Problem 24 - Precalculus: Mathematical for Calculus

Question

Amortizing a Mortgage When they bought their house, John and Mary took out a \(\$ 90,000\) mortgage at \(9 \%\) interest, repayable monthly over 30 years. Their payment is \(\$ 724.17\) per month (check this using the formula in the text). The bank gave them an amortization schedule, which is a table showing how much of each payment is interest, how much goes toward the principal, and the remaining principal after each payment. The table below shows the first few entries in the amortization schedule. $$\begin{array}{|c|c|c|c|c|} \hline \begin{array}{c} \text { Payment } \\ \text { number } \end{array} & \begin{array}{c} \text { Total } \\ \text { payment } \end{array} & \begin{array}{c} \text { Interest } \\ \text { payment } \end{array} & \begin{array}{c} \text { Principal } \\ \text { payment } \end{array} & \begin{array}{c} \text { Remaining } \\ \text { principal } \end{array} \\ \hline 1 & 724.17 & 675.00 & 49.17 & 89,950.83 \\ 2 & 724.17 & 674.63 & 49.54 & 89,901.29 \\ 3 & 724.17 & 674.26 & 49.91 & 89,851.38 \\ 4 & 724.17 & 673.89 & 50.28 & 89,801.10 \\ \hline \end{array}$$ After 10 years they have made 120 payments and are wondering how much they still owe, but they have lost the amortization schedule. (a) How much do John and Mary still owe on their mortgage? [Hint: The remaining balance is the present value of the 240 remaining payments.] (b) How much of their next payment is interest and how much goes toward the principal? [Hint: since \(9 \% \div\) \(12=0.75 \%,\) they must pay \(0.75 \%\) of the remaining principal in interest each month.]

Step-by-Step Solution

Answer

(a) They owe \( \$79,719.92\). (b) Interest: \( \$598.65\), Principal: \( \$125.52\).

Step 1: Determine Total Payments Made

John and Mary have made 120 payments since they have been paying for 10 years. As a mortgage is monthly, it equates to \( 12 \text{ payments/year} \times 10 \text{ years} = 120 \text{ payments} \).

Step 2: Calculate the Remaining Principal

The present value of the remaining 240 payments (since they have 30 years or 360 months total and have made 120 payments, leaving 240) is what they still owe. The formula for the present value of an annuity (for the remaining balance) is given by:\[PV = P \times \frac{1 - (1 + r)^{-n}}{r}\]where:- \( P = \\( 724.17 \) (monthly payment),- \( r = 0.0075 \) (monthly interest rate, \( 9 \% \div 12 \) months).- \( n = 240 \) (remaining payments).Plug in the values:\[ PV = 724.17 \times \frac{1 - (1 + 0.0075)^{-240}}{0.0075} \]\[ PV \approx 724.17 \times 110.213 \approx 79,719.92 \]John and Mary still owe approximately \( \\) 79,719.92 \).

Step 3: Calculate Interest in Next Payment

For the next payment, calculate the interest portion first. The interest for each month is calculated as a percentage of the remaining principal: \[ \text{Interest} = ext{Remaining Principal} \times ext{Monthly Interest Rate} \]\[ = 79,719.92 \times 0.0075 \]\[ \approx 598.65 \]So, approximately \( \$ 598.65 \) of their next payment is interest.

Step 4: Calculate Principal in Next Payment

The portion of their payment that goes towards the principal is the difference between the total payment and the interest payment:\[ \text{Principal Payment} = 724.17 - 598.65 \]\[ \approx 125.52 \]So, approximately \( \$ 125.52 \) of their next payment will go towards reducing the principal.

Key Concepts

Mortgage Calculation

A mortgage is a loan commonly used to purchase a home, where the property itself serves as collateral. Let's break down how you calculate a monthly mortgage payment. Each month, you're responsible for paying back both the interest and a portion of the principal.

For John and Mary, they must make monthly payments over 30 years, which amounts to 360 payments. Their payment is calculated using an amortization formula, which is designed to spread the loan costs evenly over time.

The monthly payment can be calculated using the mortgage formula: \(M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1}\) where:

• \(M\) is the monthly payment,
• \(P\) is the loan amount (\$90,000 in their case),
• \(r\) is the monthly interest rate (0.75\,\% for John and Mary),
• \(n\) is the number of payments (360).

Applying this formula ensures homeowners like John and Mary know exactly how much they will pay each month. Understanding this process is key to planning and managing finances.

Present Value of Annuity

Present value is a crucial concept in finance, which asks, "How much is a series of future payments worth today?" For John and Mary, after 10 years of paying, determining the present value of the remaining payments helps them understand what they still owe.

With 240 payments left, they calculate the present value using this formula for annuities: \(PV = P \times \frac{1 - (1 + r)^{-n}}{r}\) where:

• \(P\) is their monthly payment of \\(724.17,
• \(r\) is the monthly interest rate (0.0075 in their case),
• \(n\) is the number of remaining payments (240).

By plugging in these numbers, they find their present value is approximately \\)79,719.92. This value reflects how much they need today to cover those future payments. This is essential for financial planning as it reveals the current value of their mortgage obligation.

Interest Payment Calculation

Interest is the cost of borrowing money, and each mortgage payment includes an interest component. For John and Mary, calculating the interest portion of their monthly payment is simple. They multiply the remaining principal by the monthly interest rate: \(\textrm{Interest} = \textrm{Remaining Principal} \times \textrm{Monthly Interest Rate}\). With a remaining principal of \\(79,719.92 and a monthly interest rate of 0.0075, their interest payment is about \\)598.65.

This calculation shows how much of their next \\(724.17 monthly payment goes to interest rather than reducing the loan itself. By subtracting the interest from the total payment, they know the amount reducing their loan principal, roughly \\)125.52. Understanding this breakdown helps borrowers track their progress in reducing the loan over time.