## June 23, 2005

### What is a bubble and is this one now?

Many of those discussing the possibility of a housing market bubble seem to be taking Justice Potter Stewart's position on pornography-- they haven't defined a bubble, but they think they know it when they see it. Maybe it's useful to take a look at a formal characterization of the concept of a speculative bubble, and see how well it seems to fit the facts of the current situation.

The U.S. housing market may or may not be experiencing a bubble. But there's no doubt that there's plenty of froth in the discussion about the possibility. Calculated Risk continues to provide a useful take on each day's developments. And if that's not enough for you, The Housing Bubble 2 is but one of many blogs devoted exclusively to the topic, while Housebubble.com will lead you to a dozen new stories to peruse each day.

Let me try to contribute some formality to this public discussion, using a little math in the hopes of being most clear. Suppose that you pay a price Ht for a house right now. To keep the math as simple as possible, say you plan to stay in the house for a year and then sell it for a price Ht+1. Suppose you're able to borrow 100% of the cost of the house, taking out a loan with an annual interest rate of it, and plan to use proceeds from sale of the house to pay off the loan next year. Then a year from now you'd have to repay the principal on the loan (Ht) along with the interest (itHt).

To figure out whether or not this seems like a good deal for you, the other thing we need to know is the value to you of the housing services you'll get from the home, which we'll represent by st. One way to think of this is as the rent you'd pay if you had to rent a comparable unit rather than buy the house, so that st is the money you save yourself by buying.

So at the end of the year, how do you stand? You've got st + Ht+1 more cash in your hand than you would have had if you hadn't made the home purchase, but you owe Ht + itHt back to the bank. You'll just break even if

Suppose we take it, st, and Ht+1 as given. If the price you paid for the house is higher than the value of Ht that satisfies the above equation, you paid more than it was really worth to you. If you paid a price lower than the value Ht implied by the equation, you got a great deal. So, if we were told the values for it, st, and Ht+1, we might consider the value of Ht satisfying the above equation as the rational market value of the home.

The calculations get really simple if you assume that house prices won't change during the year that you are in the home, so that Ht = Ht+1. In that case, the only question is whether your monthly interest payments exceed the value that you place on the monthly service flow you get out of your home. To consider some illustrative numbers, suppose that in the year 2000, you would have been willing to pay \$1500 per month for the housing service flow. If you took out a 30-year fixed mortgage at that time at an 8.5% interest rate, your monthly payment on a \$200,000 house would be about \$1500. So the above equation means that, in the absence of any capital gains, \$200,000 would have been a reasonable value Ht to pay for that home back then.

Over the last 5 years, U.S. nominal GDP increased about 20%, so you might expect the value of that housing service flow to have risen about 20% as well, in which case the service flow today would be worth about \$1800/month. With a 30-year mortgage rate taken out at the 5.75% rates available now, your monthly payment on a \$300,000 house would be \$1750. In other words, if the house price had gone up 50% (which is the amount by which the average U.S. house price rose over this period), the house would be more of a bargain today than it was 5 years ago.

The trickier questions, and all the talk about bubbles, come from trying to figure out how capital gains enter into these calculations. For this purpose, it's useful to rearrange the above equation. A little algebra leads us to

Economics students will recognize this equation as saying that the value of the house can be calculated from the present value of its flow of housing services plus the present value of its resale price. The second equation tends to be how economic professors make our students figure out how much the house should be worth, whereas the first equation-- given my monthly interest payments and what I hope to sell the house for, can I really afford it-- is how most people may actually think about their personal decision. But, until the laws of algebra get repealed, the two equations are just two different ways of saying exactly the same thing.

Now, you know what interest rate the bank is charging you it, and should be able to figure out what the rent on a comparable unit st would be, but what about the third term, Ht+1? That's next year's house price, which is something that nobody knows. You'll have to make some guess about it, but where will that guess come from?

One way you might think about it is, you'll be selling the house next year to somebody who at date t+1 will be going through exactly the same calculation you are doing at date t. They'll look at what they have to pay for the house when they buy it (Ht+1), their interest payments and service value (it+1Ht+1 and st+1), and their resale value (Ht+2), and conclude that a fair price for them would be

Now suppose we substitute this value for Ht+1 back into our previous expression for Ht. Then we conclude that the fair price Ht that you should be willing to pay for a house right now is

This equation says that what you should be willing to pay for the house depends not just on the value of housing services to you st, but also the value of housing services to the next person who's going to live in the house, st+1. Even though you personally are not going to be living in the house next year, you care about st+1 because that will determine how much somebody's going to be willing to pay you for the house when you try to sell it.

It seems that all we've accomplished with this last substitution is push the uncertainty a little farther into the future, because to use the above equation to figure out the value of the house, we have to know Ht+2, what it's going to sell for two years from now. But we could go through the same sort of reckoning, using the same cash flow equations to figure out what somebody at t+2 might pay for the house, and then somebody at t+3, and so on. As long as house prices don't go up faster than interest on home loans gets compounded, the remainder term involving Ht+j is going to get smaller and smaller as j gets bigger and bigger, so that we could write

This last equation is how economists think about the "market fundamentals" value of the house, namely, the value of the house is the discounted present value of the full stream of future housing service flows. The key thing is that the house price is entirely determined by what you expect is going to happen to the value of housing service flows st+j and interest rates it+j, both now and in the future. It does not depend on what you think the person who buys the house is going to believe. You've basically assumed that they next person is going to be rational just like you, meaning they'll look at whether the value they'll get from the home justifies their monthly interest costs just like you did, in which case the time path for the housing service value and the interest rate end up being the whole story.

It's sometimes easier to see what sort of behavior this equation implies if we consider a special case in which interest rates are constant (it = i) and rents grow at a steady rate g (st+j = s(1 + g)j). In this case the last equation simplifies to

Now, the first thing to note about this equation is that a small change in either the interest rate i or the growth rate g can have a big effect on the market fundamentals house prices, and stories that try to explain the surge in house prices in the frothiest communities solely in terms of market fundamentals rely on a combination of the two effects. However, the math works just the same going the other way. If we suddenly see a big move back up in mortgage rates, or if the communities where population growth has been outstripping the construction of new housing run into a slump, the same equations predict that house prices could go down just as impressively as they went up.

This last fact is what led Alex Tabarrok at Marginal Revolution to observe that, even if the current situation is driven entirely by market fundamentals, there still is a substantial possibility of a significant decline in house prices should those fundamentals turn south.

All of which is a rather long diversion before getting to answer the promised question: what exactly is a bubble? Economists would say there's a bubble in the market if home buyers are basing their expectation about the price that they can sell the house for in the future on something other than the affordability and desirability of that home to the future buyer. In a bubble, borrowers are assuming they'll have a capital gain on their house big enough to bail them out of the deep debt they're getting into, even though it would mean an even bigger debt burden for the next folks down the road. But one of those future buyers, if we looked ahead rationally, is going to be forced to say, no sale.

Now, even if you readily believe that large numbers of home buyers are fully capable of just such miscalculation, there's another issue you'd have to come to grips with before concluding that the current situation represents a bubble rather than a response to market fundamentals. And that is the question, why are banks making loans to people who aren't going to be able to pay them back? Maybe your neighbor doesn't have the good sense not to burn his own money, but is the same also true of his bank?

If you want to come up with an answer more sophisticated than "banks are stupid, too," I think you're ultimately led to look for the real roots of the housing bubble, if you think there is one, in some kind of moral hazard argument explaining why the equity capital of lending institutions is insufficient to cover the risk in the mortgage loans they issue or hold. I earlier mentioned deposit insurance as one story that could be told along these lines, while Greg Hess and the Chicagoboyz some time back argued that Fannie Mae could be playing such a role. But it's not obvious what changed recently along these lines to only now be producing a housing bubble.

What has changed-- and stands out like a sore thumb-- is that (1) the housing "bubble" occurred at a time when mortgage rates have been at the lowest levels of the last quarter century, and (2) house prices are going up most in those communities where population and income have grown relative to the available stock of housing. Looking at the market fundamentals solution above, is it really that surprising that H goes up when i goes down and g goes up?

It may be a good idea to take a hard look at any possible moral hazard problems lurking in our present financial institutions. But economic fundamentals look to me like the more obvious place to start in trying to understand exactly what's happened to U.S. house prices over the last 5 years.

Posted by econbrowser at June 23, 2005 11:43 PM

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Tracked on August 9, 2005 07:59 AM

Bubbles are not only defined by the mathematical expressions you have elegantly set forth but also by psychology. Unfortunately, this aspect of the present real estate conundrum is ignored in your formulation.

The present loose lending environment, coupled with rising prices, have indubitably lead to speculation unrelated to growth. No economically rational individual buys a home at record prices with I/O or optional ARM mortgages because doing so is the equivalent of playing economic Russian roulette. Yet, lots of folks are spinning the cartridge chamber and pulling the trigger.

On the other hand--there is always an other hand, isn't there?--foreclosure rates are presently very low.

Posted by: pessimist at June 24, 2005 08:44 AM

The reason foreclosure rates are low is that folks are still able to sell their homes and do better than they would if they just walked away and gave the place to the bank(foreclosure).
Once the market demand proscribes preemptory sales, foreclosures will rise big time.

Posted by: realist at June 24, 2005 11:53 AM

Valid point about psychology, but we also need to take into account the non-rational source of funds: Asian central banks, and other holders of funds resultant from our CA deficit. As long as the spigot of low cost MBS flow of funds is as wide open as it is now, there will always be a greater fool to take on the risk in buying housing, especially as many of these borrowers are not sophisticated and do not understand the I/O downsides.

Posted by: fatbear at June 24, 2005 12:02 PM

Quite true, but maybe the buyers and lenders are playing chicken with the Fed. The Fed may blink first but it still may come too late as growth expectations can change much more and more rapidly than interest rates. Get ready for a wild ride.

Posted by: Lord at June 24, 2005 12:08 PM

Talk about wishful thinking! I'm seeing this in many of my normally sane friends, who search for any reason to believe their new found wealth cannot be a mirage.

I'm surprised you're unaware that accountability for the banks that approve the mortgages has been steadily drained from the system over the years. Through creative securities and repackaging with complex derivatives, the bank no longer cares as long as they can slot the loan somewhere in these packages. Believe me, banks find it hard to meet a loan they don't like.

So keep grasping at straws for creative ways to deny this bubble. It's in the best tradition of economists in every bubble.

Posted by: Spectator at June 25, 2005 01:27 AM

This is a shockingly stupid post and a shockingly stupid argument. It boils down to "there can't ever be a bubble because no one is every stupid enough to lend into one." Empirically, this is a bad argument, since, repeatedly, loans have been made into circumstances where the rewards were far lower than the risks. While on some occasions these have been from lack of "moral hazard", that is a belief that some other entity is actually bearing the risk so the risk/reward looks more attractive, it is far from clear that all such circumstances are from lack of moral hazard.

Miscalculating risk is a constant fact of economic life. The market doesn't assure that it has calculated risk correctly, merely that there is no consistent risk calculation which is superior to the market as a dominant strategy. That is, if anyone has beat the market, they will act on their information and bring prices back into line.

Posted by: Stirling Newberry at June 25, 2005 11:51 AM

Stirling, I believe you mean to say "due to moral hazard" rather than "from lack of moral hazard." Bad loans result from moral hazard, not from a lack of moral hazard.

I thought my main points were that interest rates are low and housing demand has outstripped supply in impacted communities, and that these are what a "no-bubbles" view of the current situation would say you should expect to see.

Posted by: JDH at June 25, 2005 12:32 PM

No doubt you're right in saying that if you assume the right values for i and g, you can demonstrate that any value for housing is rational. But doesn't this formulation merely replace one question ("is there a bubble?") with another question ("what are reasonable values for i and g?")?

As a prospective homeowner, I'd love to know what values for i and g are rational. And the problem is that the harder I think about this problem the more difficult it becomes for me to guess where rationality resides.

For instance, is it rational to assume that the g of the past few years will continue? Or the historically low i? Maybe. But I could make an equally strong case that it's more rational to assume a return to historic medians for both values.

Whichever way I parse the current market, it seems riddled with irrationality. As you say, if home prices are a bubble, it's difficult to figure why banks would be lending into it. But on the other hand, if home prices aren't a bubble, it's difficult to figure why so many lenders are willing to lend so much money at such low interest rates to people who can then reap much bigger returns by investing that borrowed money in real estate. It would be far more rational for the people lending the money to buy real estate directly so they can keep the rewards to themselves.

I do wonder if part of the problem is that people's expectations are fundamentally silly. As you well know, the valuation formulas you use collapse into meaninglessness in cases when g is greater than i. If you think that you can borrow money at 5% to invest in something that is sure to go up 10%, then the value of H becomes infinite and it makes sense to borrow every penny in the world to invest in it.

To me, that sounds a lot like what's going on in the real estate market right now. It may be rational in a purely mathematical sense, but it's silly in a practical sense. Isn't it?

Posted by: Ian at June 26, 2005 02:36 PM

Great comments, Ian. As far as the value of the interest rate i is concerned, if you're buying a home, you know the value of i for the next 30 years with certainty if you're taking out a conventional fixed-interest mortgage. It's true that if you instead have an adjustable rate mortgage, there's some guessing involved, and some people have argued that miscalculations about where future interest rates are headed may be part of how some housebuyers are being led astray. But the calculations that I reported showed that just using the observed decline in conventional 30-year fixed mortgage rates since 2000, you can explain the lion's share of what's happened nationally to house prices, without any guesswork or conjecture.

You're certainly also correct that one could make up some story about g that theoretically could rationalize any observed price increase. I myself would be quite reluctant to try to attribute much, if any, of the national house price increase to a perceived increase in g. Where I think perceptions of g are making a difference is in explaining why some communities have experienced so much greater house price increases than others. The thing that makes me take that hypothesis seriously is the observation that the places where population has been growing the fastest are also often the places where house prices have been going up the most.

Now, by definition, your community can't grow faster than the national average forever. Although it makes it easier to see what the formulas are saying if you do extrapolate the growth rate as if it would go on forever, the practical way that I think about it is as an approximation to how things would be if the growth continued at that rapid rate for a number of years. But your g > i example makes it clear that you don't want to go too far with that approximation.

All this aside, I agree with you completely that people may have miscalculated how long before their community's growth slows down, and when it does slow down, house prices could come crashing down. This was one of the main points that Alex Tabarrok and I were trying to make.

But, in terms of definitions, the point is that you don't need to suppose there's a bubble-- that is, you don't need to assume that house prices are determined by something other than the expected present value of the flow of housing services-- to talk about that phenomenon, and I think it helps to focus the discussion by laying out exactly what the market fundamentals no-bubble valuation of house prices should look like, and seeing how far you can go with that explanation in terms of trying to understand what has been going on.

And best of luck with your home purchase. Don't forget that one option is always to rent!

Posted by: JDH at June 26, 2005 03:52 PM

To look at the issue as a securities analyst, it is quite clear that when publicly traded homebuilders like RYL post gross profit margin %ages of 27.7 % in 1Q05, you can be absolutely certain that the supply of new houses will expand. To contrast, in the early 1990's margins of ~10% were common. When high prices make supply compelling to produce, isn't that a good indication of a bubble??

Posted by: RichL at June 26, 2005 06:27 PM

Those objecting to the math in the argument are all wet. It is fine. The problem is indeed in the formulation of expectations of growth rates of fundamentals. Robert Shiller shows in the 2nd edition of his Irrational Exuberance that we are now at all time high records for ratios of house prices to income and house prices to rent in the US. These ratios have surged in the last five years, while there have not been any noticeable increases in either the growth rates of income or rents.

Presumably a major culprit here, as implied by Jim's equations, is the decline in real mortgage interest rates. Two parts of this are the low long term nominal rates that have held despite the tightening on short term rates by the Fed, presumably because of purchases of long term US securities by Chinese and Japanese trying to prop the dollar up against their currencies in order to maintain their employment. The second part is the increase in inflation driven by the rise in oil prices that started last year. Presumably these low real mortgage rates will not remain for too much longer.

Even if one believes that many agents have rational expectations, many also have some sort of adaptive expectations. So, many agents, including mortgage lenders, are plugging in assumptions of continuing rapid price appreciation in the next few years. This justifies the 105% loans and interest only mortgages with ARMs we are seeing now. In the Washington area these have risen from about 10% of mortgages to 54% over the last three years. We have also seen a major increase in clearly speculative buyers who are "flipping" houses, the phenomenon that has led even Greenspan to worry about "froth" in the real estate markets.

Yes, Virginia, this certainly looks like a bubble, and a very big one.

Posted by: Barkley Rosser at June 27, 2005 02:08 PM

Prof. Hamilton:

Regarding "any possible moral hazard problems lurking in our present financial institutions" -- do you see any issues with the market for mortgaged-backed securities?

And I see nothing wrong with your math or definition of a bubble. But the trend towards ARMs increases the speculation inherent in discounting against an unknown future interest rate. I suspect that many home buyers are naive about the implications of that uncertainty, while those providing the financing are passing the buck (i might have said "bag") to the MBS market.

Posted by: STS at June 30, 2005 12:38 AM

reducing the NPV to a perpetuity is convenient, as JDH knows, but you've got some near-term changes coming in the denominator, to i at least. Sadly, there's no closed-form expression for what happens when rate adjustments change & interact with growth

Posted by: psh at July 5, 2005 08:16 PM

We talked about this issue some in the exchange with Ian above, psh. I presented the case where i is constant and s grows forever at rate g only as an illustration to understand some of the intuition. The formula that comes before this in my post, from which that special case is derived, is perfectly valid and perfectly general.

Posted by: JDH at July 5, 2005 08:58 PM

What does all this matter? Buy what you can realistically afford and you will never lose. This bubble speak is really geared to investors - most folks today buy because of personal reasons not so much for investment. The speculators and banks that provide the loans to them are what is pushing up the values so fast. It is driving out folks from places that they have lived all of their lives. They simply cannot afford to live in the same neighborhood because of taxes and growth. The bubble term is a myth created by folks with not folks without to make money off the hype.

Posted by: Merlin at July 14, 2005 10:06 AM

All of this discussion ignores one fact that is a big deal in the Washington DC area at least - time in residence. The DC area is a very transient area, and a lot of people here KNOW that they will not be here for more than 1, 2, or 3 years. This means they buy with the knowledge that they MUST sell in a year or two. For such a buyer, a floating rate ARM is really not that much of a risk since the buyer will not be here long enough to have to pay increased rates. The only risk is of a sharp price decline and a consequent capital loss when the property is sold. As long as Federal spending continues to increase at the rates of recent years, and local governments continue to impose increasingly restrictive zoning measures every year, it is unlikely that population growth in the capital area or the current shortage of houses will reverse soon. Of course it eventually must reverse, and when that happens someone will be left holding the bag, but current buyers can rationally bet that they won't be the bagholders, especially if they must sell in 1 or 2 years. Also, it is apparent that the housing boom has just slowed down here in the last few months, with a lot more houses being sold and prices staying flat. That was to be expected too, since there is a practical limit to how much money buyers can borrow, even with ARM loans.

Posted by: AndrewP at July 26, 2005 06:36 PM

Don't know if anyone's still reading this thread, but there's a fundamental problem with the above analysis:

"g" refers to the growth rate of rents, not house values. Current rental inflation is about 2.5%. Let's assume that continues, and that the interest rate is 6%, or 5% after tax. That leaves a discounting factor (i-g) of 2.5%. In CA, a \$1m house has a rental yield of 3.5%, or \$35,000 p.a.. Subtract property tax and upkeep, and you get about \$15,000 (1% p.a. depreciation, rising in line with rent inflation). \$15,000/2.5% yields \$560,000. That's the present value of rental cash flows on a \$1,000,000 house. Roughly half the value of the home therefore cannot be accounted for by expected rental growth rates.

BTW -- if you assume a higher rental inflation rate, think about what that means for the interest rate. Low housing inflation (42% of core CPI) is the major driver behind low interest rates.

Posted by: Scott at July 28, 2005 09:49 AM

Scott, the BLS CPI index for rent of primary residence increased at an average annual logarithmic rate of 3.4% for San Francisco, 5.5% for Los Angeles, and 6.9% for San Diego between 2000 and 2004.

Posted by: JDH at July 28, 2005 11:16 AM

Good point, JDH. S. Cal has seen some faster rent increases than the rest of the country.

The 20yr CAGR for LA/OC rents is about 3.6%, so that's probably a fair number to use since it captures the entire cycle (Which includes two booms).

Using a \$1m home with a \$15,000 net cash flow (3.5%(rent yield)-1.1%(prop tax)-1%(dep)) yields \$1.36m in PV. There you go -- real estate is a steal at these prices!

Except...not sure about using the tax adjusted WAC in the denominator since there is no cash interest expense in the numerator.

In fact, I messed up completely on the WACC. The cost of debt is relevant if the cash flows are levered. Otherwise, the WACC is the cost of equity alone. What's the cost of equity for a real estate investor? Assume 8% (low-ball opportunity cost of similar investments) for the sake of argument. That leaves a PV of \$365k.

Why not lever it? The cash flows are negative. The CF/(i-g) equation is used for "steady state" conditions, which do not apply to negative cash flows. One would have to build a two stage model with rent cashflows growing to exceed interest expense, taxes and Dep in the initial period.

Bottom line -- the unlevered calculation does tell us something about the economics of housing...

Posted by: Scott at July 28, 2005 04:03 PM