When interdependence is important, the power-law pattern frequently takes the place of the bell curve. At first glance, this may not seem very important, as the two curves do not appear to be that different. But on closer scrutiny, there is actually an enormous difference. The “tails” of a power-law curve — the regions to either side that correspond to large fluctuations — fall off very slowly in comparison with those of the bell curve. (See Exhibit 1.) These so-called fat tails imply that large events take place far more often than one would expect on the basis of “normal” statistics. In the case of market fluctuations, for example, the bell curve predicts a one-day drop of 10 percent in the valuation of a stock just about once every 500 years. The empirical power law gives a very different and more reliable estimate: about once every five years.
Large disruptive events are not only more frequent than intuition might dictate, they are also disproportionate in their effect. In any decade, a handful of the largest earthquakes do more property damage than the rest put together. Similarly, most of the total movement in any stock over a single year is often attributable to abrupt changes on a few select days. As a consequence, a plot of the typical rhythm for a power-law system shows a wild, fluctuating pattern, with a few huge peaks standing out against a background of relative quiet. In the context of evolutionary biology — where it is known that mass extinctions follow a power law, with many small events and a handful of massive cataclysms — this vision of what is normal has been referred to as “punctuated equilibrium.” Whatever the system, a power law points to a distinctive pattern of this sort, in which abrupt and violent transitions separate epochs of relative quiescence.
Power laws are normal for many complex systems. As a general rule, we should expect change to arrive not in the form of simple linear trends or dependable cycles, which naturally feed our craving for security, but in a far more erratic and unpredictable way. For a business, this perspective suggests that most of the risk it faces should be tied up with relatively infrequent and unpredictable events that alter its environment in a significant way. We might refer to this as “discontinuity” risk. Well-managed organizations are learning how to deal with it.
Reasoning with Extremes
The most famous financial mishap in recent history can be attributed to an inadequate appreciation for the consequences of power-law fluctuations. In 1994 and 1995, the hedge fund Long-Term Capital Management (LTCM) returned net profits of more than 40 percent, and by early 1998, it had increased its portfolio of assets to $1.3 billion. To many, it seemed as if LTCM, a company founded by former Salomon Brothers bond trader John Meriwether, with a board that included Nobel Prize–winning economists Myron Scholes and Robert Merton, had learned the secret for pumping money out of markets without risk.
Unfortunately for fund investors, LTCM’s assessment of the potential for overall losses was effectively built on the bell curve. A common technique in risk management is to estimate how much money the fund has a 1 percent (for example) chance of losing. This is known as the Value at Risk (VaR) and offers a rough measure of how much an investor might actually lose if things go badly.
Yet VaR estimates based on the bell curve can be wildly low. In fact, fund managers at LTCM were sophisticated enough to be aware that their bell curve estimates were probably low, yet they lacked methods for assessing the likelihood of more extreme risks. In September of 1998, “unexpected” volatility in the markets, set off by a default in Russia’s sovereign debt, led LTCM to lose more than 90 percent of its value. LTCM had borrowed more than $125 billion; the reverberations of its loss were felt across the global economy. To circumvent a more widespread collapse in the financial markets, the Federal Reserve Bank of New York organized a $3.6 billion bailout.