A person purchased a $250000 home 5 years ago. he paid 20% down and signed a 30-year mortgage at 6% compounded monthly. what is the unpaid balance of the loan (now that 5 years are gone). i need answer with explanation
To calculate the unpaid balance of the loan after 5 years, we need to break down the information provided and use the formula for the present value of a loan.
Given information:
Principal amount (home price): $250,000
Down payment: 20% of $250,000 = $50,000
Loan amount: $250,000 - $50,000 = $200,000
Interest rate: 6% per annum (compounded monthly)
Loan term: 30 years
First, let's calculate the monthly interest rate:
Monthly interest rate = Annual interest rate / 12
= 6% / 12
= 0.5% (or 0.005 as a decimal)
Next, we need to calculate the number of months for which the loan has been active:
Number of months = Loan term (in years) * 12
= 30 * 12
= 360 months
Now, we can use the present value formula to calculate the unpaid balance of the loan:
Unpaid balance = Loan amount * [(1 + r)^n - (1 + r)^t] / [(1 + r)^n - 1]
Where:
r = monthly interest rate
n = number of months
t = number of months elapsed (in this case, 5 years = 5 * 12 months)
Let's substitute these values into the formula:
Unpaid balance = $200,000 * [(1 + 0.005)^360 - (1 + 0.005)^60] / [(1 + 0.005)^360 - 1]
Now, we can calculate the value using a calculator or spreadsheet. The result will be the unpaid balance of the loan after 5 years.