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Optimal Risk Sharing for Law Invariant Monetary Utility Functions

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Author Info
Elyès Jouini () (CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - CNRS : UMR7534 - Université Paris Dauphine - Paris IX)
Walter Schachermayer (VUT - Vienna University of Technology - Technische Universität Wien)
Nizar Touzi (CREST - Centre de Recherche en Économie et Statistique - INSEE - École Nationale de la Statistique et de l'Administration Économique)

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Abstract

We consider the problem of optimal risk sharing of some given total risk between two economic agents characterized by law-invariant monetary utility functions or equivalently, law-invariant risk measures. We first prove existence of an optimal risk sharing allocation which is in addition increasing in terms of the total risk. We next provide an explicit characterization in the case where both agents' utility functions are comonotone. The general form of the optimal contracts turns out to be given by a sum of options (stop-loss contracts, in the language of insurance) on the total risk. In order to show the robustness of this type of contracts to more general utility functions, we introduce a new notion of strict risk aversion conditionally on lower tail events, which is typically satisfied by the semi-deviation and the entropic risk measures. Then, in the context of an AV@R-agent facing an agent with strict monotone preferences and exhibiting strict risk aversion conditional on lower tail events, we prove that optimal contracts again are European options on the total risk.

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Paper provided by HAL in its series Working Papers with number halshs-00176606_v1.

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Date of creation: 2007
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Handle: RePEc:hal:wpaper:halshs-00176606_v1

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Related research
Keywords: Monetary utility functions; comonotonicity; Pareto optimal allocations;

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  1. Beatrice Acciaio, 2009. "Short note on inf-convolution preserving the Fatou property," Annals of Finance, Springer, vol. 5(2), pages 281-287, March. [Downloadable!] (restricted)
  2. Michail Anthropelos & Nikolaos E. Frangos & Stylianos Z. Xanthopoulos & Athanasios N. Yannacopoulos, 2008. "On contingent claims pricing in incomplete markets: A risk sharing approach," Quantitative Finance Papers 0809.4781, arXiv.org. [Downloadable!]
  3. Michail Anthropelos & Gordan Zitkovic, 2008. "On Agents' Agreement and Partial-Equilibrium Pricing in Incomplete Markets," Quantitative Finance Papers 0803.2198, arXiv.org. [Downloadable!]
  4. Kei Fukuda & Akihiko Inoue & Yumiharu Nakano, 2007. "Optimal intertemporal risk allocation applied to insurance pricing," Quantitative Finance Papers 0711.1143, arXiv.org, revised Nov 2007. [Downloadable!]
  5. Daniel Hernandez–Hernandez & Alexander Schied, 2006. "A Control Approach to Robust Utility Maximization with Logarithmic Utility and Time-Consistent Penalties," SFB 649 Discussion Papers SFB649DP2006-061, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany. [Downloadable!]
  6. 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. [Downloadable!] (restricted)
  7. Alexander Schied & Ching-Tang Wu, 2005. "Duality theory for optimal investments under model uncertainty," SFB 649 Discussion Papers SFB649DP2005-025, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany, revised Sep 2005. [Downloadable!]
  8. Rose-Anne Dana & Cuong Le Van, 2008. "Overlapping sets of priors and the existence ofefficient allocations and equilibria for risk measures," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00308530_v1, HAL. [Downloadable!]
  9. Elyès Jouini & Walter Schachermayer & Nizar Touzi, 2006. "Law Invariant Risk Measures Have the Fatou Property," Post-Print halshs-00176522_v1, HAL. [Downloadable!]
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