Problem 31 - Precalculus Mathematics 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 \(240 \text { remaining payments. }]\) (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 approximately \$68,667.94. The next payment includes \$515.01 for interest and \$209.16 for principal.

Step 1: Calculate the Remaining Balance (Part a)

To find the remaining balance after 120 payments, we calculate the present value of the remaining 240 payments.The present value of an annuity formula is: \[PV = P \times \left(1 - (1 + r)^{-n}\right) / r\]where:- \(P\) is the monthly payment (\\(724.17)- \(r\) is the monthly interest rate (9% annual, so \(0.09/12 = 0.0075)\)- \(n\) is the number of remaining payments (240).Substitute the values:\[PV = 724.17 \times \left(1 - (1 + 0.0075)^{-240}\right) / 0.0075\]\[PV \approx 724.17 \times (1 - (1 + 0.0075)^{-240}) / 0.0075 \approx 68,667.94\]John and Mary still owe approximately \\)68,667.94.

Step 2: Determine the Interest on the Next Payment (Part b)

The interest portion of the next payment is calculated as 0.75% of the remaining principal:\[\text{Interest} = \,0.75\% \times 68,667.94 \]This results in:\[\text{Interest} \approx 0.0075 \times 68,667.94 \approx 515.01\]So, \$515.01 of the next payment is interest.

Step 3: Calculate the Principal Payment (Part b)

The principal portion of the payment is the total payment minus the interest:\[\text{Principal} = 724.17 - 515.01 = 209.16\]So, \$209.16 of the next payment goes toward the principal.

Key Concepts

Present Value of Annuity

The present value of an annuity is a key concept in understanding how mortgage payments work over time. This concept helps determine how much a series of future payments is worth in today's dollars. In the case of John and Mary's mortgage, we're looking at how much the 240 remaining payments are worth right now.
To find this, we use the formula: \[PV = P \times \left(1 - (1 + r)^{-n}\right) / r\]Here:

• \(P\) is the monthly payment, which is $724.17.
• \(r\) is the monthly interest rate. Given their annual interest rate of 9%, this becomes \(0.09/12 = 0.0075\).
• \(n\) is the number of remaining payments, which is 240.

Substituting these values into the formula will give us the present value or the remaining balance on the mortgage. This calculation effectively sums up all the remaining payments discounted back to their value today.

Monthly Interest Rate

Understanding the monthly interest rate is crucial for calculating both the interest portion of a mortgage payment and the present value of an annuity. Since mortgage interest is often provided on an annual basis, it is essential to convert this to a monthly rate to align with monthly payments.
For John and Mary's mortgage, the annual rate is 9%. By dividing this annual rate by 12, we determine the monthly interest rate:\[r = \frac{9\%}{12} = 0.0075\]The monthly interest rate of 0.75% reflects the percentage of the remaining principal paid as interest each month. This rate forms the basis for understanding how each payment distributes between interest and principal over the life of the loan.

Remaining Balance Calculation

Calculating the remaining balance of a mortgage involves understanding how much of the loan has been paid and how much still needs to be paid. For amortizing loans like mortgages, a part of each payment goes towards interest and the remainder towards reducing the principal.
After 10 years, John and Mary made 120 payments. To find the remaining balance, they need to measure the present value of the 240 payments they still owe. This value reflects the remaining principal of the loan.
By using the present value of an annuity formula (discussed earlier), we find that the remaining balance is approximately $68,667.94. This means that while they have made significant payments on their mortgage, substantial debt remains. By deducting the calculated interest portion from their total monthly payment, we arrive at the amount applied towards reducing this remaining balance.