Rent versus Buy

Rent versus Buy

Rent versus Buy: How It Works

What it does

The page follows two people who start with the same cash and spend the same amount every month. One buys a home with a mortgage; the other rents an equivalent home and invests what the buyer put into the purchase. At the end of each year it asks what each could turn into cash, which is their net worth, and reports the first year in which the buyer's is at least the renter's: the break-even year. It then shows why, by setting the money each choice loses for good side by side. Every step is printed as formula, numbers put in, answer. The page describes the two paths under the rates you enter; it does not say which to choose.

The method

Symbols. H₀ is the home price; d_p the down payment as a share of the price; b_c and s_c the buying and selling costs as shares of the price; i_y the yearly mortgage rate and i_m = i_y / 12 the monthly rate; n_m the number of monthly payments; R₀ the first monthly rent; g_h the yearly home appreciation; q the yearly rent increase; r_i the yearly investment return; p_t and m_t the yearly property tax and maintenance as shares of the home's value; I_h and I_r the yearly home and renter's insurance; t_x the marginal income-tax rate. Rates are used as decimals: 6 % is 0.06.

Cash to buy. Down payment D_p = d_p H₀; loan L₀ = H₀ − D_p; buying costs B_c = b_c H₀; cash paid today C₀ = D_p + B_c. The renter starts with C₀ invested.

Mortgage. The level monthly payment that repays L₀ in n_m payments, and the balance still owed after k payments, are

M_p = L₀ i_m (1 + i_m)ⁿ / [(1 + i_m)ⁿ − 1],   Bal_k = L₀ (1 + i_m)^k − M_p [(1 + i_m)^k − 1] / i_m

Each payment is the month's interest (balance × i_m) plus principal, the part that reduces the balance.

Monthly costs. In year y the home is worth H₀ (1 + g_h)^{y − 1} at the start of the year. Each month the owner pays M_p (while the loan lasts), one twelfth of the year's property tax and maintenance on that value, and one twelfth of the insurance, which rises with the home's value. If deductions are itemised, t_x times the month's interest and property tax is taken off as a tax saving. The renter pays the rent R₀ (1 + q)^{y − 1} and one twelfth of the renter's insurance, which rises with the rent.

The renter's account. Each month the account earns j_m = (1 + r_i)^1/12 − 1 and then receives the owner's cost less the renter's cost: a deposit when renting is cheaper, a withdrawal when it is dearer. Both people therefore spend exactly the same cash every month, and all of the difference between them shows up in net worth.

Net worth after y years. Buyer: NW_buy = H₀ (1 + g_h)^y (1 − s_c) − Bal_12y, the sale price less selling costs and the loan. Renter: NW_rent = the account. The break-even year is the first y with NW_buy ≥ NW_rent.

Unrecoverable costs. Money that buys no asset. In a year, renting costs the rent and renter's insurance. Owning costs interest + property tax + maintenance + insurance − tax saving + lost return − appreciation (1 − s_c), where the lost return is what the renter's account earned in the year: the earnings the owner gave up by putting the cash into the home. Principal is left out because it stays the owner's. The two are tied to net worth exactly: the buyer's lead over the renter changes each year by (renting cost − owning cost), starting from −(B_c + s_c H₀), the one-time costs of buying and selling.

What it assumes

Constant appreciation, rent increase and investment return for the whole period. A fixed-rate mortgage held to the sale, with no refinancing, extra payments or mortgage insurance. An equivalent home either way. Property tax and maintenance as fixed shares of the home's current value. The renter invests every unit saved and never spends it; the investment return is after fees and taxes. No tax on the gain when the home is sold. Whole years, with every cash flow at the end of its month. Simple mode fixes property tax at 1.1 %, maintenance at 1 %, home insurance at 1,800 a year, buying costs at 3 %, selling costs at 6 %, renter's insurance at 240 a year and counts no tax saving; these are round illustrative figures, not data for any place, and Advanced mode lets you replace them. When deductions are itemised the page counts all of the mortgage interest and property tax as deductible at one marginal rate: an illustration, not tax advice.

When it misleads

Home prices are not smooth. A 3 % average does not arrive as 3 % each year, and prices in one town can fall for years. Because the buyer owns a home worth several times the down payment, a small change in appreciation moves the buyer's net worth a great deal; the sensitivity table shows how much.

The investment return is a guess too, and the model assumes the renter really invests the difference every month. A renter who spends it ends with less than the page shows.

The itemised tax saving is overstated for most households. Tax rules limit these deductions, and a saving exists only for the part of itemised deductions above the standard deduction. That is why the option is off by default.

"Equivalent home" is hard to meet. Homes for sale and for rent differ in size, place and condition; the comparison is only as good as the rent entered.

A negative renter's account appears when renting is dearer for long enough to use up the starting cash. The model treats it as borrowing at the investment return, which is rarely available in practice.

Money is not the whole question. Security of tenure, freedom to move, the work and risk of owning one large asset in one place are outside the calculation.

Worked example

The default case: a 400,000 home, 20 % down, a 30-year mortgage at 6.5 %, rent of 2,200 a month for an equivalent home, a 7-year stay, 3 % appreciation, 3 % rent increases and a 6 % investment return, with the Simple-mode assumptions above.

Cash to buy: down payment 0.20 × 400,000 = 80,000; loan 320,000; buying costs 0.03 × 400,000 = 12,000; cash paid 92,000.

Mortgage: i_m = 0.065 / 12 = 0.0054167; (1 + i_m)³⁶⁰ = 6.9918; M_p = 320,000 × 0.0054167 × 6.9918 / 5.9918 = 2,022.62 a month.

First-year monthly cost: owner 2,022.62 + (0.011 + 0.01) × 400,000 / 12 + 1,800 / 12 = 2,022.62 + 700 + 150 = 2,872.62; renter 2,200 + 240 / 12 = 2,220. The renter invests the 652.62 difference.

Buyer after 7 years: home value 400,000 × 1.03⁷ = 491,949.55; (1 + i_m)⁸⁴ = 1.5742, so the balance is 320,000 × 1.5742 − 2,022.62 × 0.5742 / 0.0054167 = 289,331.98; net worth 491,949.55 × 0.94 − 289,331.98 = 173,100.59.

Renter after 7 years: the 92,000 grows to 92,000 × 1.06⁷ = 138,333.98; the 84 monthly differences with their growth add 55,173.32; net worth 193,507.31.

After 7 years the renter's net worth is higher by 20,406.72. The buyer started 36,000 behind (12,000 of buying costs and 24,000 of selling costs); over the seven years renting cost 204,127.99 that does not come back and owning 188,534.71, which recovers 15,593.28 of the gap. Followed further, the buyer's net worth passes the renter's at the end of year 12 (278,939 against 275,610), the break-even year.

The page's results carry full precision; the hand figures above are rounded, so a last digit can differ.

A calculator for education and analysis, not investment advice. Tax effects are an illustration, not tax advice.