Question

In: Advanced Math

Consider a 20-year mortgage for $282845 at an annual interest rate of 4.1%. After 6 years,...

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?

Solutions

Expert Solution

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


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