Question

3, Jimmy takes out a $50,000 mortgage on a home at a 5.6% interest rate convertible monthly. He plans to pay off the mortgage with monthly payments for 20 years. Immediately following the 60h payment, he renegotiates the loan. He agrees to make a cash payment of $7500 and will pay the remaining balance with monthly payments over 10 years at a rate of 4.1% convertible monthly. Calculate his new monthly payment.

Answer 1

according to given information Principal amount    P = $ 50000

Period   n = 20 years

Rate of interest r = 5.6 / 100 = 0.056

Compounding frequency monthly so r = 0.056 / 12 = 0.004667

Number of payments n = 20 x 12 = 240

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

PMT = [ p x r x (1+r)ⁿ ] / [(1+r)ⁿ-1]

PMT = [50000 x 0.004667 x (1+0.00467)²⁴⁰ ] / [(1+0.004667)²⁴⁰-1]

PMT = [233.35 x 3.05931] / [3.05931 - 1]

PMT = [713.8899] / [2.05931]

PMT = $346.664 ~ 346.67

So monthly payment is $ 346.67

So we have to calculate the outstanding loan balance after 60 payments by calculating the present value of the remaining installments, using the present value of an annuity formula

Pmt = Periodic monthly payment = $346.67

i = Mortgage interest rate per period = 0.004667

n = Number of remaining loan payments =240 – 60 = 180

we can use below formula

PV = PMT x [(1 - 1 / (1 + i)ⁿ)] / i

PV = 346.67 x [(1 - 1 / (1 + 0.004667)¹⁸⁰)] / (0.004667)

PV = 346.67 X [(1-1/(1.004667)¹⁸⁰)]/( 0.004667)

PV = 346.67 X [1-1/(2.31198)]/( 0.004667)

PV = 346.67 X [1-0.432529]/( 0.004667)

PV = 346.67 X [0.567471]/( 0.004667)

PV = 346.67 X 121.59224

PV = 42152.38

The balance unpaid after 60 payments made is = $42152.38

Now he agrees to paid $7500 cash payment. So the remaining balance = 42152.38 – 7500 = 34652.38

So the new principal amount p = $34652.38

Interest r = 4.1% = 4.1/100 = 0.041

Compounding period monthly so r = 0.041/12 = 0.00341667

Number of payments = 10yrs = 10 x 12 = 120

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

PMT = [ p x r x (1+r)ⁿ ] / [(1+r)ⁿ-1]

PMT = [34652.38 x 0.00341667 x (1+0.00341667)¹²⁰ ] / [(1+0.00341667)²⁴⁰-1]

PMT = [118.39574 x 1.505765] / [1.505765- 1]

PMT = [178.27625] / [0.505765]

PMT = $352.4883 ~ 352.5

So monthly payment is $ 352.2