you take out a $250,000, 30-year loan with monthly payments at a 6.5% interest rate. 4 years later, you refinance the remaaining balance at a 4% rate and pay a $1,000 fee. What is the present value of the savings from doing this?
To calculate the present value of the savings from refinancing, we need to compare the total cost of continuing with the original loan to the total cost after refinancing.
Let's break down the calculation step by step:
Step 1: Calculate the remaining balance after 4 years of the original loan.
To do this, we can use a loan amortization formula. Subtract the principal amount already paid from the original loan amount:
Loan amount = $250,000
Interest rate = 6.5% per annum (compounded monthly)
Loan term = 30 years
Number of payments made in 4 years = 4 * 12 = 48
Monthly interest rate = 6.5% / 12 = 0.00542
Using the formula for calculating remaining balance on a loan:
Remaining balance = Loan amount * ((1 + Monthly interest rate) ^ Number of payments) - (Monthly payment * (((1 + Monthly interest rate) ^ Number of payments) - 1) / Monthly interest rate)
Substituting the values:
Remaining balance = $250,000 * ((1 + 0.00542) ^ 48) - (Monthly payment * (((1 + 0.00542) ^ 48) - 1) / 0.00542)
Step 2: Calculate the total cost of the original loan.
The total cost includes the remaining balance after refinancing and the monthly payments for the remaining loan term (26 years).
We have:
Loan term = 30 years (original term)
Remaining term after 4 years = 30 years - 4 years = 26 years
Monthly payment after refinancing = Monthly payment for the original loan
Using the formula for calculating total cost of a loan:
Total cost = Remaining balance + (Monthly payment * Number of remaining payments)
Total cost = Remaining balance + (Monthly payment * (Remaining term * 12))
Step 3: Calculate the total cost after refinancing.
The total cost after refinancing includes the new loan amount, the monthly payments for the remaining loan term (30 years - 4 years = 26 years), and the fee paid.
We have:
Loan amount = Remaining balance
Interest rate after refinancing = 4% per annum (compounded monthly)
Loan term = 26 years (remaining term after 4 years)
Monthly interest rate = 4% / 12 = 0.00333
Using the formula for calculating total cost of a loan:
Total cost after refinancing = Loan amount + (Monthly payment after refinancing * Number of remaining payments) + Fee
Total cost after refinancing = Remaining balance + (Monthly payment after refinancing * (Remaining term * 12)) + $1,000 fee
Step 4: Calculate the present value of the savings from refinancing.
The present value of the savings is the difference between the total cost of the original loan and the total cost after refinancing.
Present value of savings = Total cost - Total cost after refinancing
Now that we have broken down the steps, we can plug in the values and calculate the present value of the savings using a calculator or a spreadsheet software.