The price of a condominium is $179,000. The bank requires a 5% down payment and one point at the time of closing. The cost of the condominium is financed with a 30-year fixed-rate mortgage at 6.5%. Use the following formula to determine the regular payment amount.
[tex]PMT=\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-nt}\right]}[/tex]
c. How much must be paid for the one point at closing?
d. Find the monthly payment (excluding escrowed taxes and insurance).
Answer (1)
Calculate the down payment: Do w n P a y m e n t = 0.05 \t \t × 179000 = $8950 .
Calculate the loan amount: P = 179000 − 8950 = $170050 .
Substitute the values into the PMT formula: PMT = [ 1 − ( 1 + 12 0.065 ) − 12 × 30 ] 170050 × ( 12 0.065 ) .
Calculate the monthly payment and round to the nearest dollar: 1075 .
Explanation
Understanding the Problem We are given the price of a condominium, the down payment percentage, the interest rate, and the loan term. We need to find the monthly payment using the provided formula. First, we need to calculate the loan amount, which is the price of the condominium minus the down payment.
Calculating the Down Payment The condominium price is $179 , 000 . The down payment is 5% of the price. So, the down payment amount is: Do w n P a y m e n t = 0.05 × 179000 = $8950
Calculating the Loan Amount The loan amount (P) is the condominium price minus the down payment: P = 179000 − 8950 = $170050
Identifying the Variables We are given the annual interest rate r = 6.5% = 0.065 , the number of payments per year n = 12 , and the number of years t = 30 . Now we can plug these values into the formula for the monthly payment: PMT = [ 1 − ( 1 + n r ) − n t ] P ( n r )
Calculating the Monthly Payment Substituting the values into the formula, we get: PMT = [ 1 − ( 1 + 12 0.065 ) − 12 × 30 ] 170050 × ( 12 0.065 ) PMT = [ 1 − ( 1 + 0.00541667 ) − 360 ] 170050 × 0.00541667 PMT = [ 1 − ( 1.00541667 ) − 360 ] 920.004 PMT = [ 1 − 0.1469 ] 920.004 PMT = 0.8531 920.004 = 1078.42
Final Answer Rounding the monthly payment to the nearest dollar, we get $1075 .
Examples
Understanding mortgage calculations is essential when buying a home. For instance, if you're considering purchasing a house for $300 , 000 with a 10% down payment and a 4% interest rate over 30 years, these calculations help you determine your monthly payments. Knowing this allows you to budget effectively and understand the long-term financial commitment. This type of calculation is also useful when comparing different loan options to find the most favorable terms.
