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Expectiles, Omega Ratios and Stochastic Ordering

Author

Listed:
  • Fabio Bellini

    (Università di Milano Bicocca)

  • Bernhard Klar

    (Karlsruher Institut für Technologie (KIT))

  • Alfred Müller

    (University of Siegen)

Abstract

In this paper we introduce the expectile order, defined by X ≤ e Y if e α (X) ≤e α (Y) for each α ∈ (0, 1), where e α denotes the α-expectile. We show that the expectile order is equivalent to the pointwise ordering of the Omega ratios, and we derive several necessary and sufficient conditions. In the case of equal means, the expectile order can be easily characterized by means of the stop-loss transform; in the more general case of different means we provide some sufficient conditions. In contrast with the more common stochastic orders such as ≤ s t and ≤ c x , the expectile order is not generated by a class of utility functions and is not closed with respect to convolutions. As an illustration, we compare the ≤ s t , ≤ i c x and ≤ e orders in the family of Lomax distributions and compare Lomax distributions fitted to real world data of natural disasters in the U.S. caused by different sources of weather risk like storms or floods.

Suggested Citation

  • Fabio Bellini & Bernhard Klar & Alfred Müller, 2018. "Expectiles, Omega Ratios and Stochastic Ordering," Methodology and Computing in Applied Probability, Springer, vol. 20(3), pages 855-873, September.
    • Handle: RePEc:spr:metcap:v:20:y:2018:i:3:d:10.1007_s11009-016-9527-2
    DOI: 10.1007/s11009-016-9527-2
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    References listed on IDEAS

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    Cited by:

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    2. Bellini, Fabio & Fadina, Tolulope & Wang, Ruodu & Wei, Yunran, 2022. "Parametric measures of variability induced by risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 270-284.
    3. Cascos, Ignacio & Ochoa, Maicol, 2021. "Expectile depth: Theory and computation for bivariate datasets," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    4. Bernard, Carole & Vanduffel, Steven & Ye, Jiang, 2019. "Optimal strategies under Omega ratio," European Journal of Operational Research, Elsevier, vol. 275(2), pages 755-767.
    5. Sehgal, Ruchika & Sharma, Amita & Mansini, Renata, 2023. "Worst-case analysis of Omega-VaR ratio optimization model," Omega, Elsevier, vol. 114(C).
    6. Damiano Rossello, 2022. "Performance measurement with expectiles," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 45(1), pages 343-374, June.
    7. Arab, Idir & Lando, Tommaso & Oliveira, Paulo Eduardo, 2022. "Comparison of Lp-quantiles and related skewness measures," Statistics & Probability Letters, Elsevier, vol. 183(C).
    8. Alexander Wagner & Stan Uryasev, 2019. "Portfolio Optimization with Expectile and Omega Functions," Papers 1910.14005, arXiv.org.

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