IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v75y2017icp180-188.html
   My bibliography  Save this article

The fundamental theorem of mutual insurance

Author

Listed:
  • Albrecht, Peter
  • Huggenberger, Markus

Abstract

The essence of mutual insurance is the notion that re-distributing risk in a pool of risks is more beneficial than taking the risk alone. Interpreting ‘more beneficial’ as an increase in utility and considering sequences of exchangeable risks, we are able to formalize this notion from the policyholder’s perspective and demonstrate its validity for various alternative preference functionals (e.g., expected utility, Choquet expected utility, and distortion risk measures). To obtain this result, we exploit that for a sequence of exchangeable risks the corresponding sequence of arithmetical averages is a reversed martingale.

Suggested Citation

  • Albrecht, Peter & Huggenberger, Markus, 2017. "The fundamental theorem of mutual insurance," Insurance: Mathematics and Economics, Elsevier, vol. 75(C), pages 180-188.
    • Handle: RePEc:eee:insuma:v:75:y:2017:i:c:p:180-188
    DOI: 10.1016/j.insmatheco.2017.06.002
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668716304978
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.insmatheco.2017.06.002?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Ulrich Schmidt & Horst Zank, 2008. "Risk Aversion in Cumulative Prospect Theory," Management Science, INFORMS, vol. 54(1), pages 208-216, January.
    2. Yaari, Menahem E, 1987. "The Dual Theory of Choice under Risk," Econometrica, Econometric Society, vol. 55(1), pages 95-115, January.
    3. Alain Chateauneuf & Michèle Cohen, 2008. "Cardinal extensions of EU model based on the Choquet integral," Documents de travail du Centre d'Economie de la Sorbonne v08087, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    4. Tversky, Amos & Kahneman, Daniel, 1992. "Advances in Prospect Theory: Cumulative Representation of Uncertainty," Journal of Risk and Uncertainty, Springer, vol. 5(4), pages 297-323, October.
    5. Hong, Chew Soo & Karni, Edi & Safra, Zvi, 1987. "Risk aversion in the theory of expected utility with rank dependent probabilities," Journal of Economic Theory, Elsevier, vol. 42(2), pages 370-381, August.
    6. Denneberg, Dieter, 1990. "Premium Calculation: Why Standard Deviation Should be Replaced by Absolute Deviation1," ASTIN Bulletin, Cambridge University Press, vol. 20(2), pages 181-190, November.
    7. Quiggin, John, 1982. "A theory of anticipated utility," Journal of Economic Behavior & Organization, Elsevier, vol. 3(4), pages 323-343, December.
    8. Bühlmann, Hans & Jewell, William S., 1979. "Optimal Risk Exchanges," ASTIN Bulletin, Cambridge University Press, vol. 10(3), pages 243-262, December.
    9. MOSSIN, Jan, 1968. "Aspects of rational insurance purchasing," LIDAM Reprints CORE 23, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    10. Powers, Michael R. & Venezian, Emilio C. & Juca, Iana B., 2003. "Of happy and hapless regulators: the asymptotics of ruin," Insurance: Mathematics and Economics, Elsevier, vol. 32(2), pages 317-330, April.
    11. Bryan R. Routledge & Stanley E. Zin, 2010. "Generalized Disappointment Aversion and Asset Prices," Journal of Finance, American Finance Association, vol. 65(4), pages 1303-1332, August.
    12. Acerbi, Carlo, 2002. "Spectral measures of risk: A coherent representation of subjective risk aversion," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1505-1518, July.
    13. Gul, Faruk, 1991. "A Theory of Disappointment Aversion," Econometrica, Econometric Society, vol. 59(3), pages 667-686, May.
    14. Haim Levy, 1992. "Stochastic Dominance and Expected Utility: Survey and Analysis," Management Science, INFORMS, vol. 38(4), pages 555-593, April.
    15. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
    16. Labuschagne, Coenraad C.A. & Offwood, Theresa M., 2010. "A note on the connection between the Esscher-Girsanov transform and the Wang transform," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 385-390, December.
    17. Rothschild, Michael & Stiglitz, Joseph E., 1971. "Increasing risk II: Its economic consequences," Journal of Economic Theory, Elsevier, vol. 3(1), pages 66-84, March.
    18. repec:bla:scandj:v:87:y:1985:i:3:p:463-73 is not listed on IDEAS
    19. Nadine Gatzert, 2012. "The merits of pooling claims revisited," Journal of Risk Finance, Emerald Group Publishing, vol. 13(3), pages 184-198, May.
    20. 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.
    21. R. Rockafellar & Stan Uryasev & Michael Zabarankin, 2006. "Generalized deviations in risk analysis," Finance and Stochastics, Springer, vol. 10(1), pages 51-74, January.
    22. De Giorgi, Enrico, 2005. "Reward-risk portfolio selection and stochastic dominance," Journal of Banking & Finance, Elsevier, vol. 29(4), pages 895-926, April.
    23. Goovaerts, Marc J. & Laeven, Roger J.A., 2008. "Actuarial risk measures for financial derivative pricing," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 540-547, April.
    24. Manesh, Sirous Fathi & Khaledi, Baha-Eldin & Dhaene, Jan, 2016. "Optimal allocation of policy deductibles for exchangeable risks," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 87-92.
    25. Hans Follmer & Alexander Schied, 2013. "Probabilistic aspects of finance," Papers 1309.7759, arXiv.org.
    26. Wang, Shaun S. & Young, Virginia R. & Panjer, Harry H., 1997. "Axiomatic characterization of insurance prices," Insurance: Mathematics and Economics, Elsevier, vol. 21(2), pages 173-183, November.
    27. A. Cherny, 2006. "Weighted V@R and its Properties," Finance and Stochastics, Springer, vol. 10(3), pages 367-393, September.
    28. Georges Dionne (ed.), 2013. "Handbook of Insurance," Springer Books, Springer, edition 2, number 978-1-4614-0155-1, December.
    29. 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.
    30. 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.
    31. Goovaerts, Marc J. & Kaas, Rob & Laeven, Roger J.A., 2010. "Decision principles derived from risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 294-302, December.
    32. Rothschild, Michael & Stiglitz, Joseph E., 1970. "Increasing risk: I. A definition," Journal of Economic Theory, Elsevier, vol. 2(3), pages 225-243, September.
    33. Pelsser, Antoon, 2008. "On the Applicability of the Wang Transform for Pricing Financial Risks," ASTIN Bulletin, Cambridge University Press, vol. 38(1), pages 171-181, May.
    34. Cossette, Helene & Marceau, Etienne, 2000. "The discrete-time risk model with correlated classes of business," Insurance: Mathematics and Economics, Elsevier, vol. 26(2-3), pages 133-149, May.
    35. Denuit, Michel & Vermandele, Catherine, 1998. "Optimal reinsurance and stop-loss order," Insurance: Mathematics and Economics, Elsevier, vol. 22(3), pages 229-233, July.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Michel Denuit & Jan Dhaene & Christian Y. Robert, 2022. "Risk‐sharing rules and their properties, with applications to peer‐to‐peer insurance," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 89(3), pages 615-667, September.
    2. Michel Denuit & Christian Y. Robert, 2021. "Risk sharing under the dominant peer‐to‐peer property and casualty insurance business models," Risk Management and Insurance Review, American Risk and Insurance Association, vol. 24(2), pages 181-205, June.
    3. Markus Huggenberger & Peter Albrecht, 2022. "Risk pooling and solvency regulation: A policyholder's perspective," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 89(4), pages 907-950, December.
    4. Denuit, Michel & Robert, Christian Y., 2020. "Risk reduction by conditional mean risk sharing with application to collaborative insurance," LIDAM Discussion Papers ISBA 2020024, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Matthias Nadler & Felix Bekemeier & Fabian Schar, 2022. "DeFi Risk Transfer: Towards A Fully Decentralized Insurance Protocol," Papers 2212.10308, arXiv.org.
    6. Schmeiser, Hato & Orozco-Garcia, Carolina, 2021. "The merits of pooling claims: Mutual vs. stock insurers," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 92-104.
    7. Denuit, Michel & Robert, Christian Y., 2021. "Risk sharing under the dominant peer-to-peer property and casualty insurance business models," LIDAM Discussion Papers ISBA 2021001, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    8. Boonen, Tim J., 2019. "Equilibrium recoveries in insurance markets with limited liability," Journal of Mathematical Economics, Elsevier, vol. 85(C), pages 38-45.
    9. Roland Eisen, 2021. "Vulnerability and mutual insurance," The Geneva Papers on Risk and Insurance - Issues and Practice, Palgrave Macmillan;The Geneva Association, vol. 46(2), pages 224-235, April.

    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. Markus Huggenberger & Peter Albrecht, 2022. "Risk pooling and solvency regulation: A policyholder's perspective," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 89(4), pages 907-950, December.
    2. Xia Han & Ruodu Wang & Xun Yu Zhou, 2022. "Choquet regularization for reinforcement learning," Papers 2208.08497, arXiv.org.
    3. Boonen, Tim J., 2017. "Risk Redistribution Games With Dual Utilities," ASTIN Bulletin, Cambridge University Press, vol. 47(1), pages 303-329, January.
    4. Holly Brannelly & Andrea Macrina & Gareth W. Peters, 2021. "Stochastic measure distortions induced by quantile processes for risk quantification and valuation," Papers 2201.02045, arXiv.org.
    5. Louis R. Eeckhoudt & Roger J. A. Laeven, 2021. "Probability Premium and Attitude Towards Probability," Papers 2105.00054, arXiv.org.
    6. Ghossoub, Mario, 2019. "Budget-constrained optimal insurance without the nonnegativity constraint on indemnities," Insurance: Mathematics and Economics, Elsevier, vol. 84(C), pages 22-39.
    7. Zhu, Li & Li, Haijun, 2012. "Tail distortion risk and its asymptotic analysis," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 115-121.
    8. Ruodu Wang & Yunran Wei & Gordon E. Willmot, 2020. "Characterization, Robustness, and Aggregation of Signed Choquet Integrals," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 993-1015, August.
    9. Leitner, Johannes, 2005. "Dilatation monotonous Choquet integrals," Journal of Mathematical Economics, Elsevier, vol. 41(8), pages 994-1006, December.
    10. Zvi Safra & Uzi Segal, 2005. "Are Universal Preferences Possible? Calibration Results for Non-Expected Utility Theories," Boston College Working Papers in Economics 633, Boston College Department of Economics.
    11. Trabelsi, Mohamed Ali, 2006. "Les nouveaux modèles de décision dans le risque et l’incertain : quel apport ? [The new models of decision under risk or uncertainty : What approach?]," MPRA Paper 25442, University Library of Munich, Germany.
    12. Peter Brooks & Simon Peters & Horst Zank, 2014. "Risk behavior for gain, loss, and mixed prospects," Theory and Decision, Springer, vol. 77(2), pages 153-182, August.
    13. Belles-Sampera, Jaume & Merigó, José M. & Guillén, Montserrat & Santolino, Miguel, 2013. "The connection between distortion risk measures and ordered weighted averaging operators," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 411-420.
    14. Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci, 2017. "Stochastic Dominance Analysis Without the Independence Axiom," Management Science, INFORMS, vol. 63(4), pages 1097-1109, April.
    15. Cillo, Alessandra & Delquié, Philippe, 2014. "Mean-risk analysis with enhanced behavioral content," European Journal of Operational Research, Elsevier, vol. 239(3), pages 764-775.
    16. Phillips Peter J. & Pohl Gabriela, 2018. "The Deferral of Attacks: SP/A Theory as a Model of Terrorist Choice when Losses Are Inevitable," Open Economics, De Gruyter, vol. 1(1), pages 71-85, February.
    17. Wakker, Peter P. & Yang, Jingni, 2021. "Concave/convex weighting and utility functions for risk: A new light on classical theorems," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 429-435.
    18. Trabelsi, Mohamed Ali, 2006. "Les nouveaux modèles de décision dans le risque et l’incertain : quel apport ? [The new models of decision under risk or uncertainty : What approach?]," MPRA Paper 25442, University Library of Munich, Germany.
    19. Tiantian Mao & Jun Cai, 2018. "Risk measures based on behavioural economics theory," Finance and Stochastics, Springer, vol. 22(2), pages 367-393, April.
    20. Jean Baccelli, 2018. "Risk attitudes in axiomatic decision theory: a conceptual perspective," Theory and Decision, Springer, vol. 84(1), pages 61-82, January.

    More about this item

    Keywords

    Pooling risks; Exchangeability; Reversed martingales; Choquet expected utility; Distortion risk measures;
    All these keywords.

    JEL classification:

    • D - Microeconomics
    • G - Financial Economics

    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:eee:insuma:v:75:y:2017:i:c:p:180-188. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/505554 .

    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.