Problem 31 - 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. 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? IHint: 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 $79,618.97. $597.14 of their next payment is interest, and $127.03 goes towards the principal.

Step 1: Understand Present Value of Remaining Payments

To calculate how much John and Mary still owe, we need to find the present value of the remaining 240 payments of \( \\(724.17 \) each. The present value of an annuity formula is \( PV = P \times \frac{1 - (1 + r)^{-n}}{r} \). Here, \( P = \\)724.17 \), \( n = 240 \), and \( r = \frac{0.09}{12} = 0.0075 \).

Step 2: Plug Values into Present Value Formula

Substitute the given values into the formula: \[ PV = 724.17 \times \frac{1 - (1.0075)^{-240}}{0.0075} \].

Step 3: Calculate Present Value

By calculating the above expression, we find the present value \( PV = 724.17 \times \frac{1 - (1.0075)^{-240}}{0.0075} \approx 724.17 \times 110.0946 \approx \$79,618.97 \). This is how much they still owe.

Step 4: Calculate Monthly Interest Portion

For the interest portion of the next monthly payment, use the formula \( \text{Interest} = \text{Remaining Principal} \times r \). Thus, \( \text{Interest} = 79,618.97 \times 0.0075 = \$597.14 \).

Step 5: Calculate Principal Portion of Payment

The principal portion is the remainder of their payment after subtracting the interest: \( \text{Principal} = 724.17 - 597.14 = \$127.03 \).

Key Concepts

Present Value Calculation

When dealing with a long-term financial arrangement like a mortgage, the concept of present value plays a crucial role. Present value (PV) represents the total amount needed today to match future payments made over time.
In John and Mary's case, they need to calculate the present value of their remaining 240 mortgage payments to determine how much they still owe.
The formula used is for the present value of an annuity, given by:\[PV = P \times \frac{1 - (1 + r)^{-n}}{r}\]where:

• \(P\) is the payment amount per period.
• \(r\) is the monthly interest rate.
• \(n\) is the number of remaining payments.

This equation essentially helps us to sum up all these future payments into a single lump sum value as if it were made today. For John and Mary, substituting in their values, their present value calculation yields approximately \(\$79,618.97\). This means, if they wanted to pay off their mortgage today, they'd need around this amount.

Interest and Principal in Payments

Mortgages involve regular payments, each consisting of two parts: interest and principal.
Understanding this division helps in tracking how much of your payment is reducing the debt, and how much is simply servicing the interest.Every month, the interest part of the payment is calculated based on the remaining principal.
The formula to determine the interest portion is straightforward:\[\text{Interest} = \text{Remaining Principal} \times r\]For John and Mary, with a remaining balance of approximately \(\\(79,618.97\), their next interest charge is about \(\\)597.14\).
After paying the interest, what's left of their monthly payment goes towards decreasing the principal. For their monthly payment of \(\\(724.17\), the principal portion is approximately \(\\)127.03\).
This method ensures that the loan gradually reduces over time until it's fully paid off.

Monthly Repayment Schedule

A loan’s monthly repayment schedule is an important tool for borrowers.
It provides a breakdown of each payment, showing the amounts allocated to interest and principal.
This schedule offers a roadmap for the loan’s payoff and the gradual build-up of equity.In John and Mary's scenario, each of their installments of \(\$724.17\) has a declining interest payment as the loan’s principal decreases with each payment.

• Initially, most of the payment covers interest.
• Over time, the principal portion grows, and more of the payment chips away at the debt.
• This transition is shown in the amortization schedule, though John and Mary lost theirs.

The schedule helps visualize how the debt is reducing, when more of the payment will target the principal, and it makes tracking financial progress over the life of the loan easier.