You have been living in the house you bought 6 years ago for $250,000. At that time, you took out a loan for 80% of the house at a fixed rate 25-year loan at an annual stated rate of 9.5%. You have just paid off the 72th monthly payment. Interest rates have meanwhile dropped steadily to 6.0% 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?

To determine whether or not you should refinance the remaining balance, we need to compare the savings from the lower interest rate to the cost of refinancing. Let's begin by calculating the remaining balance on your loan after 72 monthly payments.

Loan Amount: $250,000 (80% of the house's original value)

Remaining Balance = Loan Amount x (1 + Interest Rate/12)^(Total Number of Payments) - Monthly Payment x [((1 + Interest Rate/12)^(Total Number of Payments) - 1) / (Interest Rate/12)]

Where:
- Interest Rate: Annual Stated Rate / 100
- Total Number of Payments: Loan Term in years x 12
- Monthly Payment: Loan Amount x (Monthly Interest Rate x (1 + Monthly Interest Rate)^(Total Number of Payments)) / ((1 + Monthly Interest Rate)^(Total Number of Payments) - 1)

Let's calculate the remaining balance:

Annual Stated Rate = 9.5%
Monthly Interest Rate = Annual Stated Rate / 12 / 100

Loan Term = 25 years

Total Number of Payments = 25 x 12 = 300

Monthly Payment = $250,000 x (Monthly Interest Rate x (1 + Monthly Interest Rate)^300) / ((1 + Monthly Interest Rate)^300 - 1)

Now, we can calculate the remaining balance after 72 monthly payments:

Remaining Balance = $250,000 x (1 + Monthly Interest Rate)^300 - Monthly Payment x [((1 + Monthly Interest Rate)^300 - 1) / (Monthly Interest Rate)]

Now that we have the remaining balance, let's determine the monthly payment and remaining term if you refinance at the new interest rate of 6.0%.

Interest Rate (refinance): 6.0%

New Monthly Payment = Remaining Balance x (Monthly Interest Rate (refinance) x (1 + Monthly Interest Rate (refinance))^Remaining Term) / ((1 + Monthly Interest Rate (refinance))^Remaining Term - 1)

Remaining Term = 300 - 72 = 228

Now, let's calculate the savings if you decide to refinance:

Savings = (Old Monthly Payment - New Monthly Payment) x Remaining Term

Old Monthly Payment = Monthly Payment in the original loan

Finally, let's compare the savings to the cost of refinancing:

Net Benefit = Savings - Refinancing Fee

If the Net Benefit is positive, then it would be financially beneficial to refinance the remaining balance. Otherwise, it may not be worth refinancing.

Let's calculate the remaining balance, new monthly payment, savings, and net benefit.

To determine whether you should refinance the remaining balance and how much you would save or lose, we need to calculate the remaining balance after 72 monthly payments and compare it to the cost of refinancing.

First, let's calculate the remaining balance on your loan after 72 monthly payments. Since it is a fixed-rate loan, the monthly payment amount remains the same throughout the loan term. We can use the loan amortization formula to find the remaining balance.

Loan amount = $250,000
Loan term = 25 years
Annual interest rate = 9.5%

To calculate the monthly interest rate, divide the annual interest rate by 12:
Monthly interest rate = 9.5% / 12 = 0.00792

To calculate the monthly payment, we can use the loan amortization formula:

Monthly payment = (loan amount * monthly interest rate)/(1 - (1 + monthly interest rate)^(-n))

Where n is the total number of payments. In this case, n = 25 years * 12 months/year = 300 months.

Plugging in the values:
Monthly payment = (250,000 * 0.00792)/(1 - (1 + 0.00792)^(-300))
Monthly payment ≈ $2,147.29

Now, we calculate the remaining balance after 72 monthly payments using the formula:

Remaining balance = (Loan amount * (1 + monthly interest rate)^n) - (monthly payment * ((1 + monthly interest rate)^n - 1)/monthly interest rate)

Plugging in the values:
Remaining balance = (250,000 * (1 + 0.00792)^72) - (2,147.29 * ((1 + 0.00792)^72 - 1)/0.00792)
Remaining balance ≈ $173,550.41

Now, let's consider the option to refinance. The interest rate has dropped to 6.0% per year, and the fee to refinance is $4,000.

To calculate the new monthly payment with the lower interest rate, we'll use the remaining balance of $173,550.41, a loan term of 25 years (300 months), and the new annual interest rate.

New annual interest rate = 6.0%
New monthly interest rate = 6.0% / 12 = 0.005

Using the loan amortization formula, we can calculate the new monthly payment:

New monthly payment = (remaining balance * new monthly interest rate)/(1 - (1 + new monthly interest rate)^(-n))

Plugging in the values:
New monthly payment = (173,550.41 * 0.005)/(1 - (1 + 0.005)^(-300))
New monthly payment ≈ $1,042.80

Now, let's calculate the total cost of the new loan by multiplying the new monthly payment by the total number of payments:

Total cost of new loan = New monthly payment * n
Total cost of new loan ≈ $1,042.80 * 300 ≈ $312,840

To determine how much you would save/lose by refinancing, we need to compare the total cost of the new loan ($312,840) with the remaining balance on the current loan ($173,550.41) plus the refinancing fee ($4,000).

Total cost of refinancing = Remaining balance + Refinancing fee
Total cost of refinancing ≈ $173,550.41 + $4,000 ≈ $177,550.41

Savings/Loss = Total cost of current loan - Total cost of refinancing
Savings/Loss ≈ $312,840 - $177,550.41 = $135,289.59

Therefore, if you decide to refinance, you would save approximately $135,289.59 over the remaining term of the loan.