IDEAS home Printed from https://ideas.repec.org/p/cte/wsrepe/ws130101.html
   My bibliography  Save this paper

Multivariate risk measures : a constructive approach based on selections

Author

Listed:
  • Cascos Fernández, Ignacio
  • Molchanov, Ilya

Abstract

Since risky positions in multivariate portfolios can be offset by various choices of capital requirements that depend on the exchange rules and related transaction costs, it is natural to assume that the risk measures of random vectors are set-valued. Furthermore, it is reasonable to include the exchange rules in the argument of the risk and so consider risk measures of set-valued portfolios. This situation includes the classical Kabanov's transaction costs model, where the set-valued portfolio is given by the sum of a random vector and an exchange cone, but also a number of further cases of additional liquidity constraints. The definition of the selection risk measure is based on calling a set-valued portfolio acceptable if it possesses a selection with all individually acceptable marginals. The obtained risk measure is coherent (or convex), law invariant and has values being upper convex closed sets. We describe the dual representation of the selection risk measure and suggest efficient ways of approximating it from below and from above. In case of Kabanov's exchange cone model, it is shown how the selection risk measure relates to the set-valued risk measures considered by Kulikov (2008), Hamel and Heyde (2010), and Hamel et al. (2013)

Suggested Citation

  • Cascos Fernández, Ignacio & Molchanov, Ilya, 2013. "Multivariate risk measures : a constructive approach based on selections," DES - Working Papers. Statistics and Econometrics. WS ws130101, Universidad Carlos III de Madrid. Departamento de Estadística.
    • Handle: RePEc:cte:wsrepe:ws130101
    as

    Download full text from publisher

    File URL: https://e-archivo.uc3m.es/bitstream/handle/10016/16112/ws130101.pdf?sequence=1
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Alfred Galichon & Ivar Ekeland & Marc Henry, 2009. "Comonotonic measures of multivariates risks," Working Papers hal-00401828, HAL.
    2. Alexander Cherny & Pavel Grigoriev, 2007. "Dilatation monotone risk measures are law invariant," Finance and Stochastics, Springer, vol. 11(2), pages 291-298, April.
    3. repec:dau:papers:123456789/9738 is not listed on IDEAS
    4. M. Kaina & L. Rüschendorf, 2009. "On convex risk measures on L p -spaces," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(3), pages 475-495, July.
    5. Embrechts, Paul & Puccetti, Giovanni, 2006. "Bounds for functions of multivariate risks," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 526-547, February.
    6. Ignacio Cascos & Ilya Molchanov, 2007. "Multivariate risks and depth-trimmed regions," Finance and Stochastics, Springer, vol. 11(3), pages 373-397, July.
    7. 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.
    8. repec:hal:wpspec:info:hdl:2441/5rkqqmvrn4tl22s9mc4b1h6b4 is not listed on IDEAS
    9. repec:dau:papers:123456789/353 is not listed on IDEAS
    10. repec:dau:papers:123456789/2278 is not listed on IDEAS
    11. Walter Schachermayer, 2004. "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time," Mathematical Finance, Wiley Blackwell, vol. 14(1), pages 19-48, January.
    12. repec:hal:spmain:info:hdl:2441/5rkqqmvrn4tl22s9mc4b1h6b4 is not listed on IDEAS
    13. Y.M. Kabanov, 1999. "Hedging and liquidation under transaction costs in currency markets," Finance and Stochastics, Springer, vol. 3(2), pages 237-248.
    14. Elyés Jouini & Moncef Meddeb & Nizar Touzi, 2004. "Vector-valued coherent risk measures," Finance and Stochastics, Springer, vol. 8(4), pages 531-552, November.
    15. Patrick Cheridito & Tianhui Li, 2009. "Risk Measures On Orlicz Hearts," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 189-214, April.
    16. Burgert, Christian & Ruschendorf, Ludger, 2006. "Consistent risk measures for portfolio vectors," Insurance: Mathematics and Economics, Elsevier, vol. 38(2), pages 289-297, April.
    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. 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.
    2. Zachary Feinstein & Birgit Rudloff, 2013. "A comparison of techniques for dynamic multivariate risk measures," Papers 1305.2151, arXiv.org, revised Jan 2015.

    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. 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.
    2. Zachary Feinstein & Birgit Rudloff, 2013. "A comparison of techniques for dynamic multivariate risk measures," Papers 1305.2151, arXiv.org, revised Jan 2015.
    3. Emmanuel Lepinette & Ilya Molchanov, 2016. "Risk Arbitrage and Hedging to Acceptability under Transaction Costs," Papers 1605.07884, arXiv.org, revised Apr 2020.
    4. Shuo Gong & Yijun Hu & Linxiao Wei, 2022. "Risk measurement of joint risk of portfolios: a liquidity shortfall aspect," Papers 2212.04848, arXiv.org.
    5. Cosimo Munari, 2020. "Multi-utility representations of incomplete preferences induced by set-valued risk measures," Papers 2009.04151, arXiv.org.
    6. Zachary Feinstein & Birgit Rudloff, 2018. "Scalar multivariate risk measures with a single eligible asset," Papers 1807.10694, arXiv.org, revised Feb 2021.
    7. Cascos Fernández, Ignacio & Molchanov, Ilya, 2006. "Multivariate risks and depth-trimmed regions," DES - Working Papers. Statistics and Econometrics. WS ws063815, Universidad Carlos III de Madrid. Departamento de Estadística.
    8. Cousin, Areski & Di Bernardino, Elena, 2014. "On multivariate extensions of Conditional-Tail-Expectation," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 272-282.
    9. Ludger Rüschendorf, 2012. "Worst case portfolio vectors and diversification effects," Finance and Stochastics, Springer, vol. 16(1), pages 155-175, January.
    10. Zachary Feinstein & Birgit Rudloff, 2017. "A recursive algorithm for multivariate risk measures and a set-valued Bellman’s principle," Journal of Global Optimization, Springer, vol. 68(1), pages 47-69, May.
    11. Zachary Feinstein & Birgit Rudloff, 2015. "Multi-portfolio time consistency for set-valued convex and coherent risk measures," Finance and Stochastics, Springer, vol. 19(1), pages 67-107, January.
    12. Yanhong Chen & Yijun Hu, 2019. "Set-Valued Law Invariant Coherent And Convex Risk Measures," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(03), pages 1-18, May.
    13. Wei, Linxiao & Hu, Yijun, 2014. "Coherent and convex risk measures for portfolios with applications," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 114-120.
    14. Cosimo Munari, 2021. "Multi-utility representations of incomplete preferences induced by set-valued risk measures," Finance and Stochastics, Springer, vol. 25(1), pages 77-99, January.
    15. Torres, Raúl & Lillo, Rosa E. & Laniado, Henry, 2015. "A directional multivariate value at risk," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 111-123.
    16. Areski Cousin & Elena Di Bernardino, 2013. "On Multivariate Extensions of Conditional-Tail-Expectation," Working Papers hal-00877386, HAL.
    17. Zachary Feinstein & Birgit Rudloff, 2013. "Time consistency of dynamic risk measures in markets with transaction costs," Quantitative Finance, Taylor & Francis Journals, vol. 13(9), pages 1473-1489, September.
    18. Ignacio Cascos & Ilya Molchanov, 2006. "Multivariate risks and depth-trimmed regions," Papers math/0606520, arXiv.org, revised Nov 2006.
    19. 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.
    20. Zachary Feinstein & Birgit Rudloff, 2015. "A Supermartingale Relation for Multivariate Risk Measures," Papers 1510.05561, arXiv.org, revised Jan 2018.

    More about this item

    Statistics

    Access and download statistics

    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:cte:wsrepe:ws130101. 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: Ana Poveda (email available below). General contact details of provider: http://portal.uc3m.es/portal/page/portal/dpto_estadistica .

    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.