IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v322y2025i2p554-572.html
   My bibliography  Save this article

Differential quantile-based sensitivity in discontinuous models

Author

Listed:
  • Pesenti, Silvana M.
  • Millossovich, Pietro
  • Tsanakas, Andreas

Abstract

Differential sensitivity measures provide valuable tools for interpreting complex computational models, as used in applications ranging from simulation to algorithmic prediction. Taking the derivative of the model output in direction of a model parameter can reveal input–output relations and the relative importance of model parameters and input variables. Nonetheless, it is unclear how such derivatives should be taken when the model function has discontinuities and/or input variables are discrete. We present a general framework for addressing such problems, considering derivatives of quantile-based output risk measures, with respect to distortions to random input variables (risk factors), which impact the model output through step-functions. We prove that, subject to weak technical conditions, the derivatives are well-defined and we derive the corresponding formulas. We apply our results to the sensitivity analysis of compound risk models and to a numerical study of reinsurance credit risk in a multi-line insurance portfolio.

Suggested Citation

  • Pesenti, Silvana M. & Millossovich, Pietro & Tsanakas, Andreas, 2025. "Differential quantile-based sensitivity in discontinuous models," European Journal of Operational Research, Elsevier, vol. 322(2), pages 554-572.
    • Handle: RePEc:eee:ejores:v:322:y:2025:i:2:p:554-572
    DOI: 10.1016/j.ejor.2024.12.008
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0377221724009500
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.ejor.2024.12.008?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Paul Embrechts & Marius Hofert, 2013. "A note on generalized inverses," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(3), pages 423-432, June.
    2. Koike, Takaaki & Saporito, Yuri & Targino, Rodrigo, 2022. "Avoiding zero probability events when computing Value at Risk contributions," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 173-192.
    3. Zhiyong Chen & Paul Glasserman, 2008. "Sensitivity estimates for portfolio credit derivatives using Monte Carlo," Finance and Stochastics, Springer, vol. 12(4), pages 507-540, October.
    4. Andreas Tsanakas & Pietro Millossovich, 2016. "Sensitivity Analysis Using Risk Measures," Risk Analysis, John Wiley & Sons, vol. 36(1), pages 30-48, January.
    5. Lamboni, M. & Iooss, B. & Popelin, A.-L. & Gamboa, F., 2013. "Derivative-based global sensitivity measures: General links with Sobol’ indices and numerical tests," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 87(C), pages 45-54.
    6. Mathias Lindholm & Ronald Richman & Andreas Tsanakas & Mario V. Wuthrich, 2022. "A Discussion of Discrimination and Fairness in Insurance Pricing," Papers 2209.00858, arXiv.org.
    7. Guangwu Liu & Liu Jeff Hong, 2009. "Kernel estimation of quantile sensitivities," Naval Research Logistics (NRL), John Wiley & Sons, vol. 56(6), pages 511-525, September.
    8. 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.
    9. Mara, Thierry A. & Tarantola, Stefano, 2012. "Variance-based sensitivity indices for models with dependent inputs," Reliability Engineering and System Safety, Elsevier, vol. 107(C), pages 115-121.
    10. Mark Broadie & Paul Glasserman, 1996. "Estimating Security Price Derivatives Using Simulation," Management Science, INFORMS, vol. 42(2), pages 269-285, February.
    11. L. Jeff Hong, 2009. "Estimating Quantile Sensitivities," Operations Research, INFORMS, vol. 57(1), pages 118-130, February.
    12. Borgonovo, Emanuele & Ghidini, Valentina & Hahn, Roman & Plischke, Elmar, 2023. "Explaining classifiers with measures of statistical association," Computational Statistics & Data Analysis, Elsevier, vol. 182(C).
    13. Nikita Kozodoi & Johannes Jacob & Stefan Lessmann, 2021. "Fairness in Credit Scoring: Assessment, Implementation and Profit Implications," Papers 2103.01907, arXiv.org, revised Jun 2022.
    14. Rüschendorf, Ludger & de Valk, Vincent, 1993. "On regression representations of stochastic processes," Stochastic Processes and their Applications, Elsevier, vol. 46(2), pages 183-198, June.
    15. Lindholm, M. & Richman, R. & Tsanakas, A. & Wüthrich, M.V., 2022. "Discrimination-Free Insurance Pricing," ASTIN Bulletin, Cambridge University Press, vol. 52(1), pages 55-89, January.
    16. Isadora Antoniano‐Villalobos & Emanuele Borgonovo & Sumeda Siriwardena, 2018. "Which Parameters Are Important? Differential Importance Under Uncertainty," Risk Analysis, John Wiley & Sons, vol. 38(11), pages 2459-2477, November.
    17. Fissler, Tobias & Pesenti, Silvana M., 2023. "Sensitivity measures based on scoring functions," European Journal of Operational Research, Elsevier, vol. 307(3), pages 1408-1423.
    18. Lamboni, Matieyendou & Kucherenko, Sergei, 2021. "Multivariate sensitivity analysis and derivative-based global sensitivity measures with dependent variables," Reliability Engineering and System Safety, Elsevier, vol. 212(C).
    19. L. Jeff Hong & Guangwu Liu, 2009. "Simulating Sensitivities of Conditional Value at Risk," Management Science, INFORMS, vol. 55(2), pages 281-293, February.
    20. Borgonovo, Emanuele & Plischke, Elmar, 2016. "Sensitivity analysis: A review of recent advances," European Journal of Operational Research, Elsevier, vol. 248(3), pages 869-887.
    21. Michael C. Fu & L. Jeff Hong & Jian-Qiang Hu, 2009. "Conditional Monte Carlo Estimation of Quantile Sensitivities," Management Science, INFORMS, vol. 55(12), pages 2019-2027, December.
    22. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
    23. Kozodoi, Nikita & Jacob, Johannes & Lessmann, Stefan, 2022. "Fairness in credit scoring: Assessment, implementation and profit implications," European Journal of Operational Research, Elsevier, vol. 297(3), pages 1083-1094.
    24. Lamboni, Matieyendou, 2022. "Efficient dependency models: Simulating dependent random variables," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 199-217.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Francesca Centrone & Emanuela Rosazza Gianin, 2025. "Capital Allocation Rules and Generalized Collapse to the Mean: Theory and Practice," Mathematics, MDPI, vol. 13(6), pages 1-14, March.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Silvana M. Pesenti & Pietro Millossovich & Andreas Tsanakas, 2023. "Differential Quantile-Based Sensitivity in Discontinuous Models," Papers 2310.06151, arXiv.org, revised Oct 2024.
    2. Pesenti, Silvana M. & Millossovich, Pietro & Tsanakas, Andreas, 2019. "Reverse sensitivity testing: What does it take to break the model?," European Journal of Operational Research, Elsevier, vol. 274(2), pages 654-670.
    3. Makam, Vaishno Devi & Millossovich, Pietro & Tsanakas, Andreas, 2021. "Sensitivity analysis with χ2-divergences," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 372-383.
    4. Pesenti, Silvana M. & Tsanakas, Andreas & Millossovich, Pietro, 2018. "Euler allocations in the presence of non-linear reinsurance: Comment on Major (2018)," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 29-31.
    5. L. Jeff Hong & Sandeep Juneja & Jun Luo, 2014. "Estimating Sensitivities of Portfolio Credit Risk Using Monte Carlo," INFORMS Journal on Computing, INFORMS, vol. 26(4), pages 848-865, November.
    6. Isadora Antoniano‐Villalobos & Emanuele Borgonovo & Sumeda Siriwardena, 2018. "Which Parameters Are Important? Differential Importance Under Uncertainty," Risk Analysis, John Wiley & Sons, vol. 38(11), pages 2459-2477, November.
    7. Millossovich, Pietro & Tsanakas, Andreas & Wang, Ruodu, 2024. "A theory of multivariate stress testing," European Journal of Operational Research, Elsevier, vol. 318(3), pages 851-866.
    8. L. Jeff Hong & Yi Yang & Liwei Zhang, 2011. "Sequential Convex Approximations to Joint Chance Constrained Programs: A Monte Carlo Approach," Operations Research, INFORMS, vol. 59(3), pages 617-630, June.
    9. He, Zhijian, 2022. "Sensitivity estimation of conditional value at risk using randomized quasi-Monte Carlo," European Journal of Operational Research, Elsevier, vol. 298(1), pages 229-242.
    10. Guangwu Liu & L. Jeff Hong, 2011. "Kernel Estimation of the Greeks for Options with Discontinuous Payoffs," Operations Research, INFORMS, vol. 59(1), pages 96-108, February.
    11. Xi Chen & Kyoung-Kuk Kim, 2016. "Efficient VaR and CVaR Measurement via Stochastic Kriging," INFORMS Journal on Computing, INFORMS, vol. 28(4), pages 629-644, November.
    12. Huang, Zhenzhen & Kwok, Yue Kuen & Xu, Ziqing, 2024. "Efficient algorithms for calculating risk measures and risk contributions in copula credit risk models," Insurance: Mathematics and Economics, Elsevier, vol. 115(C), pages 132-150.
    13. Katja Schilling & Daniel Bauer & Marcus C. Christiansen & Alexander Kling, 2020. "Decomposing Dynamic Risks into Risk Components," Management Science, INFORMS, vol. 66(12), pages 5738-5756, December.
    14. Yijie Peng & Michael C. Fu & Bernd Heidergott & Henry Lam, 2020. "Maximum Likelihood Estimation by Monte Carlo Simulation: Toward Data-Driven Stochastic Modeling," Operations Research, INFORMS, vol. 68(6), pages 1896-1912, November.
    15. Peng, Yijie & Fu, Michael C. & Hu, Jiaqiao & L’Ecuyer, Pierre & Tuffin, Bruno, 2025. "Generalized likelihood ratio method for stochastic models with uniform random numbers as inputs," European Journal of Operational Research, Elsevier, vol. 321(2), pages 493-502.
    16. Guangxin Jiang & Michael C. Fu, 2015. "Technical Note—On Estimating Quantile Sensitivities via Infinitesimal Perturbation Analysis," Operations Research, INFORMS, vol. 63(2), pages 435-441, April.
    17. Peter W. Glynn & Yijie Peng & Michael C. Fu & Jian-Qiang Hu, 2021. "Computing Sensitivities for Distortion Risk Measures," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1520-1532, October.
    18. Bernd Heidergott & Warren Volk-Makarewicz, 2016. "A Measure-Valued Differentiation Approach to Sensitivities of Quantiles," Mathematics of Operations Research, INFORMS, vol. 41(1), pages 293-317, February.
    19. Andreas Tsanakas & Pietro Millossovich, 2016. "Sensitivity Analysis Using Risk Measures," Risk Analysis, John Wiley & Sons, vol. 36(1), pages 30-48, January.
    20. Koike, Takaaki & Saporito, Yuri & Targino, Rodrigo, 2022. "Avoiding zero probability events when computing Value at Risk contributions," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 173-192.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:ejores:v:322:y:2025:i:2:p:554-572. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/eor .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.