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 need to use the formulas for loan payments, loan repayment periods, and compound interest. Let's break it down step by step.
a. How long will it take to repay the loan?
To find the loan repayment period, we can use the formula for the number of periods required to repay a loan. The formula is:
n = -(log(1 - (r * P) / A) / log(1 + r))
Where:
n = number of periods (in months)
r = monthly interest rate (decimal)
P = principal amount (loan amount)
A = monthly payment amount
First, let's convert the annual interest rate to a monthly interest rate:
r = 4% / 100 / 12 = 0.00333
Now, substituting the given values into the formula:
n = -(log(1 - (0.00333 * 200000) / 3000) / log(1 + 0.00333))
Using a calculator or spreadsheet, calculate the value of n. The result will give you the number of months it will take to repay the loan.
b. How much will be the final payment?
To calculate the final payment, we need to determine if there is any remaining balance after making all the equally monthly payments. If there is a remaining balance, the final payment will be the remaining balance plus any accumulated interest.
To find the remaining balance after n months, we can use the formula for the future value of an ordinary annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = future value (remaining balance)
P = principal amount (loan amount)
r = monthly interest rate (decimal)
n = number of periods (in months)
Substituting the given values into the formula:
FV = 200000 * ((1 + 0.00333)^n - 1) / 0.00333
Calculate the value of FV, which will give you the remaining balance after n months. The final payment will be the remaining balance plus any accumulated interest.
c. Determine how much interest she will pay for her loan.
To calculate the total interest paid, subtract the loan amount from the total amount repaid:
Total Interest = (A * n) - P
Where:
A = monthly payment amount
n = number of periods (in months)
P = principal amount (loan amount)
Substitute the given values into the formula to calculate the total interest paid.
These calculations will give you the answers to all three questions.