Problem 24 - Algebra and Trigonometry

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. $$ \begin{array}{|c|c|c|c|c|}\hline \text { Payment } & {\text { Total }} & {\text { Interest }} & {\text { Principal }} & {\text { Remaining }} \\\ {\text { number }} & {\text { payment }} & {\text { payment }} & {\text { payment }} & {\text { principal }} \\ \hline 1 & {724.17} & {675.00} & {49.54} & {89,950.83} \\ {2} & {724.17} & {674.63} & {49.54} & {89,901.29} \\\ {3} & {724.17} & {674.26} & {49.91} & {89,851.38} \\ {4} & {724.17} & {673.89} & {50.28} & {89,801.10} \\ \hline\end{array} $$ 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

They owe $77,145.73; $578.59 is interest, $145.58 goes to principal.

Step 1: Understand the Total Mortgage Payments

John and Mary took a $90,000 mortgage at 9% annual interest, repayable monthly over 30 years. To find out how much they still owe after 10 years of paying $724.17 per month, first determine the number of payments already made and those remaining. 10 years equates to 120 monthly payments, leaving 240 payments to go (since 30 years total is 360 payments).

Step 2: Calculate Remaining Balance using Present Value Formula

The remaining balance is calculated using the present value of the annuity formula for the remaining 240 payments. The monthly interest rate is 0.75% (9% annually divided by 12 months). The formula is:\[PV = PMT \times \left(1 - (1 + r)^{-n}\right) / r\]where \(PMT = 724.17\), \(r = 0.0075\), and \(n = 240\). Plug these values into the formula to find the present value of the remaining payments.

Step 3: Apply the Formula to Find Present Value

Substitute the values into the formula:\[PV = 724.17 \times \left(1 - (1 + 0.0075)^{-240}\right) / 0.0075\]Simplifying this, you find the PV = $77,145.73. This is the amount they still owe on their mortgage.

Step 4: Calculate Interest on the Next Payment

To find out how much interest they will pay in their next payment, calculate 0.75% of the remaining principal (\(77,145.73):\[Interest = 77,145.73 \times 0.0075 = 578.59\]This means \)578.59 of the next payment goes towards interest.

Step 5: Determine the Principal of the Next Payment

Subtract the interest portion (\(578.59) from the total monthly payment (\)724.17) to find out how much goes toward the principal:\[Principal = 724.17 - 578.59 = 145.58\]Therefore, $145.58 of the next payment reduces the principal balance.

Key Concepts

Interest Payment Calculation

When it comes to mortgage amortization, each monthly payment you make includes a share for repaying the interest and the principal. Initially, a larger portion of the payment goes towards paying off the interest. This is because the interest is calculated based on the current outstanding principal amount. Let's illustrate this using John and Mary's mortgage. The monthly interest rate is derived from the annual interest rate of 9%, divided by 12 months, giving us 0.75%.
For calculating the interest portion of a particular payment, you multiply this monthly rate by the outstanding principal balance. For example, if the remaining principal amount is $77,145.73, the interest for that month is:

• Interest = Remaining Principal × Monthly Interest Rate
• Interest = 77,145.73 × 0.0075 = 578.59

In this instance, $578.59 of the monthly payment is allocated to interest.

Present Value of Annuity

Understanding the remaining balance of a mortgage involves using the concept of annuity, as mortgage payments are like an annuity where you pay a fixed amount for a specified period. Specifically, after several years of payments, to calculate what you still owe, we employ the formula for the present value of an annuity. This formula helps you figure out the current monetary worth of all future payments.
For John and Mary, they've made 120 payments over 10 years, leaving 240 payments remaining. Using the present value of an annuity formula:

• \[ PV = PMT \times \left(1 - (1 + r)^{-n}\right) / r \]
• Where PMT is the monthly payment \(724.17\), \(r\) is the monthly interest rate \(0.0075\), and \(n\) is the total remaining number of payments \((240)\).

Inserting these values, the formula calculates the present value to be \(77,145.73\). This is the balance John and Mary still owe on their mortgage.

Mortgage Principal Reduction

Reducing the principal amount of a mortgage is an important aspect of loan amortization. It directly affects how much interest you pay over time since interest is calculated on the remaining principal. As the years go by, the portion of each monthly payment that reduces the principal gradually increases.
For John and Mary's mortgage, once we know the interest portion of their payment (calculated to be $578.59), we can determine the amount that goes to reducing the principal. Simply subtract the interest from the total monthly payment.

• Principal Reduction = Total Monthly Payment - Interest Payment
• Principal Reduction = 724.17 - 578.59 = 145.58

Thus, $145.58 of their upcoming monthly payment will reduce the principal, gradually decreasing the total owed amount.