Problem 24 - Algebra and Trigonometry

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\(\frac{3}{4} \%\) 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 but a lower total cost.

Step 1: Understand the Formula

The monthly mortgage payment can be calculated using the formula: \[ m = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \]where \( P \) is the principal amount (\$300,000), \( r \) is the monthly interest rate, and \( n \) is the total number of payments. We need to calculate this for both mortgage options.

Step 2: Calculate Monthly Interest Rates and Payments for 30-year Mortgage

For the 30-year mortgage, the annual interest rate is 6.5%. First, convert the annual interest rate to a monthly rate: \[ r = \frac{6.5\%}{12} = \frac{0.065}{12} \approx 0.0054167 \].The total number of monthly payments is \( n = 30 \times 12 = 360 \).Plug these into the formula:\[ m_{30} = 300,000 \frac{0.0054167(1 + 0.0054167)^{360}}{(1 + 0.0054167)^{360} - 1} \]Calculate this to find the monthly payment \( m_{30} \).

Step 3: Calculate Monthly Interest Rates and Payments for 15-year Mortgage

For the 15-year mortgage, the annual interest rate is 5.75%. Convert this to a monthly rate: \[ r = \frac{5.75\%}{12} = \frac{0.0575}{12} \approx 0.0047917 \].The total number of monthly payments is \( n = 15 \times 12 = 180 \).Plug these into the formula:\[ m_{15} = 300,000 \frac{0.0047917(1 + 0.0047917)^{180}}{(1 + 0.0047917)^{180} - 1} \]Calculate this to find the monthly payment \( m_{15} \).

Step 4: Compare Monthly Payments

Compare the monthly payments \( m_{30} \) and \( m_{15} \). The larger value indicates the mortgage with the higher monthly payment.

Step 5: Calculate Total Payment for 30-year Mortgage

The total amount paid over the life of the 30-year mortgage is the monthly payment times the number of payments: \[ \, \text{Total for 30-year} = m_{30} \times 360 \].

Step 6: Calculate Total Payment for 15-year Mortgage

The total amount paid over the life of the 15-year mortgage is calculated similarly: \[ \, \text{Total for 15-year} = m_{15} \times 180 \].

Step 7: Compare Total Payments

Compare the total payments calculated in steps 5 and 6. The smaller total payment indicates which mortgage is less expensive over its lifetime.

Key Concepts

Monthly Payment Formula

Understanding how to calculate the monthly mortgage payment is crucial for making informed financial decisions. Mortgages typically require borrowers to repay the principal amount plus interest in monthly installments. The formula used to determine this monthly payment is: \[ m = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] where:

• \( m \) is the monthly payment,
• \( P \) is the loan principal, which is the total amount borrowed,
• \( r \) is the monthly interest rate (annual rate divided by 12),
• \( n \) is the total number of payments, which is typically the number of years of the mortgage multiplied by 12.

By using this formula, borrowers can calculate how much they need to pay every month over the entire loan period.

Interest Rate Conversion

When dealing with mortgage interest rates, it's important to convert the annual rate into a monthly rate for accurate calculations. Annual interest rates are typically provided by lenders, but since payments are made monthly, the conversion is necessary. For our examples:

• 30-year mortgage: Convert an annual interest rate of 6.5% to a monthly rate by dividing by 12, which yields approximately 0.0054167.
• 15-year mortgage: Similarly, for a 5.75% annual rate, the monthly rate is approximately 0.0047917.

This small conversion step is crucial as it plays a big role in determining the monthly payment and ultimately affects the loan's total cost.

Loan Term Comparison

Choosing between different loan terms is a major part of deciding on a mortgage. Comparing a 30-year mortgage to a 15-year one involves looking at both the monthly payments and the total payment over the life of the loan. Longer terms, like a 30-year mortgage, usually offer lower monthly payments but might end up more expensive overall due to interest:

• Advantage: Lower monthly payments
• Disadvantage: More interest paid over the lifetime of the loan

In contrast, shorter terms like 15-year mortgages typically require higher monthly payments but result in less total interest paid:

• Advantage: Less interest paid overall
• Disadvantage: Higher monthly payments

The choice depends on the borrower's financial goals, budget, and cash flow considerations.

Total Payment Calculation

Calculating the total cost of a mortgage is essential to understanding the financial commitment you're making. To find this number, multiply the monthly payment by the total number of payments. Using the formula results we obtained previously, we determine:

• 30-year mortgage: Total payment is the monthly payment \(m_{30}\) times 360, as there are 360 payments in total.
• 15-year mortgage: Total payment is the monthly payment \(m_{15}\) times 180, since this loan has 180 payments.

By comparing these totals, borrowers can see the full financial impact of different loan terms and make a more informed decision. Generally, shorter term loans, although with higher monthly payments, lead to less overall interest paid and therefore lower total payment.