Question

Use PMT=(P(frac (r)/(n)))((1-(1+(r)/(n))^-n)) to determine the regular payment amount, rounded to the nearest dollar. The price of a home is 127.000 The bank requires a 20% down payment and three points at the time of closing. The cost of the home is financed with a 30-year faxed-rate mortgage at 7% Complete parts(a) through (d) below a. How much is the required down payment? square b. What is the morigage amount? square c. Using the formula above, what is the monthly payment? square d. What is the total cost of interest over 30 years? square 101,600 25,400 58,890 118,110 676 740 845 141,760

Answer 1

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**