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Postmodern Portfolio Theory pp 237–245Cite as

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Going to Extremes: Leptokurtosis as an Epistemic Threat

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Abstract

“Time present and time past/are both present in time future/and time future contained in time past.”1 In projecting forward rather than backward over time, economic forecasting cannot escape “timeless” moments of a different sort: the mathematical moments of the distribution of financial returns.2 Of the four moments of greatest interest to financial institutions and their regulators3—mean, variance, skewness, and kurtosis—it is the fourth moment, kurtosis, that should pose the deepest epistemic concern. Kurtosis eludes detection where it counts most—in its fat tails. Our expectations and perceptions may underestimate the most extreme risks by a significant margin. The overarching goal in financial responses to leptokurtosis and “fat tails” is the accurate forecasting of extreme events. Simple accuracy in description, if attainable and attained, would be a fantastic accomplishment.

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Notes

  1. 1.

    T.S. Eliot, Burnt Norton, in Four Quartets 13–22, 13 (Harcourt, Brace & Co. 1971; 1st ed. 1943).

  2. 2.

    Cf. Eliot, Little Gidding, in Four Quartets, supra note 1, at 49–59, 58 (“for history is a pattern/Of timeless moments”).

  3. 3.

    See https://en.wikipedia.org/wiki/Method_of_moments_(statistics).

  4. 4.

    Campbell R. Harvey & Akhtar Siddique, Conditional Skewness in Asset Pricing Tests, 55 J. Fin. 1263–1295, 1265 (2000).

  5. 5.

    See Campbell R. Harvey, Drivers of Expected Returns in International Markets, 1:1 Emerging Mkts. Q. 32–49, 38 (Fall 2000).

  6. 6.

    In banking, VaR supplies the primary measure of risk in the VaR-based portfolio insurance strategy, distinct from the competing constant proportion portfolio insurance strategy. See, e.g., Lan-chih Ho, John Cadle, Michael Theobald, An Analysis of Risk-Based Asset allocation and Portfolio Insurance Strategies, 36 Rev. Quant. Fin. & Acctg. 247–267 (2011); Chonghui Jiang, Yongkai Ma & Yunbi An, The Effectiveness of the VaR-Based Portfolio Insurance Strategy: An Empirical Analysis, 18 Int’l Rev. Fin. Analysis 185–197 (2009); Koichi Matsumoto, Portfolio Insurance with Liquidity Risk, 14 Asia-Pac. Fin. Mkts. 363–386 (2007). In addition, Leo Constantino Tay appears to have written “A Value-at-Risk Capital Asset Pricing Model” as a working paper of the economics department of the Ateneo de Manila University, the Philippines. This paper is not available online, but is cited in Leo Constantino Tay, A Mean Value-at-Risk Framework for Speculating and Hedging with Options (March 8, 2001) (available at http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.27.9281&rep=rep1&type=pdf). Cf. Harvey, supra note 5, at 38 (observing that assets with “a highly negative VaR,” if that quantity is treated by markets as a measure of risk, should be expected “to command a high average return”). See generally Leslie A. Balzer, Investment Risk: A Unified Approach to Upside and Downside Returns, in Managing Downside Risk in Financial Markets: Theory, Practice and Implementation 103–155, 113–15 (Frank A. Sortino & Stephen E. Satchell eds., 2001) (evaluating the completeness of VaR and expected shortfall as risk measures).

  7. 7.

    https://en.wikipedia.org/wiki/Kurtosis.

  8. 8.

    The terms platykurtic and leptokurtic are traceable to Karl R. Pearson: Skew Variation: A Rejoinder, 4 Biometrika 169–212 (1905).

  9. 9.

    https://en.wikipedia.org/wiki/Kurtosis.

  10. 10.

    See Eugene F. Schuster, Classification of Probability Laws by Tail Behavior, 79 J. Am. Stat. Ass’n 936–939 (1984).

  11. 11.

    Freeman J. Dyson, A Note on Kurtosis, 106 J. Royal Stat. Soc’y: Series B 360–361 (1943); accord Kevin P. Balanda & H.L. MacGillivray, Kurtosis: A Critical Review, 42 Am. Statistician 111–119, 111 (1988).

  12. 12.

    See, e.g., Per Bak, How Nature Works: The Science of Self-Organized Criticality (1996); Simon A. Levin, Fragile Dominion: Complexity and the Commons 55 (1999); Manfred Schroeder, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise 103–119 (1991); M.E.J. Newman, Power Laws, Pareto Distributions and Zipf’s Law, 46 Contemp. Physics 323–351, 327–330 (2005).

  13. 13.

    Albert-László Barabási, Linked: The New Science of Networks 68 n.1 (2002).

  14. 14.

    See George Kinsley Zipf, Human Behavior and the Principle of Least Effort: An Introduction to Human Ecology (1949); George Kinsley Zipf, Selective Studies and the Principle of Relative Frequency in Language (1932).

  15. 15.

    See, e.g., Pablo Marquet, Of Predators, Prey, and Power Laws, 295 Science 2229–2230, 2229 (2002) (hailing the “vast number of biological power laws”).

  16. 16.

    See Donald L. Turcotte, Fractals and Chaos in Geology and Geophysics (1997); Didier Sornette, Leon Knopoff, Y.Y. Kagan & C. Vanneste, Rank-Ordering Statistics of Extreme Events: Application to the Distribution of Large Earthquakes, 101 J. Geophys. Research 13,884-13,893 (1996).

  17. 17.

    See Paul Krugman, Development, Geography, and Economic Theory 42–46 (1997); Felix Auerbach, Das Gesetz der Bevölkerungskonzentration, 59 Petermanns Geographische Mitteilungen 73–76 (1913); Xavier Gabaix, Zipf’s Law for Cities: An Explanation, 114 Q.J. Econ. 739–767 (1999).

  18. 18.

    See A.Z. Mekjian, Model of a Fragmentation Process and Its Power-Law Behavior, 64 Phys. Rev. Letters 2125–2128 (1990).

  19. 19.

    See Vilfredo Pareto, Cours d’Economie Politique (1896); David G. Champernowne, A Model of Income Distribution, 63 Econ. J. 318–357 (1953).

  20. 20.

    See Robert L. Axtell, Zipf Distribution of U.S. Firm Sizes, 293 Science 1818–1820 (2001); Michael H.R. Stanley, Sergey V. Buldyrev, Shlomo Havlin, Rosario N. Mantegna & H. Eugene Stanley, Zipf’s Plots and the Size Distribution of Firms, 49 Econ. Letters 453–457 (1995).

  21. 21.

    See Bernardo A. Huberman, The Laws of the Web: Patterns in the Ecology of Information 19–31 (2001); see also Bernardo A. Huberman & Lada A. Adamic, Growth Dynamics of the World-Wide Web, 401 Nature 131 (1999); Réka Albert, Hawoong Jeong & Albert-László Barabási, Diameter of the World-Wide Web, 401 Nature 130–131 (1999).

  22. 22.

    See David G. Post & Michael B. Eisen, How Long Is the Coastline of the Law? Thoughts on the Fractal Nature of Legal Systems, 29 J. Leg. Stud. 545–584 (2000); Daniel A. Farber, Earthquakes and Tremors in Statutory Interpretation: An Empirical Study of the Dynamics of Interpretation, 89 Minn. L. Rev. 848–889 (2005).

  23. 23.

    See Ian A. Hatton, Kevin S. McCann, John M. Fryxell, T. Jonathan Davies, Matteo Smerlak, Anthony R.E. Sinclair & Michel Loreau, The Predator–Prey Power Law: Biomass Scaling Across Terrestrial and Aquatic Biomes, 349 Science 1070 (2015) (full article at 349 Science (6252), aac6284); cf. Just Cebrian, Energy Flows in Ecosystems, 349 Science 1053–1054 (2015).

  24. 24.

    See Xavier Gabaix, Parameswaran Gopikrishnan, Vasiliki Plerou & H. Eugene Stanley, A Theory of Power-Law Distributions in Financial Market Fluctuations, 423 Nature 267–270 (2003).

  25. 25.

    See Peter Fulde & Richard A. Ferrell, Superconductivity in a Strong-Spin Exchange Field, 135 Phys. Rev. A 550 (1964). The stretched exponential distribution, among other possibilities, offers one finite alternative to power-law distributions. See L. Laherrère & D. Sornette, Stretched Exponential Distributions in Nature and Economy: “Fat Tails” with Characteristic Scales, 2 Eur. Phys. J. B 525 (1998); cf. James Ming Chen, Modeling Citation and Download Data in Legal Scholarship (Feb. 2, 2015) (available at http://ssrn.com/abstract=905316) (modeling citations and downloads in legal scholarship as an ordinary exponential distribution).

  26. 26.

    See generally Nicolas Champagnat, Madalina Deaconu, Antoine Lejay, Nicolas Navet & Souhail Boukherouaa, An Empirical Analysis of Heavy-Tails Behavior of Financial Data: The Case for Power Laws (Aug. 14, 2013) (available at https://hal.inria.fr/hal-00851429).

  27. 27.

    Martin L. Weitzman, A Review of The Stern Review on the Economics of Climate Change, 45 J. Econ. Lit. 703–724, 723 (2007).

  28. 28.

    Daniel A. Farber, Uncertainty, 99 Geo. L.J. 901–959, 926 (2011).

  29. 29.

    Id.

  30. 30.

    Id. at 925.

  31. 31.

    See, e.g., Nicholas Barberis, The Psychology of Tail Events; Progress and Challenges, 103 Am. Econ. Rev. 611–616 (2013); Craig R. Fox & Amos Tversky, A Belief-Based Account of Decision Under Uncertainty, 44 Mgmt. Sci. 879–95 (2004).

  32. 32.

    This is a distinction originating in Frank Knight, Risk, Uncertainty, and Profit 231–235 (1921); accord, e.g., Farber, Uncertainty, supra note 28, at 903 & n.5.

  33. 33.

    Farber, Uncertainty, supra note 28, at 906.

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  1. College of Law, Michigan State University, East Lansing, Michigan, USA

    James Ming Chen

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Chen, J.M. (2016). Going to Extremes: Leptokurtosis as an Epistemic Threat. In: Postmodern Portfolio Theory. Quantitative Perspectives on Behavioral Economics and Finance. Palgrave Macmillan, New York. https://doi.org/10.1057/978-1-137-54464-3_12

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