Arm New Payment 89055E

1. **State the problem:** George took a $200,000 loan for 30 years at 8% annual interest compounded monthly. After 5 years, the interest rate changes to 5.5% compounded monthly, and we need to find the new monthly payment. 2. **Identify the formulas:** The monthly payment for a loan is given by the amortization formula: $$ M = P \frac{r(1+r)^n}{(1+r)^n - 1} $$ where: - $M$ is the monthly payment, - $P$ is the loan principal, - $r$ is the monthly interest rate (annual rate divided by 12), - $n$ is the total number of monthly payments. 3. **Calculate the remaining loan balance after 5 years at the original rate:** - Original monthly interest rate: $r_1 = \frac{8\%}{12} = \frac{0.08}{12} = 0.0066667$ - Total payments originally: $n = 30 \times 12 = 360$ - Payments made: $5 \times 12 = 60$ First, calculate the original monthly payment $M_1$: $$ M_1 = 200000 \times \frac{0.0066667(1+0.0066667)^{360}}{(1+0.0066667)^{360} - 1} $$ Calculate powers: $$ (1+0.0066667)^{360} = (1.0066667)^{360} \approx 10.935 $$ So, $$ M_1 = 200000 \times \frac{0.0066667 \times 10.935}{10.935 - 1} = 200000 \times \frac{0.0729}{9.935} = 200000 \times 0.00734 = 1468.72 $$ 4. **Calculate the remaining balance after 60 payments:** The remaining balance $B$ after $k$ payments is: $$ B = P (1+r)^k - M \frac{(1+r)^k - 1}{r} $$ Substitute values: $$ B = 200000 (1.0066667)^{60} - 1468.72 \times \frac{(1.0066667)^{60} - 1}{0.0066667} $$ Calculate powers: $$ (1.0066667)^{60} \approx 1.489 $$ Calculate numerator: $$ 1.489 - 1 = 0.489 $$ Calculate fraction: $$ \frac{0.489}{0.0066667} = 73.35 $$ Calculate balance: $$ B = 200000 \times 1.489 - 1468.72 \times 73.35 = 297800 - 107700 = 190100 $$ 5. **Calculate new monthly payment with new interest rate and remaining term:** - New monthly interest rate: $r_2 = \frac{5.5\%}{12} = \frac{0.055}{12} = 0.0045833$ - Remaining term: $30 - 5 = 25$ years, so $n_2 = 25 \times 12 = 300$ months Use amortization formula with $P = 190100$, $r = 0.0045833$, $n = 300$: $$ M_2 = 190100 \times \frac{0.0045833 (1+0.0045833)^{300}}{(1+0.0045833)^{300} - 1} $$ Calculate powers: $$ (1.0045833)^{300} \approx 3.946 $$ Calculate numerator: $$ 0.0045833 \times 3.946 = 0.01808 $$ Calculate denominator: $$ 3.946 - 1 = 2.946 $$ Calculate fraction: $$ \frac{0.01808}{2.946} = 0.006136 $$ Calculate new monthly payment: $$ M_2 = 190100 \times 0.006136 = 1166.68 $$ 6. **Final answer:** The new monthly payment after the interest rate reset is approximately **$1166.68**.