1. The monthly payment on the mortgage is determined by the formula
`M=P*(R/(1-(1+R)^(-N)))` , where P is the principal (P = $200,000), R is the monthly rate, expressed as a decimal (R can be found as annual rate divided by 12:
`R = 0.04/12 = 1/300` ), and N is the number of months in the terms of the mortgage. In the given example, we know the monthly payment M = $3,000, and we need to find how long it will take Vanna to pay off the loan (N).
Plugging everything given in the formula results in
`3000 = 200,000((1/300)/(1-(1+1/300)^(-N)))`
This simplifies to
`4.5 = 1/(1-(301/300)^(-N))`
`1- (301/300)^(-N) =2/9 `
`(301/300)^(-N) = 7/9`
`N=-log_(301/300) (7/9) = 75.52`
It will take 76 months (rounding up) or 6 years and 4 months for Vanna to repay the loan.
To find the exact number of months and the exact final payment, prepare the amortization schedule. See the amortization schedule for the first several months in the attached file. In the B column, the remaining principal is shown after the payment is made each month. This is the amount from which the interest is taken. Some of the monthly payment goes to pay this interest, and that is the amount shown in column C. It is calculated as a monthly rate of 0.04/12 of the remaining principal. The part of the monthly payment that goes to reduce the principal is then calculated as the monthly payment $3,000 minus the interest portion. This amount is shown in column D, and it gets subtracted from the remaining principal. Then, the new remaining principal is entered in column B in the next row.
3. Assuming Vanna made payments for 75 months, the total amount paid would be
75*$3000 = $225,000. The interest she paid on the $200,000 loan is
$225,000 - $200,000 = $25,000. This is equivalent to 12.5% simple interest rate.
5. If she made a 15% down payment, the amount she borrowed would be 85% of $200,000, or $170,000. Repeating the calculations from question 1 for P = $170,000 results in N = 62.9, or 63 months. This means she would repay the loan one year sooner and pay less interest.
7. When writing the concluding statement, make sure to highlight that one should look for the least interest rate and try to pay as much down payment as feasible when taking out a loan!