Hey guys! really need help with this one!!!
On December 31, 1995, a house is purchased with the buyer taking out a 30-year $90,000 mortgage at 9% interest compounded monthly. The mortgage payments are made at the end of each month. Calculate:
(A) the unpaid balance of the loan on December 31,2005, just after the 120th payment.
(B) the interest that will be paid during January 2006.
Thanks in advance!
first you need the monthly payment
PV = 90000
i = .09/12 = .0075
n = 12x30 = 360
90000 = Payment [ 1 - 1.0075^-360]/.0075
payment = 724.16
balance right after 120th payment
= 90000(1.0075)^120 - 724.16( 1.0075^120 - 1)/.0075
= 80486.77
interest on next month = 80486.77(.0075) = 603.65
To calculate the unpaid balance of the loan on December 31, 2005, just after the 120th payment, we need to use the concept of a loan amortization schedule.
An amortization schedule breaks down each monthly payment into two parts: the principal payment and the interest payment. The principal payment reduces the outstanding balance of the loan, while the interest payment covers the interest charged on the remaining balance.
Let's break down the steps to calculate the unpaid balance:
Step 1: Calculate the monthly interest rate:
First, we need to convert the annual interest rate to a monthly rate. Since the interest is compounded monthly, we divide the annual interest rate by 12. In this case, the annual rate is 9%, so the monthly rate is 9% / 12 = 0.75%.
Step 2: Calculate the number of payments remaining:
Since the mortgage was taken out for 30 years, there are a total of 30 * 12 = 360 monthly payments. On December 31, 2005, after the 120th payment, the number of payments remaining is 360 - 120 = 240 payments.
Step 3: Calculate the monthly payment:
To determine the monthly payment amount, we can use the formula for calculating a fixed monthly payment on a loan:
P = (r * PV) / (1 - (1 + r)^(-n))
Where:
- P is the monthly payment
- r is the monthly interest rate (in decimal form)
- PV is the present value or the loan amount
- n is the total number of payments
In our case, r = 0.75% (0.0075 as a decimal), PV = $90,000, and n = 360.
By substituting these values into the formula, we can calculate the monthly payment.
Step 4: Calculate the unpaid balance:
To calculate the unpaid balance after the 120th payment on December 31, 2005, we need to determine the remaining balance on the loan. We start with the original loan amount and subtract the principal payments made up to that point.
Now let's calculate each step:
Step 1: Monthly interest rate = 9%/12 = 0.75%
Step 2: Number of payments remaining = 360 - 120 = 240 payments
Step 3: Monthly payment:
Using the loan formula mentioned above, we can calculate the monthly payment:
P = (0.0075 * $90,000) / (1 - (1 + 0.0075)^(-240))
P ≈ $690.58
Step 4: Unpaid balance after the 120th payment:
To calculate the unpaid balance, we start with the original loan amount and subtract the principle payments made up to that point. Since we have 120 payments and each payment is for $690.58, the total principal paid after 120 payments is 120 * $690.58 = $82,869.60.
Therefore, the unpaid balance on December 31, 2005, after the 120th payment is:
Unpaid balance = Original loan amount - Principal paid
Unpaid balance = $90,000 - $82,869.60 = $7,130.40
(A) The unpaid balance of the loan on December 31, 2005, just after the 120th payment is approximately $7,130.40.
To calculate the interest that will be paid during January 2006, we need to find the interest portion of the next payment after the 120th payment.
Step 1: Calculate the monthly interest:
The monthly interest is calculated by multiplying the unpaid balance by the monthly interest rate.
Monthly interest = Unpaid balance * Monthly interest rate
Monthly interest = $7,130.40 * 0.0075
Monthly interest ≈ $53.48
(B) The interest that will be paid during January 2006 is approximately $53.48.