Vanna has just financed the purchase of a home for $200 000. She agreed to repay the loan by making equal monthly blended payments of $3000 each at 4%/a, compounded monthly.
a. How long will it take to repay the loan?
b. How much will be the final payment?
c. Determine how much interest she will pay for her loan.
To answer these questions, we will need to use the formula for a loan repayment known as the annuity formula. The annuity formula is given by:
P = (PMT * (1 - (1 + r)^(-n))) / r
Where:
P = principal amount (loan amount)
PMT = monthly payment
r = interest rate per compounding period
n = total number of compounding periods
a. How long will it take to repay the loan?
To find the time it will take to repay the loan, we need to solve for n in the annuity formula. Plugging in the given values:
P = $200,000
PMT = $3,000
r = 4% per year (0.04/12 per month)
n = unknown
We can rearrange the formula to solve for n:
n = [log((PMT / r) / ((PMT / r) - P))] / log(1 + r)
Plugging in the values:
n = [log(($3,000 / (0.04/12)) / (($3000 / (0.04/12)) - $200,000))] / log(1 + (0.04/12))
Using a calculator to solve this equation, we get n ≈ 100.31 months.
Therefore, it will take approximately 100.31 months to repay the loan.
b. How much will be the final payment?
The final payment will be the same as the monthly payment since the loan is completely repaid in the given number of months. So, the final payment will be $3,000.
c. Determine how much interest she will pay for her loan.
To find the total interest paid, we can subtract the principal (loan amount) from the total amount repaid. The total amount repaid is the monthly payment multiplied by the total number of months.
Total interest = (PMT * n) - P
Plugging in the values:
Total interest = ($3,000 * 100.31) - $200,000
Using a calculator to solve this equation, we get a total interest of approximately $100,300.
Therefore, Vanna will pay approximately $100,300 in interest for her loan.