Problem 46 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Sarah secured a bank loan of $$\$ 200,000$$ for the purchase of a house. The mortgage is to be amortized through monthly payments for a term of \(15 \mathrm{yr}\), with an interest rate of \(6 \% /\) year compounded monthly on the unpaid balance. She plans to sell her house in 5 yr. How much will Sarah still owe on her house?

Step-by-Step Solution

Answer

After selling her house in 5 years, Sarah still owes approximately \$142,182.69 on her mortgage.

Step 1: Convert the annual interest rate into a monthly interest rate

First, we need to find the equivalent monthly interest rate from the given annual interest rate of 6%. To do this, divide the annual interest rate by 12: \(i = \frac{6\%}{12} = \frac{0.06}{12} = 0.005\)

Step 2: Calculate the total number of monthly payments

Next, determine the total number of monthly payments to be made for the 15-year term: \(n = 15 \text{years} * 12 \text{months/year} = 180 \text{months}\)

Step 3: Determine the monthly mortgage payment

Now, calculate the monthly mortgage payment using the loan amount (\(\$200,000\)), the monthly interest rate (0.005), and the total number of monthly payments (180). Use the mortgage payment formula: \(M = P \frac{i(1 + i)^{n}}{(1 + i)^{n} - 1}\) Substitute the values: \(M = 200{,}000 \frac{0.005(1 + 0.005)^{180}}{(1 + 0.005)^{180} - 1}\) Calculate the result: \( M = \approx\$ 1,678.61\)

Step 4: Determine the number of payments made after selling the house in 5 years

Find out the total number of monthly payments made after Sarah sells her house in 5 years: \(payments = 5 \text{years} * 12 \text{months/year} = 60 \text{months}\)

Step 5: Calculate the remaining balance after 5 years

Now, determine the remaining balance after 5 years using the same mortgage payment formula, while replacing \(n\) with the remaining number of payments (remaining after 5 years): \(remaining \text{ balance} = M \frac{(1 + i)^{n} - (1 + i)^{payments}}{i}\) Substitute the values: \(remaining \text{ balance} = 1,678.61 \frac{(1 + 0.005)^{180} - (1 + 0.005)^{60}}{0.005}\) Calculate the result: \(remaining \text{ balance} \approx \$142,182.69\) After selling her house in 5 years, Sarah still owes approximately \$142,182.69 on her mortgage.

Key Concepts

Monthly Interest Rate

When dealing with mortgage amortization, understanding how the interest rate is applied is crucial. Interest rates on mortgages are typically given on an annual basis, but they are often compounded monthly.
To convert the annual interest rate to a monthly one, you simply divide by 12, since there are 12 months in a year. In Sarah's case, the annual interest rate was 6%.
So, to find the monthly interest rate, we calculate:

• Annual rate = 6% or 0.06 in decimal form
• Monthly rate = \( \frac{0.06}{12} = 0.005 \)

This 0.005 or 0.5% is the monthly interest rate that will be applied to the remaining balance each month.

Mortgage Payment Formula

The mortgage payment formula is a mathematical equation used to determine the monthly payment required to pay off a mortgage loan. This formula considers the principal amount, the monthly interest rate, and the number of payments.The formula is represented as:\[M = P \frac{i(1 + i)^{n}}{(1 + i)^{n} - 1}\]Where:

• **M** is the monthly payment
• **P** is the principal loan amount
• **i** is the monthly interest rate
• **n** is the total number of payments

Applying known values:

• P = \(200,000
• i = 0.005 (5% monthly interest rate)
• n = 180 (15 years \( \times \) 12 months)

Substituting these into the formula, Sarah's monthly payment becomes approximately \( \\)1,678.61 \). This is the amount Sarah would pay every month to amortize her mortgage over the agreed term.

Remaining Balance Calculation

After making payments for a certain period, it's useful to calculate the remaining balance of a mortgage, especially if you're considering selling your home early.
The remaining balance is the leftover amount that still needs to be repaid after making partial payments.To find it, we use a variant of the mortgage payment formula:\[\text{Remaining Balance} = M \frac{(1 + i)^{n} - (1 + i)^{p}}{i}\]Where:

• \( M \) is the monthly payment, already calculated as \( \\(1,678.61 \)
• \( i \) is the monthly interest rate, which is 0.005
• \( n \) is the total period, 180 months
• \( p \) is the number of payments made, which is 60 in Sarah's 5-year scenario

Using these values, Sarah's remaining balance after 5 years is approximately \( \\)142,182.69 \). This figure represents what Sarah will still owe on her mortgage when she decides to sell her house.