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A one-sided Vysochanskii-Petunin inequality with financial applications


  • Mercadier, Mathieu
  • Strobel, Frank


We derive a one-sided Vysochanskii-Petunin inequality, providing probability bounds for random variables analogous to those given by Cantelli’s inequality under the additional assumption of unimodality, potentially relevant for applied statistical practice across a wide range of disciplines. As a possible application of this inequality in a financial context, we examine refined bounds for the individual risk measure of Value-at-Risk, providing a potentially useful alternative benchmark with interesting regulatory implications for the Basel multiplier.

Suggested Citation

  • Mercadier, Mathieu & Strobel, Frank, 2021. "A one-sided Vysochanskii-Petunin inequality with financial applications," European Journal of Operational Research, Elsevier, vol. 295(1), pages 374-377.
  • Handle: RePEc:eee:ejores:v:295:y:2021:i:1:p:374-377
    DOI: 10.1016/j.ejor.2021.02.041

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    References listed on IDEAS

    1. Babat, Onur & Vera, Juan C. & Zuluaga, Luis F., 2018. "Computing near-optimal Value-at-Risk portfolios using integer programming techniques," European Journal of Operational Research, Elsevier, vol. 266(1), pages 304-315.
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    3. Meng, Xiaochun & Taylor, James W., 2020. "Estimating Value-at-Risk and Expected Shortfall using the intraday low and range data," European Journal of Operational Research, Elsevier, vol. 280(1), pages 191-202.
    4. James W. Taylor, 2019. "Forecasting Value at Risk and Expected Shortfall Using a Semiparametric Approach Based on the Asymmetric Laplace Distribution," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(1), pages 121-133, January.
    5. Mercadier, Mathieu & Lardy, Jean-Pierre, 2019. "Credit spread approximation and improvement using random forest regression," European Journal of Operational Research, Elsevier, vol. 277(1), pages 351-365.
    6. Leung, Melvern & Li, Youwei & Pantelous, Athanasios A. & Vigne, Samuel A., 2021. "Bayesian Value-at-Risk backtesting: The case of annuity pricing," European Journal of Operational Research, Elsevier, vol. 293(2), pages 786-801.
    7. Barrieu, Pauline & Scandolo, Giacomo, 2015. "Assessing financial model risk," European Journal of Operational Research, Elsevier, vol. 242(2), pages 546-556.
    8. Staino, Alessandro & Russo, Emilio, 2020. "Nested Conditional Value-at-Risk portfolio selection: A model with temporal dependence driven by market-index volatility," European Journal of Operational Research, Elsevier, vol. 280(2), pages 741-753.
    9. Markus Leippold & Nikola Vasiljević, 2020. "Option-Implied Intrahorizon Value at Risk," Management Science, INFORMS, vol. 66(1), pages 397-414, January.
    10. P. M. Hartigan, 1985. "Computation of the Dip Statistic to Test for Unimodality," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 34(3), pages 320-325, November.
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