Quantile-based risk sharing with heterogeneous beliefs

  • Paul Embrechts
  • Haiyan Liu
  • Tiantian Mao
  • Ruodu Wang
Full Length Paper Series B


We study risk sharing problems with quantile-based risk measures and heterogeneous beliefs, motivated by the use of internal models in finance and insurance. Explicit forms of Pareto-optimal allocations and competitive equilibria are obtained by solving various optimization problems. For Expected Shortfall (ES) agents, Pareto-optimal allocations are shown to be equivalent to equilibrium allocations, and the equilibrium pricing measure is unique. For Value-at-Risk (VaR) agents or mixed VaR and ES agents, a competitive equilibrium does not exist. Our results generalize existing ones on risk sharing problems with risk measures and belief homogeneity, and draw an interesting connection to early work on optimization properties of ES and VaR.


Risk sharing Competitive equilibrium Belief heterogeneity Quantiles Non-convexity Risk measures 

Mathematics Subject Classification

91A06 91B50 46N10 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  • Paul Embrechts
    • 1
    • 2
  • Haiyan Liu
    • 3
    • 4
  • Tiantian Mao
    • 5
  • Ruodu Wang
    • 6
  1. 1.RiskLab, Department of MathematicsETH ZurichZurichSwitzerland
  2. 2.Swiss Finance InstituteGenevaSwitzerland
  3. 3.Department of MathematicsMichigan State UniversityMichiganUSA
  4. 4.Department of Statistics and ProbabilityMichigan State UniversityMichiganUSA
  5. 5.Department of Statistics and Finance, School of ManagementUniversity of Science and Technology of ChinaHefeiChina
  6. 6.Department of Statistics and Actuarial ScienceUniversity of WaterlooWaterlooCanada

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