To solve this problem, we'll go through each part step by step, starting with your provided values.
Given:
- Price of the home = $127,000
- Required down payment = 20% of the home price
- Mortgage term = 30 years
- Interest rate = 7% annually
- Points = 3 at closing (which typically gets calculated as a percent of the loan amount, but we will deal with it later if necessary)
a. Required Down Payment
The down payment is calculated as follows:
\[ \text{Down Payment} = \text{Price of Home} \times \text{Down Payment Percentage} \]
\[ \text{Down Payment} = 127,000 \times 0.20 = 25,400 \]
b. Mortgage Amount
The mortgage amount is the price of the home minus the down payment:
\[ \text{Mortgage Amount} = \text{Price of Home} - \text{Down Payment} \]
\[ \text{Mortgage Amount} = 127,000 - 25,400 = 101,600 \]
c. Monthly Payment Calculation Using the PMT Formula
To calculate the monthly payment (PMT), we will use the formula you provided and the values we derived.
- P = Mortgage Amount = $101,600
- r = annual interest rate = 7% = 0.07
- n = number of payments (30 years × 12 months) = 360
- Monthly interest rate = r / 12 = 0.07 / 12 ≈ 0.0058333
Now we can apply the PMT formula:
\[ PMT = P \left( \frac{r/n}{1 - (1 + r/n)^{-n}} \right) \]
Substituting the values:
\[ PMT = 101,600 \left( \frac{0.0058333}{1 - (1 + 0.0058333)^{-360}} \right) \]
Calculating the denominator:
\[ 1 + 0.0058333 = 1.0058333 \] \[ (1.0058333)^{-360} \approx 0.059239 \] \[ 1 - 0.059239 ≈ 0.940761 \]
So,
\[ PMT = 101,600 \left( \frac{0.0058333}{0.940761} \right) \approx 101,600 \times 0.006196 \approx 629.40 \]
Rounded to the nearest dollar: Monthly payment = $629
d. Total Cost of Interest Over 30 Years
To find the total cost of interest, we need to find the total amount paid over the life of the loan and subtract the principal (mortgage amount).
First, calculate the total payments made:
\[ \text{Total Payments} = \text{PMT} \times n = 629 \times 360 = 226,440 \]
Now, subtract the mortgage amount:
\[ \text{Total Interest} = \text{Total Payments} - \text{Mortgage Amount} = 226,440 - 101,600 = 124,840 \]
Therefore, the total cost of interest over 30 years is $124,840.
Summary of Results:
- a. Required Down Payment: $25,400
- b. Mortgage Amount: $101,600
- c. Monthly Payment: $629
- d. Total Cost of Interest: $124,840