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Quantile-based risk sharing with heterogeneous beliefs

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

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.

Keywords

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

Mathematics Subject Classification

91A06 91B50 46N10 

References

  1. 1.
    Acciaio, B., Svindland, G.: Optimal risk sharing with different reference probabilities. Insur. Math. Econ. 44(3), 426–433 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Anthropelos, M., Kardaras, C.: Equilibrium in risk-sharing games. Finance Stoch. 21(3), 815–865 (2017)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    BCBS Standards: Minimum capital requirements for Market Risk. January 2016. Basel Committee on Banking Supervision. Bank for International Settlements, Basel (2016)Google Scholar
  5. 5.
    Barrieu, P., El Karoui, N.: Inf-convolution of risk measures and optimal risk transfer. Finance Stoch. 9(2), 269–298 (2005)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, H., Joslin, S., Tran, N.-K.: Rare disasters and risk sharing with heterogeneous beliefs. Rev. Financ. Stud. 25(7), 2189–2224 (2012)CrossRefGoogle Scholar
  7. 7.
    Dana, R.-A., Le Van, C.: Overlapping sets of priors and the existence of efficient allocations and equilibria for risk measures. Math. Finance 20(3), 327–339 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Embrechts, P.: A Darwinian view on internal models. J. Risk 20(1), 1–21 (2017)CrossRefGoogle Scholar
  9. 9.
    Embrechts, P., Liu, H., Wang, R.: Quantile-based risk sharing. Oper. Res. (2017). https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2744142 (forthcoming)
  10. 10.
    Embrechts, P., Wang, R.: Seven proofs for the subadditivity of expected shortfall. Depend. Modeling 3, 126–140 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Föllmer, H., Schied, A.: Stochastic Finance. An Introduction in Discrete Time, 4th edn. Walter de Gruyter, Berlin (2016)CrossRefMATHGoogle Scholar
  12. 12.
    Jouini, E., Schachermayer, W., Touzi, N.: Optimal risk sharing for law invariant monetary utility functions. Math. Finance 18(2), 269–292 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    McNeil, A.J., Frey, R., Embrechts, P.: Quantitative Risk Management: Concepts, Techniques and Tools, Revised edn. Princeton University Press, Princeton (2015)MATHGoogle Scholar
  14. 14.
    Pflug, G.C.: Some remarks on the value-at-risk and the conditional value-at-risk. In: Uryasev, S. (ed.) Probabilistic Constrained Optimization, pp. 272–281. Springer, Dordrecht (2000)CrossRefGoogle Scholar
  15. 15.
    Pflug, G.C.: Subdifferential representations of risk measures. Math. Program. Ser. B 108(2–3), 339–354 (2006)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Pflug, G.C., Pichler, A.: Multistage Stochastic Optimization. Springer, Berlin (2014)CrossRefMATHGoogle Scholar
  17. 17.
    Pflug, G.C., Pichler, A., Wozabal, D.: The \(1/n\) investment strategy is optimal under high model ambiguity. J. Bank. Finance 36(2), 410–417 (2012)CrossRefGoogle Scholar
  18. 18.
    Pflug, G.C., Pohl, M.: A review on ambiguity in stochastic portfolio optimization. Set-Valued Var. Anal. (2017).  https://doi.org/10.1007/s11228-017-0458-z Google Scholar
  19. 19.
    Pflug, G.C., Römisch, W.: Modeling, Measuring and Managing Risk. World Scientific, Singapore (2007)CrossRefMATHGoogle Scholar
  20. 20.
    Pflug, G.C., Wozabal, D.: Ambiguity in portfolio selection. Quant. Finance 7(4), 435–442 (2007)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Puccetti, G., Wang, R.: Extremal dependence concepts. Stat. Sci. 30(4), 485–517 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–42 (2000)CrossRefGoogle Scholar
  23. 23.
    Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26(7), 1443–1471 (2002)CrossRefGoogle Scholar
  24. 24.
    Rüschendorf, L.: Mathematical Risk Analysis Dependence, Risk Bounds, Optimal Allocations and Portfolios. Springer, Heidelberg (2013)MATHGoogle Scholar
  25. 25.
    Starr, R.M.: General Equilibrium Theory: An Introduction, 2nd edn. Cambridge University Press, Cambridge (2011)CrossRefMATHGoogle Scholar
  26. 26.
    Xiong, W.: Bubbles, crises, and heterogeneous beliefs. In: Fouque, J.-P., Langsam, J. (eds.) Handbook for Systemic Risk, pp. 663–713. Cambridge University Press, Cambridge (2013)CrossRefGoogle Scholar

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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