Problem 28 - Calculus and its applications

Question

Comparing loan options. The Aubrys plan to finance a new home through an amortized loan of \(\$ 275,000\). The lender offers two options: (1) a 30-yr term at an annual interest rate of \(4 \%,\) compounded monthly, and (2) a 20-yr term at an annual interest rate of \(5 \%\), compounded monthly. a) Find the monthly payments for options 1 and 2 . b) Assume that the Aubrys make every monthly payment. Find their total payments for options 1 and 2 c) Assume that the Aubrys intend to make every monthly payment. Which option will result in less interest paid, and by how much?

Step-by-Step Solution

Answer

Option 2 has a lower monthly payment (\(\$1,813.74\)) than option 1 (\(\$1,313.38\)), and results in \(\$37,519.20\) less interest paid over the loan term.

Step 1: Calculate Monthly Payment for Option 1

First, we need to calculate the monthly payment for a 30-year loan at a 4% annual interest rate. We'll use the amortization formula for monthly payments:\[ M = \frac{P \cdot \frac{r}{n}}{1 - \left(1 + \frac{r}{n}\right)^{-nt}} \]where:- \(P = 275,000\) is the principal amount,- \(r = 0.04\) is the annual interest rate,- \(n = 12\) is the number of compounding periods per year (monthly),- \(t = 30\) is the loan term in years.Plugging in the values:\[ M = \frac{275,000 \cdot \frac{0.04}{12}}{1 - \left(1 + \frac{0.04}{12}\right)^{-12 \times 30}} \]Calculating:\[ \frac{0.04}{12} = 0.003333 \]\[ M = \frac{275,000 \times 0.003333}{1 - (1.003333)^{-360}} \]\[ M \approx \frac{916.575}{1 - 0.3085} \]\[ M \approx \frac{916.575}{0.6915} \]\[ M \approx 1,313.38 \]So, the monthly payment for option 1 is approximately \(\$1,313.38\).

Step 2: Calculate Monthly Payment for Option 2

Now, calculate the monthly payment for a 20-year loan at a 5% annual interest rate using the same formula. The parameters are as follows:- \(P = 275,000\),- \(r = 0.05\),- \(n = 12\),- \(t = 20\).Substitute these values:\[ M = \frac{275,000 \cdot \frac{0.05}{12}}{1 - \left(1 + \frac{0.05}{12}\right)^{-12 \times 20}} \]Calculating:\[ \frac{0.05}{12} = 0.004167 \]\[ M = \frac{275,000 \times 0.004167}{1 - (1.004167)^{-240}} \]\[ M \approx \frac{1,145.825}{1 - 0.3777} \]\[ M \approx \frac{1,145.825}{0.6223} \]\[ M \approx 1,813.74 \]Thus, the monthly payment for option 2 is approximately \(\$1,813.74\).

Step 3: Calculate Total Payments for Option 1

To find the total cost of the loan for option 1, multiply the monthly payment by the total number of payments:\[ \text{Total Payments} = M \times n \times t \]With \(M = 1,313.38\), \(n = 12\), and \(t = 30\):\[ \text{Total Payments} = 1,313.38 \times 12 \times 30 \]\[ \text{Total Payments} \approx 472,816.80 \]Thus, the total amount paid for option 1 is approximately \(\$472,816.80\).

Step 4: Calculate Total Payments for Option 2

Calculate the total payments for option 2 by multiplying the monthly payment by the total number of payments:\[ \text{Total Payments} = M \times n \times t \]With \(M = 1,813.74\), \(n = 12\), and \(t = 20\):\[ \text{Total Payments} = 1,813.74 \times 12 \times 20 \]\[ \text{Total Payments} \approx 435,297.60 \]Therefore, the total amount paid for option 2 is approximately \(\$435,297.60\).

Step 5: Calculate Interest Paid for Both Options

The interest paid can be found by subtracting the principal from the total payments.- For option 1: \[ \text{Interest Paid} = 472,816.80 - 275,000 = 197,816.80 \]- For option 2: \[ \text{Interest Paid} = 435,297.60 - 275,000 = 160,297.60 \]

Step 6: Compare Interest for Both Options

Compare the interest paid for both options:- Interest paid in option 1: \(197,816.80\)- Interest paid in option 2: \(160,297.60\)The difference in interest paid is:\[ 197,816.80 - 160,297.60 = 37,519.20 \]Option 2 results in \(\$37,519.20\) less interest paid than option 1.

Key Concepts

Monthly Payment Calculation

When deciding on a loan, determining the monthly payment is one of the most crucial aspects. It helps borrowers understand how much they need to budget each month to meet their repayment obligations. Applying the amortization formula, we derive the monthly payment for each loan option.

For option 1, which has a 30-year term with an annual interest rate of 4%, the formula is simplified to:

• \[ M = \frac{275,000 \times \frac{0.04}{12}}{1 - \left(1 + \frac{0.04}{12}\right)^{-360}} \]
• This comes out to approximately \( \\(1,313.38 \) per month.

In comparison, option 2, with a shorter 20-year term and a higher annual interest rate of 5%, gives us:

• \[ M = \frac{275,000 \times \frac{0.05}{12}}{1 - \left(1 + \frac{0.05}{12}\right)^{-240}} \]
• Resulting in about \( \\)1,813.74 \) monthly.

Understanding these values is foundational for making a budget-friendly decision that aligns with one's financial capacity.

Total Loan Cost

The total cost of the loan is a fundamental factor when considering financing options. It represents the full amount you will have paid by the end of the loan term, which includes both the principal and interest. To find the total loan cost, you multiply the monthly payment by the total number of payments over the life of the loan.

For option 1, the total payments are:

• \[ \text{Total Payments} = 1,313.38 \times 12 \times 30 \]
• This gives a total cost of approximately \( \\(472,816.80 \).

For option 2, with its higher monthly payments but shorter duration, the calculation is:

• \[ \text{Total Payments} = 1,813.74 \times 12 \times 20 \]
• Yielding a total cost of around \( \\)435,297.60 \).

Comparing the total loan costs helps identify which option leads to overall savings or fits better with long-term financial goals.

Interest Comparison

An essential part of deciding between loan options is comparing the total interest paid over the life of the loan. Interest is the cost of borrowing money and will contribute significantly to the total loan cost. Comparing how much you pay in interest can help in selecting the most cost-effective option.

For option 1, the interest paid would be calculated by subtracting the principal from the total payments:

• \[ \text{Interest Paid Option 1} = 472,816.80 - 275,000 = 197,816.80 \]

Meanwhile, option 2 results in:

• \[ \text{Interest Paid Option 2} = 435,297.60 - 275,000 = 160,297.60 \]

In comparing the two, you'll find option 2 is advantageous because it results in \( \$37,519.20 \) less interest paid compared to option 1. This realization is crucial when aiming to minimize the cost of borrowing.