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Tail Conditional Expectation for vector-valued Risks

Author

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  • Imen Bentahar

Abstract

In his paper we introduce a quantile-based risk measure for multivariate financial positions "the vector-valued Tail-conditional-expectation (TCE)". We adopt the framework proposed by Jouini, Meddeb, and Touzi [9] to deal with multi-assets portfolios when one accounts for frictions in the financial market. In this framework, the space of risks formed by essentially bounded random vectors, is endowed with some partial vector preorder >= accounting for market frictions. In a first step we provide a definition for quantiles of vector-valued risks which is compatible with the preorder >=. The TCE is then introduced as a natural extension of the "classical" real-valued tail-conditional-expectation. Our main result states that for continuous distributions TCE is equal to a coherent vector-valued risk measure. We also provide a numerical algorithm for computing vector-valued quantiles and TCE.

Suggested Citation

  • Imen Bentahar, 2006. "Tail Conditional Expectation for vector-valued Risks," SFB 649 Discussion Papers SFB649DP2006-029, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    • Handle: RePEc:hum:wpaper:sfb649dp2006-029
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    File URL: http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2006-029.pdf
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    References listed on IDEAS

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    1. Cheridito, Patrick & Delbaen, Freddy & Kupper, Michael, 2004. "Coherent and convex monetary risk measures for bounded càdlàg processes," Stochastic Processes and their Applications, Elsevier, vol. 112(1), pages 1-22, July.
    2. Elyés Jouini & Moncef Meddeb & Nizar Touzi, 2004. "Vector-valued coherent risk measures," Finance and Stochastics, Springer, vol. 8(4), pages 531-552, November.
    3. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
    4. 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.
    5. Stefan Jaschke & Uwe Küchler, 2001. "Coherent risk measures and good-deal bounds," Finance and Stochastics, Springer, vol. 5(2), pages 181-200.
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    Cited by:

    1. Klaus Herrmann & Marius Hofert & Melina Mailhot, 2017. "Multivariate Geometric Expectiles," Papers 1704.01503, arXiv.org, revised Jan 2018.
    2. Areski Cousin & Elena Di Bernadino, 2011. "On Multivariate Extensions of Value-at-Risk," Papers 1111.1349, arXiv.org, revised Apr 2013.
    3. Çağin Ararat & Andreas H. Hamel & Birgit Rudloff, 2017. "Set-Valued Shortfall And Divergence Risk Measures," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(05), pages 1-48, August.
    4. Areski Cousin & Elena Di Bernadino, 2013. "On Multivariate Extensions of Value-at-Risk," Working Papers hal-00638382, HAL.
    5. c{C}au{g}{i}n Ararat & Andreas H. Hamel & Birgit Rudloff, 2014. "Set-valued shortfall and divergence risk measures," Papers 1405.4905, arXiv.org, revised Sep 2017.
    6. Zachary Feinstein & Birgit Rudloff, 2013. "A comparison of techniques for dynamic multivariate risk measures," Papers 1305.2151, arXiv.org, revised Jan 2015.

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    More about this item

    Keywords

    Risk measures; vector-valued risk measures; coherent risk-measures; quantiles; tail-conditional-expectation;
    All these keywords.

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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