Use PMT $=\frac{ P \left(\frac{ r }{ n }\right)}{\left[1-\left(1+\frac{ r }{ n }\right)^{- nt }\right]}$ to determine the regular payment amount, rounded to the nearest dollar. The price of a home is $174,000. 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 10%. Complete parts (a) through (e) below.
b. Find the amount of the mortgage.
$139,200
c. How much must be paid for the three points at closing?
$4176 (Round to the nearest dollar as needed.)
d. Find the monthly payment (excluding escrowed taxes and insurance).
$ \square (Round to the nearest dollar as needed.)
Answer (1)
Identify the given values: P = $139,200, r = 0.10, n = 12, t = 30.
Substitute the values into the PMT formula: PMT = [ 1 − ( 1 + 12 0.10 ) − ( 12 × 30 ) ] 139200 × ( 12 0.10 ) .
Calculate the monthly payment: PMT = 1222.07.
Round the monthly payment to the nearest dollar: 1222 .
Explanation
Understanding the Problem We are given the formula for calculating the monthly payment (PMT) of a mortgage: PMT = [ 1 − ( 1 + n r ) − n t ] P ( n r ) We are also given the following information:
Price of the home: $174,000
Down payment: 20%
Amount of the mortgage (P): $139,200
Points at closing: 3
Cost of three points: $4176
Mortgage term (t): 30 years
Fixed interest rate (r): 10% or 0.10
Number of payments per year (n): 12 (monthly payments) Our objective is to find the monthly payment (PMT) using the given formula and provided information.
Identifying the Values First, we need to identify the values for P, r, n, and t:
P = $139,200 (mortgage amount)
r = 0.10 (annual interest rate)
n = 12 (number of payments per year)
t = 30 (number of years)
Substituting the Values Now, we substitute the values into the PMT formula: PMT = [ 1 − ( 1 + 12 0.10 ) − ( 12 × 30 ) ] 139200 × ( 12 0.10 )
Calculating the Monthly Payment Next, we calculate the monthly payment (PMT): PMT = [ 1 − ( 1 + 0.0083333 ) − 360 ] 139200 × ( 0.0083333 ) PMT = [ 1 − ( 1.0083333 ) − 360 ] 1160 PMT = [ 1 − 0.05158 ] 1160 PMT = 0.94842 1160 PMT = 1222.07
Rounding the Result Finally, we round the monthly payment to the nearest dollar: $1222
Final Answer Therefore, the monthly payment (excluding escrowed taxes and insurance) is $1222.
Examples
Understanding mortgage calculations is essential for anyone planning to buy a home. For example, suppose you want to buy a house for $200,000 and need to take out a mortgage. By understanding how the interest rate, loan term, and down payment affect your monthly payments, you can budget effectively and make informed financial decisions. Knowing these calculations allows you to compare different loan offers and choose the one that best fits your financial situation. For instance, a lower interest rate or a larger down payment can significantly reduce your monthly payments and the total amount you pay over the life of the loan. This knowledge empowers you to plan your finances responsibly and achieve your homeownership goals.
