To determine George's new monthly payment after the reset of his ARM loan, we first calculate the remaining balance using the original terms and then apply the new interest rate of 2.5%, compounded monthly, using the annuity formula for the remaining term of the loan.
George secured an adjustable-rate mortgage (ARM) loan with an initial balance of $300,000 at a 9% interest rate, compounded monthly.
After 5 years, the interest rate on his ARM has decreased to 2.5% per year, compounded monthly.
To calculate the new monthly payment due to the interest rate reset, we must first determine the remaining balance on the loan after 5 years of payments at the initial interest rate and then calculate the monthly payment using the new interest rate.
Firstly, we use the initial loan terms to find the monthly payment for the 9% interest rate:
Initial monthly interest rate = 9% / 12 months = 0.75% or 0.0075 in decimal.
The initial monthly payment (M) can be calculated using the formula for an annuity:
M =
where P is the loan principal, r is the monthly interest rate, and n is the total number of monthly payments.
For George's loan:
P = $300,000 r = 0.0075 n = 30 years * 12 months/year = 360 months
To calculate the remaining balance after 5 years, we need to know how many payments have already been made (5 years * 12 months/year = 60 payments) and how many are left (300 total payments remaining).
With the new rate of 2.5%, the new monthly payment can be recalculated similarly with n being the remaining number of payments (300) and r = 2.5% / 12 = 0.00208333.
Plugging these values into the annuity formula gives us George's new monthly payment, which needs to be rounded to the nearest cent.