Hostname: page-component-54dcc4c588-smtgx Total loading time: 0 Render date: 2025-10-04T02:41:00.936Z Has data issue: false hasContentIssue false
  • English
  • Français

Pareto-optimal risk exchange in a continuous-time economy: Application to target benefit pension

Published online by Cambridge University Press:  15 September 2025

Cheng Tao Open the ORCID record for Cheng Tao [Opens in a new window] ,
Yang Shen  and
Tak Kuen Siu
Cheng Tao*
Affiliation:
Nankai-Taikang College of Insurance and Actuarial Science, Nankai University, Tianjin, PR China
Yang Shen
Affiliation:
School of Risk and Actuarial Studies, UNSW Business School, University of New South Wales, Sydney, Australia
Tak Kuen Siu
Affiliation:
Department of Actuarial Studies and Business Analytics, Macquarie Business School, Macquarie University, Sydney, Australia
*
Corresponding author: Cheng Tao; Email: taocheng@nankai.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies a long-standing problem of risk exchange and optimal resource allocation among multiple entities in a continuous-time pure risk-exchange economy. We establish a novel risk exchange mechanism that allows entities to share and transfer risks dynamically over time. To achieve Pareto optimality, we formulate the problem as a stochastic control problem and derive explicit solutions for the optimal investment, consumption, and risk exchange strategies using a martingale method. To highlight practical applications of the solution to the proposed problem, we apply our results to a target benefit pension plan, featuring the potential benefits of risk sharing within this pension system. Numerical examples show the sensitivity of investment portfolios, the adjustment item, and allocation ratios to specific parameters. It is observed that an increase in the aggregate endowment process results in a rise in the adjustment item. Furthermore, the allocation ratios exhibit a positive correlation with the weights of the agents.

Keywords

Pareto-optimal risk exchangecontinuous-time economyinvestment and consumptiontarget benefit pensionlongevity risk

Information

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 55 , Special Issue 3: Risk Sharing , September 2025 , pp. 615 - 643
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

Aase, K.K. (2002) Perspectives of risk sharing. Scandinavian Actuarial Journal, 2002(2), 73–128.CrossRefGoogle Scholar
Borch, K. (1960a) Reciprocal reinsurance treaties. ASTIN Bulletin, 1(4), 170–191.CrossRefGoogle Scholar
Borch, K. (1960b) Reciprocal reinsurance treaties seen as a two-person co-operative game. Scandinavian Actuarial Journal, 1960(1-2), 29–58.CrossRefGoogle Scholar
Borch, K. (1960c) The safety loading of reinsurance premiums. Scandinavian Actuarial Journal, 1960(3-4), 163–184.CrossRefGoogle Scholar
Borch, K. (1962) Equilibrium in a reinsurance market. Econometrica, 30, 424–444.CrossRefGoogle Scholar
Bühlmann, H. and Jewell, W.S. (1979) Optimal risk exchanges. ASTIN Bulletin, 10, 243–262.CrossRefGoogle Scholar
Cachon, G.P. and Lariviere, M.A. (2005) Supply chain coordination with revenue-sharing contracts: Strengths and limitations. Management Science, 51(1), 30–44.CrossRefGoogle Scholar
Canadian Institute of Actuaries (2015) Report of the task force on target benefit plans.Google Scholar
Chen, A. and Rach, M. (2021) Current developments in German pension schemes: What are the benefits of the new target pension? European Actuarial Journal, 11(1), 21–47.CrossRefGoogle Scholar
Chen, L., Li, D., Wang, Y. and Zhu, X. (2022) The optimal cyclical design for a target benefit pension plan. Journal of Pension Economics and Finance, 22(3), 284–303.CrossRefGoogle Scholar
Chen, P., Yao, H., Yang, H. and Zhu, D. (2023) Target benefit versus defined contribution scheme: A multi-period framework. ASTIN Bulletin, 53(3), 545–579.CrossRefGoogle Scholar
Cheung, K.C., Rong, Y. and Yam, S.C.P. (2014) Borch’s theorem from the perspective of comonotonicity. Insurance: Mathematics and Economics, 54, 144–151.Google Scholar
Cox, S.H., Lin, Y. and Liu, S. (2021) Optimal longevity risk transfer and investment strategies. North American Actuarial Journal, 25(S1), S40–S65.CrossRefGoogle Scholar
Gerber, H.U. (1975) A geometric proof of Borch’s theorem. Mitteilungen/Vereinigung Schweizerischer Versicherungsmathematiker, 75, 183–187.Google Scholar
Gerber, H.U. (1978) Pareto-optimal risk exchanges and related decision problems. ASTIN Bulletin, 10, 25–33.CrossRefGoogle Scholar
Gerber, H.U. and Pafum, G. (1998) Utility functions: From risk theory to finance. North American Actuarial Journal, 2(3), 74–91.CrossRefGoogle Scholar
He, Z. (2009) Optimal executive compensation when firm size follows geometric Brownian motion. The Review of Financial Studies, 22(2), 859–892.CrossRefGoogle Scholar
Iwaki, H. (2002) An economic premium principle in a continuous-time economy. Journal of the Operations Research Society of Japan, 45(4), 346–361.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S.E. (1998) Methods of Mathematical Finance. New York: Springer.Google Scholar
Moore, M.A., Boardman, A.E. and Vining, A.R. (2013) More appropriate discounting: The rate of social time preference and the value of the social discount rate. Journal of Benefit-Cost Analysis, 4(1), 1–16.CrossRefGoogle Scholar
Parker, J.A. (2003) Consumption risk and expected stock returns. The American Economic Review, 93, 376–382.CrossRefGoogle Scholar
Rong, X., Tao, C. and Zhao, H. (2023) Target benefit pension plan with longevity risk and intergenerational equity. ASTIN Bulletin, 53, 84–103.CrossRefGoogle Scholar
Tao, C., Rong, X. and Zhao, H. (2025) Target benefit pension with longevity risk and stochastic interest rate valuation. Insurance: Mathematics and Economics, 120, 285–301.Google Scholar
Wang, S. and Lu, Y. (2019) Optimal investment strategies and risk-sharing arrangements for a hybrid pension plan. Insurance: Mathematics and Economics, 89, 46–62.Google Scholar
Wang, S., Lu, Y. and Sanders, B. (2018) Optimal investment strategies and intergenerational risk sharing for target benefit pension plans. Insurance: Mathematics and Economics, 80, 1–14.Google Scholar
Zhu, X., Hardy, M. and Saunders, D. (2021) Fair transition from defined benefit to target benefit. ASTIN Bulletin, 51(3), 873–904.CrossRefGoogle Scholar
Supplementary material: File

Tao et al. supplementary material

Tao et al. supplementary material
Download Tao et al. supplementary material(File)
File 352.4 KB