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?

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%?

Answer 1

To determine the price range of houses the Johnsons can consider for a 40-year mortgage, we first need to determine the maximum monthly payment they can afford based on their income and other financial obligations.

Minimum monthly payment: $1,000
Maximum monthly payment: $1,800

Using the formula for the present value of an annuity, we can calculate the maximum loan amount they can afford based on their monthly payment:

PV = PMT x [(1 - (1 / (1 + r)^n)) / r]

Where:
PMT = monthly payment = $1,800
r = monthly interest rate = 9.5%/12 = 0.00792
n = number of payments = 40 x 12 = 480

PV = $1,800 x [(1 - (1 / (1 + 0.00792)^480)) / 0.00792] = $307,809.36

This means the maximum price of the house they can consider is $307,809.36 plus the $25,000 down payment they have already saved, which equals $332,809.36.

If they decide to secure a 20-year mortgage instead of a 40-year mortgage at a local mortgage rate of 9%, we can use the same formula to calculate the maximum loan amount they can afford:

PMT = $1,800
r = 9%/12 = 0.0075
n = 20 x 12 = 240

PV = $1,800 x [(1 - (1 / (1 + 0.0075)^240)) / 0.0075] = $363,618.13

This means the maximum price of the house they can consider is $363,618.13 plus the $25,000 down payment they have already saved, which equals $388,618.13.