In: Advanced Math
Consider a 20-year mortgage for $282845 at an annual interest rate of 4.1%. After 6 years, the mortgage is refinanced to an annual interest rate of 2.5%. What are the monthly payments after refinancing?
According to given information first we need find the monthly payment .
Principal amount P = $282845
Period t = 20 x 12 = 240 monthly payments
Rate of interest r = 4.1%
Rate of interest r = 4.1 / 100 = 0.041 => 0.041/12 = 0.00341667
Now we can use the below formula to find the monthly payment (PMT)
PMT = [ p x r x (1+r)t ] / [(1+r)t-1]
PMT = [ 282845 x 0.00341667 x (1+0.00341667)240 ] / [(1+0.00341667)240-1]
PMT = [966.388 x 2.26733] / [2.26733 - 1]
PMT = [2191.1205] / [1.26733]
PMT = 1728.9265 ~ 1729
So MONTHLY PAYMENT amount is $1729
So we have to calculate the outstanding loan balance after 6 years which is72 payments by calculating the present value of the remaining installments, using the present value of an annuity formula
Pmt = Periodic monthly payment = $1729
i = Mortgage interest rate per period = 0.00341667
n = Number of remaining loan payments =240 – 72 = 168
we can use below formula
PV = Pmt x [(1 - 1 / (1 + i)n)] / i
PV = 1729 x [(1 - 1 / (1 + 0.00341667)168)] / (0.00341667)
PV = 1729 X [(1-1/(1.00341667)168)]/( 0.00341667)
PV = 1729 X [1-1/(1.77361)]/( 0.00341667)
PV = 1729 X [1-0.56382]/( 0.00341667)
PV = 1729 X [0.43618]/( 0.00341667)
PV = 1729 X 127.6623
PV = 220728.141 ~ 220728.14
The balance unpaid after 72 payments made is = $220728.14
Now this amount is refinance with the rate of interest is 2.5% for remaining 14 years.
So rate r = 2.5/100 = 0.025 / 12 = 0.0020833
And number of payments = 14 x 12 = 168
PMT = [ p x r x (1+r)t ] / [(1+r)t-1]
PMT = [220728.14 x 0.0020833x (1+0.0020833)240 ] / [(1+0.0020833)240-1]
PMT = [459.8429 x 1.41854] / [1.41854 - 1]
PMT = [652.3055] / [0.41854]
PMT = 1558.5261 ~ 1558.5
So MONTHLY PAYMENT amount is $1558.5