The Johnsons have accumulated a nest egg of $25,000 that they intend to use as a down payment toward the purchase of a new house. Because their present gross income has placed them in a relatively high tax bracket, they have decided to invest a minimum of $1,000/month in monthly payments (to take advantage of their tax deductions) toward the purchase of their house. However, because of other financial obligations, their monthly payments should not exceed $1,800. If local mortgage rates are 9.5%/year compounded monthly for a conventional 40-year mortgage, what is the price range of houses that they should consider?
If the Johnsons decide to secure a 20-year mortgage instead of a 40-year mortgage, what is the price range of houses they should consider when the local mortgage rate for this type of loan is 9%?

A) The price of houses for 40-yr mortgage lies between $5,461,292 and $9,810,325;
For 20-yr mortgage - between $692,887 and $1,227,196.
B) The price of houses for 40-yr mortgage lies between $148,447 and $247,205;
For 20-yr mortgage - between $136,145 and $225,061.
C) The price of houses for 40-yr mortgage lies between $136,145 and $225,061;
For 20-yr mortgage - between $148,447 and $247,205.

Question

The Johnsons have accumulated a nest egg of $25,000 that they intend to use as a down payment toward the purchase of a new house. Because their present gross income has placed them in a relatively high tax bracket, they have decided to invest a minimum of $1,000/month in monthly payments (to take advantage of their tax deductions) toward the purchase of their house. However, because of other financial obligations, their monthly payments should not exceed $1,800. If local mortgage rates are 9.5%/year compounded monthly for a conventional 40-year mortgage, what is the price range of houses that they should consider? ​
If the Johnsons decide to secure a 20-year mortgage instead of a 40-year mortgage, what is the price range of houses they should consider when the local mortgage rate for this type of loan is 9%?
​
A) The price of houses for 40-yr mortgage lies between $5,461,292 and $9,810,325;
For 20-yr mortgage - between $692,887 and $1,227,196.
B) The price of houses for 40-yr mortgage lies between $148,447 and $247,205;
For 20-yr mortgage - between $136,145 and $225,061.
C) The price of houses for 40-yr mortgage lies between $136,145 and $225,061;
For 20-yr mortgage - between $148,447 and $247,205.

Quizplus · Accepted Answer

The mortgage payment calculation involves the formula for a fixed-rate mortgage, which is P = [r*PV] / [1 - (1 + r)^-n], where P is the monthly payment, r is the monthly interest rate (annual rate divided by 12), PV is the principal amount (loan amount), and n is the total number of payments (months). For the 40-year mortgage at 9.5% annual interest, compounded monthly, the monthly interest rate is 0.095/12. For the 20-year mortgage at 9% annual interest, the monthly interest rate is 0.09/12. By substituting the minimum and maximum monthly payments they can afford ($1,000 and $1,800) into the formula and solving for PV (the loan amount), and then adding their down payment of $25,000 to the calculated loan amounts, we can determine the price range of houses they should consider. The correct price ranges must take into account the different terms and interest rates of the mortgages, leading to the conclusion in choice B, which correctly reflects the impact of these variables on the total price range of houses they can afford.

The Johnsons Have Accumulated a Nest Egg of $25,000 That