Congratulations Your offer for a house has been accepted You

\"Congratulations! Your offer for a house has been accepted. You will need to borrow $330,000. The bank can finance the loan through two options: a 15-year mortgage at 4.06% APR and a 30-year mortgage at 4.65% APR. Both mortgages have monthly compounding. What is the difference in monthly payments between these two options? Express your answer in terms of positive dollars.\"

Solution

Monthly payment = Loan amount / PVIFA(r%, N) where

r: Monthly interest rate

N: Number of months

PVIFA(r%, N) = [1 - (1 + r)^-N] / r

(1) APR = 4.06, 15-year

r = 4.06% / 12 = 0.3383%

N = 15 x 12 = 180

Monthly payment ($) = 330,000 / PVIFA(0.3383%, 180) = 330,000 / 134.6574** = 2,451

(2) APR = 4.65, 30-year

r = 4.65% / 12 = 0.3875%

N = 30 x 12 = 360

Monthly payment ($) = 330,000 / PVIFA(0.3875%, 360) = 330,000 / 193.9350^# = 1,702

(3) Difference in monthly payment ($) = 2,451 - 1,702 = 749

**PVIFA(0.3383%, 180) = [1 - (1.003383)^-180] / 0.003383 = (1 - 0.5445) / 0.003383 = 0.4555 / 0.003383 = 134.6574

#PVIFA(0.3875%, 360) = [1 - (1.003875)^-360] / 0.003875 = (1 - 0.2485) / 0.003875 = 0.7515 / 0.003875 = 193.9350