Problem 14 - Calculus and its applications

Question

Mortgages. Andre purchases an office building for \(\$ 450,000 .\) He makes a \(30 \%\) down payment and finances the remainder through a 15-yr mortgage at an annual interest rate of \(4.15 \%,\) compounded monthly. a) Find Andre's monthly mortgage payment. b) Assume that Andre makes every payment for the life of the loan. Find his total payments. c) How much interest does Andre pay over the life of the loan?

Step-by-Step Solution

Answer

Andre's monthly mortgage payment is $2,356.08, total payments are $424,094.40, and the interest paid is $109,094.40.

Step 1: Calculate Down Payment

Andre makes a 30% down payment on the office building. To find the down payment amount, we calculate 30% of $450,000:\[\text{Down Payment} = 0.30 \times 450,000 = 135,000\]

Step 2: Determine the Loan Amount

After the down payment, the remaining cost is financed through a mortgage. We subtract the down payment from the total purchase price:\[\text{Loan Amount} = 450,000 - 135,000 = 315,000\]

Step 3: Calculate Monthly Interest Rate

The annual interest rate is 4.15%. To find the monthly interest rate, divide by 12:\[\text{Monthly Interest Rate} = \frac{4.15}{100 \times 12} = 0.0034583\text{ (approx)}\]

Step 4: Find the Number of Total Payments

For a 15-year mortgage with monthly payments, the total number of payments is:\[\text{Total Payments} = 15 \times 12 = 180\]

Step 5: Calculate Monthly Mortgage Payment

The monthly payment can be calculated using the formula for an amortizing loan:\[M = P \frac{r(1+r)^n}{(1+r)^n - 1}\]Where:- \(M\) is the monthly payment,- \(P\) is the loan amount (\(315,000\)),- \(r\) is the monthly interest rate (\(0.0034583\)),- \(n\) is the number of payments (\(180\)).Substituting the values:\[M = 315,000 \times \frac{0.0034583(1+0.0034583)^{180}}{(1+0.0034583)^{180} - 1}\]Computing this gives approximately:\[M \approx 2,356.08\]

Step 6: Calculate Total Payments Over Loan Term

The total amount paid over 15 years is the monthly payment multiplied by the total number of payments.\[\text{Total Payments} = 2,356.08 \times 180 = 424,094.40\]

Step 7: Calculate Total Interest Paid

The total interest paid is the total payment amount minus the amount financed.\[\text{Total Interest} = 424,094.40 - 315,000 = 109,094.40\]

Key Concepts

Understanding Down Payment

A down payment is a crucial upfront part of purchasing a property, like Andre's office building. It's the initial amount that the buyer pays out of pocket towards the purchase. In many cases, a certain percentage of the total property price is set as a down payment. Andre, for example, paid 30% of the total purchase price of $450,000. This equates to: \[ \text{Down Payment} = 0.30 \times 450,000 = 135,000 \] Here are some key points about down payments:

• They reduce the amount you need to finance, thereby lowering the mortgage balance due.
• A higher down payment could lead to lower interest rates, as it reduces risk for lenders.
• It's often a necessary part of securing a mortgage loan.

Understanding the importance and role of a down payment in the larger context of a mortgage can help buyers prepare for their property purchase effectively.

Monthly Mortgage Payment Breakdown

Buying a property often involves taking out a mortgage, where you'll need to make monthly payments over a set period. These payments typically consist of principal and interest. Andre's office building's remaining cost after the down payment was financed through a mortgage. To budget correctly, calculating his monthly mortgage payment was essential.Here's how the monthly mortgage payments are calculated:

• Identify your total loan amount after the down payment. For Andre, this was \(315,000.
• Determine the monthly interest rate by dividing the annual rate by 12. Andre's rate was 0.0034583 (approximately).
• Use the mortgage formula: \[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \] where \(M\) is the monthly payment, \(P\) is the loan amount (\)315,000), \(r\) is the monthly interest rate, and \(n\) is the number of payments.

Andre's calculation results in a monthly payment of about $2,356.08. Understanding how these payments are calculated can help predict monthly financial obligations accurately.

Interest Calculation Over Time

Interest is the cost of borrowing money, and it accumulates over the life of a loan or mortgage. In Andre’s situation, determining the total interest paid during the loan term is necessary to understand the true cost of the office purchase.For Andre:

• Compute total payments by multiplying the monthly payment by the total number of payments: \[ \text{Total Payments} = 2,356.08 \times 180 = 424,094.40 \]
• Subtract the principal amount (loan amount) from the total payments to find the total interest paid: \[ \text{Total Interest} = 424,094.40 - 315,000 = 109,094.40 \]

The total interest of $109,094.40 over 15 years shows how interest can significantly increase the cost of borrowing.Knowing how to calculate interest helps in assessing the competitiveness of different loan offers and choosing the most favourable terms for your situation.