A person purchased a $205,107 home 10 years ago by paying 10% down and signing a 30-year mortgage at 8.1% compounded monthly. Interest rates have dropped and the owner wants to refinance the unpaid balance by signing a new 15-year mortgage at 4.5% compounded monthly. How much interest will refinancing save
To find out how much interest refinancing will save, we need to compare the total amount of interest paid under the original mortgage with the total amount of interest that would be paid under the new mortgage.
Let's break down the steps on how to calculate the interest savings:
Step 1: Calculate the original mortgage amount
The person purchased a $205,107 home by paying 10% down, so the original mortgage amount would be 90% of the purchase price:
Original mortgage amount = $205,107 * (1 - 0.10) = $184,596.30
Step 2: Calculate the unpaid balance on the original mortgage
Since 10 years have passed, the remaining balance after 10 years will be:
Unpaid balance = Original mortgage amount * (1 + monthly interest rate)^total number of payments
We need to convert the annual interest rate to a monthly interest rate:
Monthly interest rate = (1 + 0.081)^(1/12) - 1 = 0.006615
The total number of payments over 30 years is 30 * 12 = 360 months.
Unpaid balance = $184,596.30 * (1 + 0.006615)^360 = $109,919.44
Step 3: Calculate the total interest paid on the original mortgage
To determine the total interest paid on the original mortgage, subtract the original mortgage amount from the sum of all monthly payments made over 30 years:
Total interest paid = (Monthly payment * total number of payments) - Original mortgage amount
The monthly payment can be calculated using the formula for the monthly payment on a mortgage:
Monthly payment = P * r * (1 + r)^n / ((1 + r)^n - 1)
Where:
P = Principal amount (unpaid balance on the mortgage)
r = Monthly interest rate
n = Total number of payments (30 years * 12 months per year)
Monthly payment = $109,919.44 * 0.006615 * (1 + 0.006615)^360 / ((1 + 0.006615)^360 - 1)
Monthly payment = $1,424.85
Total interest paid = ($1,424.85 * 360) - $184,596.30
Total interest paid = $393,743.00 - $184,596.30
Total interest paid = $209,146.70
Step 4: Calculate the new mortgage payments and total interest paid
Now, we need to calculate the new mortgage payments and total interest paid under the new 15-year mortgage with a 4.5% interest rate.
New mortgage amount = Unpaid balance from step 2 = $109,919.44
To calculate the new monthly payment, we can use the same formula as in step 3, but with the adjusted values:
Monthly payment = P * r * (1 + r)^n / ((1 + r)^n - 1)
Monthly payment = $109,919.44 * 0.00375 * (1 + 0.00375)^180 / ((1 + 0.00375)^180 - 1)
Monthly payment = $873.32
Total interest paid = (Monthly payment * total number of payments) - New mortgage amount
Total interest paid = ($873.32 * 180) - $109,919.44
Total interest paid = $157,197.60 - $109,919.44
Total interest paid = $47,278.16
Step 5: Calculate the interest savings
Interest savings = Total interest paid under original mortgage - Total interest paid under new mortgage
Interest savings = $209,146.70 - $47,278.16
Interest savings = $161,868.54
So, refinancing the unpaid balance with the new 15-year mortgage at 4.5% compounded monthly would save approximately $161,868.54 in interest.