Question

The problem describes a debt to be amortized. (Round your answers to the nearest cont.) A man buys a house for $310,000. He makes a $150,000 down payment and amortizes the rest of the purchase price with semiannual payments over the next 12 years. The interest rate on the debt is 8%, compounded semiannually. (a Find the size of each payment. $ (b) Find the total amount paid for the purchase. $ (c) Find the total interest paid over the wife of the loan $

Answer 1

man purchase a new home with $310000

Down payment = $150000

Mortgage amount = 310000 – 150000 = 160000

Mortgage amount = $160000

Total term = 12 years

Rate of interest = 8% = 8/100 = 0.08

Compounding frequency = semi annually = 2

Now we can use the below formula to find the payment (PMT)

PMT = [ P x (r/n) x (1+(r/n)^nt ] / [(1+(r/n))^nt -1]

P = principal amount = $160000

Rate of interest r = 0.08

Compounding frequency n = 2 semi annually

Number of years t = 12

PMT = monthly payment = $?

PMT = [160000 x (0.08/2) x (1+(0.08/2))^{2 (12)} ] / [(1+(0.08/2)²⁽¹²⁾-1]

PMT = [160000 x (0.04) x (1+(0.04))⁽²⁴⁾ ] / [(1+(0.04)⁽²⁴⁾-1]

PMT = [6400 x (1.04))⁽²⁴⁾ ] / [(1.04)⁽²⁴⁾-1]

PMT = [6400 x (2.5633)] / [(2.5633)-1]

PMT = [16405.12] / [(1.5633)]

PMT = 10493.9

So regular semi annual payemt = $10493.9

b) now the total payment during loan period = number of payments x PMT

                                                                                = 24 x 10493.9 = $251853.6

So total amount paid to house = down payment + total payment during loan

                                                                = 150000 + 251853.6 = $401853.6

c) Now the total interest = total loan payment – principal amount

                                                = $251853.6 - $160000 = $91853.6

So total interest = $91853.6

Now we can use the below formula to find the principal amount(P)

PMT = [ P x (r/n) x (1+(r/n)^nt ] / [(1+(r/n))^nt -1]

P = principal amount = $?

Rate of interest r = 6% = 6/100 = 0.06

Compounding frequency n = 12 monthly

Number of years t = 2

PMT = monthly payment = $300 (maximum)

300 = [P x (0.06/12) x (1+(0.06/12))^{12 (2)} ] / [(1+(0.06/12)¹²⁽²⁾-1]

300 = [P x (0.005) x (1+(0.005))⁽²⁴⁾ ] / [(1+(0.05)⁽²⁴⁾-1]

300 = [P x (0.005) x (1.005))⁽²⁴⁾ ] / [(1.05)⁽²⁴⁾-1]

300 = [P x (0.005) x (1.12715] / [(1.12715)-1]

300 = [P x (0.005635)] / [(0.12715)]

300 = [P x 0.044323]

P = 300 / 0.044323 = 6768.49 ~ 6768.5

So home owner can afford only $6768.5