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Distributionally robust inference for extreme Value-at-Risk

Author

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  • Yuen, Robert
  • Stoev, Stilian
  • Cooley, Daniel

Abstract

Under general multivariate regular variation conditions, the extreme Value-at-Risk of a portfolio can be expressed as an integral of a known kernel with respect to a generally unknown spectral measure supported on the unit simplex. The estimation of the spectral measure is challenging in practice and virtually impossible in high dimensions. This motivates the problem studied in this work, which is to find universal lower and upper bounds of the extreme Value-at-Risk under practically estimable constraints. That is, we study the infimum and supremum of the extreme Value-at-Risk functional, over the infinite dimensional space of all possible spectral measures that meet a finite set of constraints. We focus on extremal coefficient constraints, which are popular and easy to interpret in practice. Our contributions are twofold. First, we show that optimization problems over an infinite dimensional space of spectral measures are in fact dual problems to linear semi-infinite programs (LSIPs) – linear optimization problems in Euclidean space with an uncountable set of linear constraints. This allows us to prove that the optimal solutions are in fact attained by discrete spectral measures supported on finitely many atoms. Second, in the case of balanced portfolia, we establish further structural results for the lower bounds as well as closed form solutions for both the lower- and upper-bounds of extreme Value-at-Risk in the special case of a single extremal coefficient constraint. The solutions unveil important connections to the Tawn–Molchanov max-stable models. The results are illustrated with two applications: a real data example and closed-form formulae in a market plus sectors framework.

Suggested Citation

  • Yuen, Robert & Stoev, Stilian & Cooley, Daniel, 2020. "Distributionally robust inference for extreme Value-at-Risk," Insurance: Mathematics and Economics, Elsevier, vol. 92(C), pages 70-89.
    • Handle: RePEc:eee:insuma:v:92:y:2020:i:c:p:70-89
    DOI: 10.1016/j.insmatheco.2020.03.003
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    References listed on IDEAS

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    1. Johanna F. Ziegel, 2016. "Coherence And Elicitability," Mathematical Finance, Wiley Blackwell, vol. 26(4), pages 901-918, October.
    2. M. A. Goberna & M. A. López, 2018. "Recent contributions to linear semi-infinite optimization: an update," Annals of Operations Research, Springer, vol. 271(1), pages 237-278, December.
    3. Molchanov, Ilya & Strokorb, Kirstin, 2016. "Max-stable random sup-measures with comonotonic tail dependence," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2835-2859.
    4. Barbe, Philippe & Fougères, Anne-Laure & Genest, Christian, 2006. "On the Tail Behavior of Sums of Dependent Risks," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 361-373, November.
    5. Henry Lam & Clementine Mottet, 2017. "Tail Analysis Without Parametric Models: A Worst-Case Perspective," Operations Research, INFORMS, vol. 65(6), pages 1696-1711, December.
    6. Hult, Henrik & Lindskog, Filip, 2005. "Extremal behavior of regularly varying stochastic processes," Stochastic Processes and their Applications, Elsevier, vol. 115(2), pages 249-274, February.
    7. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    8. Puccetti Giovanni & Rüschendorf Ludger & Manko Dennis, 2016. "VaR bounds for joint portfolios with dependence constraints," Dependence Modeling, De Gruyter, vol. 4(1), pages 1-14, December.
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    2. Chia-Sheng Tu & Chung-Yuen Yang & Ming-Tang Tsai, 2020. "An Optimal Phase Arrangement of Distribution Transformers under Risk Assessment," Energies, MDPI, vol. 13(21), pages 1-16, November.

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