Problem 24 - Precalculus: Mathematics for Calculus

Question

Mortgage \(A\) 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 \(5 \frac{1}{4} \%\) 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 but lower total cost.

Step 1: Determine the Monthly Payment Formula

The monthly payment \( M \) for a loan is calculated using the formula: \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] where:- \( P \) is the loan principal,- \( r \) is the monthly interest rate (annual rate divided by 12),- \( n \) is the total number of payments (months).

Step 2: Calculate the Monthly Payment for the 30-year Mortgage

Convert the annual interest rate to a monthly rate: \( 6.5\% \) annually is \( \frac{6.5}{100 \times 12} = 0.005417 \) per month. For 30 years, \( n = 30 \times 12 = 360 \) months. Substitute these into the formula:\[ M = 300,000 \times \frac{0.005417(1 + 0.005417)^{360}}{(1 + 0.005417)^{360} - 1} \]Compute the value of \( M \).

Step 3: Calculate the Monthly Payment for the 15-year Mortgage

Convert the annual interest rate to a monthly rate: \( 5.25\% \) annually is \( \frac{5.25}{100 \times 12} = 0.004375 \) per month. For 15 years, \( n = 15 \times 12 = 180 \) months. Substitute these into the formula:\[ M = 300,000 \times \frac{0.004375(1 + 0.004375)^{180}}{(1 + 0.004375)^{180} - 1} \]Compute the value of \( M \).

Step 4: Compare the Monthly Payments

Calculate the monthly payments for both mortgages based on the computations from Step 2 and Step 3 and determine which is higher.

Step 5: Calculate Total Payment for the 30-year Mortgage

Multiply the monthly payment for the 30-year mortgage by the total number of payments (360 months). This gives the total amount paid over the life of the loan.

Step 6: Calculate Total Payment for the 15-year Mortgage

Multiply the monthly payment for the 15-year mortgage by the total number of payments (180 months). This gives the total amount paid over the life of the loan.

Step 7: Compare Total Payments

Compare the total payments calculated in Step 5 and Step 6 to determine which mortgage has the lower total payment over its lifetime.

Key Concepts

Monthly Payment Formula

In mortgage calculations, the monthly payment formula is essential for determining how much you will pay each month towards your loan. This formula considers several factors:

• The loan principal, which is the original amount of money you borrowed.
• The monthly interest rate, derived from the annual interest rate divided by 12.
• The total number of monthly payments over the life of the loan, typically calculated in months for either a 15-year or a 30-year period.

The formula itself is given by:\[M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}\]Here, \( M \) represents the monthly payment, \( P \) is the principal, \( r \) is the monthly interest rate, and \( n \) is the number of payments. By replacing these values in the formula, you can calculate the monthly payment that fits your loan's specific terms.

Loan Principal

Loan principal is the initial amount of money that a borrower applies for from a lender to purchase a home. For mortgage calculations, the principal amount is crucial since it directly influences the monthly payment and total interest paid over the life of the loan. In our example, the couple needs a mortgage principal of $300,000.How does the principal affect calculations?

• A higher principal means more money will be borrowed, which typically leads to higher monthly payments and more interest accrued over time.
• The principal is used in the monthly payment formula as \( P \), defining the base financial obligation without interest.

Understanding your principal amount helps in planning your financial capabilities and structuring an affordable loan that suits your needs.

Annual Interest Rate

The annual interest rate determines how much you will pay each year, as a percentage, on the outstanding balance of your mortgage. For the monthly payment formula, it's important to convert this rate to a monthly rate:\[r = \frac{\text{Annual Interest Rate}}{100 \times 12}\]This conversion simplifies the calculation of the borrower’s monthly obligation. In our example, two rates are given: 6.5% for the 30-year mortgage and 5.25% for the 15-year mortgage, which are converted into monthly rates. Why is understanding the interest rate important?

• A lower interest rate reduces your monthly payment and the total interest paid over time.
• Interest rates often vary based on the economy, the lender, and the duration of the loan, making it critical to comprehend their impact on total loan cost.

Choosing a mortgage with an optimized interest rate can save significant amounts over the loan's lifetime.

Total Payment Comparison

When selecting a mortgage, comparing total payments over the lifetime of different loan options is essential. This involves calculating the total amount paid, which includes both the principal and interest. To find the total payment:

• Multiply the monthly payment by the total number of payments. For a 30-year mortgage: 360 payments; for a 15-year mortgage: 180 payments.

In our example, you would calculate and compare these values: - **30-year Mortgage:** A lower monthly payment but substantially higher total interest due to the longer duration. - **15-year Mortgage:** A higher monthly payment, yet lower total cost due to less interest accumulation. Why compare total payments?

• A lower total payment could free funds for other investments or savings.
• Understanding the long-term financial commitment helps in selecting the best mortgage option.

Striking a balance between monthly affordability and overall cost can improve financial security and satisfaction with your mortgage choice.