# Vector-valued coherent risk measures

## Author Info

• Elyés Jouini

()

• Moncef Meddeb

()

• Nizar Touzi

()

## Abstract

We define (d,n)-coherent risk measures as set-valued maps from $L^\infty_d$ into $\mathbb{R}^n$ satisfying some axioms. We show that this definition is a convenient extension of the real-valued risk measures introduced by Artzner et al. [2]. We then discuss the aggregation issue, i.e., the passage from $\mathbb{R}^d-$ valued random portfolio to $\mathbb{R}^n-$ valued measure of risk. Necessary and sufficient conditions of coherent aggregation are provided. Copyright Springer-Verlag Berlin/Heidelberg 2004

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## Bibliographic Info

Article provided by Springer in its journal Finance and Stochastics.

Volume (Year): 8 (2004)
Issue (Month): 4 (November)
Pages: 531-552

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Handle: RePEc:spr:finsto:v:8:y:2004:i:4:p:531-552

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## Related research

Keywords: Coherent risk measures; liquidity risk; risk aggregation;

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## Citations

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Cited by:
1. Areski Cousin & Elena Di Bernadino, 2013. "On Multivariate Extensions of Value-at-Risk," Working Papers hal-00638382, HAL.
2. Ignacio Cascos, 2006. "The Expected Convex Hull Trimmed Regions Of A Sample," Statistics and Econometrics Working Papers, Universidad Carlos III, Departamento de EstadÃ­stica y EconometrÃ­a ws066919, Universidad Carlos III, Departamento de Estadística y Econometría.
3. Ivar Ekeland & Alfred Galichon & Marc Henry, 2012. "Comonotonic measures of multivariate risks," Sciences Po publications info:hdl:2441/5rkqqmvrn4t, Sciences Po.
4. Davide La Torre & Marco Maggis, 2012. "A Goal Programming Model with Satisfaction Function for Risk Management and Optimal Portfolio Diversification," Papers 1201.1783, arXiv.org, revised Sep 2012.
5. Andreas Hamel & Andreas Löhne & Birgit Rudloff, 2014. "Benson type algorithms for linear vector optimization and applications," Journal of Global Optimization, Springer, Springer, vol. 59(4), pages 811-836, August.
6. Giuseppe Benedetti & Luciano Campi, 2011. "Multivariate utility maximization with proportional transaction costs and random endowment," Working Papers hal-00586377, HAL.
7. Cousin, Areski & Di Bernardino, Elena, 2014. "On multivariate extensions of Conditional-Tail-Expectation," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 272-282.
8. Areski Cousin & Elena Di Bernardinoy, 2013. "On Multivariate Extensions of Conditional-Tail-Expectation," Working Papers hal-00877386, HAL.
9. Andreas H. Hamel & Birgit Rudloff & Mihaela Yankova, 2012. "Set-valued average value at risk and its computation," Papers 1202.5702, arXiv.org, revised Jan 2013.
10. Areski Cousin & Elena Di Bernadino, 2011. "On Multivariate Extensions of Value-at-Risk," Papers 1111.1349, arXiv.org, revised Apr 2013.
11. Burgert, Christian & Ruschendorf, Ludger, 2006. "Consistent risk measures for portfolio vectors," Insurance: Mathematics and Economics, Elsevier, vol. 38(2), pages 289-297, April.
12. Zachary Feinstein & Birgit Rudloff, 2012. "Time consistency of dynamic risk measures in markets with transaction costs," Papers 1201.1483, arXiv.org, revised Dec 2012.
13. Ignacio Cascos & Ilya Molchanov, 2006. "Multivariate Risks And Depth-Trimmed Regions," Statistics and Econometrics Working Papers, Universidad Carlos III, Departamento de EstadÃ­stica y EconometrÃ­a ws063815, Universidad Carlos III, Departamento de Estadística y Econometría.
14. repec:spo:wpecon:info:hdl:2441/5rkqqmvrn4tl22s9mc4b1h6b4 is not listed on IDEAS
15. Ignacio Cascos & Ilya Molchanov, 2006. "Multivariate risks and depth-trimmed regions," Papers math/0606520, arXiv.org, revised Nov 2006.
16. Bakshi, Gurdip & Panayotov, George, 2010. "First-passage probability, jump models, and intra-horizon risk," Journal of Financial Economics, Elsevier, Elsevier, vol. 95(1), pages 20-40, January.
17. Wei, Linxiao & Hu, Yijun, 2014. "Coherent and convex risk measures for portfolios with applications," Statistics & Probability Letters, Elsevier, Elsevier, vol. 90(C), pages 114-120.
18. \c{C}a\u{g}\in Ararat & Andreas H. Hamel & Birgit Rudloff, 2014. "Set-valued shortfall and divergence risk measures," Papers 1405.4905, arXiv.org.
19. Cousin, Areski & Di Bernardino, Elena, 2013. "On multivariate extensions of Value-at-Risk," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 119(C), pages 32-46.
20. Burren, Daniel, 2013. "Insurance demand and welfare-maximizing risk capital—Some hints for the regulator in the case of exponential preferences and exponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 551-568.
21. Imen Bentahar, 2006. "Tail Conditional Expectation for vector-valued Risks," SFB 649 Discussion Papers SFB649DP2006-029, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.

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