Use PMT = [tex]$\frac{ P \left(\frac{ r }{ n }\right)}{\left[1-\left(1+\frac{ r }{ n }\right)^{- nt }\right]}$[/tex] to determine the regular payment amount, rounded to the nearest dollar. The price of a home is $218,000. 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 7%. Complete parts (a) through (e) below.
a. Find the required down payment.
$43600
b. Find the amount of the mortgage.
$
Answer (2)
Calculate the down payment: Do w n P a y m e n t = 0.20 × $218 , 000 = $43 , 600 .
Calculate the mortgage amount: M or t g a g e A m o u n t = $218 , 000 − $43 , 600 = $174 , 400 .
Calculate the monthly payment using the PMT formula: PMT = [ 1 − ( 1 + 12 0.07 ) − 12 × 30 ] 174400 ( 12 0.07 ) ≈ $1160.29 .
Round the monthly payment to the nearest dollar: $1160 .
Explanation
Understanding the Problem We are given the price of a home, the down payment percentage, the interest rate, and the mortgage term. We need to find the amount of the mortgage and the monthly payment using the given formula.
Calculating the Down Payment First, we calculate the down payment amount. The down payment is 20% of the home price, which is $218,000. So, the down payment is: Do w n P a y m e n t = 0.20 × $218 , 000 = $43 , 600
Calculating the Mortgage Amount Next, we calculate the amount of the mortgage. The mortgage amount is the price of the home minus the down payment: M or t g a g e A m o u n t = $218 , 000 − $43 , 600 = $174 , 400
Calculating the Monthly Payment Now, we calculate the monthly payment using the formula: PMT = [ 1 − ( 1 + n r ) − n t ] P ( n r ) Where:
P = Mortgage amount = $174,400
r = Annual interest rate = 7% = 0.07
n = Number of payments per year = 12 (monthly)
t = Number of years = 30
Plugging in the values, we get: PMT = [ 1 − ( 1 + 12 0.07 ) − 12 × 30 ] 174400 ( 12 0.07 ) PMT = 1 − ( 1 + 0.07/12 ) − 360 174400 × ( 0.07/12 ) PMT = 1 − ( 1.005833 ) − 360 1017.3333 PMT = 1 − 0.128875 1017.3333 PMT = 0.871125 1017.3333 PMT = 1167.83 Rounding to the nearest dollar, the monthly payment is $1168.
Final Answer The amount of the mortgage is $174,400 and the monthly payment is approximately $1160.
Examples
Understanding mortgage calculations is essential for making informed decisions when buying a home. For instance, knowing how the down payment affects the mortgage amount and how the interest rate and loan term influence the monthly payment can help you budget effectively. Suppose you're comparing two homes: one with a higher price but lower interest rate, and another with a lower price but higher interest rate. By calculating the monthly payments for each scenario, you can determine which option is more affordable for you over the long term. This analysis ensures you're not just looking at the initial price but also the total cost of the home, including interest paid over the life of the loan. This is a practical application of understanding and using the PMT formula.
The down payment required for the home is $43,600, and the amount of the mortgage after the down payment is $174,400.
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