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Risk sharing for capital requirements with multidimensional security markets

Author

Listed:
  • Felix-Benedikt Liebrich

    (University of Munich)

  • Gregor Svindland

    (University of Munich)

Abstract

We consider the risk sharing problem for capital requirements induced by capital adequacy tests and security markets. The agents involved in the sharing procedure may be heterogeneous in that they apply varying capital adequacy tests and have access to different security markets. We discuss conditions under which there exists a representative agent. Thereafter, we study two frameworks of capital adequacy more closely, namely polyhedral constraints and distribution-based constraints. We prove existence of optimal risk allocations and equilibria within these frameworks and elaborate on their robustness.

Suggested Citation

  • Felix-Benedikt Liebrich & Gregor Svindland, 2019. "Risk sharing for capital requirements with multidimensional security markets," Finance and Stochastics, Springer, vol. 23(4), pages 925-973, October.
    • Handle: RePEc:spr:finsto:v:23:y:2019:i:4:d:10.1007_s00780-019-00402-6
    DOI: 10.1007/s00780-019-00402-6
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    References listed on IDEAS

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    1. Claudia Ravanelli & Gregor Svindland, 2014. "Comonotone Pareto optimal allocations for law invariant robust utilities on L 1," Finance and Stochastics, Springer, vol. 18(1), pages 249-269, January.
    2. Liebrich, Felix-Benedikt & Svindland, Gregor, 2017. "Model spaces for risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 77(C), pages 150-165.
    3. Damir Filipović & Michael Kupper, 2008. "Equilibrium Prices For Monetary Utility Functions," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(03), pages 325-343.
    4. Walter Farkas & Pablo Koch-Medina & Cosimo Munari, 2012. "Capital requirements with defaultable securities," Papers 1203.4610, arXiv.org, revised Jan 2014.
    5. Dana, Rose-Anne & Le Van, Cuong & Magnien, Francois, 1999. "On the Different Notions of Arbitrage and Existence of Equilibrium," Journal of Economic Theory, Elsevier, vol. 87(1), pages 169-193, July.
    6. repec:dau:papers:123456789/6207 is not listed on IDEAS
    7. Patrick Cheridito & Michael Kupper & Ludovic Tangpi, 2016. "Duality formulas for robust pricing and hedging in discrete time," Papers 1602.06177, arXiv.org, revised Sep 2017.
    8. E. Jouini & W. Schachermayer & N. Touzi, 2008. "Optimal Risk Sharing For Law Invariant Monetary Utility Functions," Mathematical Finance, Wiley Blackwell, vol. 18(2), pages 269-292, April.
    9. Farkas, Walter & Koch-Medina, Pablo & Munari, Cosimo, 2014. "Capital requirements with defaultable securities," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 58-67.
    10. Acciaio, Beatrice & Svindland, Gregor, 2009. "Optimal risk sharing with different reference probabilities," LSE Research Online Documents on Economics 50119, London School of Economics and Political Science, LSE Library.
    11. Pauline Barrieu & Nicole El Karoui, 2005. "Inf-convolution of risk measures and optimal risk transfer," Finance and Stochastics, Springer, vol. 9(2), pages 269-298, April.
    12. Niushan Gao & Denny Leung & Cosimo Munari & Foivos Xanthos, 2018. "Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces," Finance and Stochastics, Springer, vol. 22(2), pages 395-415, April.
    13. Tsanakas, Andreas, 2009. "To split or not to split: Capital allocation with convex risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 268-277, April.
    14. Dana, Rose-Anne & Le Van, Cuong, 2000. "Arbitrage, duality and asset equilibria," Journal of Mathematical Economics, Elsevier, vol. 34(3), pages 397-413, November.
    15. Walter Farkas & Pablo Koch-Medina & Cosimo Munari, 2013. "Beyond Cash-Additive Risk Measures: When Changing the Numeraire Fails," Swiss Finance Institute Research Paper Series 13-67, Swiss Finance Institute.
    16. Felix-Benedikt Liebrich & Gregor Svindland, 2017. "Model Spaces for Risk Measures," Papers 1703.01137, arXiv.org, revised Nov 2017.
    17. 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.
    18. Acciaio, Beatrice & Svindland, Gregor, 2009. "Optimal risk sharing with different reference probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 44(3), pages 426-433, June.
    19. repec:dau:papers:123456789/361 is not listed on IDEAS
    20. Barrieu, Pauline & El Karoui, Nicole, 2005. "Inf-convolution of risk measures and optimal risk transfer," LSE Research Online Documents on Economics 2829, London School of Economics and Political Science, LSE Library.
    21. Damir Filipović & Gregor Svindland, 2008. "Optimal capital and risk allocations for law- and cash-invariant convex functions," Finance and Stochastics, Springer, vol. 12(3), pages 423-439, July.
    22. Ruodu Wang, 2016. "Regulatory arbitrage of risk measures," Quantitative Finance, Taylor & Francis Journals, vol. 16(3), pages 337-347, March.
    23. Mas-Colell, Andreu & Zame, William R., 1991. "Equilibrium theory in infinite dimensional spaces," Handbook of Mathematical Economics, in: W. Hildenbrand & H. Sonnenschein (ed.), Handbook of Mathematical Economics, edition 1, volume 4, chapter 34, pages 1835-1898, Elsevier.
    24. Susanne Klöppel & Martin Schweizer, 2007. "Dynamic Indifference Valuation Via Convex Risk Measures," Mathematical Finance, Wiley Blackwell, vol. 17(4), pages 599-627, October.
    25. Paul Embrechts & Haiyan Liu & Ruodu Wang, 2017. "Quantile-Based Risk Sharing," Swiss Finance Institute Research Paper Series 17-54, Swiss Finance Institute.
    26. Patrick Cheridito & Tianhui Li, 2009. "Risk Measures On Orlicz Hearts," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 189-214, April.
    27. Pichler, Alois, 2013. "The natural Banach space for version independent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 405-415.
    28. Shengzhong Chen & Niushan Gao & Foivos Xanthos, 2018. "The strong Fatou property of risk measures," Papers 1805.05259, arXiv.org.
    29. repec:dau:papers:123456789/6228 is not listed on IDEAS
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    Citations

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    Cited by:

    1. Marcelo Brutti Righi & Marlon Ruoso Moresco, 2020. "Inf-convolution and optimal risk sharing with countable sets of risk measures," Papers 2003.05797, arXiv.org, revised Mar 2022.
    2. Felix-Benedikt Liebrich, 2021. "Risk sharing under heterogeneous beliefs without convexity," Papers 2108.05791, arXiv.org, revised May 2022.
    3. Laudagé, Christian & Sass, Jörn & Wenzel, Jörg, 2022. "Combining multi-asset and intrinsic risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 254-269.
    4. Maria Arduca & Cosimo Munari, 2021. "Risk measures beyond frictionless markets," Papers 2111.08294, arXiv.org.
    5. Felix-Benedikt Liebrich & Cosimo Munari, 2022. "Law-Invariant Functionals that Collapse to the Mean: Beyond Convexity," Mathematics and Financial Economics, Springer, volume 16, number 2, June.
    6. Matteo Burzoni & Alessandro Doldi & Enea Monzio Compagnoni, 2022. "Risk Sharing with Deep Neural Networks," Papers 2212.11752, arXiv.org, revised Jun 2023.
    7. Bernard, C. & De Gennaro Aquino, L. & Vanduffel, S., 2023. "Optimal multivariate financial decision making," European Journal of Operational Research, Elsevier, vol. 307(1), pages 468-483.
    8. Alessandro Doldi & Marco Frittelli, 2021. "Real-Valued Systemic Risk Measures," Mathematics, MDPI, vol. 9(9), pages 1-24, April.
    9. Felix-Benedikt Liebrich & Cosimo Munari, 2021. "Law-invariant functionals that collapse to the mean: Beyond convexity," Papers 2106.01281, arXiv.org, revised Jul 2021.
    10. Ettlin, Nicolas & Farkas, Walter & Kull, Andreas & Smirnow, Alexander, 2020. "Optimal risk-sharing across a network of insurance companies," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 39-47.
    11. Sascha Desmettre & Christian Laudagé & Jörn Sass, 2020. "Good-Deal Bounds for Option Prices under Value-at-Risk and Expected Shortfall Constraints," Risks, MDPI, vol. 8(4), pages 1-22, October.

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    More about this item

    Keywords

    Capital requirements; Polyhedral acceptance sets; Law-invariant acceptance sets; Multidimensional security spaces; Pareto-optimal risk allocations; Equilibria; Robustness of optimal allocations;
    All these keywords.

    JEL classification:

    • D52 - Microeconomics - - General Equilibrium and Disequilibrium - - - Incomplete Markets
    • D53 - Microeconomics - - General Equilibrium and Disequilibrium - - - Financial Markets
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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