Finance and Stochastics

, Volume 23, Issue 1, pp 173–207 | Cite as

A paradox in time-consistency in the mean–variance problem?

  • Alain Bensoussan
  • Kwok Chuen Wong
  • Sheung Chi Phillip Yam


We establish new conditions under which a constrained (no short-selling) time-consistent equilibrium strategy, starting at a certain time, will beat the unconstrained counterpart, as measured by the magnitude of their corresponding equilibrium mean–variance value functions. We further show that the pure strategy of solely investing in a risk-free bond can sometimes simultaneously dominate both constrained and unconstrained equilibrium strategies. With numerical experiments, we also illustrate that the constrained strategy can dominate the unconstrained one for most of the commencement dates (even more than 90%) of a prescribed planning horizon. Under a precommitment approach, the value function of an investor increases with the size of the admissible sets of strategies. However, this may fail to be true under the game-theoretic paradigm, as the constraint of time-consistency itself affects the value function differently when short-selling is and is not prohibited.


Time-consistency Mean–variance State-dependent risk-aversion Equilibrium strategy Short-selling prohibition 

Mathematics Subject Classification (2010)

60J25 91G10 91G80 

JEL Classification

C72 C73 G11 



We are very grateful to various participants for their valuable discussions and comments after the talks based on the present work. We thank our colleague John Wright for his suggestions on enriching the presentation of our paper. We express our sincere gratitude to the editors and anonymous referees for their very useful suggestions and inspiring comments which much enhanced our article. The first author acknowledges the financial support from the National Science Foundation under grant DMS-1612880, and the Research Grant Council of Hong Kong Special Administrative Region under grant GRF 11303316. The second author acknowledges the financial support from ERC (279582) and SFI (16/IA/4443,16/SPP/3347), and the present work constitutes a part of his work for his postgraduate dissertation. The third author acknowledges the financial support from HKGRF-14300717 with the project title: New Kinds of Forward–Backward Stochastic Systems with Applications, HKSAR-GRF-14301015 with title: Advance in Mean Field Theory, Direct Grant for Research 2014/15 with project code: 4053141 offered by CUHK. He also thanks Columbia University for the kind invitation to be a visiting faculty member in the Department of Statistics during his sabbatical leave. The third author also recalls the unforgettable moments and the happiness shared with his beloved father during the drafting of the present article at their home. Although he just lost his father with the deepest sadness at the final stage of the review of this work, his father will never leave the heart of Phillip Yam; and he used this work in memory of his father’s brave battle against liver cancer.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Alain Bensoussan
    • 1
    • 2
  • Kwok Chuen Wong
    • 3
  • Sheung Chi Phillip Yam
    • 4
  1. 1.International Center for Decision and Risk Analysis, Naveen Jindal School of ManagementThe University of Texas at DallasRichardsonUSA
  2. 2.Department of Systems Engineering and Engineering ManagementCity University of Hong KongHong KongChina
  3. 3.School of Mathematical SciencesDublin City UniversityDublinIreland
  4. 4.Department of StatisticsChinese University of Hong KongHong KongChina

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