Question

Consider a 30-year mortgage for $386,936 at an annual interest rate of 5.1%. After 12 years, the mortgage is refinanced to an annual interest rate of 3.7%. How much interest is paid on this mortgage?

Answer 1

Since it is not clear whether you want interest paid on old mortgage or new one, so I will calculate interest paid for old mortgage and new one.

So this is what we are going to do, First we need to calculate monthly payment so that we can calculate the balance of this mortgage after 12 years. From this balance we will again calculate monthly payment for the new mortgage with 3.7% interest rate. Then we can calculate interest paid on both mortgages.

We can use present value of annuity formula to find the monthly payment.

Where,
PVA = Present Value of Annuity / Amount of loan
A = Annuity or Payment
i = rate of interest
n = number of years
a = number of payments per year
na = number of payments

Substituting the values in the formula, we get:

So monthly payment on the first mortgage is $2100.87

Now we can find the balance of loan after 12 years, ie after 144th (12*12) payment.

Where,
PV = Present value / original balance
A = Annuity / Payment
i = rate of interest
a = number of payments per year
n = number of years
na = total number of payments

Therefore the balance of loan after 12 years = $296,546.48

On this balance there is a refinancing at lower rate of 3.7%.

Again we have to find the monthly payment.

We can use the same formula of present value of annuity.

Now we have monthly payment of both mortgages.

So let us first calculate interest paid on first mortgage.

Amount of loan =  $386,936.00

Principal Balance at 12 years = $296,546.48

Therefore, Principal part paid =  $386,936.00 - $296,546.48

=  $90,389.52

Total amount paid = $2100.87 * 144 = $302,525.01

Therefore, Interest paid = Total payment - Principal part

=$302,525.01 - $90,389.52

= $212,135.50

Now let us find the interest paid after refinancing.

Monthly payment after refinancing = $1,882.49

Total number of payments = (30-12) * 12 = 216

Total payment after refinancing = $1,882.49 * 216 = $406,617.97

Balance of loan at the time of refinancing =  $296,546.48

Therefore, Interest paid on new mortgage = Total payment - Balance of loan at refinancing

=$406,617.97 - $296,546.48

=$110,071.49