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?
$4,647
To determine whether you should refinance the remaining balance and how much you would save or lose, we need to compare the cost of refinancing to the potential savings from the lower interest rate.
First, let's calculate the remaining balance on your loan after 10 years of payments. Since you had a 15-year loan, you would have made 12 * 10 = 120 monthly payments.
To calculate the remaining balance, we need to find the present value of the loan. This can be done using the formula for the present value of an annuity:
Present Value = Loan Amount * (1 - (1 + r)^(-n)) / r
Where:
- Loan Amount is the original loan amount, which is 80% of the house price purchased for $300,000 = 0.8 * $300,000 = $240,000
- r is the monthly interest rate, which is the annual interest rate divided by 12 months = 9% / 12 = 0.75%
- n is the number of months remaining, which is 180 months (15 years) - 120 months = 60 months
By plugging in the values into the formula, we can calculate the present value of the loan after 10 years:
Present Value = $240,000 * (1 - (1 + 0.0075)^(-60)) / 0.0075 ≈ $187,577.94
Now, let's calculate the monthly payment on the remaining balance at the current interest rate of 6%. We can use the formula for the monthly payment of a loan:
Monthly Payment = Loan Amount * r / (1 - (1 + r)^(-n))
Where:
- Loan Amount is the remaining balance, $187,577.94
- r is the monthly interest rate, 6% / 12 = 0.5%
- n is the number of months remaining, 60 months
By substituting the values into the formula, we can calculate the monthly payment:
Monthly Payment = $187,577.94 * 0.005 / (1 - (1 + 0.005)^(-60)) ≈ $3,520.73
Now, let's compare the monthly payment and total costs for the remaining loan with the current interest rate and the refinanced loan with the reduced interest rate of 6%.
1. Remaining loan:
- Monthly payment: $3,520.73
- Total cost: $3,520.73 * 60 months = $211,243.80
2. Refinanced loan:
- Monthly payment: $187,577.94 * 0.005 / (1 - (1 + 0.005)^(-60)) ≈ $3,091.55
- Total cost: $3,091.55 * 60 months + $4,000 refinancing fee = $188,493.00
To determine whether you should refinance, compare the total costs of the two options:
Remaining loan cost ($211,243.80) vs. Refinanced loan cost ($188,493.00)
Based on the calculations, refinancing the remaining balance would result in savings since the total cost of the refinanced loan is lower. The potential savings from refinancing would be approximately $211,243.80 - $188,493.00 = $22,750.80.