Wednesday, August 26, 2009

The Normality of Surprises

Rationalizing away the possibility of extreme events (or surprises) under conditions of probabilistic normality is an interesting behavioral economics phenomenon in society. Such behavior is signaled by attributions of “uncertainty,” “anomalies,” “shocks,” or some other lofty but equally pretentious notion of time-space deception. The truth is that denying the possibility of extreme outcomes defies the logic of normality, as such outcomes are certainly possible (or even imminent) under normal conditions.

Carl Friedrich Gauss (1777–1855) discovered the specific characteristics of what we now know as normality. The conceptual framework for describing normality is the normal frequency distribution, or “bell curve,” also known as a “Gaussian” distribution. Normality is a common assumption of many of nature’s phenomena, and a central concept in probability theory.

Extreme values in the universe (or population) of outcomes occur naturally and more frequently than many presume. Under conditions of normality, 1 in 22 observations will deviate by twice the standard deviation (which is the square root of the variance) from the mean, 1 in 370 will deviate by three times the standard deviation, and up to 5 in 1,000 observations will deviate from the mean by three or more times the standard deviation. Note especially that extreme outcomes can fall well beyond the mean (to infinity).

We as analysts have a duty to educate decision-makers about how to use probability theory to advance the cause of modern finance in society. That includes emphasizing the counter-intuitive possibilities of extreme events in the economy. To assume away the normality of such surprises would be naïve.


Max said...

One of the lessons of recent financial tribulations seems to be that extreme events are not well characterized by normal distributions. Actual distributions of market prices have thicker tails (more extreme surprises) than predicted by the mathematically simpler, and widely used risk models. See "The Black Swan" by Naseem Taleb for more. My analyses of errors in measurements of the physical constants, and analysis of forecasts by DOE Energy Information Administration show similar thick tailed distributions. But, if we recognize these, it is quite possible to do useful risk analysis if we are willing to let go of the familiar Gaussian.
Max Henrion

Lorenzo said...

I could not agree more with Max. Whoever has tried his hands at network science knows that a lot of complex real systems are dominated by fluctuations and that the relevant statistics is absolutely far away from a Gaussian.

nina said...

Two comments:
First, unlike the natural sciences, there is no guarantee that the future behavior of people will resemble the past, and this non-stationarity (and also stationary but non-Markovian path-dependence effects) is in addition to non-Gaussianity of the past; also your past historical time series may not be long enough to have a good statistical sense of the tail thickness.
Second, there are many alternatives to Gauss, many of which fit history better. For example, EVT fits the tail separately, and the Tukey g-and-h ( reference )
tweaks a Gaussian to include skew and kurtosis.
All this is univariate - it gets much messier with more than one variable and you may need to use some copula other than Gaussian.

Stracia said...

With financial data series, the problem is typically not that "past historical time series may not be long enough" to contain non-normal events. The problem is typically quite different -- that the past contains plenty of non-normal events, but that Gaussian distributions simply fail to capture these. THAT's the problem. Extreme Value Theory and transformations a la Tukey are all well and good, but no more address a lack of appropriate representational history than your original complaint against Gaussian curves! (Which can likewise be tweaked to represent assumptions making them non-normal.)

Emperical sampling is a simple starting point for an approach too rarely discussed. The other methods mentioned throw out history in favor of theory. How elegant, how sloppy.

Jackson said...

The problem of fat tails in security returns has been a recognized concern in academic finance for at least the last 45 years. Nassim Taleb did not discover this issue nor is it a new lesson from recent events.

Obviously practitioners in option pricing, risk management, statistical arbitrage, etc. should use a more appropriate distribution but I question whether the average investor cares much. The principles of diversification and risk/return tradeoff isn't affected by slight improvements in the underlying return distribution representation.

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