Question: Manuel purchases a house and gets a 20-year mortgage for $215,000 at 5% APR. In addition to the monthly payment, the lender requires him to pay into an escrow account for the homeowners insurance and property tax. His homeowners insurance is $1400 per year and the property tax is $2300 per year. Determine the monthly payment to the lender that includes the insurance and property tax. Round your answer to the nearest cent.
Formulas
Regular Payment for Fixed Installment Loans
PMT = (P\cdot r/n) / [1 - (1 + r/n)^{-nt}]
Number of Fixed Payments Required to Pay Off Credit Card Debt
R = -\log[1 - (r/n)(A/PMT)] / \log(1 + r/n)
1. **State the problem:** Manuel has a $215,000 mortgage for 20 years at 5% APR. We need to find his total monthly payment including homeowners insurance and property tax.
2. **Identify variables:**
- Principal $P = 215000$
- Annual interest rate $r = 0.05$
- Number of payments per year $n = 12$
- Total years $t = 20$
- Homeowners insurance per year = $1400$
- Property tax per year = $2300$
3. **Calculate the monthly mortgage payment using the formula:**
$$PMT = \frac{P \cdot \frac{r}{n}}{1 - (1 + \frac{r}{n})^{-nt}}$$
4. **Calculate components:**
- Monthly interest rate $\frac{r}{n} = \frac{0.05}{12} = 0.0041667$
- Total number of payments $nt = 12 \times 20 = 240$
5. **Calculate denominator:**
$$1 - (1 + 0.0041667)^{-240} = 1 - (1.0041667)^{-240}$$
Calculate $(1.0041667)^{-240}$:
$$ (1.0041667)^{240} \approx 2.71264 \implies (1.0041667)^{-240} = \frac{1}{2.71264} \approx 0.3680 $$
So denominator:
$$1 - 0.3680 = 0.6320$$
6. **Calculate numerator:**
$$P \cdot \frac{r}{n} = 215000 \times 0.0041667 = 895.83$$
7. **Calculate monthly mortgage payment:**
$$PMT = \frac{895.83}{0.6320} \approx 1417.53$$
8. **Calculate monthly insurance and tax payments:**
- Insurance monthly: $\frac{1400}{12} = 116.67$
- Property tax monthly: $\frac{2300}{12} = 191.67$
9. **Calculate total monthly payment:**
$$1417.53 + 116.67 + 191.67 = 1725.87$$
10. **Final answer:**
Manuel's total monthly payment including mortgage, insurance, and property tax is **$1725.87**.