Question

Emma has not yet decided whether she wants to rent or buy a property here in Oxford and she is asking your team to help her make this financial decision. Her monthly budget is $1,500 to cover any housing expenses including rent or owners’ costs (example: mortgage, hazard insurance, property taxes, and Home Owner Association fees). A good start of the analysis is by applying financial concepts such as the “time value of money.” Emma’s contract is for three years and is renewable for three more. Her plan is to stay in the Oxford area for no more than 8 years. Even if her job continues to work out well, she will try to move from the location she selects to a bigger house.

Estimate the maximum house value she can afford to buy. Assume the mortgage is fixed rate, 30 years maturity, 80% LTV, with no points. The interest rate that she was quoted is 4.8% with monthly payments. The property tax rate in the city of Oxford is 0.7% per year of the property value; the hazard insurance premium is 0.5% per year, and that on average you should consider $50 per month for maintenance. Determine the required monthly payment for the mortgage and the maximum house value she can afford if she buys. With this maximum house value select one that Emma can afford and follow the instruction for the case.

Answer 1

Let's say that A be the mortgage amount Emma is able to pay per month to service the mortgage.

Frequency = monthly, Period = month

Interest rate per period, R = Interest rate per month = 4.8% / 12 = 0.40%

Number of periods, N = nos. of months in 30 years = 12 x 30 = 360

Hence, the present value of mortgage payments = Loan amount = PV of annuities, A = A / R x [1 - (1 + R)^-N]

= A / 0.4% x [1 - (1 + 0.4%)^-360] =  190.60A

LTV = 80%

Hence, maximum value of house = Loan amount / LTV = 190.60A / 80% = 238.25A

Property tax rate in the city of Oxford = 0.7% per year of the property value = 0.7% x 238.25A = 1.67A

Monthly property tax = 1.67A / 12 = 0.14A

The hazard insurance premium = 0.5% per year

Hence monthly payment = 0.5% x 238.25A / 12 = 0.10A

$50 per month for maintenance

Monthly budget towards house = $ 1,500

Hence, Monthly mortgage payment + monthly property tax + monthly hazard insurance premium + monthly maintenance expense = 1,500

Hence, A + 0.14A + 0.10A + 50 = 1,500

Or, 1.24A = 1,500 - 50 = 1,450

Hence, A = 1,450 / 1.24 = $ 1,169.35

Hence, the required monthly payment for the mortgage, A = $ 1,169.35

and the maximum house value she can afford if she buys =  238.25A = $ 278,599 (can be rounded off to $ 278,600 as well)