At another level, market crashes constitute beautiful examples of events that we would all like to forecast. The arrow of time is inexorably projecting us toward the undetermined future. Predicting the future captures the imagination of all and is perhaps the greatest challenge.

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Faster-than-exponential growth also occurs in computing power, as measured by the evolution of the number of MIPS per $1,000 of computer from 1900 to 1997. Thus the so-called Moore's law is incorrect, since it implies only an exponential growth. This faster than exponential acceleration has been argued to lead to a transition to a new era, around 2030, corresponding to the epoch when we will have the technological means to create superhuman intelligence.

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It is difficult to assess how much this gambling spirit is active in the minds of individual investors. If it is, even to a small degree, it is relevant to our discussion since it makes investors prone to imitation and herding because they invest on little information. It may also explain the anomalously large volatility of prices and their potential instabilities.

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The same basic ingredients are found repeatably: fueled by initially well founded economic fundamentals, investors develop a self-fulfilling enthusiasm by an imitative process or crowd behavior that leads to an unsustainable accelerating overvaluation.

Indeed, the frequency of crashes in the Monte Carlo simulations was much smaller than the frequency of crashes in the real data: if one of the most frequently used benchmarks of the industry is incapable of reproducing the observed frequency of crashes, this indeed means that there is something to explain that may require new concepts and methods.

According to the academic world view that markets are efficient, only the revelation of a dramatic piece of information can cause a crash, yet in reality even the most thorough post-mortem analyses are typically inconclusive as to what this piece of information might have been. For traders and investors, the fear of a crash is a perpetual source of stress, and the onset of the event itself always ruins the lives of some of them.

The concept of a random walk is simple but rich for its many applications, not only in finance but also in physics and the description of natural phenomena. It is arguably one of the most founding concepts in modern physics as well as in finance, as it underlies the theories of elementary particles, which are the building blocks of our universe, as well as those describing the complex organization of matter around us.