1. **State the problem:** Jack and Jill bought a house for 465000 and made a down payment of 50000. They amortize the remaining principal over 20 years with monthly payments. The initial interest rate is 2.35% compounded semi-annually for 5 years. After 5 years, they renew for 4 years at 2.15%. We need to find the new monthly payment after renewal. 2. **Calculate the initial loan principal:** $$\text{Principal} = 465000 - 50000 = 415000$$ 3. **Convert the initial interest rate to an effective monthly rate:** Given 2.35% annual interest compounded semi-annually, the effective annual rate (EAR) is: $$EAR = \left(1 + \frac{0.0235}{2}\right)^2 - 1 = (1.01175)^2 - 1 = 0.0236$$ The monthly interest rate is: $$i = (1 + EAR)^{\frac{1}{12}} - 1 = (1.0236)^{\frac{1}{12}} - 1 \approx 0.001945$$ 4. **Calculate the number of monthly payments in 5 years:** $$n = 5 \times 12 = 60$$ 5. **Calculate the monthly payment for the initial 5-year term using the amortization formula:** $$P = \frac{i \times PV}{1 - (1 + i)^{-n}}$$ where $PV=415000$, $i=0.001945$, $n=60$. Calculate denominator: $$1 - (1 + 0.001945)^{-60} = 1 - (1.001945)^{-60} \approx 1 - 0.886 = 0.114$$ Calculate numerator: $$0.001945 \times 415000 = 807.175$$ Monthly payment: $$P = \frac{807.175}{0.114} \approx 7077.4$$ 6. **Calculate the remaining balance after 5 years:** Remaining balance formula: $$B = PV \times (1 + i)^n - P \times \frac{(1 + i)^n - 1}{i}$$ Calculate: $$(1 + i)^n = (1.001945)^{60} \approx 1.127$$ $$B = 415000 \times 1.127 - 7077.4 \times \frac{1.127 - 1}{0.001945}$$ Calculate second term denominator: $$\frac{0.127}{0.001945} = 65.3$$ Calculate second term: $$7077.4 \times 65.3 = 462,000$$ Calculate first term: $$415000 \times 1.127 = 467,000$$ Remaining balance: $$B = 467,000 - 462,000 = 5,000$$ 7. **Calculate the new monthly interest rate for the 4-year renewal at 2.15% compounded semi-annually:** $$EAR = \left(1 + \frac{0.0215}{2}\right)^2 - 1 = (1.01075)^2 - 1 = 0.0217$$ Monthly rate: $$i_{new} = (1 + 0.0217)^{\frac{1}{12}} - 1 \approx 0.00179$$ 8. **Calculate the number of payments for the 4-year term:** $$n_{new} = 4 \times 12 = 48$$ 9. **Calculate the new monthly payment for the remaining balance $B=5000$ over 4 years:** $$P_{new} = \frac{i_{new} \times B}{1 - (1 + i_{new})^{-n_{new}}}$$ Calculate denominator: $$1 - (1 + 0.00179)^{-48} = 1 - (1.00179)^{-48} \approx 1 - 0.918 = 0.082$$ Calculate numerator: $$0.00179 \times 5000 = 8.95$$ Monthly payment: $$P_{new} = \frac{8.95}{0.082} \approx 109.15$$ **Final answer:** The new monthly payment after renewal is approximately **109.15**.