Answer :
To find the total monthly payment for the mortgage involving a house valued at [tex]$200,000 with a $[/tex]4.5\%[tex]$ interest rate, a $[/tex]5\%[tex]$ down payment, a $[/tex]3.5\%[tex]$ local tax rate, and $[/tex]600 per year for homeowners insurance, we'll follow these steps:
1. Calculate the down payment amount:
Since the down payment is [tex]$5\%$[/tex] of the house's value:
[tex]\[ \text{Down payment} = 200,000 \times 0.05 = 10,000 \][/tex]
2. Determine the loan amount:
Subtract the down payment from the house value:
[tex]\[ \text{Loan amount} = 200,000 - 10,000 = 190,000 \][/tex]
3. Calculate the monthly mortgage payment (P + I):
We use the fixed-rate mortgage formula for monthly payments:
[tex]\[ M = P\frac{r(1+r)^n}{(1+r)^n - 1} \][/tex]
where:
- \(P\) is the loan amount ($190,000)
- \(r\) is the monthly interest rate (annual rate / 12 = 0.045 / 12)
- \(n\) is the total number of payments (loan term in years \(\times\) 12 = 30 \times 12)
Calculate the monthly interest rate:
[tex]\[ r = \frac{0.045}{12} = 0.00375 \][/tex]
Calculate the number of payments:
[tex]\[ n = 30 \times 12 = 360 \][/tex]
Plug these values into the formula:
[tex]\[ M = 190,000 \times \frac{0.00375(1+0.00375)^{360}}{(1+0.00375)^{360} - 1} \][/tex]
Solving this gives:
[tex]\[ M \approx 962.70 \][/tex]
4. Calculate the monthly property tax:
The annual property tax is [tex]$3.5\%$[/tex] of the house value:
[tex]\[ \text{Annual property tax} = 200,000 \times 0.035 = 7,000 \][/tex]
Monthly property tax:
[tex]\[ \text{Monthly property tax} = \frac{7,000}{12} \approx 583.33 \][/tex]
5. Calculate monthly homeowners insurance:
The annual homeowners insurance is $600:
[tex]\[ \text{Monthly homeowners insurance} = \frac{600}{12} = 50 \][/tex]
6. Calculate PMI (Private Mortgage Insurance):
Since the down payment is $5\%, the loan-to-value (LTV) ratio is:
[tex]\[ \text{LTV} = \frac{190,000}{200,000} = 0.95 \Rightarrow 95\% \][/tex]
According to the PMI table, for a loan-to-value ratio within the range of [tex]$95.01\%$[/tex] to [tex]$97\%$[/tex] over a 30-year fixed mortgage, the PMI rate is [tex]$0.90\%$[/tex].
[tex]\[ \text{Annual PMI} = 190,000 \times 0.009 = 1,710 \][/tex]
Monthly PMI:
[tex]\[ \text{Monthly PMI} = \frac{1,710}{12} \approx 142.50 \][/tex]
7. Calculate the total monthly payment:
Add the monthly mortgage payment, property tax, homeowners insurance, and PMI:
[tex]\[ \text{Total monthly payment} = 962.70 + 583.33 + 142.50 + 50 = 1,738.54 \][/tex]
Based on the detailed step-by-step calculations, the total monthly payment is approximately \$1,738.54.
The correct answer is:
None of the options (A. [tex]$2330.20$[/tex], B. [tex]$2654.23$[/tex], C. [tex]$2202.72$[/tex], D. [tex]$2894.71$[/tex]) match the calculated total monthly payment.
However, the monthly payment calculated here is correct as per the detailed breakdown provided.
Given the exact answer choices provided in the problem, the correct total monthly payment is [tex]$\$[/tex] 1738.54$.
1. Calculate the down payment amount:
Since the down payment is [tex]$5\%$[/tex] of the house's value:
[tex]\[ \text{Down payment} = 200,000 \times 0.05 = 10,000 \][/tex]
2. Determine the loan amount:
Subtract the down payment from the house value:
[tex]\[ \text{Loan amount} = 200,000 - 10,000 = 190,000 \][/tex]
3. Calculate the monthly mortgage payment (P + I):
We use the fixed-rate mortgage formula for monthly payments:
[tex]\[ M = P\frac{r(1+r)^n}{(1+r)^n - 1} \][/tex]
where:
- \(P\) is the loan amount ($190,000)
- \(r\) is the monthly interest rate (annual rate / 12 = 0.045 / 12)
- \(n\) is the total number of payments (loan term in years \(\times\) 12 = 30 \times 12)
Calculate the monthly interest rate:
[tex]\[ r = \frac{0.045}{12} = 0.00375 \][/tex]
Calculate the number of payments:
[tex]\[ n = 30 \times 12 = 360 \][/tex]
Plug these values into the formula:
[tex]\[ M = 190,000 \times \frac{0.00375(1+0.00375)^{360}}{(1+0.00375)^{360} - 1} \][/tex]
Solving this gives:
[tex]\[ M \approx 962.70 \][/tex]
4. Calculate the monthly property tax:
The annual property tax is [tex]$3.5\%$[/tex] of the house value:
[tex]\[ \text{Annual property tax} = 200,000 \times 0.035 = 7,000 \][/tex]
Monthly property tax:
[tex]\[ \text{Monthly property tax} = \frac{7,000}{12} \approx 583.33 \][/tex]
5. Calculate monthly homeowners insurance:
The annual homeowners insurance is $600:
[tex]\[ \text{Monthly homeowners insurance} = \frac{600}{12} = 50 \][/tex]
6. Calculate PMI (Private Mortgage Insurance):
Since the down payment is $5\%, the loan-to-value (LTV) ratio is:
[tex]\[ \text{LTV} = \frac{190,000}{200,000} = 0.95 \Rightarrow 95\% \][/tex]
According to the PMI table, for a loan-to-value ratio within the range of [tex]$95.01\%$[/tex] to [tex]$97\%$[/tex] over a 30-year fixed mortgage, the PMI rate is [tex]$0.90\%$[/tex].
[tex]\[ \text{Annual PMI} = 190,000 \times 0.009 = 1,710 \][/tex]
Monthly PMI:
[tex]\[ \text{Monthly PMI} = \frac{1,710}{12} \approx 142.50 \][/tex]
7. Calculate the total monthly payment:
Add the monthly mortgage payment, property tax, homeowners insurance, and PMI:
[tex]\[ \text{Total monthly payment} = 962.70 + 583.33 + 142.50 + 50 = 1,738.54 \][/tex]
Based on the detailed step-by-step calculations, the total monthly payment is approximately \$1,738.54.
The correct answer is:
None of the options (A. [tex]$2330.20$[/tex], B. [tex]$2654.23$[/tex], C. [tex]$2202.72$[/tex], D. [tex]$2894.71$[/tex]) match the calculated total monthly payment.
However, the monthly payment calculated here is correct as per the detailed breakdown provided.
Given the exact answer choices provided in the problem, the correct total monthly payment is [tex]$\$[/tex] 1738.54$.
