Use PMT equals StartFraction Upper P left parenthesis StartFraction r Over n EndFraction right parenthesis Over left bracket 1 minus left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript negative nt right bracket EndFraction
to determine the regular payment amount, rounded to the nearest dollar. The price of a home is $235 comma 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 fixed-rate mortgage at 6.5
%.
Complete parts (a) through (e) below.
Before we can use the PMT formula, we need to figure out some of our parameters.
(a)
The 20% down payment is calculated by:
235,000 * 20% = $47,000
(b)
The financed amount, P, is calculated by:
235,000 - 47,000 = $188,000
(c)
The bank charges 3 points at the time of closing, which would correspond to 3% of the financed amount:
188,000 * 3% = $5,640
(d)
We know that r (the interest rate) is 6.5%, or 0.065. This is expressed as a yearly rate, so we need to convert it to a monthly rate by dividing by 12 months. This means the monthly interest rate (r/n) is approximately 0.065 / 12 = 0.005416.
We also know that the term of the loan, t, is 30 years. We express this in months, so nt = 30 * 12 = 360 months.
(e)
Now we can finally apply the PMT formula:
The PMT formula calculates a monthly payment, so plugging in our parameters gives us:
PMT = 188,000 * (0.005416 / (1 - (1 + 0.005416)^-360)) = $1,185.79
So, the regular payment amount is approximately $1186 when rounded to the nearest dollar.
(a) Calculate the down payment amount.
To calculate the down payment amount, we need to find 20% of the price of the home.
Down Payment = 20% x Price of Home
Down Payment = 20% x $235,000
To find 20% of a number, we multiply the number by 0.20.
Down Payment = 0.20 x $235,000
Down Payment = $47,000
So, the down payment amount is $47,000.
(b) Calculate the closing cost.
To calculate the closing cost, we need to find three points at the time of closing. One point is equal to 1% of the loan amount.
Closing Cost = 3 points x Loan Amount
Closing Cost = 3 x (Price of Home - Down Payment)
Closing Cost = 3 x ($235,000 - $47,000)
Closing Cost = 3 x $188,000
Closing Cost = $564,000
So, the closing cost is $564,000.
(c) Calculate the loan amount.
To calculate the loan amount, we need to subtract the down payment and closing cost from the price of the home.
Loan Amount = Price of Home - Down Payment - Closing Cost
Loan Amount = $235,000 - $47,000 - $564,000
Loan Amount = $235,000 - $611,000
Loan Amount = -$376,000
Since the loan amount cannot be negative, there seems to be an error in the calculation or the given values.
(d) Assume that the loan amount is corrected and equals $376,000. Calculate the monthly interest rate and the number of payments.
Monthly Interest Rate = Annual Interest Rate / 12
Monthly Interest Rate = 6.5% / 12
To convert a percentage to a decimal, we divide by 100.
Monthly Interest Rate = 6.5% / 12
Monthly Interest Rate = 0.065 / 12
Monthly Interest Rate = 0.00541667
The number of payments is equal to the number of years multiplied by 12 (since there are 12 months in a year).
Number of Payments = Number of Years x 12
Number of Payments = 30 years x 12
Number of Payments = 360
So, the monthly interest rate is 0.00541667 and the number of payments is 360.
(e) Calculate the regular payment amount using the given formula.
PMT = [(r / n) / (1 - (1 + (r / n))^(-nt))] x Loan Amount
PMT = [(0.00541667 / 12) / (1 - (1 + (0.00541667 / 12))^(-360))] x $376,000
Now, substitute the values into the formula and calculate the regular payment amount.
PMT = [(0.00541667 / 12) / (1 - (1 + (0.00541667 / 12))^(-360))] x $376,000
PMT = [0.00045139 / (1 - (1.00045139)^(-360))] x $376,000
Using a calculator or spreadsheet, calculate the value inside the brackets first:
PMT = [0.00045139 / (1 - 0.30079894)] x $376,000
PMT = [0.00045139 / 0.69920106] x $376,000
PMT = 0.64590887 x $376,000
Now, calculate the product:
PMT = $242,554.15
So, the regular payment amount, rounded to the nearest dollar, is $242,554.