IDEAS home Printed from https://ideas.repec.org/p/hal/wpaper/hal-01742629.html
   My bibliography  Save this paper

Inf-Convolution of Choquet Integrals and Applications in Optimal Risk Transfer

Author

Listed:
  • Nabil Kazi-Tani

    (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

Abstract

Motivated by reinsurance optimization, we study in this paper some particular optimal risk transfer problems, between two economic agents who do not share the same risk vision and anticipation. More precisely, we conduct an analysis of Choquet integrals, as non necessarily law invariant monetary risk measures. We first establish a new representation result of convex comonotone risk measures, then we give a representation result of Choquet integrals by introducing the notion of local distortion. This allows us to compute in an explicit manner the inf-convolution of two Choquet integrals, with examples illustrating the impact of the absence of the law invariance property.

Suggested Citation

  • Nabil Kazi-Tani, 2018. "Inf-Convolution of Choquet Integrals and Applications in Optimal Risk Transfer," Working Papers hal-01742629, HAL.
    • Handle: RePEc:hal:wpaper:hal-01742629
    Note: View the original document on HAL open archive server: https://hal.science/hal-01742629
    as

    Download full text from publisher

    File URL: https://hal.science/hal-01742629/document
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
    2. Jean-Marc Tallon & Alain Chateauneuf, 2002. "Diversification, convex preferences and non-empty core in the Choquet expected utility model," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 19(3), pages 509-523.
    3. Jean-Marc Tallon & Alain Chateauneuf, 2002. "Diversification, convex preferences and non-empty core in the Choquet expected utility model," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 19(3), pages 509-523.
    4. Chateauneuf, Alain & Dana, Rose-Anne & Tallon, Jean-Marc, 2000. "Optimal risk-sharing rules and equilibria with Choquet-expected-utility," Journal of Mathematical Economics, Elsevier, vol. 34(2), pages 191-214, October.
    5. 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.
    6. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
    7. Alain Chateauneuf & Fabio Maccheroni & Massimo Marinacci & Jean-Marc Tallon, 2005. "Monotone continuous multiple priors," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(4), pages 973-982, November.
    8. Hipp, Christian & Vogt, Michael, 2003. "Optimal Dynamic XL Reinsurance," ASTIN Bulletin, Cambridge University Press, vol. 33(2), pages 193-207, November.
    9. 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.
    10. Michael I. Taksar, 2000. "Optimal risk and dividend distribution control models for an insurance company," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 51(1), pages 1-42, February.
    11. Carlier Guillaume & Dana Rose-Anne, 2006. "Law invariant concave utility functions and optimization problems with monotonicity and comonotonicity constraints," Statistics & Risk Modeling, De Gruyter, vol. 24(1/2006), pages 1-26, July.
    12. Wang, Shaun S., 2002. "A Universal Framework for Pricing Financial and Insurance Risks," ASTIN Bulletin, Cambridge University Press, vol. 32(2), pages 213-234, November.
    13. Dow, James & Werlang, Sergio Ribeiro da Costa, 1992. "Uncertainty Aversion, Risk Aversion, and the Optimal Choice of Portfolio," Econometrica, Econometric Society, vol. 60(1), pages 197-204, January.
    14. repec:dau:papers:123456789/5392 is not listed on IDEAS
    15. Chateauneuf, Alain & Dana, Rose-Anne & Tallon, Jean-Marc, 2000. "Optimal risk-sharing rules and equilibria with Choquet-expected-utility," Journal of Mathematical Economics, Elsevier, vol. 34(2), pages 191-214, October.
    16. 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.
    17. Elyès Jouini & Walter Schachermayer & Nizar Touzi, 2006. "Law Invariant Risk Measures Have the Fatou Property," Post-Print halshs-00176522, HAL.
    18. repec:dau:papers:123456789/361 is not listed on IDEAS
    19. 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.
    20. repec:dau:papers:123456789/342 is not listed on IDEAS
    21. Jürgen Eichberger & David Kelsey, 1999. "E-Capacities and the Ellsberg Paradox," Theory and Decision, Springer, vol. 46(2), pages 107-138, April.
    22. repec:dau:papers:123456789/5461 is not listed on IDEAS
    23. Carlier, G. & Dana, R.-A., 2007. "Are generalized call-spreads efficient?," Journal of Mathematical Economics, Elsevier, vol. 43(5), pages 581-596, June.
    24. Hipp, Christian & Taksar, Michael, 2010. "Optimal non-proportional reinsurance control," Insurance: Mathematics and Economics, Elsevier, vol. 47(2), pages 246-254, October.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Denuit Michel & Dhaene Jan & Goovaerts Marc & Kaas Rob & Laeven Roger, 2006. "Risk measurement with equivalent utility principles," Statistics & Risk Modeling, De Gruyter, vol. 24(1), pages 1-25, July.
    2. Mao, Tiantian & Hu, Jiuyun & Liu, Haiyan, 2018. "The average risk sharing problem under risk measure and expected utility theory," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 170-179.
    3. Acciaio, Beatrice & Svindland, Gregor, 2009. "Optimal risk sharing with different reference probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 44(3), pages 426-433, June.
    4. Madhav Chandrasekher & Mira Frick & Ryota Iijima & Yves Le Yaouanq, 2022. "Dual‐Self Representations of Ambiguity Preferences," Econometrica, Econometric Society, vol. 90(3), pages 1029-1061, May.
    5. Knispel, Thomas & Laeven, Roger J.A. & Svindland, Gregor, 2016. "Robust optimal risk sharing and risk premia in expanding pools," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 182-195.
    6. Marcelo Brutti Righi & Marlon Ruoso Moresco, 2024. "Inf-convolution and optimal risk sharing with countable sets of risk measures," Annals of Operations Research, Springer, vol. 336(1), pages 829-860, May.
    7. Pazdera, Jaroslav & Schumacher, Johannes M. & Werker, Bas J.M., 2017. "The composite iteration algorithm for finding efficient and financially fair risk-sharing rules," Journal of Mathematical Economics, Elsevier, vol. 72(C), pages 122-133.
    8. Bensalem, Sarah & Santibáñez, Nicolás Hernández & Kazi-Tani, Nabil, 2020. "Prevention efforts, insurance demand and price incentives under coherent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 369-386.
    9. Sarah Bensalem & Nicolás Hernández Santibáñez & Nabil Kazi-Tani, 2019. "Prevention efforts, insurance demand and price incentives under coherent risk measures," Working Papers hal-01983433, HAL.
    10. Laeven, R.J.A. & Stadje, M.A., 2011. "Entropy Coherent and Entropy Convex Measures of Risk," Discussion Paper 2011-031, Tilburg University, Center for Economic Research.
    11. Mario Ghossoub & Qinghua Ren & Ruodu Wang, 2024. "Counter-monotonic risk allocations and distortion risk measures," Papers 2407.16099, arXiv.org.
    12. Tim J. Boonen & Fangda Liu & Ruodu Wang, 2021. "Competitive equilibria in a comonotone market," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 72(4), pages 1217-1255, November.
    13. Rose-Anne Dana & Cuong Le Van, 2009. "No-arbitrage, overlapping sets of priors and the existence of efficient allocations and equilibria in the presence of risk and ambiguity," Post-Print halshs-00281582, HAL.
    14. Li, Peng & Lim, Andrew E.B. & Shanthikumar, J. George, 2010. "Optimal risk transfer for agents with germs," Insurance: Mathematics and Economics, Elsevier, vol. 47(1), pages 1-12, August.
    15. Chateauneuf, Alain & Ventura, Caroline, 2010. "The no-trade interval of Dow and Werlang: Some clarifications," Mathematical Social Sciences, Elsevier, vol. 59(1), pages 1-14, January.
    16. 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.
    17. Burgert, Christian & Rüschendorf, Ludger, 2008. "Allocation of risks and equilibrium in markets with finitely many traders," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 177-188, February.
    18. Rose-Anne Dana & Cuong Le Van, 2008. "No-arbitrage, overlapping sets of priors and the existence of efficient allocations and equilibria in the presence of risk and ambiguity," Documents de travail du Centre d'Economie de la Sorbonne b08039, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne, revised Nov 2009.
    19. Araujo, Aloisio & Chateauneuf, Alain & Faro, José Heleno, 2018. "Financial market structures revealed by pricing rules: Efficient complete markets are prevalent," Journal of Economic Theory, Elsevier, vol. 173(C), pages 257-288.
    20. Liu, Peng & Wang, Ruodu & Wei, Linxiao, 2020. "Is the inf-convolution of law-invariant preferences law-invariant?," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 144-154.

    More about this item

    Keywords

    Capacity; Choquet Integrals; Risk Measures; Inf-convolution; Risk transfer;
    All these keywords.

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:wpaper:hal-01742629. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.