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 it makes financial sense to refinance the remaining balance, we need to calculate the total cost of both options: continuing with the current loan or refinancing.
First, let's find out the remaining balance on the loan after paying off the 72nd monthly payment. Since it's a 25-year loan, the loan term in months is 25 * 12 = 300 months. Therefore, the remaining number of payments is 300 - 72 = 228 months.
To calculate the remaining balance, we use the formula for the present value of an ordinary annuity:
Remaining Balance = Monthly Payment * (1 - (1 + Monthly Interest Rate)^(-Remaining Payments))) / Monthly Interest Rate
The monthly interest rate is the stated annual rate divided by 12 and expressed as a decimal, so it is 9.5% / 12 = 0.00792.
Let's assume the monthly payment stays the same throughout the loan term. To find the monthly payment, we can use the present value of an ordinary annuity formula and solve for PMT (payment):
Monthly Payment = Remaining Balance * (Monthly Interest Rate) / (1 - (1 + Monthly Interest Rate)^(-Remaining Payments))
Now we can calculate the remaining balance and the monthly payment:
Remaining Balance = 250,000 * (1 + (1 + 0.00792)^-228) / 0.00792 = $204,172.84 (rounded to the nearest cent)
Monthly Payment = 204,172.84 * 0.00792 / (1 - (1 + 0.00792)^-228) = $2,137.18 (rounded to the nearest cent)
Now that we have the remaining balance and the monthly payment on the current loan, let's calculate the cost of refinancing.
The new loan would be at an interest rate of 6.0% per year, and we assume the remaining loan term remains the same at 228 months. Using the same formula shown earlier, we can calculate the new monthly payment:
New Monthly Payment = Remaining Balance * (Monthly Interest Rate) / (1 - (1 + Monthly Interest Rate)^(-Remaining Payments))
New Monthly Payment = 204,172.84 * 0.005 / (1 - (1 + 0.005)^-228) = $1,286.05 (rounded to the nearest cent)
The total cost of refinancing is the sum of the new monthly payment multiplied by the remaining number of payments and the refinancing fee:
Total Cost of Refinancing = (New Monthly Payment * Remaining Payments) + Refinancing Fee
Total Cost of Refinancing = ($1,286.05 * 228) + $4,000 = $293,158.40 (rounded to the nearest cent)
Finally, to determine if refinancing is beneficial, we compare the total cost of the current loan (remaining balance * remaining payments) with the total cost of refinancing.
Total Cost of Current Loan = Monthly Payment * Remaining Payments = $2,137.18 * 228 = $486,695.04 (rounded to the nearest cent)
Savings/Loss = Total Cost of Current Loan - Total Cost of Refinancing
Savings/Loss = $486,695.04 - $293,158.40 = $193,536.64 (rounded to the nearest cent)
If you decide to refinance, you would save $193,536.64 (rounded to the nearest cent).