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Quantile-based risk sharing with heterogeneous beliefs

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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.

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Notes

  1. ES is also called CVaR, AVaR or TVaR in various contexts. In particular, CVaR is common in the optimization literature, e.g. [14] and [22, 23]. In this paper, we stick to the term ES following the risk management literature, e.g. [13] and [9].

  2. Amongst others [1, 5, 7, 12, 24] and [2] studied risk sharing problems with convex risk measures and expected utilities, different from the setting of quantile-based risk measures in this paper.

  3. Following the tradition in the literature of risk sharing, we refer to a participant in the risk sharing problem, such as an investor or a firm, as an agent.

  4. In this paper, all random future positions are already discounted, and that is why expectations correspond to prices.

  5. Roughly speaking, the first FTWE states that, under some conditions, an equilibrium allocation is Pareto-optimal, and the second FTWE states that, under some conditions, a Pareto-optimal allocation is an equilibrium allocation.

  6. Following the risk management literature, a functional \(f:\mathcal {X}\rightarrow \mathbb {R}\) is called translation invariant if \(f(X+c)=f(X)+c\) for all \(X\in \mathcal {X}\) and \(c\in \mathbb {R}\).

  7. This remark is based on a very valuable suggestion of an anonymous referee.

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Author information

Authors and Affiliations

  1. RiskLab, Department of Mathematics, ETH Zurich, Zurich, Switzerland

    Paul Embrechts

  2. Swiss Finance Institute, Geneva, Switzerland

    Paul Embrechts

  3. Department of Mathematics, Michigan State University, Michigan, USA

    Haiyan Liu

  4. Department of Statistics and Probability, Michigan State University, Michigan, USA

    Haiyan Liu

  5. Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, China

    Tiantian Mao

  6. Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Canada

    Ruodu Wang

Authors
  1. Paul Embrechts
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  2. Haiyan Liu
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  4. Ruodu Wang
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Corresponding author

Correspondence to Tiantian Mao.

Additional information

The authors thank the two anonymous referees, the Guest Editor, and the Editor for valuable comments that improved the content and the presentation of the paper. P. Embrechts would like to thank the Swiss Finance Institute for financial support. Part of this paper was written while he was Hung Hing Ying Distinguished Visiting Professor at the Department of Statistics and Actuarial Science of the University of Hong Kong. H. Liu thanks the University of Science and Technology of China for supporting her visit in Fall 2017. T. Mao was supported by the NNSF of China (Nos. 71671176, 11371340). R. Wang acknowledges support from the Natural Sciences and Engineering Research Council of Canada (RGPIN-435844-2013).

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Embrechts, P., Liu, H., Mao, T. et al. Quantile-based risk sharing with heterogeneous beliefs. Math. Program. 181, 319–347 (2020). https://doi.org/10.1007/s10107-018-1313-1

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