Question

The Johnsons have accumulated a nest egg of $50,000 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 $2900/month in monthly payments (to take advantage of the tax deduction) toward the purchase of their house. However, because of other financial obligations, their monthly payments should not exceed $3500. If local mortgage rates are 5.5%/year compounded monthly for a conventional 30-year mortgage, what is the price range of houses that they should consider? (Round your answers to the nearest cent.)

Least expensive

most expensive

Answer 1

Minimum monthly mortgage = $2,900 per month

Maximum monthly mortgage = $3,500 per month

Monthly mortgage is calculated using the below formula:

Monthly mortgage = [P * (R/n) * (1+R/n)^nT] / {[(1+R/n)^nT] - 1}

where P = Loan amount or principal

R = mortgage rate

n = number of compoundings

T = number of years

Thus for monthly mortgage = $2,900:

2,900 = [P * (0.055/12) * (1 + 0.055/12)^(12*30)] / {[(1 + 0.055/12)^(12*30)] - 1}

2,900 = [P * (0.004583) * (1.004583)^360] / [(1.004583)^360 - 1]

2,900 = (P * 0.004583 * 5.18677) / (4.18677)

12,141.63 = P * 0.004583 * 5.18677

12,141.63 = P * 0.02377

P = 12,141.63 / 0.02377

= $510,739 (rounded off)

Thus for monthly mortgage = $3,500:

3,500 = [P * (0.055/12) * (1 + 0.055/12)^(12*30)] / {[(1 + 0.055/12)^(12*30)] - 1}

3,500 = [P * (0.004583) * (1.004583)^360] / [(1.004583)^360 - 1]

3,500 = (P * 0.004583 * 5.18677) / (4.18677)

14,653.69 = P * 0.004583 * 5.18677

14,653.69 = P * 0.02377

P = 14,653.69 / 0.02377

= $616,409 (rounded off)

Least expensive house = $510,739

Most expensive house = $616,409