Hostname: page-component-54dcc4c588-hp6zs Total loading time: 0 Render date: 2025-09-24T04:06:33.993Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal design of fixed and variable costs in peer-to-peer insurance with heterogeneous risk

Published online by Cambridge University Press:  22 September 2025

Tim J. Boonen Open the ORCID record for Tim J. Boonen [Opens in a new window] ,
Ze Chen  and
Wentao Hu Open the ORCID record for Wentao Hu [Opens in a new window]
Tim J. Boonen
Affiliation:
Department of Statistics and Actuarial Science, School of Computing and Data Science, The University of Hong Kong, Hong Kong, People’s Republic of China
Ze Chen*
Affiliation:
Nankai-Taikang College of Insurance and Actuarial Science, Nankai University, Tianjin 300350, PR China China Institute of Insuance & School Finance, Renmin University of China, Beijing, People’s Republic of China
Wentao Hu
Affiliation:
School of Finance Renmin University of China Beijing, People’s Republic of China Faculty of Economics and Business, KU Leuven, Leuven, Belgium
*
Corresponding author: Ze Chen; Email: zechen@ruc.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper examines the optimal design of peer-to-peer (P2P) insurance models, which combines outside insurance purchases with P2P risk sharing and heterogeneous risk. Participants contribute deposits to collectively cover the premium for group-based insurance against tail risks and to share uncovered losses. We analyze the cost structure by decomposing it into a fixed premium for outside coverage and a variable component for shared losses, the latter of which may be partially refunded if aggregate losses are sufficiently low. We derive closed-form solutions to the optimal sharing rule that maximizes a mean-variance objective from the perspective of a central or social planner, and we characterize its theoretical properties. Building on this foundation, we further investigate the choice of deposit for the common fund. Finally, we also provide numerical illustrations.

Keywords

Risk sharingpeer-to-peer insuranceoptimal risk pooling

Information

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , First View , pp. 1 - 26
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The International Actuarial Association

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abdikerimova, S., Boonen, T.J. and Feng, R. (2024) Multi-period peer-to-peer risk sharing. Journal of Risk and Insurance, 91(4), 943–982.CrossRefGoogle Scholar
Abdikerimova, S. and Feng, R. (2022) Peer-to-peer multi-risk insurance and mutual aid. European Journal of Operational Research, 299(2), 735–749.CrossRefGoogle Scholar
Bäuerle, N. and Glauner, A. (2018) Optimal risk allocation in reinsurance networks. Insurance: Mathematics and Economics, 82, 37–47.Google Scholar
Boonen, T.J., Tan, K.S. and Zhuang, S.C. (2016) Pricing in reinsurance bargaining with comonotonic additive utility functions. ASTIN Bulletin: The Journal of the IAA, 46(2), 507–530.CrossRefGoogle Scholar
Borch, K.K. (1960) An attempt to determine the optimum amount of stop loss reinsurance. Transactions of the 16th International Congress of Actuaries, I, 597–610.Google Scholar
Cai, J., Liu, H. and Wang, R. (2017) Pareto-optimal reinsurance arrangements under general model settings. Insurance: Mathematics and Economics, 77, 24–37.Google Scholar
Charpentier, A., Kouakou, L., Löwe, M., Ratz, P. and Vermet, F. (2022) Collaborative insurance sustainability and network structure. https://arxiv.org/abs/2107.02764.Google Scholar
Chen, S., Liu, Y. and Weng, C. (2019) Dynamic risk-sharing game and reinsurance contract design. Insurance: Mathematics and Economics, 86, 216–231.Google Scholar
Chen, Z. and Feng, R. (2021) A theory of decentralized insurance. Working paper.Google Scholar
Chen, Z., Feng, R., Hu, W. and Mao, Y. (2023) Optimal risk pooling of peer-to-peer insurance. Working paper.CrossRefGoogle Scholar
Chen, Z., Feng, R., Wei, L. and Zhao, J. (2020) Cost-effectiveness, unfairness and adverse selection in mutual aid. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3925904.Google Scholar
Cheung, K.C., Yam, S.C.P. and Zhang, Y. (2019) Risk-adjusted bowley reinsurance under distorted probabilities. Insurance: Mathematics and Economics, 86, 64–72.Google Scholar
Chi, Y., Lin, X.S. and Tan, K.S. (2017) Optimal reinsurance under the risk-adjusted value of an insurer’s liability and an economic reinsurance premium principle. North American Actuarial Journal, 21(3), 417–432.CrossRefGoogle Scholar
Clemente, G.P. and Marano, P. (2020) The broker model for peer-to-peer insurance: An analysis of its value. The Geneva Papers on Risk and Insurance-Issues and Practice, 45, 457–481.CrossRefGoogle Scholar
Denuit, M. (2019) Size-biased transform and conditional mean risk sharing, with application to p2p insurance and tontines. ASTIN Bulletin, 49(3), 591–617.CrossRefGoogle Scholar
Denuit, M. and Dhaene, J. (2012) Convex order and comonotonic conditional mean risk sharing. Insurance: Mathematics and Economics, 51(2), 265–270.Google Scholar
Denuit, M., Dhaene, J. and Robert, C.Y. (2022) Risk-sharing rules and their properties, with applications to peer-to-peer insurance. Journal of Risk and Insurance, 89(3), 615–667.CrossRefGoogle Scholar
Denuit, M. and Robert, C.Y. (2021a) Risk sharing under the dominant peer-to-peer property and casualty insurance business models. Risk Management and Insurance Review, 24(2), 181–205.CrossRefGoogle Scholar
Denuit, M. and Robert, C.Y. (2021b) Stop-loss protection for a large p2p insurance pool. Insurance: Mathematics and Economics, 100, 210–233.Google Scholar
Denuit, M. and Robert, C.Y. (2022) Collaborative insurance with stop-loss protection and team partitioning. North American Actuarial Journal, 26(1), 143–160.CrossRefGoogle Scholar
Feng, R., Liu, C. and Taylor, S. (2023) Peer-to-peer risk sharing with an application to flood risk pooling. Annals of Operations Research, 321(1-2), 813–842.CrossRefGoogle Scholar
Jiao, Z., Kou, S., Liu, Y. and Wang, R. (2023) An axiomatic theory for anonymized risk sharing. arXiv preprint arXiv: 2208.07533.Google Scholar
Levantesi, S. and Piscopo, G. (2022) Mutual peer-to-peer insurance: The allocation of risk. Journal of Co-operative Organization and Management, 10(1), 100154.CrossRefGoogle Scholar
Li, D., Rong, X. and Zhao, H. (2014) Optimal reinsurance–investment problem for maximizing the product of the insurer’s and the reinsurer’s utilities under a cev model. Journal of Computational and Applied Mathematics, 255, 671–683.CrossRefGoogle Scholar
Li, J., Fang, Z., Nie, C. and Chen, S. (2023) Optimal design for network mutual aid. Probability in the Engineering and Informational Sciences, 37(2), 567–596.CrossRefGoogle Scholar
Tang, Q., Tong, Z. and Xun, L. (2022) Portfolio risk analysis of excess of loss reinsurance. Insurance: Mathematics and Economics, 102, 91–110.Google Scholar
Von Bieberstein, F., Feess, E., Fernando, J.F., Kerzenmacher, F. and Schiller, J. (2019) Moral hazard, risk sharing, and the optimal pool size. Journal of Risk and Insurance, 86(2), 297–313.CrossRefGoogle Scholar
Zhao, Y. and Zeng, Y. (2023) Optimal commissions and subscriptions in mutual aid platforms. ASTIN Bulletin: The Journal of the IAA, 53(3), 658–683.CrossRefGoogle Scholar