(15 points) You have been living in the house you bought 10 years ago for $300,000. At that time, you took out a loan for 80% of the house at a fixed rate 15-year loan at an annual stated rate of 9%. You have just paid off the 120th monthly payment. Interest rates have meanwhile dropped steadily to 6% per year, and you think it is finally time to refinance the remaining balance. But there is a catch. The fee to refinance your loan is $4,000. Should you refinance the remaining balance? How much would you save/lose if you decided to refinance?

120 monthly payments = $240,000

That leaves $60,000 yet to be paid
$60,000 at 9% = $60,000 + 5,400 in interest total $65,400
$60,000 at 6% = $3,600 in interest or $63,600 then add the fee and the total is $$67,600

9% = $65,400
6% = $67,600
You would lose $2,200 if you refinanced.

To determine whether you should refinance the remaining balance, let's break down the calculations and compare the savings or losses if you decide to refinance.

1. Calculate the original loan amount:
The house was bought for $300,000, and you took out a loan for 80% of the house price. Therefore, the original loan amount was 0.8 * $300,000 = $240,000.

2. Determine the remaining balance after 10 years and 120 monthly payments:
To find the remaining balance after 10 years, we need to calculate the number of payments made. Since you've made 120 monthly payments (10 years), there are 15 * 12 = 180 total monthly payments for the loan.

Using the future value of an annuity formula, we can calculate the remaining balance. The formula is:

Remaining Balance = P * ((1 + r)^n - (1 + r)^m) / ((1 + r)^n - 1)

Where:
P is the original loan amount (in this case, $240,000).
r is the monthly interest rate, which is the annual rate divided by 12 and expressed as a decimal (9% = 0.09, so r = 0.09 / 12 = 0.0075).
n is the total number of payments (180).
m is the number of payments made (120).

Using these values, we can calculate the remaining balance:

Remaining Balance = $240,000 * ((1 + 0.0075)^180 - (1 + 0.0075)^120) / ((1 + 0.0075)^180 - 1)

Using a calculator or spreadsheet, this calculation yields a remaining balance of approximately $130,084.

3. Calculate the new monthly payment with the lower interest rate:
Next, let's determine the new monthly payment if you refinance at a 6% interest rate. The loan term will remain the same since you have already paid off 10 years.

Using the present value of an annuity formula, the new monthly payment can be calculated. The formula is:

New Monthly Payment = PV * (r * (1 + r)^n) / ((1 + r)^n - 1)

Where:
PV is the remaining balance after 10 years ($130,084).
r is the new monthly interest rate, which is the annual rate divided by 12 and expressed as a decimal (6% = 0.06, so r = 0.06 / 12 = 0.005).
n is the number of remaining monthly payments (60).

Using these values, we can calculate the new monthly payment:

New Monthly Payment = $130,084 * (0.005 * (1 + 0.005)^60) / ((1 + 0.005)^60 - 1)

Using a calculator or spreadsheet, this calculation yields a new monthly payment of approximately $1,317.

4. Calculate the total cost of refinancing:
To determine the total cost of refinancing, we need to consider the fee of $4,000 and the new monthly payment.

Total Cost of Refinancing = $4,000 + ($1,317 * 60)

This calculation gives a total cost of refinancing of $81,820.

5. Calculate the total cost without refinancing:
To determine the total cost if you do not refinance and continue with the original loan, calculate the remaining monthly payments at the current interest rate of 9% for the next five years.

Remaining Monthly Payments = (180 - 120) = 60

Total Cost without Refinancing = ($1,420 * 60)

This calculation gives a total cost without refinancing of $85,200.

6. Compare the savings/losses:
To find out if you would save or lose money by refinancing, subtract the total cost with refinancing from the total cost without refinancing.

Savings/Losses = Total Cost without Refinancing - Total Cost with Refinancing

Savings/Losses = $85,200 - $81,820

In this case, refinancing would result in savings of approximately $3,380.

Therefore, based on the calculations, you should refinance the remaining balance as it would save you approximately $3,380.