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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
Article
  • 97 Downloads

Abstract

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.

Keywords

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 

Notes

Acknowledgements

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.

References

  1. 1.
    Basak, S., Chabakauri, G.: Dynamic mean–variance asset allocation. Rev. Financ. Stud. 23, 2970–3016 (2010) CrossRefGoogle Scholar
  2. 2.
    Bellman, R.E.: Dynamic Programming. Princeton University Press, Princeton (1957) zbMATHGoogle Scholar
  3. 3.
    Bensoussan, A., Wong, K.C., Yam, S.C.P., Yung, S.P.: Time-consistent portfolio selection under short-selling prohibition: from discrete to continuous setting. SIAM J. Financ. Math. 5, 153–190 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Björk, T., Khapko, M., Murgoci, A.: Time inconsistent stochastic control in continuous time: theory and examples. Working Paper (2016). Available online at arXiv:1612.03650
  5. 5.
    Björk, T., Murgoci, A.: A theory of Markovian time inconsistent stochastic control in discrete time. Finance Stoch. 18, 545–592 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Björk, T., Murgoci, A., Zhou, X.Y.: Mean–variance portfolio optimization with state-dependent risk aversion. Math. Finance 24, 1–24 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cui, X.Y., Li, D., Shi, Y.: Self-coordination in time inconsistent stochastic decision problems: a planner-doer game framework. J. Econ. Dyn. Control 75, 91–113 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Czichowsky, C.: Time-consistent mean–variance portfolio selection in discrete and continuous time. Finance Stoch. 17, 227–271 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ekeland, I., Mbodj, O., Pirvu, T.A.: Time-consistent portfolio management. SIAM J. Financ. Math. 3, 1–32 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ekeland, I., Pirvu, T.A.: Investment and consumption without commitment. Math. Financ. Econ. 2, 57–86 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Forsyth, P.A., Wang, J.: Continuous time mean variance asset allocation: a time-consistent strategy. Eur. J. Oper. Res. 209, 184–201 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Harris, C., Laibson, D.: Instantaneous gratification. Q. J. Econ. 128, 205–248 (2013) CrossRefzbMATHGoogle Scholar
  13. 13.
    Karp, L.S.: Non-constant discounting in continuous time. J. Econ. Theory 132, 557–568 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kronborg, M.T., Steffensen, M.: Inconsistent investment and consumption problems. Appl. Math. Optim. 71, 473–515 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Li, D., Zhou, X.Y.: Continuous-time mean–variance portfolio selection: a stochastic LQ framework. Appl. Math. Optim. 42, 19–33 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Marín-Solano, J., Navas, J.: Consumption and portfolio rules for time-inconsistent investors. Eur. J. Oper. Res. 201, 860–872 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Markowitz, H.: Portfolio selection. J. Finance 7, 77–91 (1952) Google Scholar
  18. 18.
    Pedersen, J.L., Peskir, G.: Optimal mean–variance portfolio selection. Math. Financ. Econ. 11, 137–160 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Phelps, E.S., Pollak, R.A.: On second-best national saving and game-equilibrium growth. Rev. Econ. Stud. 35, 185–199 (1968) CrossRefGoogle Scholar
  20. 20.
    Pollak, R.A.: Consistent planning. Rev. Econ. Stud. 35, 201–208 (1968) CrossRefGoogle Scholar
  21. 21.
    Peleg, B., Yaari, M.E.: On the existence of a consistent course of action when tastes are changing. Rev. Econ. Stud. 40, 391–401 (1973) CrossRefzbMATHGoogle Scholar
  22. 22.
    Schweizer, M.: Mean–variance hedging. In: Cont, R. (ed.) Encyclopedia of Quantitative Finance, pp. 1177–1181. Wiley, New York (2010) Google Scholar
  23. 23.
    Strotz, R.H.: Myopia and inconsistency in dynamic utility maximization. Rev. Econ. Stud. 23, 165–180 (1955) CrossRefGoogle Scholar
  24. 24.
    Vigna, E.: Tail optimality and preferences consistency for intertemporal optimization problems. Working Paper, Collegio Carlo Alberto 502 (2017). Available online at https://www.carloalberto.org/assets/working-papers/no.502.pdf

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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