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Derivative of Reduced Cumulative Distribution Function and Applications

Author

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  • Kevin Maritato

    (Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794, USA
    Department of Mathematics, Computer and Information Science, SUNY Old Westbury, Old Westbury, NY 11568, USA)

  • Stan Uryasev

    (Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794, USA)

Abstract

The reduced cumulative distribution function (rCDF) is the maximal lower bound for the cumulative distribution function (CDF). It is equivalent to the inverse of the conditional value at risk (CVaR), or one minus the buffered probability of exceedance (bPOE). This paper introduces the reduced probability density function (rPDF), the derivative of rCDF. We first explore the relation between rCDF and other risk measures. Then we describe three means of calculating rPDF for a distribution, depending on what is known about the distribution. For functions with a closed-form formula for bPOE, we derive closed-form formulae for rPDF. Further, we describe formulae for rPDF based on a numerical bPOE when there is a closed-form formula for CVaR but no closed-form formula for bPOE. Finally, we give a method for numerically calculating rPDF for an empirical distribution, and compare the results with other methods for known distributions. We conducted a case study and used rPDF for sensitivity analysis and parameter estimation with a method similar to the maximum likelihood method.

Suggested Citation

  • Kevin Maritato & Stan Uryasev, 2023. "Derivative of Reduced Cumulative Distribution Function and Applications," JRFM, MDPI, vol. 16(10), pages 1-24, October.
    • Handle: RePEc:gam:jjrfmx:v:16:y:2023:i:10:p:450-:d:1262456
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    References listed on IDEAS

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    1. Matthew Norton & Valentyn Khokhlov & Stan Uryasev, 2018. "Calculating CVaR and bPOE for Common Probability Distributions With Application to Portfolio Optimization and Density Estimation," Papers 1811.11301, arXiv.org, revised Feb 2019.
    2. Andreas Krause, 2003. "Exploring the Limitations of Value at Risk: How Good Is It in Practice?," Journal of Risk Finance, Emerald Group Publishing Limited, vol. 4(2), pages 19-28, January.
    3. 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.
    4. Rockafellar, R. Tyrrell & Uryasev, Stanislav, 2002. "Conditional value-at-risk for general loss distributions," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1443-1471, July.
    5. Zinoviy Landsman & Emiliano Valdez, 2003. "Tail Conditional Expectations for Elliptical Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(4), pages 55-71.
    6. Matthew Norton & Valentyn Khokhlov & Stan Uryasev, 2021. "Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation," Annals of Operations Research, Springer, vol. 299(1), pages 1281-1315, April.
    7. Mulvey, John M. & Erkan, Hafize G., 2006. "Applying CVaR for decentralized risk management of financial companies," Journal of Banking & Finance, Elsevier, vol. 30(2), pages 627-644, February.
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