a person purchased a $142,819 home 10 years ago by paying 15% down and signing a 30-year mortgage at 10.5% compounded monthly. Interest rates have dropped and the owner wants to refinance the upaid balance by signing a new 15-year mortgage at 6.3% compounded monthly. How much interest will refinancing save?
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To calculate the interest saved by refinancing, we need to determine the total interest paid under the current mortgage and compare it to the total interest that would be paid under the new mortgage.
First, let's calculate the current mortgage balance. Since the person paid a 15% down payment on a $142,819 home, the original loan amount was 85% of the home price:
Loan amount = $142,819 * 0.85 = $121,396.15
Next, we need to calculate the remaining balance on the mortgage after 10 years of payments. To do this, we'll use the formula for the future value of an ordinary annuity:
Future Value = P * [(1 + r)^n - 1] / r
where:
P = Monthly payment
r = Monthly interest rate
n = Number of payments
First, let's calculate the current monthly payment using the current mortgage details:
Monthly interest rate = (10.5% / 100) / 12 = 0.00875
Number of payments = 30 years * 12 months/year = 360
Using the formula, we can calculate the current mortgage balance:
Current mortgage balance = P * [(1 + r)^n - 1] / r
Since the goal is to find the remaining balance after 10 years, we need to solve for P. Rearranging the formula:
P = (Current mortgage balance * r) / [(1 + r)^n - 1]
Now, let's substitute the values into the formula:
Current mortgage balance = ($121,396.15 * 0.00875) / [(1 + 0.00875)^360 - 1]
Current mortgage balance ≈ $92,925.99
Thus, the remaining balance on the current mortgage after 10 years is approximately $92,925.99.
To calculate the total interest paid under the current mortgage, we subtract the original loan amount from the remaining balance:
Total interest paid = Current mortgage balance - Loan amount
Total interest paid = $92,925.99 - $121,396.15
Total interest paid ≈ $28,470.16
Now, let's calculate the interest paid under the new mortgage.
Since the new mortgage is for 15 years, the number of payments will be 15 years * 12 months/year = 180.
Using the same formula as before, we can calculate the total interest paid under the new mortgage:
New mortgage interest rate = (6.3% / 100) / 12 = 0.00525
New mortgage total interest paid = (Loan amount * r) / [(1 + r)^n - 1]
Substituting the values into the formula:
New mortgage total interest paid = ($92,925.99 * 0.00525) / [(1 + 0.00525)^180 - 1]
New mortgage total interest paid ≈ $19,090.16
Therefore, the interest saved by refinancing is:
Interest saved = Total interest paid under the current mortgage - Total interest paid under the new mortgage
Interest saved = $28,470.16 - $19,090.16
Interest saved ≈ $9,380.00
Thus, refinancing the unpaid balance will save approximately $9,380.00 in interest.