Question

8. 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 gains from selling the house should be at least equal to the present value of rent payments

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

Present value of rent payments

Here, the periodic cost of capital is monthly, since rent payments are made monthly. Monthly cost of capital = 6% / 12 = 0.5%

Present value of rent payments = ($1000 / (1 + 0.5%)¹) + ($1000 / (1 + 0.5%)²) + ($1000 / (1 + 0.5%)³) + ..............+ ($1000 / (1 + 0.5%)¹²)

Present value of rent payments = $11,619

Present value of gains from selling house

Cost of house (including fees)  = $200,000 * (1 + 1%) = $202,000

Let us say the sale price of house is X. Then, net proceeds after commission = X * (1 - 5%) = 0.95X

Gains from selling house = 0.95X - $202,000

Present value of gains = (0.95X - $202,000) / (1 + 6%)¹ = ((0.95X - $202,000) / 1.06)

For her to be better off buying than renting, the present value of gains from selling the house should be at least equal to the present value of rent payments

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

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

X = $225,596

Appreciation required in house value (in $) =  $225,596 - $200,000 = $25,596

Appreciation required in house value (in %) =  $25,596 / $200,000 = 12.80%