A one-sided Vysochanskii-Petunin inequality with financial applications. (English) Zbl 1487.60045

Summary: 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.


60E15 Inequalities; stochastic orderings
26D15 Inequalities for sums, series and integrals
91G70 Statistical methods; risk measures
Full Text: DOI


[1] Babat, O.; Vera, J. C.; Zuluaga, L. F., Computing near-optimal value-at-risk portfolios using integer programming techniques, European Journal of Operational Research, 266, 1, 304-315 (2018) · Zbl 1403.91303
[2] Barrieu, P.; Scandolo, G., Assessing financial model risk, European Journal of Operational Research, 242, 2, 546-556 (2015) · Zbl 1341.91086
[3] Basel Committee on Banking Supervision, Revisions to the Basel II market risk framework (2009), Bank for International Settlements
[4] Berkowitz, J.; Christoffersen, P.; Pelletier, D., Evaluating value-at-risk models with desk-level data, Management Science, 57, 12, 2213-2227 (2011)
[5] Cantelli, F. P., Sui confini della probabilità, Atti del Congresso Internazionale dei Matematici, 6, 47-59 (1928) · JFM 58.0544.03
[6] Chebyshev, P. L., Des valeurs moyennes, Journal de Mathématiques Pures et Appliquées, 12, 177-184 (1867)
[7] Hartigan, J. A.; Hartigan, P. M., The dip test of unimodality, Annals of Statistics, 13, 1, 70-84 (1985) · Zbl 0575.62045
[8] Jorion, P., Value at Risk: The new benchmark for managing financial risk (2001)
[9] Leippold, M.; Vasiljevic, N., Option-implied intrahorizon value at risk, Management Science, 66, 1, 397-414 (2020)
[10] (in press) · Zbl 1487.91166
[11] Meng, X.; Taylor, J. W., Estimating value-at-risk and expected shortfall using the intraday low and range data, European Journal of Operational Research, 280, 1, 191-202 (2020) · Zbl 1431.91444
[12] Mercadier, M.; Lardy, J.-P., Credit spread approximation and improvement using random forest regression, European Journal of Operational Research, 277, 1, 351-365 (2019) · Zbl 1431.91415
[13] Pukelsheim, F., The three sigma rule, The American Statistician, 48, 88-91 (1994)
[14] Shapiro, S. S.; Wilk, M. B., An analysis of variance test for normality (complete samples), Biometrika, 52, 3-4, 591-611 (1965) · Zbl 0134.36501
[15] Stahl, G., Three cheers, Risk Magazine, 10, 67-69 (1997)
[16] Staino, A.; Russo, E., Nested conditional value-at-risk portfolio selection: A model with temporal dependence driven by market-index volatility, European Journal of Operational Research, 280, 2, 741-753 (2020) · Zbl 1431.91369
[17] Taylor, J. W., Forecasting value at risk and expected shortfall using a semiparametric approach based on the asymmetric Laplace distribution, Journal of Business & Economic Statistics, 37, 1, 121-133 (2019)
[18] Vysochanskii, D.; Petunin, Y., Justification of the \(3 \sigma\) rule for unimodal distributions, Theory of Probability and Mathematical Statistics, 21, 25-36 (1980)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.