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Time-consistent annuitization and asset allocation under the mean-variance criterion

Part of: Mathematical finance Game theory

Published online by Cambridge University Press:  18 November 2025

Xiaoqing Liang Open the ORCID record for Xiaoqing Liang [Opens in a new window]  and
Virginia R. Young
Xiaoqing Liang*
Affiliation:
Hebei University of Technology
Virginia R. Young*
Affiliation:
University of Michigan
*
*Postal address: Department of Statistics, School of Sciences, Hebei University of Technology, Tianjin 300401, P. R. China. Email: liangxiaoqing115@hotmail.com
**Postal address: Department of Mathematics, University of Michigan, Ann Arbor, Michigan, 48109. Email: vryoung@umich.edu
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Abstract

We study two continuous-time, time-inconsistent problems for an individual who purchases life annuities and invests her wealth in a risky asset under the mean-variance criterion. In the first problem, the buyer may only purchase life annuities at a bounded, continuous rate, while in the second problem, the buyer may purchase any amount of life annuity income at any time, which results in a singular control problem. We find the individual’s time-consistent equilibrium control strategies explicitly for the two life-annuity problems by solving the corresponding extended Hamilton–Jacobi–Bellman systems of equations. We also discuss the effects of parameters on the equilibrium strategies of the two life-annuity problems.

Keywords

Life annuitiesinvestmentmean-variance criteriontime-consistent equilibrium

MSC classification

Primary: 91A15: Stochastic games
Secondary: 91G10: Portfolio theory

Information

Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 20
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Basak, S. and Chabakauri, G. (2010). Dynamic mean-variance asset allocation. Rev. Financial Studies 23, 2970–3016.10.1093/rfs/hhq028CrossRefGoogle Scholar
Bayraktar, E. and Young, V. R. (2009). Minimizing the probability of lifetime ruin with deferred life annuities. North Amer. Actuarial J. 13, 141–154.10.1080/10920277.2009.10597543CrossRefGoogle Scholar
Björk, T. and Murgoci, A. (2010). A general theory of Markovian time inconsistent stochastic control problems. Working paper, available at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1694759.Google Scholar
Björk, T., Murgoci, A. and Zhou, X. Y. (2014). Mean-variance portfolio optimization with state-dependent risk aversion. Math. Finance 24, 1–24.10.1111/j.1467-9965.2011.00515.xCrossRefGoogle Scholar
Borodin, A. N. and Salminen, P. (2012). Handbook of Brownian Motion – Facts and Formulae, 2nd edn. Birkhäuser, Basel.Google Scholar
Dai, M., Jia, Y. and Jin, H. (2024). Dynamic mean-variance portfolio selection with transaction costs. Working paper, available at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4958481.Google Scholar
Davis, M. H. A. and Norman, A. R. (1990). Portfolio selection with transaction costs. Math. Operat. Res. 15, 676–713.10.1287/moor.15.4.676CrossRefGoogle Scholar
Finkelstein, A. and Poterba, J. (2002). Selection effects in the United Kingdom individual annuities market. Econom. J. 112, 28–50.Google Scholar
Goedde-Menke, M., Lehmensiek-Starke, M. and Nolte, S. (2014). An empirical test of competing hypotheses for the annuity puzzle. J. Econom. Psychol. 43, 75–91.10.1016/j.joep.2014.04.001CrossRefGoogle Scholar
Guan, G., Liang, Z. and Ma, X. (2024). Optimal annuitization and asset allocation under linear habit formation. Insurance Math. Econom. 114, 176–191.10.1016/j.insmatheco.2023.11.007CrossRefGoogle Scholar
Landriault, D., Li, B., Li, D. and Young, V. R. (2018). Equilibrium strategies for the mean-variance investment problem over a random horizon. SIAM J. Financial Math. 9, 1046–1073.10.1137/17M1153479CrossRefGoogle Scholar
Liang, X. and Young, V. R. (2023). Annuitizing at a bounded, absolutely continuous rate to minimize the probability of lifetime ruin. Insurance Math. Econom. 112, 80–96.10.1016/j.insmatheco.2023.06.003CrossRefGoogle Scholar
Liang, Z., Luo, X. and Yuan, F. (2024). Equilibria for time-inconsistent singular control problems. SIAM J. Control. Optimization 62, 3213–3238.10.1137/23M1609701CrossRefGoogle Scholar
Markowitz, H. (1952). Portfolio selection. J. Finance 7, 77–91.Google Scholar
Milevsky, M. A. and Young, V. R. (2007). Annuitization and asset allocation. J. Econom. Dynam. Control 31, 3138–3177.10.1016/j.jedc.2006.11.003CrossRefGoogle Scholar
Protter, P. E. (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer, New York.Google Scholar
Wang, T. and Young, V. R. (2012). Maximizing the utility of consumption with commutable life annuities. Insurance Math. Econom. 51, 352–369.10.1016/j.insmatheco.2012.06.002CrossRefGoogle Scholar