Financial Mathematics Text

Sunday, January 13, 2013

What is a Financial Bubble? Part I

The idea that drives this metaphor is blowing a bubble. One can blow it larger and larger but eventually it pops. Some economists have denied the existence of bubbles while others have expressed skepticism that the concept is well-defined enough to be scientifically meaningful. Eugene Fama, a prominent supporter of the Efficient Markets Hypothesis, stated as much in an interview in The New Yorker.

I share these reservations over defining a bubble but I still suspect it's a useful concept. As a result, I'd like to explore the issue a bit. I'll be doing so in a series looking at various examples and aspects of financial bubbles in order to get a better grip on whether or not we can give it a meaningful definition.

Bubble as Deviation from Trend

Today I'll start by looking at GMO's bubble criterion. In particular, I'll be relying on one of Jeremy Grantham's quarterly articles: Pavlov's Bulls. (You can actually sign up at for free to access all of their interesting articles).

The working definition for bubble here seems to be "deviation from trend". It can be illustrated quite nicely (see Exhibits 4-8). I've attempted to reconstruct some of these because there's a major question that needs to be decided upon: How do we establish the trend?

So let's look at some examples (and these all can be found in Exhibit 7 from Grantham's letter).

Stock Bubbles

Figure 1
S&P 500 Bubble 1920-1932

Here I set 1925 price to 1.0.

Figure 2

S&P 500 Bubble 1946-1984

Here I set 1954 to 1.0.

Figure 3

S&P 500 Bubble 1992-2009

Here I set 1994 to 1.0.

Now how do I arrive at those selections? Well, I basically made the choice based on a glance at GMO's exhibits. My goal was to replicate their results (which I did decently well... if I you look at Figure 1,  you'll notice that my bubble is higher than GMO's suggesting that there is something different in our data and methods).

So how is this constructed?

Grantham notes that the S&P 500 graphs are constructed by using the "Detrended Real Price". This is defined as:
the price index divided by CPI+2%, since the long-term trend increase in the price of the S&P 500 has been on the order of 2% real.
What's being referenced is amounts to fitting an exponential curve to S&P 500 price series adjusted for inflation:

Figure 4

S&P 500 Bubble Exponential Trend

Using Robert Shiller's data, I fitted an exponential function to the real price. If you look in the upper part of Figure 4, you'll see the equation. It indicates the the growth rate is 1.73% which is pretty close to Grantham's 2% figure.

But how trendy are these trends?

As far as I can tell, deviation from the trend is the norm not the other way around:

Figure 5

S&P 500 Bubble Detrended Price

The red lines correspond to the trends in Figures 1-3. While the 1920-1932 and the 1946-1984 have about the same trend lines, the trend line for the 1992-2009 period is slightly higher. We could also ask what's going on in the period between 1880-1915; the "trend" level is clearly higher here.

There is clearly a good amount of room for making a "judgement call" and it's not entirely clear what criterion we ought to use to make that call. 

Currency and Commodity Bubbles

Now Exhibit 7 from Grantham's article has a few other examples. I'll take a look at two examples, one currency and one commodity.

We'll start with the dollar. I used the US dollar trade weighted index from FRED. My results were somewhat different than GMO's but I have no idea what index they were using (I set 1979 to 1.0).

Figure 6

US Dollar Bubble

The red area is the area identified in GMO's exhibit. I'm presuming they're assuming the trend should be constant. The black line represents my best guess of what GMO has in mind regarding the trend (their picture doesn't actually have a line). Just doing a quick visual on the graph, the green line represents my best guess as to what the "trend" would be presuming it's linear and there is even a trend to identify. So I have some reservations on this one as well.

Next, we'll look at our old friend gold.

Figure 7

Gold Bubble

The numbers will be different compared with the GMO exhibit as I'm not entirely sure which year those dollars are inflation adjusted with (I used 2012 $ \$ $). But I'm presuming they're assuming gold (and other commodities) should maintain their value and keep up with inflation. What the correct value is, I'm not sure. I put it at the $ \$ $500 mark because that's where the "trend" appears to be.

So does that mean we're in another gold bubble? I'm not sure. I guess one thing that would have to be questioned is why gold should keep up with CPI. Why can't gold outpace CPI? After all, it's possible for growth in the supply of gold to lag growth in productivity gains elsewhere in the economy in which case gold might outpace CPI.

But I am somewhat sympathetic to the idea. I had noted earlier in a post Modeling Gold Returns that when gold outpaces CPI, subsequent real returns tend to be poor. So there may be some argument for that.

But that doesn't imply that I have any idea what gold ought to be worth right now. Should it be $ \$ $500/oz as the above trend suggests? If gold were to revert to a mean Gold/CPI level of about 3, that would put gold around $ \$ $700/oz. But why should we believe that?

So I guess I have somewhat more sympathy for the gold trend over the currency trend but still have some reservations over how we decide what the trend ought to be.

Housing Bubbles

While this was not included in GMO's piece, it would only make sense to add this one. I've already discussed the issue of housing as an investment here. Historically housing has, more or less, kept up with inflation. Here's a chart based on Robert Shiller's data.

Figure 8

Housing Bubble

Throughout history, home prices remain more or less flat in real terms. But there are a couple of moves. For example, I have drawn three "trend" lines. Were housing prices really cheap during the 1920's and 30's? And the trend seems to be higher in more recent years.

Concluding Remarks

I'm not entirely clear on one goes deciding whether or not a trend exists and what that trend is. As a result, I'm a bit reluctant to accept the "deviation from trend" criterion for bubbles. At the very least I have some reservations.

Now I do think that the "trends" that were used had some reality grounded in "fundamentals". My next post in this series will be looking at that. For example, in the case of the S&P 500, the idea that the price will rise via compounding growth is grounded in fundamentals regarding earning, dividend and cash flow growth. But there's no reason to suppose such growth need to be a smooth constant rate. As such, the real fundamental "trend" might be more lumpy.

Regarding currencies, I do believe there's room to argue for fundamentals here as well. But I'm not sure why constant value would be the correct "trend". There's obviously different dynamics that can contribute to the rise and fall of currency prices.

As already noted before, I'm sympathetic to the idea that gold should be grounded to CPI in some way. But I'm not sure in what way it should be grounded. I see no reason to suppose that gold can't outpace CPI and still make sense when broader considerations are factored in. So if there is a "trend" that needs to be fleshed out.

Overall I think that if we were to defend the "deviation from trend" idea, we'd end up advocating "deviation from fundamentals". So that will be the next step in our examination.

For those interested in reproducing the S&P 500 detrended price, the following equation is what I used:

$$\text{Detrended Price} = C p e^{-2\%t}$$

where C is some constant (selected so that the year you want to be equal to 1.0 is equal to 1.0), p is the inflation adjusted price of the S&P 500 and t is the time in years.


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