Tiny (or huge) bubbles

The financial markets continue to be panicky, waiting for another shoe to drop. I saw something by Nick Rowe of Carleton University on how we got into this whole crisis in the first place, the emergence of the housing "bubble". How did we get into a situation where a) house prices were going up dramatically and b) then not only stopped going up, but collapsed? (Said collapse triggering defaults and destroying much of the value of many mortgage-backed securities, which in turn generated much of the current mess.)

Nothing particularly dramatic happened in August 2007, the point at which this all started to go south. So how come home prices collapsed and defaults started to go up?

Let's start with how you would value a hosue, if you were going to sell it one year later. You'd set the price today according to the following:

P(Today) = Annual Rent + P(Year Later)/(1+r)

The price today is equal to what you'd earn by renting out the house (or, equivalently, the amount of rent you don't have to pay for a year) PLUS the price of the house in one year from now, discounted. What is this discounting term, 1/(1+r)? This comes from the fact that you are waiting one year to sell. If you could sell the house for $1000 in one year, then you'd be willing to take less than $1000 today, just because you don't have to wait to get the money. The higher is r, the stronger this discounting.

Now, let's assume that you aren't planning to sell the house any time soon. In fact, you will hold the house until an infinite number of years from now (I know, dumb, but not much different mathematically from holding it for 30 years). You expect that both rents and the price of the house will go up at G percent per year. So now the value of the house is

P(Today) = Value of Rents to Infinity + P(Infinity) / (1+r)(to infinity power)

Given the growth rate of G, this works out to

P(Today) = Today's Rent / (r-G) + P[(1+G)/(1+r)](to infinity power)

So what do we have? The price of houses today depends on the rental value divided by this term (r-G). This is just valuing the stream of rental payments (or implicit payments to yourself) that the house generates. The second term depends on the ratio of (1+G) to (1+r), raised to the infinity power.

Now, as long as G < r, or the growth rate of house prices is less than the interest rate you can earn investing "safely", then the price of your house today depends ONLY on the rents you earn. Why? Because if G < r then (1+G)/(1+r) < 1 and any fraction to the infinity power is essentially equal to zero. In contrast, let's consider the origin of a bubble in housing. What happens here? The assumed growth rate of house prices, G, goes up for some reason (perhaps lots of people with subprime mortgages bid up the price of houses). At the same time, the "safe" return rate, r, is pretty low thanks to a) a Fed that is averting a recession in 2001/2002 and b) a glut of global savings. This means that G is almost equal to r. The ratio of Rent/(r-G) blows up, and therefore so does the price of a house. (Note that if r-G is very small, then Rents divided by something very small is a very large number). Any small changes in Rents results in really dramatic changes in price levels for houses. What if our assumed G goes up above r? Then the valuation of a house is P(Today) = Infinity Now, obviously house prices didn't go to infinity, but it seems pretty clear that house prices were way larger than made sense. In other words, there are only two outcomes that can happen when we get G > r. 1) House prices actually DO go to infinity or 2) we were wrong about G and eventually we will come to our senses and reset our assumed G to less than r and get finite values for houses.

So let's go back to the case where G < r, but really really close to r. House values still go WAY up, but not to infinity. Now in August 2007 some of the subprime mortgagees who had bid up house prices (and hence made us believe that G was really big) default, and everyone realizes that the rents they could earn on their house just went down (even by just a little). A small drop in rents, given that G is almost as big as r, means a huge drop in the price of houses.

A big drop in house prices convinces everyone that G is in fact lower than we had assumed. The drop in G then lowers the price of houses even more, and we have ourselves a crash.

Ultimately, the origins of this bubble are a shrinking of the difference between G and r. The crash happens when this difference widens out again. Why does this gap shrink? We assume that a surge in house prices today thanks to higher demand is indicative of higher G forever (as opposed to a temporary blip). Also, the risk free rate, r, is coming down at the same time.

Why does the gap open up again? First, when defaults show that rents (or the implied value of your house) are lower, and second, when this causes us to realize that G isn't as high as we thought.