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“Hindsight is 20/20.” This somewhat clichéd adage aptly applies to financial bubbles. Frequently we hear professionals and laymen alike, debate about whether certain securities or sectors are currently in a bubble. The discussion usually unfolds in the following way: the optimists insist that “this time it’s different”, while the opposition stubbornly retaliates by stating that “prices have significantly deviated from their fundamental value”. We have heard these arguments on numerous occasions, but whom are we supposed to believe and how can we make our own sound judgments? Considering the enormous growth certain sectors of the economy are currently experiencing, we decided to undertake the difficult task of shedding some light on a frequently misinterpreted concept.

History and Definition

A financial bubble results from a drastic overvaluation of certain asset classes. Opinions are divided over what actually causes this rapid increase in prices, but two different schools of thought predominate. The classical-liberal perspective blames central banks for causing macroeconomic distortions, via interest rate interventions, that lead to artificially high investment and inevitably speculation. On the other hand, Keynesians refer to “animal spirits” or simply put, human instincts that enable a herd mentality to develop. Investors are spurred on by a positive feedback mechanism until some segments of the market lose faith and the entire framework collapses. An analogous concept would be that of “irrational exuberance”, which both Alan Greenspan, former Chairman of the Federal Reserve, and Robert Shiller, a Yale economist, used to describe investor overconfidence during the dot-com bubble and the subprime mortgage crisis respectively.

Market bubbles are not a recent phenomenon. A well-known example of a historical speculative bubble is the Dutch tulip mania. At the peak of its hysteria, it was not uncommon for a single tulip bulb to trade for several times the annual income of a skilled craftsman. It is also interesting to note that concurrently exchanges and rudimentary futures markets (which were later forcibly transformed into options markets after a parliamentary decree was imposed by avaricious government officials who wanted to limit potential losses on their bets) developed, whereby florists entered into notarized contracts with tulip growers that specified future tulip purchases. This allowed for year-round trading and mass speculation in numerous Dutch towns. Eventually, following an unexpected default on a contract by a trader in Haarlem in early 1637 (allegedly caused by an outbreak of the bubonic plague), prices collapsed and many an unwary investor lost a fortune.

A combination of two factors limited the supply of certain tulips (like the highly coveted Semper Augustus pictured above). Firstly, it takes approximately seven to twelve years for a seed to grow into a flowering bulb. Secondly, only if the tulip’s bulb were infected with a mosaic virus (called the “Tulip breaking virus”), would the tulip develop the variegated petals that were in high demand.

This graph uses a log scale to depict the steep rise and fall in prices for tulip bulbs. However, it is important to note that price data from this time period is highly inconsistent since it was gathered from several sources including notarized futures contracts, spot markets and estate sales. Moreover, some of the data stems from anti-speculative pamphlets, which were widely distributed after the catastrophic collapse, thereby casting further doubt on the extent of the variability.

Identification and Detection of Bubbles

One obvious consequence of the tremendous detriment that bubbles have caused historically is the need for appropriate means capable of spotting excessive valuations in the market. After the credit crisis and the recession of 2008, the economist Hyman P. Minsky’s theory of Financial Instability has been widely recognized as one of the key concepts in order to understand the process behind bubbles.

In his book Stabilizing an Unstable Economy (1986) five different stages are presented, which accurately describe the general behavior of the boom and bust cycle. Starting from a “displacement” that captures the attention of investors (e.g. in the years before the explosion of the dot-com bubble, nobody expected that an online toy retailer could outperform a physical toy store), prices start rising sharply and gain solid momentum, which characterizes the so-called “boom”. What Minsky labels as “euphoria” soon follows. A stage in which caution is completely abandoned and new ways of evaluating are made up to claim that there is a rational behind what is happening. Consequently prices skyrocket beyond any measure. However this could mean that the bubble is close to an end and the fear of missing out on what might be a huge opportunity is substituted by the wisdom of smart money selling its positions and gaining as much as possible before the storm will rage. This is the phase of “profit taking”. Having perceived the danger, investors “panic” and demand is entirely dismantled. The positive trend inverts and prices plummet at a higher speed than they had been growing at. Investors holding the good or security in question are willing to liquidate it at any price, but nobody wants to buy it and prices go down even further.

Charles Kindleberger, an economic historian, built a revised model in the late 1970s based on Minsky’s theories, which saw a tremendous rise in popularity immediately after the dot-com bubble.

Is Mathematics the Answer?

Numerous experts have elaborated explanations similar to Minsky’s five-stage approach. However some went even further. Among them, it is worth mentioning the mathematical model presented by Robert Jarrow, Younes Kchia and Philip Protter in 2011. According to their paper, which was published in the SIAM Journal of Financial Mathematics, the use of statistical tools enables a specialist to determine – with enough empirical evidence – whether or not a specific asset is undergoing a bubble.

The key concept behind their reasoning is that during a financial bubble, the prices of assets rise far above their real value. Therefore, in order to detect the presence of a bubble, it is necessary to find out what the volatility of the asset’s price is.

The authors define it through a standard stochastic differential equation driven by Brownian motion, a natural process that characterizes the random movement of small particles in gas or liquids. After having estimated the volatility for various price levels from tick data, a special tool called “Reproducing Kernel Hilbert Spaces (RKHS)” is applied. It aims to extrapolate the expected volatility for large values that cannot be inferred by real data. Through this extrapolation process, the volatility growth rate can be calculated as long as the price gets arbitrarily large. In the end it all comes down to the speed at which the increase in volatility occurs. If it is not fast enough then – with a high degree of certainty – the asset is not in a bubble.

The diagram pictured above explains the concept behind the replication of Kernel Hilbert space through the use of probability density functions.

Analysis of growth rates for many stocks during the dot-com bubble has proven that the method is effective, even if it did not work with all the securities taken into consideration.

Despite these drawbacks, the fact that mathematics might be a useful tool to detect such phenomena as bubbles is truly encouraging.

Other Widely Used Measures

According to Didier Sornette, director of the Financial Crisis Observatory at the Swiss Federal Institute of Technology Zurich, a rise in prices is usually spaced out by moments of panicky selling. In order to monitor the behavior of thousands of assets, Mr. Sornette has been assisted by the use of a supercomputer. On the other hand, Robert Schiller has used the median P/E ratio to take into account that prices tend to overstate the real value of assets during bubbles. After dividing the market price for large U.S. stocks by their ten-year average of earnings after inflation during different time periods, he noticed that in late 1800s the median value was 16. As a matter of fact, during the dot-com bubble it was 44. His research on the topic led him to win the Nobel Prize in 2013.

The “Shiller PE ratio” or “cyclically adjusted PE ratio” is based on the works of the founders of value investing, Benjamin Graham and David Dodd. Today it is a widely used measure to predict future returns, but it has also proven to be a useful indicator of market crashes (as can be seen from the extraordinarily high values experienced in 1929 and 2000).


Although all these techniques might be useful, it is still notoriously difficult to predict when bubbles will pop. What makes this phenomenon so complex is the role that peoples’ emotions play. Market trends largely reflect what is assumed to be popular opinion. Therefore even stock market veterans still require a lot of effort and luck to spot bubbles and walk away during the “profit taking” stage. A memorable quote by John Maynard Keynes, one of the fathers of the imperfect market theory, summarizes the dilemma, “Markets can stay irrational longer than you can stay solvent”. The past has proven it and the future is about to do it again.




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