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Portfolio optimization under convex incentive schemes

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  • Maxim Bichuch
  • Stephan Sturm

Abstract

We consider the terminal wealth utility maximization problem from the point of view of a portfolio manager who is paid by an incentive scheme, which is given as a convex function g of the terminal wealth. The manager’s own utility function U is assumed to be smooth and strictly concave; however, the resulting utility function U∘g fails to be concave. As a consequence, the problem considered here does not fit into the classical portfolio optimization theory. Using duality theory, we prove wealth-independent existence and uniqueness of the optimal portfolio in general (incomplete) semimartingale markets as long as the unique optimizer of the dual problem has a continuous law. In many cases, this existence and uniqueness result is independent of the incentive scheme and depends only on the structure of the set of equivalent local martingale measures. As examples, we discuss (complete) one-dimensional models as well as (incomplete) lognormal mixture and popular stochastic volatility models. We also provide a detailed analysis of the case where the unique optimizer of the dual problem does not have a continuous law, leading to optimization problems whose solvability by duality methods depends on the initial wealth of the investor. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Maxim Bichuch & Stephan Sturm, 2014. "Portfolio optimization under convex incentive schemes," Finance and Stochastics, Springer, vol. 18(4), pages 873-915, October.
    • Handle: RePEc:spr:finsto:v:18:y:2014:i:4:p:873-915
    DOI: 10.1007/s00780-014-0236-9
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    Cited by:

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    3. Escobar-Anel, M. & Havrylenko, Y. & Zagst, R., 2020. "Optimal fees in hedge funds with first-loss compensation," Journal of Banking & Finance, Elsevier, vol. 118(C).
    4. Yang Liu & Zhenyu Shen, 2024. "PSAHARA Utility Family: Modeling Non-monotone Risk Aversion and Convex Compensation in Incomplete Markets," Papers 2406.00435, arXiv.org, revised Nov 2024.
    5. Thai Nguyen & Mitja Stadje, 2018. "Optimal investment for participating insurance contracts under VaR-Regulation," Papers 1805.09068, arXiv.org, revised Jul 2019.
    6. Chen, An & Hieber, Peter & Nguyen, Thai, 2019. "Constrained non-concave utility maximization: An application to life insurance contracts with guarantees," European Journal of Operational Research, Elsevier, vol. 273(3), pages 1119-1135.
    7. Zongxia Liang & Yang Liu & Litian Zhang, 2021. "A Framework of State-dependent Utility Optimization with General Benchmarks," Papers 2101.06675, arXiv.org, revised Dec 2023.
    8. Alain Bensoussan & Abel Cadenillas & Hyeng Keun Koo, 2015. "Entrepreneurial Decisions on Effort and Project with a Nonconcave Objective Function," Mathematics of Operations Research, INFORMS, vol. 40(4), pages 902-914, October.
    9. Zongxia Liang & Yang Liu & Ming Ma & Rahul Pothi Vinoth, 2021. "A Unified Formula of the Optimal Portfolio for Piecewise Hyperbolic Absolute Risk Aversion Utilities," Papers 2107.06460, arXiv.org, revised Oct 2023.
    10. John Armstrong & Damiano Brigo & Alex S. L. Tse, 2024. "The importance of dynamic risk constraints for limited liability operators," Annals of Operations Research, Springer, vol. 336(1), pages 861-898, May.
    11. Dong, Yinghui & Zheng, Harry, 2020. "Optimal investment with S-shaped utility and trading and Value at Risk constraints: An application to defined contribution pension plan," European Journal of Operational Research, Elsevier, vol. 281(2), pages 341-356.
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    13. Christian Dehm & Thai Nguyen & Mitja Stadje, 2020. "Non-concave expected utility optimization with uncertain time horizon," Papers 2005.13831, arXiv.org, revised Oct 2021.
    14. Bi, Xiuchun & Cui, Zhenyu & Fan, Jiacheng & Yuan, Lvning & Zhang, Shuguang, 2023. "Optimal investment problem under behavioral setting: A Lagrange duality perspective," Journal of Economic Dynamics and Control, Elsevier, vol. 156(C).

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    More about this item

    Keywords

    Portfolio optimization; Fund manager’s problem; Incentive scheme; Convex duality; Delegated portfolio management; 91G10; 90C26; G11;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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