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The World Is Not Normal, the World Is Not Heavy-Tailed,
the World Is What It Is
the World Is What It Is
T he normal distribution was meant to describe the possible manifestations of financial reality — until it turned out not to work. The response was heavy-tailed distributions, meant to better capture extreme values. The problem is that these new concepts grew out of the very same assumptions as the ones before them — the randomness of financial phenomena.
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Random Walk and Normal Distribution
Taking the random walk process as the point of departure for describing changes in asset value ultimately leads to a picture of the phenomenon viewed as a normal distribution. Value thus moves within the range of variability carried by the normal distribution, and on the probability map dictated by that same distribution. That this is not true was demonstrated by the 2008 crisis. Actual losses exceeded the insurmountable levels indicated by the normal distribution (0.999 in credit and operational risk, 0.99 in market risk). The blame was placed on the phenomenon of correlation among exposures — it was this that broke normality. The behaviors of individual entities, in an extreme situation, are not independent. A rupture occurs in the intersubjective plane — in the shared conviction underlying the pricing of value. Panic, shared by all, dissolves the assumption of independence on which the normal distribution was derived. This problem was already formulated, as early as 1963, by Mandelbrot (1963). He described it as a two-phase system. Once the crisis phase sets in, all the assumptions that held in "normal" reality cease to hold — we are dealing with a different mechanics altogether. The distribution, then, is not normal (Colander et al. (2009) addressed the question of choosing the theoretical distribution meant to represent an institution's risk profile within the market risk concept: "Normality of asset price changes which can be rejected at a confidence level of 99.9999 […] Anyone who has attended a course in first-year statistics can do this within minutes.").
Fat-Tailed Distributions Are Built on the Same Foundation
This idea, together with several others, laid the foundation for the development of a new concept — heavy-tailed distributions. This class of distributions was intended to address the discontinuities between the normal situation and the extreme one. A new, statistical way of describing the events that indifferent reality brings had thus been found. The foundations on which these theories were built, however, were not called into question — the very idea of a probabilistic description of financial phenomena, the random walk model, and the normal distribution arising from it. The new concepts grew out of those already in existence. Heavy-tailed distributions are simply an extension of them, resting on the very same foundational concepts and statistical mechanics.
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This is not, of course, to say that these concepts are false in the mathematical sense. They are true, and elegant. No one calls into question the proofs from which they were derived (though it should be borne in mind that these theories rest on numerous assumptions that, in real-world applications of the models, are seldom satisfied). The point, rather, is that it remains unclear whether these concepts describe the real world. This widely applied probabilistic conceptual map is, after all, the result of adopting, sometime toward the end of the nineteenth century, a certain convention regarding the way in which the world is to be described. We do not, however, know the answers to the following questions: Are changes in value a random process?
Is the random walk process an accurate description of the world of value? Are all the assumptions satisfied that make it possible to derive the normal distribution from this process? Every manifestation of reality as it actually unfolds (such as the 2008 crisis) results merely in a modification of processes, methods, and techniques, within the formulas already adopted and applied — corrections to distributions, heavy-tailedness, the shift from VaR to Expected Shortfall. What it does not produce, however, is reflection at the fundamental, paradigmatic level, concerning the correspondence of these formulas with the real world. ■
Maciej Buczak
Risk management professional with 8+ years validating financial models in banking, currently teaching risk management and econometrics in Warsaw. His research questions the paradigmatic foundations of financial risk architecture — from VaR to AMA and IRB — drawing on extreme value theory.
References Academic sources cited throughout this article — click to expand the full list
Colander, D., Föllmer, H., Haas, A., Goldberg, M., Juselius, K., Kirman, A., Lux, T., Sloth, B. (2009), The Financial Crisis and the Systemic Failure of Academic Economics, Kiel Working Paper No. 1489, Kiel Institute for the World Economy.
Mandelbrot, B. (1963), The variation of certain speculative prices, The Journal of Business, 36(4), pp. 394-419.
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