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Multivariate stress scenarios and solvency

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  • McNeil, Alexander J.
  • Smith, Andrew D.

Abstract

We show how the probabilistic concepts of half-space trimming and depth may be used to define convex scenario sets Qα for stress testing the risk factors that affect the solvency of an insurance company over a prescribed time period. By choosing the scenario in Qα which minimizes net asset value at the end of the time period, we propose the idea of the least solvent likely event (LSLE) as a solution to the forward stress testing problem. By considering the support function of the convex scenario set Qα, we establish theoretical properties of the LSLE when financial risk factors can be assumed to have a linear effect on the net assets of an insurer. In particular, we show that the LSLE may be interpreted as a scenario causing a loss equivalent to the Value-at-Risk (VaR) at confidence level α, provided the α-quantile is a subadditive risk measure on linear combinations of the risk factors. In this case, we also show that the LSLE has an interpretation as a per-unit allocation of capital to the underlying risk factors when the overall capital is determined according to the VaR. These insights allow us to define alternative scenario sets that relate in similar ways to coherent measures, such as expected shortfall. We also introduce the most likely ruin event (MLRE) as a solution to the problem of reverse stress testing.

Suggested Citation

  • McNeil, Alexander J. & Smith, Andrew D., 2012. "Multivariate stress scenarios and solvency," Insurance: Mathematics and Economics, Elsevier, vol. 50(3), pages 299-308.
  • Handle: RePEc:eee:insuma:v:50:y:2012:i:3:p:299-308
    DOI: 10.1016/j.insmatheco.2011.12.005
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    References listed on IDEAS

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    11. Kretzschmar, Gavin & McNeil, Alexander J. & Kirchner, Axel, 2010. "Integrated models of capital adequacy - Why banks are undercapitalised," Journal of Banking & Finance, Elsevier, vol. 34(12), pages 2838-2850, December.
    12. Goovaerts, Marc J. & Kaas, Rob & Laeven, Roger J.A., 2011. "Worst case risk measurement: Back to the future?," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 380-392.
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    Cited by:

    1. Daniel Grigat & Fabio Caccioli, 2017. "Reverse stress testing interbank networks," Papers 1702.08744, arXiv.org, revised Mar 2017.
    2. Koliai, Lyes, 2016. "Extreme risk modeling: An EVT–pair-copulas approach for financial stress tests," Journal of Banking & Finance, Elsevier, vol. 70(C), pages 1-22.
    3. Yi He & John H. J. Einmahl, 2017. "Estimation of extreme depth-based quantile regions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 449-461, March.
    4. Giorgi, Emanuele & McNeil, Alexander J., 2016. "On the computation of multivariate scenario sets for the skew-t and generalized hyperbolic families," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 205-220.
    5. Hofert, Marius & McNeil, Alexander J., 2015. "Subadditivity of Value-at-Risk for Bernoulli random variables," Statistics & Probability Letters, Elsevier, vol. 98(C), pages 79-88.
    6. Peter Grundke & Kamil Pliszka, 2018. "A macroeconomic reverse stress test," Review of Quantitative Finance and Accounting, Springer, vol. 50(4), pages 1093-1130, May.
    7. Colin Ellis, 2017. "Scenario-based stress tests: are they painful enough?," Contemporary Economics, University of Economics and Human Sciences in Warsaw., vol. 11(2), June.
    8. Klaus Herrmann & Marius Hofert & Melina Mailhot, 2017. "Multivariate Geometric Expectiles," Papers 1704.01503, arXiv.org, revised Jan 2018.
    9. 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.

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    More about this item

    Keywords

    Stress testing; Solvency II; Risk measures; Convex analysis; Scenario sets;
    All these keywords.

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General

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