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


  • Zachary Feinstein


  • Birgit Rudloff



Equivalent characterizations of multi-portfolio time consistency are deduced for closed convex and coherent set-valued risk measures on $L^{p}({\varOmega,\mathcal{F},\mathbb{P}; \mathbb{R}^{d}})$ with image space in the power set of $L^{p}({\varOmega,\mathcal{F}_{t},\mathbb{P}; \mathbb{R}^{d}})$ . 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 is shown to have in markets with proportional transaction costs the stability property and to satisfy in markets with convex transaction costs 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. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • 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.
  • Handle: RePEc:spr:finsto:v:19:y:2015:i:1:p:67-107
    DOI: 10.1007/s00780-014-0247-6

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    References listed on IDEAS

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    Cited by:

    1. c{C}au{g}{i}n Ararat & Zachary Feinstein, 2019. "Set-Valued Risk Measures as Backward Stochastic Difference Inclusions and Equations," Papers 1912.06916,, revised Sep 2020.
    2. Francesca Centrone & Emanuela Rosazza Gianin, 2020. "Capital Allocation For Set-Valued Risk Measures," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 23(01), pages 1-16, February.
    3. 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.
    4. Emmanuel Lepinette, 2020. "Random optimization on random sets," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(1), pages 159-173, February.
    5. Zachary Feinstein & Birgit Rudloff, 2018. "Time consistency for scalar multivariate risk measures," Papers 1810.04978,, revised Jul 2019.
    6. Emmanuel Lepinette & Ilya Molchanov, 2016. "Risk Arbitrage and Hedging to Acceptability under Transaction Costs," Papers 1605.07884,, revised Apr 2020.
    7. Zachary Feinstein & Birgit Rudloff, 2018. "Scalar multivariate risk measures with a single eligible asset," Papers 1807.10694,, revised Jun 2020.
    8. Barigou, Karim & Chen, Ze & Dhaene, Jan, 2019. "Fair dynamic valuation of insurance liabilities: Merging actuarial judgement with market- and time-consistency," Insurance: Mathematics and Economics, Elsevier, vol. 88(C), pages 19-29.
    9. Zachary Feinstein & Birgit Rudloff, 2015. "A recursive algorithm for multivariate risk measures and a set-valued Bellman's principle," Papers 1508.02367,, revised Jul 2016.
    10. Zachary Feinstein & Birgit Rudloff, 2015. "A Supermartingale Relation for Multivariate Risk Measures," Papers 1510.05561,, revised Jan 2018.

    More about this item


    Dynamic risk measures; Transaction costs; Set-valued risk measures; Time consistency; Multi-portfolio time consistency; Stability; 91B30; 46A20; 46N10; 26E25; G32; C61; G15; G28;

    JEL classification:

    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • G15 - Financial Economics - - General Financial Markets - - - International Financial Markets
    • G28 - Financial Economics - - Financial Institutions and Services - - - Government Policy and Regulation


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