Problem 31 - Precalculus: Mathematics for Calculus

Question

DISCOVER: 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? IHint: 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

John and Mary still owe \( \$82,918.20 \). The next payment has \( \$621.89 \) in interest and \( \$102.28 \) toward principal.

Step 1: Calculate Monthly Interest Rate

The annual interest rate is 9%. To find the monthly interest rate, divide the annual rate by 12: \( \frac{9\%}{12} = 0.75\% \). Convert to a decimal form for calculations: \( 0.0075 \).

Step 2: Determine Remaining Principal After 120 Payments

Use the present value formula for annuities to determine the remaining principal after 120 payments. The formula is: \[ P = PMT \times \left(\frac{1 - (1 + r)^{-n}}{r}\right) \] where \( PMT = 724.17 \) (monthly payment), \( r = 0.0075 \) (monthly interest rate), and \( n = 240 \) (remaining payments).Plug in the values: \[ P = 724.17 \times \left(\frac{1 - (1 + 0.0075)^{-240}}{0.0075}\right) \] Calculate \( (1 + 0.0075)^{-240} \approx 0.1416 \),then \( 1 - 0.1416 = 0.8584 \),\[ P = 724.17 \times \left(\frac{0.8584}{0.0075}\right) \approx 724.17 \times 114.453 \approx 82,918.20 \]. Therefore, the remaining principal is approximately \( $82,918.20 \).

Step 3: Calculate Interest for Next Payment

Interest on the next payment is calculated using \( 0.0075 \) times the remaining principal \( 82,918.20 \).\[ \text{Interest} = 0.0075 \times 82,918.20 \approx 621.89 \].

Step 4: Find Principal for Next Payment

Subtract the calculated interest from the total monthly payment to find the principal portion:\[ \text{Principal} = 724.17 - 621.89 = 102.28 \].

Step 5: Confirm and Reflect on Results

After recalculating, the amounts are verified. After 120 payments, John and Mary still owe approximately \( \(82,918.20 \). For the next payment, \( \)621.89 \) is interest, and \( $102.28 \) is applied to the principal.

Key Concepts

Mortgage Calculation

Understanding how to calculate a mortgage is essential for anyone considering buying a home. A mortgage calculation typically involves determining the amount you borrow, known as the loan principal, and the interest rate applied to this amount. In our example, John and Mary borrowed $90,000 with an interest rate of 9% over a 30-year period.
To calculate monthly payments, a specific formula for annuities is used. The formula takes into account the principal, the interest rate, and the number of payment periods. This allows you to find out how much you need to pay each month to eventually repay the loan fully by its term end. Through this calculation, John and Mary have determined their monthly payment to be $724.17.
Knowing how to perform these calculations will help ensure your budget can accommodate long-term loan obligations, providing clarity and foresight into your financial planning.

• Principal: The initial amount of the loan.
• Interest Rate: The percentage charged on the principal.
• Term: Total duration over which the loan is repaid.

Interest Rate

The interest rate is a crucial component of mortgage costs. It represents the lender's charge for borrowing money and is typically expressed as an annual percentage. In John and Mary's mortgage, the annual rate is 9%, but mortgage payments are monthly, so you need to convert this to a monthly rate.
To find the monthly interest rate, divide the annual rate by 12 months, giving 0.75% per month or 0.0075 in decimal form. This monthly rate is used to calculate both the mortgage payment and the amount of interest in each payment.
The calculation of interest for each payment is straightforward. For any given month, the interest portion is the monthly interest rate multiplied by the remaining principal balance. Eventually, as the principal decreases, the interest portion of each payment also declines, while the principal portion increases.

• Annual Rate vs. Monthly Rate: Convert by dividing by 12.
• Interest Payment: Rate multiplied by remaining principal.

Principal Payment

The principal payment is the portion of your monthly mortgage payment that goes directly towards reducing the remaining balance of the loan. Over time, as you continue to make payments, this balance decreases.
Initially, a larger portion of your payment goes towards interest, but as the balance reduces, the interest cost goes down, and more of your payment is applied towards the principal.
In John and Mary's case, after 120 payments, they still owed around $82,918.20 on their principal. For their next payment, the principal portion can be calculated by deducting the interest payment ($621.89) from the total payment ($724.17), resulting in a principal payment of $102.28. Understanding how principal payments work will give you a clearer picture of how quickly you're paying down your loan.

• Reduce Loan Balance: Each payment reduces the principal owed.
• Over Time: Gradually, the principal portion of payments will increase.

Monthly Payments

Monthly payments are an essential part of the mortgage repayment process. They are a combination of the principal payment and the interest payment. Throughout the life of a mortgage, the fixed monthly payment remains constant, but the distribution between principal and interest shifts.
In a standard amortization schedule, like the one John and Mary received, the monthly payment formula ensures that their payment will remain $724.17 each month.
However, each payment includes different portions of interest and principal. Initially, the payment bulk goes towards interest. As time progresses, more goes towards paying off the principal. This constant monthly payment structure helps plan and adhere to your budget.

• Fixed Monthly Amount: Ensures payment remains the same over time.
• Adjustable Portions: Begins with more interest, gradually shifts to more principal.