Derek borrows $255,011.00 to buy a house. He has a 30-year mortgage with a rate of 5.16%. After making 144.00 payments, how much does he owe on the mortgage?

Answer 1

To determine how much Derek owes on his mortgage after making 144 payments, we need to calculate the remaining balance on the mortgage after these payments have been made. This can be done using the formula for the remaining balance on a fixed-rate mortgage.

The formula for the remaining balance \( B \) after \( n \) payments can be expressed as:

\[ B = P \times \frac{(1 + r)^N - (1 + r)^n}{(1 + r)^N - 1} \]

Where:

• \( B \) is the remaining balance
• \( P \) is the original loan amount (principal)
• \( r \) is the monthly interest rate (annual rate / 12)
• \( N \) is the total number of payments (loan term in months)
• \( n \) is the number of payments already made

Given:

• \( P = 255,011.00 \)
• Annual interest rate = 5.16%, thus the monthly interest rate \( r = \frac{5.16%}{12} = \frac{0.0516}{12} \approx 0.0043 \)
• Total number of payments \( N = 30 \times 12 = 360 \)
• Number of payments made \( n = 144 \)

Now, we can plug in the values into the formula.

Step 1: Calculate the monthly interest rate 'r'

\[ r = \frac{5.16}{100} \div 12 \approx 0.0043 \]

Step 2: Substitute into the balance formula

Now substituting the values into the formula:

\[ B = 255,011.00 \times \frac{(1 + 0.0043)^{360} - (1 + 0.0043)^{144}}{(1 + 0.0043)^{360} - 1} \]

Step 3: Calculate \( (1 + r)^{360} \) and \( (1 + r)^{144} \)

1. Calculate \( (1 + r)^{360} \): \[ (1 + 0.0043)^{360} \approx 4.4676 \]
2. Calculate \( (1 + r)^{144} \): \[ (1 + 0.0043)^{144} \approx 1.7033 \]

Step 4: Calculate the remaining balance

Now substituting these values:

\[ B = 255,011.00 \times \frac{4.4676 - 1.7033}{4.4676 - 1} \]

Calculating the numerator and denominator:

• Numerator: \( 4.4676 - 1.7033 \approx 2.7643 \)
• Denominator: \( 4.4676 - 1 \approx 3.4676 \)

Thus:

\[ B = 255,011.00 \times \frac{2.7643}{3.4676} \approx 255,011.00 \times 0.7973 \approx 203,144.92 \]

Final Answer

After making 144 payments, Derek owes approximately $203,144.92 on his mortgage.

Answer 2

What values do I enter for N, i/y, PV, PMT, and FV to solve on a finance calculator?

Answer 3

To use a financial calculator to determine how much Derek owes on his mortgage after making 144 payments, you can set the values for \( N \), \( i/y \), \( PV \), \( PMT \), and \( FV \) as follows:

1. N (Total Number of Payments):

   • This is the total number of payments over the life of the mortgage. For a 30-year mortgage with monthly payments: \[ N = 30 \times 12 = 360 \]

2. i/y (Interest Rate per Period):

   • This is the monthly interest rate. Given an annual interest rate of 5.16%: \[ i/y = \frac{5.16}{100} \div 12 \approx 0.0043 \text{ (or 0.43%)} \]

3. PV (Present Value / Loan Amount):

   • This is the original amount of the loan (the principal), which is: \[ PV = -255,011.00 \] (It’s entered as a negative number since it represents cash outflow/investment.)

4. PMT (Payment):

   • First, you need to calculate the monthly payment amount using the \( N \), \( i/y \), and \( PV \). The formula for calculating the monthly payment \( PMT \) can be done using the same financial calculator or through the formula:

   \[ PMT = PV \cdot \frac{r(1+r)^N}{(1+r)^N - 1} \] Where \( r \) is the monthly interest rate. You can also use the calculator to derive this value before inputting it for the remaining balance calculation.

5. FV (Future Value):

   • Since you're calculating the remaining balance after making 144 payments, the future value after 360 payments would be: \[ FV = ? \] When solving for the balance after 144 payments, you will generally leave it as 0 since you're interested in the remaining balance.

Steps on the Calculator:

1. Calculate Monthly Payment (PMT):

   • Input \( N = 360 \)
   • Input \( i/y = 0.43 \)
   • Input \( PV = -255011 \)
   • Calculate \( PMT \) (this will give you the monthly payment amount).

2. Calculate Remaining Balance after 144 Payments:

   • Input \( N = 360 \) (total payments)
   • Input \( i/y = 0.43 \) (monthly interest)
   • Input \( PV = -255011 \) (the original loan)
   • Input \( PMT \) (the amount calculated in the previous step)
   • Input \( N = 144 \) (the number of payments already made)
   • Calculate \( FV \) (this will give you the remaining balance).

Conclusion:

1. To find the monthly payment, use \( N = 360, i/y = 0.43, PV = -255011 \) to find \( PMT \).
2. Then, to find \( FV \) after 144 payments, set \( N = 216 \) (remaining payments), \( i/y = 0.43, PV = 0, PMT = [value from previous calculation] \).

This process gives you the remaining balance as \( FV \).