Monday, June 2, 2008

Present Discounted Value and Real Estate

Last week, a writer at the Times described his changing view of buying versus renting by making reference to the "real estate market's version of the price-earnings ratio," or the price of a unit of housing divided by the annual total rent for an equivalent unit.

This is a useful summary measure because in equilibrium, it should be equal to 1 / [1 - 1/(1+i)], where i is the annual (nominal) interest rate on a mortgage. This is because the price of a housing unit should equal the present discounted value of its income stream, a.k.a. its rent. If you own housing, you're "paying yourself" the rent by enjoying it. (Or you could rent it out yourself.)

The math of present discounted value suggests that in equilibrium, the price of housing, P, should be equal to

P = R / [1 - 1/(1+i)]

where R is the annual rent. Or equivalently,

P/R = 1 / [1 - 1/(1+i)]

Suppose i = 7.5%. Then P/R should be about 14.3. Suppose you live in New York and rent at $2000 per month. Then the purchase price of that unit should be about $344,000. If P exceeds that, renting is a better deal.

Sound low? We've assumed that the rent, R, stays the same over time. If R were rising, as it probably is in most markets, say at some rate r, then the equation becomes:

P/R = 1 / [1 - (1+r)/(1+i)]

And you might expect to see a P/R ratio around 19.5 or so. The deductibility of interest payments also lowers the after-tax interest rate you actually pay, to something below i.