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Multi-portfolio time consistency for set-valued convex and coherent risk measures

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  • Zachary Feinstein
  • Birgit Rudloff
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    Abstract

    Equivalent characterizations of multi-portfolio time consistency are deduced for closed convex and coherent set-valued risk measures on $L_d^p(\Omega,\mathcal F_T,\mathbb P)$ with image space in the power set of $L_d^p(\Omega,\mathcal F_t,\mathbb P)$. In the convex case, multi-portfolio time consistency is equivalent to a cocycle condition on the sum of minimal penalty functions. In the coherent case, multi-portfolio time consistency is equivalent to a generalized version of stability of the dual variables. As examples, the set-valued entropic risk measure with constant risk aversion coefficient is shown to satisfy the cocycle condition for its minimal penalty functions, the set of superhedging portfolios in markets with proportional transaction costs is shown to have the stability property and in markets with convex transaction costs is shown to satisfy the composed cocycle condition, and a multi-portfolio time consistent version of the set-valued average value at risk, the composed AV@R, is given and its dual representation deduced.

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    File URL: http://arxiv.org/pdf/1212.5563
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1212.5563.

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    Date of creation: Dec 2012
    Date of revision: May 2014
    Handle: RePEc:arx:papers:1212.5563

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    9. Cheridito, Patrick & Stadje, Mitja, 2009. "Time-inconsistency of VaR and time-consistent alternatives," Finance Research Letters, Elsevier, vol. 6(1), pages 40-46, March.
    10. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
    11. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    12. Patrick Cheridito & Michael Kupper, 2011. "Composition Of Time-Consistent Dynamic Monetary Risk Measures In Discrete Time," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(01), pages 137-162.
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