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New Classes of Distortion Risk Measures and Their Estimation

Author

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  • Jungsywan H. Sepanski

    (Department of Statistics, Actuarial and Data Sciences, Central Michigan University, Mount Pleasant, MI 48859, USA)

  • Xiwen Wang

    (Citigroup, Tampa, FL 33610, USA)

Abstract

In this paper, we present a new method to construct new classes of distortion functions. A distortion function maps the unit interval to the unit interval and has the characteristics of a cumulative distribution function. The method is based on the transformation of an existing non-negative random variable whose distribution function, named the generating distribution, may contain more than one parameter. The coherency of the resulting risk measures is ensured by restricting the parameter space on which the distortion function is concave. We studied cases when the generating distributions are exponentiated exponential and Gompertz distributions. Closed-form expressions for risk measures were derived for uniform, exponential, and Lomax losses. Numerical and graphical results are presented to examine the effects of the parameter values on the risk measures. We then propose a simple plug-in estimate of risk measures and conduct simulation studies to compare and demonstrate the performance of the proposed estimates. The plug-in estimates appear to perform slightly better than the well-known L-estimates, but also suffer from biases when applied to heavy-tailed losses.

Suggested Citation

  • Jungsywan H. Sepanski & Xiwen Wang, 2023. "New Classes of Distortion Risk Measures and Their Estimation," Risks, MDPI, vol. 11(11), pages 1-21, November.
    • Handle: RePEc:gam:jrisks:v:11:y:2023:i:11:p:194-:d:1277752
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    References listed on IDEAS

    as
    1. Wei Wang & Huifu Xu, 2023. "Preference robust distortion risk measure and its application," Mathematical Finance, Wiley Blackwell, vol. 33(2), pages 389-434, April.
    2. Furman, Edward & Wang, Ruodu & Zitikis, Ričardas, 2017. "Gini-type measures of risk and variability: Gini shortfall, capital allocations, and heavy-tailed risks," Journal of Banking & Finance, Elsevier, vol. 83(C), pages 70-84.
    3. Lynn Wirch, Julia & Hardy, Mary R., 1999. "A synthesis of risk measures for capital adequacy," Insurance: Mathematics and Economics, Elsevier, vol. 25(3), pages 337-347, December.
    4. Kevin Dowd & John Cotter & Ghulam Sorwar, 2008. "Spectral Risk Measures: Properties and Limitations," Journal of Financial Services Research, Springer;Western Finance Association, vol. 34(1), pages 61-75, August.
    5. 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.
    6. Fadal A.A. Aldhufairi & Jungsywan H. Sepanski, 2020. "New families of bivariate copulas via unit weibull distortion," Journal of Statistical Distributions and Applications, Springer, vol. 7(1), pages 1-20, December.
    7. Acerbi, Carlo, 2002. "Spectral measures of risk: A coherent representation of subjective risk aversion," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1505-1518, July.
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    Cited by:

    1. Jonas Šiaulys & Sylwia Lewkiewicz & Remigijus Leipus, 2024. "The Random Effect Transformation for Three Regularity Classes," Mathematics, MDPI, vol. 12(24), pages 1-20, December.

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