Two years ago, you purchased a $18,000 car, putting $3,500 down and borrowing the rest. Your loan was a 48-month fixed rate loan at a stated rate of 6.0% per year. You paid a non-refundable application fee of $100 at that time in cash. Interest rates have fallen during the last two years and a new bank now offers to refinance your car by lending you the balance due at a stated rate of 4.5% per year. You will use the proceeds of this loan to pay off the old loan. Suppose the new loan over the residual loan life requires a $200 non-refundable application fee. Given all this information, should you refinance? How much do you gain/lose if you do?
To determine whether you should refinance your car loan and calculate the gain/loss, we need to compare the costs of the existing loan and the new loan.
Existing Loan:
Principal amount = $18,000 - $3,500 (down payment) = $14,500
Loan term = 48 months
Stated interest rate = 6.0% per year
Non-refundable application fee = $100
To find the monthly payment for the existing loan, we can use the formula for a fixed-rate loan:
Monthly interest rate = (6.0% / 12) = 0.005
Number of months = 48
Monthly payment for the existing loan:
P = r * PV / (1 - (1 + r)^(-n))
where P is the monthly payment, r is the monthly interest rate, PV is the present value, and n is the number of months.
P = 0.005 * 14,500 / (1 - (1 + 0.005)^(-48))
P ≈ $342.93
Now, let's calculate the total cost of the existing loan over the remaining loan term:
Total cost of existing loan = Monthly payment * Number of months
Total cost of existing loan = $342.93 * 48
Total cost of existing loan ≈ $16,474.64
New Loan:
Balance due on existing loan after 2 years = Total cost of the existing loan so far
To find the total cost of the existing loan so far, we need to know how much you have paid towards the principal in the past 2 years. Let's calculate that:
Principal paid in the first 2 years = (Monthly payment * 12) * 2
Principal paid in the first 2 years = $342.93 * 12 * 2
Principal paid in the first 2 years ≈ $8,229.12
Balance due on existing loan after 2 years = Principal amount - Principal paid in the first 2 years
Balance due on existing loan after 2 years = $14,500 - $8,229.12
Balance due on existing loan after 2 years ≈ $6,270.88
Now, let's calculate the new monthly payment for the remaining balance using the new stated interest rate and the remaining loan term of 48 months:
Principal amount = Balance due on existing loan after 2 years
Loan term = 48 months
Stated interest rate = 4.5% per year
Non-refundable application fee = $200
To find the monthly payment for the new loan, we'll use the same formula as before, but with the new interest rate:
Monthly interest rate = (4.5% / 12) = 0.00375
Number of months = 48
Monthly payment for the new loan:
P = r * PV / (1 - (1 + r)^(-n))
P = 0.00375 * 6,270.88 / (1 - (1 + 0.00375)^(-48))
P ≈ $147.16
Total cost of the new loan = Monthly payment * Number of months
Total cost of the new loan = $147.16 * 48
Total cost of the new loan ≈ $7,060.48
To determine whether you should refinance, compare the total cost of the existing loan to the total cost of the new loan:
Cost difference = Total cost of the existing loan - Total cost of the new loan
Cost difference = $16,474.64 - $7,060.48
Cost difference ≈ $9,414.16
If the cost difference is positive, it means you would save money by refinancing. If the cost difference is negative, it means you would lose money.
Since the cost difference is positive ($9,414.16), you would save money by refinancing.