over the life of each loan. Which mortgage has the lower total payment over its lifetime?

Step-by-Step Solution

Answer

The 15-year mortgage has a larger monthly payment. It also has a lower total payment over its lifetime.

Step 1: Determine the monthly interest rates

First, convert the annual interest rates to monthly rates. For the 30-year mortgage at \(6 \frac{1}{2} \%\), the monthly interest rate, \(r\), is \(\frac{6.5\%}{12} = 0.5417\%\) or \(0.005417\) in decimal. For the 15-year mortgage at \(5 \frac{3}{4} \%\), \(r = \frac{5.75\%}{12} = 0.4792\%\) or \(0.004792\) in decimal.

Step 2: Find the number of total payments

For the 30-year mortgage, the number of monthly payments \(n = 30 \times 12 = 360\). For the 15-year mortgage, the number of monthly payments \(n = 15 \times 12 = 180\).

Step 3: Calculate the monthly payment for each mortgage

Use the formula for the monthly payment \(M\) of a mortgage: \[ M = P \frac{r (1 + r)^n}{(1 + r)^n - 1} \]where \(P\) is the principal loan amount, \(r\) is the monthly interest rate, and \(n\) is the total number of payments.For the 30-year mortgage: \[ M = 300,000 \times \frac{0.005417 (1 + 0.005417)^{360}}{(1 + 0.005417)^{360} - 1} \approx \\(1,896.20 \]For the 15-year mortgage: \[ M = 300,000 \times \frac{0.004792 (1 + 0.004792)^{180}}{(1 + 0.004792)^{180} - 1} \approx \\)2,489.11 \]

Step 4: Compare monthly payments

The monthly payment for the 30-year mortgage is approximately \\(1,896.20, while for the 15-year mortgage, it is approximately \\)2,489.11. Hence, the 15-year mortgage has the larger monthly payment.

Step 5: Calculate total payment over the life of each loan

Calculate the total payments by multiplying the monthly payment by the number of payments for each mortgage.For the 30-year mortgage: \[ \text{Total} = 1,896.20 \times 360 = \\(682,632 \]For the 15-year mortgage: \[ \text{Total} = 2,489.11 \times 180 = \\)448,039.80 \]

Step 6: Compare total payments

The total amount paid with the 30-year mortgage is \\(682,632, whereas the total amount paid with the 15-year mortgage is \\)448,039.80. Therefore, the 15-year mortgage has the lower total payment over its lifetime.

Key Concepts

Monthly Payments

When you take out a mortgage, you typically make monthly payments until the loan is fully paid. These payments comprise both principal and interest. The principal is the original loan amount, and interest is the cost of borrowing the money.

To calculate the monthly payment, you'll need to know the loan amount, the interest rate, and the total number of payments across the years. Using the formula \( M = P \frac{r (1 + r)^n}{(1 + r)^n - 1} \), you can determine the monthly payment amount:

• \(P\) is the principal, or the initial amount borrowed.
• \(r\) is the monthly interest rate, derived by dividing the annual rate by 12.
• \(n\) is the number of total payments (years multiplied by 12).

Using this formula, you can calculate the monthly payments for different loan terms and interest rates. For instance, in the 30-year mortgage example, the monthly payment comes to approximately \(\\(1,896.20\), whereas, for the 15-year mortgage, it's \(\\)2,489.11\). The shorter term 15-year mortgage naturally has higher monthly payments.

Interest Rates

Interest rates are critical in determining your loan's cost and the monthly payment amount. They are expressed as an annual percentage, influencing how much you'll pay in addition to the principal.

It's important to convert the annual interest rate to a monthly rate for mortgage calculations since payments are monthly. You divide the annual rate by 12 to get the monthly interest rate.

• For example, if the annual interest rate is \(6.5\%\), the monthly rate is \(\frac{6.5\%}{12} = 0.5417\%\).
• In decimal form, this rate would be \(0.005417\).

A lower interest rate makes a significant difference over the term of the mortgage. The 15-year mortgage in our exercise, despite having a lower annual interest rate of \(5.75\%\), results in higher monthly payments due to the shorter term but saves substantial money over the life of the loan compared to a longer-term loan.

Loan Total Payments

Total payments are the sum of all monthly payments over the life of the loan, encapsulating both the principal and total interest paid.

To find this, you multiply the monthly payment by the total number of payments. This amount reflects the entire cost of the mortgage:

• For the 30-year mortgage with monthly payments of \(\\(1,896.20\), the total cost is \(1,896.20 \times 360 = \\)682,632\).
• For the 15-year mortgage with monthly payments of \(\\(2,489.11\), it's \(2,489.11 \times 180 = \\)448,039.80\).

Comparing the two, the 15-year mortgage, despite having larger monthly payments, results in a lower total payment. This is primarily due to less interest being paid over a shorter time. Therefore, while the 30-year mortgage offers lower monthly payments, it costs significantly more overall. Choosing the mortgage term often depends on balancing monthly affordability against long-term savings.