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The Nonexistent Relationship between Worlds: the Expected → the Extreme
V alue at Risk models describe both everyday life and extreme situations using a single, shared probability distribution — ordinary observations in its body, extremes in its tail, at distant quantiles such as 0.99 or 0.999. This article argues that this is a flawed assumption: extreme events do not arise from the same mechanics as everyday life, but belong to an entirely different category of phenomena. Even Extreme Value Theory, though it comes closest to addressing the problem, still derives its distributions from that same shared foundation — never truly separating these two worlds.
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Everyday Life and the Extreme World on a Single Distribution
The sanctioned and widely applied approaches to describing economic phenomena, including risk models, do not identify any distinction between the expected world and the extreme one. Models based on Value at Risk methods, whose task is to determine the extremity level at which reality may materialize — models that are, de facto, extreme value models — are, in fact, models of the total picture of reality. They see both everyday life and the world of extreme events alike. They see this comprehensive picture by means of a probability distribution — very often a normal distribution — in which everyday life is visible within the body of the distribution, and extreme situations within its tail, at the extreme quantiles (e.g., 0.99, 0.999). For the most part, this picture is built on data drawn from everyday experience; data on extremes are either lacking, or scarce and difficult to adapt to current reality (most often, these are distant historical observations). Such an approach means that extreme situations — the ones that truly matter, that these models are meant to address — emerge merely as a certain extension, a tail, of everyday life. This can be described as a rare picture of everyday life — say, at the 0.99 or 0.999 quantile.
Reinhart and Rogoff (2009), navigating within this very conceptual map, state that analysts, policymakers, and academics alike display an "unfortunate tendency to view recent experience through the narrow window opened by standard data sets, typically based on a narrow range of experience in terms of countries and time periods." The authors argue that financial crises operate on cycles considerably longer than the several-, or even dozen-or-so-year data sets typically used for such purposes. This reasoning thus assumes that longer data series will adequately capture the entire spectrum of the phenomenon — that this "complete" phenomenon can then be displayed on a single map, a single probability distribution. On such a distribution, one would find both the images of everyday life and extreme, crisis events alike. This particular way of viewing the matter is, of course, a widely accepted axiom of the statistical perspective on phenomena.
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Meanwhile, it would seem, extreme events belong to a separate category of events altogether — drawn, as it were, from a different shelf. Their mechanics cannot be derived from the very same picture as that of everyday life, merely by shifting that everyday life further along the quantile. These are, most often, events that could not occur within everyday life at all — even after rescaling them, on the same distribution, downward, toward everyday life. They are brought about by a different set of processes and mechanics altogether. They ought, therefore, to be set apart as a separate class of description.
The methods coming closest to such an approach are those derived from Extreme Value Theory (EVT). It is here, within this field, that attempts have been made to describe such extreme manifestations of reality separately. In the block maxima method, on which EVT I was built (Fisher and Tippett, 1928), only maxima are subjected to modeling — the maximum manifestations of reality within a block (a period). In the Peaks Over Threshold method, meanwhile — EVT II (Balkema and de Haan, 1974; Pickands, 1975) — extreme values are modeled, understood as those lying above a certain, arbitrarily established threshold. Only such values, designated as extreme, are modeled by means of these methods. As noted earlier, these methods come closest to the mark, yet do not, strictly speaking, constitute the separate modeling of extreme events they might seem to. Both variants of EVT, after all, concern limiting distributions derived from the ordinary distributions of the phenomenon itself. In EVT I, we have limiting distributions for maxima drawn from the phenomenon's distribution — say, a normal one; in EVT II, limiting distributions for values located above the threshold. These, however, remain limiting distributions emerging from the ordinary distribution of the phenomenon — the distribution of everyday life. They remain, in a certain sense, anchored within that everyday life. At the philosophical level, then, these methods are, in essence, no different from models based on the distributions of everyday life, and the extremity readings taken from them — for instance, by way of the VaR concept.
They constitute merely a certain refinement of the tail region of these ordinary distributions — an attempt to apply a magnifying glass to that tail, allowing for a more precise view of that particular region, and thus offering hope of better, more accurate estimation of distant quantiles. It is difficult, however, to defend the claim that approaches of this kind have fully separated everyday reality from extreme reality. This, however, is precisely what is postulated with respect to risk models. Extreme values, after all, cannot be derived from the mechanics that govern everyday life. ■
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
Balkema, A.A., de Haan, L. (1974), Residual Life Time at Great Age, The Annals of Probability, Vol. 2, No. 5, pp. 792-804.
Fisher, R.A., Tippett, L.H.C. (1928), Limiting forms of the frequency distribution of the largest or smallest member of a sample, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 24, No. 2, pp. 180-190.
Pickands, J. (1975), Statistical Inference Using Extreme Order Statistics, The Annals of Statistics, Vol. 3, No. 1, pp. 119-131.
Reinhart, C.M., Rogoff, K.S. (2009), This Time Is Different: Eight Centuries of Financial Folly, Princeton University Press, Princeton.
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