Question

A friend, Ms. Michelle, was renting a house for $1,000 per month, but recently purchased a comparable home for $200,000 plus 1% for various fees, inspections, etc. Ms. Michelle’s opportunity cost of capital is 6% per year and she will have to pay a 5% commission when she sells the house. Assuming that she has to move and sell the house in one year, how much the house appreciate in value for her to be better off than renting?

Answer 1

For her to be better off buying than renting, the present value of the gains from selling the house should be higher than the present value of rent expense

Present value = future value / (1 + periodic rate)^{number of periods}

Here, the periodic rate = 6% / 12 = 0.5% (converting annual rate into monthly rate)

Present value of rent = ($1000 / 1.005¹) + ($1000 / 1.005²) + ($1000 / 1.005³) + ............+ ($1000 / 1.005¹²)

Present value of rent = $11,619

Let us say the house is sold for X. Then, net amount received after commission = X * (1 - 5%) = 0.95X

Present value of net amount = 0.95X / 1.06

Purchase price of home = $200,000 * (1 + 1%) = $202,000

The present value of rent should equal the present value of gains from selling the house.

Present value of gains from selling the house = Present value of net amount - purchase price

Solving for X, we get :

$11,619 = (0.95X / 1.06) - $202,000

X = $238,354

The sale price of house should be $238,354

Appreciation required = ($238,354 - $200,000) / $200,000 = 19.18%