Question

A borrower takes out a 30-year adjustable rate mortgage loan for $200,000 with monthly payments. The first two years of the loan have a "teaser" rate of 2%, after that, the rate can reset with a 7% annual payment cap. On the reset date, the composite rate is 6%. Assume that the loan allows for negative amortization. What would be the outstanding balance on the loan at the end of Year 3?

Answer 1

Sol:

Loan Principal balance at year 0 (P) = $200,000

Interest rate (r)= 2%, Monthly = 2%/12 = 0.1667%

Loan period (n) = 30 years, Monthly = 30 x 12 = 360

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

PMT = $200,000 x 0.1667% x (1 + 0.1667%)^360 / (1 + 0.1667%)^360 -1

PMT = $200,000 x 0.001667 x (1 + 0.001667)^360 / (1 + 0.001667)^360 -1

PMT = $200,000 x 0.001667 x (1.001667)^360 / (1.001667)^360 -1 = $739.24

Loan Principal balance at year 2 = P x (1 + r)^n - PMT x (1 + r)^n -1 / r

Loan Principal balance at year 2 = 200000 x (1 + 0.1667%)^24 - 739.24 x (1 + 0.1667%)^24 -1 / 0.1667%

Loan Principal balance at year 2 = 200000 x (1 + 0.001667)^24 - 739.24 x (1 + 0.001667)^24 -1 / 0.001667

Loan Principal balance at year 2 = 200000 x (1.001667)^24 - 739.24 x (1.001667)^24 -1 / 0.001667

Loan Principal balance at year 2 = $190,069.22

Outstanding Principal (P) = $190,069.22

Interest rate at beginning of year 3 (r) = (previous rate + 7%, composite rate)

Interest rate at beginning of year 3 (r) = (2% + 7%,6%) = 6%, Monthly = 6%/12 = 0.5%

No of outstanding payments (n) = 28 x 2 = 336

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

PMT = 190,069.22 x 0.5% x (1 + 0.5%)^336 / (1 + 0.5%)^336 - 1

PMT = 190,069.22 x 0.005 x (1 + 0.005)^336 / (1 + 0.005)^336 - 1

PMT = 190,069.22 x 0.005 x (1.005)^336 / (1.005)^336 -1 = $1169.16

Outstanding Principal balance at end of year 3 = P x (1 + r)^n - PMT x (1 + r)^n -1 / r

Outstanding Principal balance at end of year 3 = 190,069.22 x (1 + 0.5%)^12 - 1169.16 x (1 + 0.5%)^12 -1 / 0.05%

Outstanding Principal balance at end of year 3 = 190,069.22 x (1 + 0.005)^12 - 1169.16 x (1 + 0.005)^12 -1 / 0.005

Outstanding Principal balance at end of year 3 = 190,069.22 x (1.005)^12 - 1169.16 x (1.005)^12 -1 / 0.005

Outstanding Principal balance at end of year 3 = $187,370.03

Therefore outstanding balance on the loan at the end of Year 3 will be $187,370.03