A realtor sold a house on August 31, 1997 for $150k to a buyer in which a 20% down payment was made. The buyer took a 15yr mortgage on the property with an effective interest rate of 8% per annum. The buyer intends to pay off the mortgage owed in yearly payments starting on August 31, 1998.
a) How much of the mortgage will still be owed after the payment due in August 31, 2004, has been made?
b) Solve the same problem by separating the interest and the principal amounts.
Can you please show them step by step format so I can learn. thank you.
So the loan (B0) was 150*.8 = 120K
The balance after 1 year and 1 payment would be B1=B0*(1.08)-P.
B2=(B1*(1.08)-P = B0*(1.08)^2 - P*(1.08) - P
....
B15 = B0(1.08)^15 - sum(P*(1.08)^i)
Solve for the annual payment P such that B15=0.
An Excel spreadsheet can handle this otherwise tedious calculation rather quickly.
Sure, I can help you with that! Let's start with part a) - finding the amount of the mortgage still owed after the payment due on August 31, 2004.
We know that the original sale price of the house was $150,000 and a 20% down payment was made. So, the mortgage amount would be $150,000 - (20% of $150,000).
Step 1: Calculate the down payment
Down payment = 20% * $150,000 = $30,000
Step 2: Calculate the mortgage amount
Mortgage amount = $150,000 - $30,000 = $120,000
Now, let's move on to part b) - separating the interest and principal amounts.
To calculate the yearly mortgage payment, we can use the formula for calculating the payment amount on a mortgage.
Mortgage Payment = Principal * (r * (1 + r)^n) / ((1 + r)^n - 1)
Where:
Principal = $120,000 (mortgage amount)
r = monthly interest rate (8% per annum, so 8%/12 = 0.6666667% or 0.006666667 in decimal form)
n = total number of payments (15 years, so 15 * 12 = 180 monthly payments)
Step 1: Calculate the monthly interest rate
Monthly interest rate = 8% / 12 = 0.6666667%
Step 2: Calculate the total number of payments
Number of payments = 15 years * 12 months = 180 monthly payments
Step 3: Calculate the mortgage payment
Mortgage Payment = $120,000 * (0.006666667 * (1 + 0.006666667)^180) / ((1 + 0.006666667)^180 - 1)
Using a calculator, the mortgage payment is approximately $1,144.02.
Now, let's calculate how much of the mortgage will still be owed after the payment due on August 31, 2004.
Step 4: Calculate the number of payments made by August 31, 2004
Number of payments made = 2004 - 1998 = 6 payments
Step 5: Calculate the remaining balance on the mortgage
Remaining balance = Mortgage Payment * ((1 + r)^n - (1 + r)^(number of payments made)) / ((1 + r)^n - 1)
Remaining balance = $1,144.02 * ((1 + 0.006666667)^180 - (1 + 0.006666667)^6) / ((1 + 0.006666667)^180 - 1)
Using a calculator, the remaining balance on the mortgage after the payment due on August 31, 2004, is approximately $107,984.04.
So, after the payment due in August 31, 2004, the remaining balance on the mortgage will be approximately $107,984.04.
To solve this problem, we need to calculate the remaining mortgage owed after the specified payment dates. Let's break down the steps for both parts of the question:
a) How much of the mortgage will still be owed after the payment due on August 31, 2004, has been made?
Step 1: Calculate the outstanding balance after each yearly payment.
To do this, we'll use the formula for calculating the remaining balance on a fixed-rate mortgage:
Outstanding Balance = Principal * (1 + r)^n - Payment * [(1 + r)^n - 1] / r
Where:
- Principal is the initial loan amount ($150,000 - 20% down payment)
- r is the monthly interest rate (8% per annum / 12 months)
- n is the number of payments made
Step 2: Determine the number of payments made from August 31, 1998, to August 31, 2004.
Since the buyer intends to make yearly payments, there would be 6 payments made (1998, 1999, 2000, 2001, 2002, and 2003).
Step 3: Plug the values into the formula and calculate the remaining balance after the 6th payment.
b) Solve the same problem by separating the interest and the principal amounts.
Step 1: Calculate the annual interest payment.
To do this, we'll multiply the outstanding balance at the beginning of each year by the annual interest rate (8%).
Step 2: Calculate the principal payment for each year.
To find the principal payment, we subtract the interest payment from the total payment made each year.
Step 3: Calculate the outstanding balance at the end of each year.
To find the remaining balance, we subtract the principal payment from the previous year's outstanding balance.
Step 4: Repeat Steps 1-3 for each year until the 6th payment in 2004.
By following these steps, you can calculate the remaining mortgage owed and separate the interest and principal amounts for each year.