Problem 13 - Calculus and its applications

Question

Home mortgages. The Hogansons purchase a new home for \(\$ 195,000\). They make a \(25 \%\) down payment and finance the remainder with a 30 -yr mortgage at an annual interest rate of \(5.2 \%,\) compounded monthly. a) Find the Hogansons' monthly mortgage payment. b) Assume that the Hogansons make every payment for the life of the loan. Find their total payments. c) How much interest do the Hogansons pay over the life of the loan?

Step-by-Step Solution

Answer

Monthly payment: \(\$806.12\); Total payments: \(\$290,203.20\); Total interest: \(\$143,953.20\).

Step 1: Calculate the Down Payment

The Hogansons make a 25% down payment on the home purchase price of \( \\(195,000 \). The down payment is calculated as \( 0.25 \times 195,000 = \\)48,750 \).

Step 2: Determine the Loan Amount

After the down payment, the amount to be financed (loan amount) is the purchase price minus the down payment: \( 195,000 - 48,750 = \$146,250 \).

Step 3: Calculate the Monthly Interest Rate

The annual interest rate is 5.2%, compounded monthly. Convert this annual rate to a monthly interest rate by dividing by 12: \( \frac{5.2}{12} = 0.4333\% \). As a decimal, this is \( \frac{0.4333}{100} = 0.004333 \).

Step 4: Determine the Number of Payments

Since it is a 30-year mortgage with monthly payments, calculate the total number of payments: \( 30 \times 12 = 360 \) payments.

Step 5: Use the Loan Payment Formula

The formula for the monthly mortgage payment \( M \) is \[ M = P \frac{r(1+r)^n}{(1+r)^n-1} \] where \( P \) is the loan amount (\$146,250), \( r \) is the monthly interest rate (0.004333), and \( n \) is the number of payments (360).

Step 6: Calculate the Monthly Mortgage Payment

Substitute the values into the formula: \[ M = 146,250 \frac{0.004333(1+0.004333)^{360}}{(1+0.004333)^{360}-1} \]. Doing the calculation gives a monthly payment \( M \approx \$806.12 \).

Step 7: Find Total Payments

Multiply the monthly payment by the number of payments: \( 806.12 \times 360 = \$290,203.20 \). This is the total amount they pay over the 30 years.

Step 8: Calculate Total Interest Paid

Subtract the original loan amount from the total payments to find the total interest paid: \( 290,203.20 - 146,250 = \$143,953.20 \).

Key Concepts

Amortization Schedule

Understanding how to manage a mortgage over its lifespan is crucial, and an amortization schedule is a handy tool for this. An amortization schedule is a table that details each periodic payment on an amortizing loan. It shows how much of each payment goes towards interest and how much goes towards the principal balance.
For the Hogansons' 30-year mortgage, each monthly payment is initially composed largely of interest. Over time, more of each payment chips away at the principal. Here’s why this matters:

• At the start of the mortgage, your payments primarily go towards interest.
• Over time, as the principal decreases, the interest portion of your payment gets smaller.
• By the end of your mortgage term, the bulk of your payment reduces the principal.

Visualizing this through an amortization schedule helps you understand the financial journey of your loan, ensuring that you stay on track with payments and can plan for future financial decisions. It removes guesswork by detailing exactly how each payment is applied over time.

Compound Interest

Compound interest is a powerful force in the world of finance, and it plays a crucial role in mortgage calculations. It refers to the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods.
For a mortgage like the Hogansons', interest is compounded monthly. Meaning, each month the interest is calculated not only on the original amount borrowed but also on the interest that has been added up to that point. Here’s how it benefits or affects:

• Initially, you will see a considerable chunk of your payments going towards interest.
• The impact of compounding means that interest can accumulate rapidly if not managed.
• Understanding this can motivate you to make additional payments, if possible, to reduce total interest paid.

Familiarity with compound interest in your mortgage helps in visualizing the cost of borrowing in realistic terms, fostering decisions such as refinancing or early payments to mitigate the impact of accrued interest.

Loan Payment Formula

Calculating your monthly mortgage payment accurately is essential for financial planning. The formula used can initially seem daunting but is straightforward when broken down. Let’s demystify it with what's used in the Hogansons' case.
The formula is:\[ M = P \frac{r(1+r)^n}{(1+r)^n-1} \]Where:

• \(M\) is the total monthly mortgage payment.
• \(P\) is the principal loan amount ($146,250 ext{ for the Hogansons}).
• \(r\) is the monthly interest rate (0.004333 or 0.4333%).
• \(n\) is the number of payments (360 months).

This formula calculates how much you need to pay monthly to cover both principal and interest over the period of the loan. By dissecting this formula, borrowers can fully understand their payments, ensuring greater control over their financial commitments and enabling more strategic decisions to be made for any additional payments to lessen total interest paid.