4. 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 9%/a, compounded monthly.
c. How long will it take to repay the loan?
d. How much will be the final payment?
e. Determine how much interest she will pay for her loan.
g. How much sooner would the loan be paid if she made a 15% down payment?
h. How much would Vanna have saved if she had obtained a loan at 7%/a, compounded monthly?
c. To find out how long it will take to repay the loan, we need to use the formula for the present value of an annuity:
PV = PMT * ((1 - (1 + r)^(-n)) / r)
Where PV is the present value or loan amount, PMT is the monthly payment, r is the monthly interest rate, and n is the number of months.
In this case, the loan amount (PV) is $200,000, the monthly payment (PMT) is $3000, and the monthly interest rate (r) is 9%/12 = 0.75%. We need to solve for n.
Plugging in these values into the formula, we get:
200,000 = 3000 * ((1 - (1 + 0.0075)^(-n)) / 0.0075)
We solve this equation to find the value of n. One way to do this is through trial and error or using a financial calculator or spreadsheet program.
d. The final payment is the last payment made to repay the loan. Since we have determined the number of months it will take to repay the loan in the previous calculation, we can multiply that number by the monthly payment to get the final payment.
Final payment = Number of months * Monthly payment
e. To determine the total interest paid for the loan, we need to subtract the loan amount (PV) from the total amount repaid, which is the monthly payment multiplied by the number of months.
Total interest = Total amount repaid - Loan amount
g. If Vanna makes a 15% down payment, she will need to borrow $200,000 - 15% of $200,000 = $170,000.
Using the same formula as in part c, we can calculate how long it will take to repay the loan with a down payment of $170,000 instead of $200,000.
h. To determine how much Vanna would save if she obtained a loan at 7%/a compounded monthly, we need to calculate the new monthly payment using the same formula as in part c with a monthly interest rate of 7%/12 = 0.58%.
Once we have the new monthly payment, we can subtract it from the original monthly payment and multiply it by the number of months to find the amount saved.