1. **State the problem:** Calculate the monthly payment on a $500,000 mortgage at 4.5% annual interest compounded semi-annually for 25 years. 2. **Formula used:** The mortgage payment can be found using the amortization formula: $$P = \frac{r \times PV}{1 - (1 + r)^{-n}}$$ where: - $P$ is the monthly payment, - $PV$ is the loan principal ($500,000$), - $r$ is the monthly interest rate, - $n$ is the total number of monthly payments. 3. **Important rules:** - Interest is compounded semi-annually, so we must convert the nominal annual rate to an effective monthly rate. - Number of payments $n = 25 \times 12 = 300$ months. 4. **Convert nominal annual rate to effective monthly rate:** The nominal annual rate is 4.5% compounded semi-annually, so the semi-annual rate is $\frac{4.5}{2} = 2.25\% = 0.0225$. The effective annual rate (EAR) is: $$EAR = (1 + 0.0225)^2 - 1 = 1.0225^2 - 1 = 1.04550625 - 1 = 0.04550625$$ The effective monthly rate $r$ is: $$r = (1 + EAR)^{\frac{1}{12}} - 1 = (1.04550625)^{\frac{1}{12}} - 1 \approx 0.0037$$ 5. **Calculate monthly payment:** $$P = \frac{0.0037 \times 500000}{1 - (1 + 0.0037)^{-300}}$$ Calculate denominator: $$1 - (1 + 0.0037)^{-300} = 1 - (1.0037)^{-300}$$ Calculate $(1.0037)^{-300}$: $$= \frac{1}{(1.0037)^{300}} \approx \frac{1}{3.030} = 0.330$$ So denominator: $$1 - 0.330 = 0.670$$ Calculate numerator: $$0.0037 \times 500000 = 1850$$ Finally: $$P = \frac{1850}{0.670} \approx 2761.19$$ **Answer:** The monthly payment is approximately $2761.19$.