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Sagot :
To determine Antonia's total monthly payments on her mortgage, we need to consider several components: the monthly mortgage payment (principal and interest), the PMI (Private Mortgage Insurance) payment, property tax, and homeowners insurance.
Let's break down each of these components.
### 1. Monthly Mortgage Payment (Principal and Interest)
Antonia's house price is [tex]$210,000, and she made a 5% down payment. \[ \text{Down Payment} = 210,000 \times 0.05 = 10,500 \] The loan amount (principal) is therefore: \[ \text{Loan Amount} = 210,000 - 10,500 = 199,500 \] The mortgage is a 15-year fixed-rate mortgage with an annual interest rate of 4.75%. To find the monthly mortgage payment, we need the monthly interest rate and the total number of payments. \[ \text{Monthly Interest Rate} = \frac{4.75\%}{12} = 0.3958\% = 0.003958 \] \[ \text{Number of Monthly Payments} = 15 \times 12 = 180 \] The formula for the monthly mortgage payment (M) is: \[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \] where: - \( P \) is the loan amount ($[/tex]199,500),
- [tex]\( r \)[/tex] is the monthly interest rate (0.003958),
- [tex]\( n \)[/tex] is the number of monthly payments (180).
Plugging in these values:
[tex]\[ M = 199,500 \times \frac{0.003958 (1+0.003958)^{180}}{(1+0.003958)^{180}-1} \][/tex]
After calculation, the monthly mortgage payment is approximately:
[tex]\[ M = 1551.77 \][/tex]
### 2. Monthly PMI (Private Mortgage Insurance) Payment
Since Antonia made a 5% down payment, her loan-to-value ratio is 95%, which falls into the 90.01% to 95% range for PMI premiums according to the table. The PMI rate is 0.26% annually for a 15-year fixed-rate loan.
To find the monthly PMI payment:
[tex]\[ \text{Annual PMI} = 199,500 \times 0.0026 = 518.70 \][/tex]
[tex]\[ \text{Monthly PMI} = \frac{518.70}{12} \approx 43.22 \][/tex]
### 3. Monthly Property Tax
The annual tax rate is 3.5%, and the house's assessed value is [tex]$205,000. \[ \text{Annual Property Tax} = 205,000 \times 0.035 = 7175 \] \[ \text{Monthly Property Tax} = \frac{7175}{12} \approx 597.92 \] ### 4. Monthly Homeowners Insurance The annual homeowners insurance cost is $[/tex]480.
[tex]\[ \text{Monthly Insurance} = \frac{480}{12} = 40 \][/tex]
### 5. Total Monthly Payment
The total monthly payment includes the monthly mortgage payment, PMI, property tax, and homeowners insurance.
[tex]\[ \text{Total Monthly Payment} = 1551.77 + 43.22 + 597.92 + 40 \][/tex]
[tex]\[ \text{Total Monthly Payment} \approx 2232.92 \][/tex]
So, Antonia's total monthly payments are approximately:
[tex]\[ \boxed{2232.92} \][/tex]
Let's break down each of these components.
### 1. Monthly Mortgage Payment (Principal and Interest)
Antonia's house price is [tex]$210,000, and she made a 5% down payment. \[ \text{Down Payment} = 210,000 \times 0.05 = 10,500 \] The loan amount (principal) is therefore: \[ \text{Loan Amount} = 210,000 - 10,500 = 199,500 \] The mortgage is a 15-year fixed-rate mortgage with an annual interest rate of 4.75%. To find the monthly mortgage payment, we need the monthly interest rate and the total number of payments. \[ \text{Monthly Interest Rate} = \frac{4.75\%}{12} = 0.3958\% = 0.003958 \] \[ \text{Number of Monthly Payments} = 15 \times 12 = 180 \] The formula for the monthly mortgage payment (M) is: \[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \] where: - \( P \) is the loan amount ($[/tex]199,500),
- [tex]\( r \)[/tex] is the monthly interest rate (0.003958),
- [tex]\( n \)[/tex] is the number of monthly payments (180).
Plugging in these values:
[tex]\[ M = 199,500 \times \frac{0.003958 (1+0.003958)^{180}}{(1+0.003958)^{180}-1} \][/tex]
After calculation, the monthly mortgage payment is approximately:
[tex]\[ M = 1551.77 \][/tex]
### 2. Monthly PMI (Private Mortgage Insurance) Payment
Since Antonia made a 5% down payment, her loan-to-value ratio is 95%, which falls into the 90.01% to 95% range for PMI premiums according to the table. The PMI rate is 0.26% annually for a 15-year fixed-rate loan.
To find the monthly PMI payment:
[tex]\[ \text{Annual PMI} = 199,500 \times 0.0026 = 518.70 \][/tex]
[tex]\[ \text{Monthly PMI} = \frac{518.70}{12} \approx 43.22 \][/tex]
### 3. Monthly Property Tax
The annual tax rate is 3.5%, and the house's assessed value is [tex]$205,000. \[ \text{Annual Property Tax} = 205,000 \times 0.035 = 7175 \] \[ \text{Monthly Property Tax} = \frac{7175}{12} \approx 597.92 \] ### 4. Monthly Homeowners Insurance The annual homeowners insurance cost is $[/tex]480.
[tex]\[ \text{Monthly Insurance} = \frac{480}{12} = 40 \][/tex]
### 5. Total Monthly Payment
The total monthly payment includes the monthly mortgage payment, PMI, property tax, and homeowners insurance.
[tex]\[ \text{Total Monthly Payment} = 1551.77 + 43.22 + 597.92 + 40 \][/tex]
[tex]\[ \text{Total Monthly Payment} \approx 2232.92 \][/tex]
So, Antonia's total monthly payments are approximately:
[tex]\[ \boxed{2232.92} \][/tex]
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