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Merton Problem with Taxes: Characterization, computation and Approximation

Author

Listed:
  • Imen Ben Tahar

    (CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris sciences et lettres - CNRS - Centre National de la Recherche Scientifique)

  • Nizar Touzi

    (CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique - X - École polytechnique - CNRS - Centre National de la Recherche Scientifique)

  • Mete H. Soner

    (D-MATH - Department of Mathematics [ETH Zurich] - ETH Zürich - Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich])

Abstract

We formulate a computationally tractable extension of the classical Merton optimal consumptioninvestment problem to include the capital gains taxes. This is the continuous-time version of the model introduced by Dammon, Spatt, and Zhang [Rev. Financ. Stud., 14 (2001), pp. 583-616]. In this model the tax basis is computed as the average cost of the stocks in the investor's portfolio. This average rule introduces only one additional state variable, namely the tax basis. Since the other tax rules such as the first in first out rule require the knowledge of all past transactions, the average model is computationally much easier. We emphasize the linear taxation rule, which allows for tax credits when capital gains losses are experienced. In this context wash sales are optimal, and we prove it rigorously. Our main contributions are a first order explicit approximation of the value function of the problem and a unique characterization by means of the corresponding dynamic programming equation. The latter characterization builds on technical results isolated in the accompanying paper [I. Ben Tahar, H. M. Soner, and N. Touzi, SIAM J. Control Optim., 46 (2007), pp. 1779-1801]. We also suggest a numerical computation technique based on a combination of finite differences and the Howard iteration algorithm. Finally, we provide some numerical results on the welfare consequences of taxes and the quality of the first order approximation.

Suggested Citation

  • Imen Ben Tahar & Nizar Touzi & Mete H. Soner, 2010. "Merton Problem with Taxes: Characterization, computation and Approximation," Post-Print hal-00703102, HAL.
    • Handle: RePEc:hal:journl:hal-00703102
    DOI: 10.1137/080742178
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    Citations

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    Cited by:

    1. Yaoting Lei & Ya Li & Jing Xu, 2020. "Two Birds, One Stone: Joint Timing of Returns and Capital Gains Taxes," Management Science, INFORMS, vol. 66(2), pages 823-843, February.
    2. Koo, Ja Eun & Lim, Byung Hwa, 2021. "Consumption and life insurance decisions under hyperbolic discounting and taxation," Economic Modelling, Elsevier, vol. 94(C), pages 288-295.
    3. Ali Al-Aradi & Sebastian Jaimungal, 2020. "A Variational Analysis Approach to Solving the Merton Problem," Papers 2003.08450, arXiv.org.
    4. Feyzullah Egriboyun & H. Soner, 2010. "Optimal investment strategies with a reallocation constraint," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 71(3), pages 551-585, June.
    5. Jiatu Cai & Xinfu Chen & Min Dai, 2018. "Portfolio Selection with Capital Gains Tax, Recursive Utility, and Regime Switching," Management Science, INFORMS, vol. 64(5), pages 2308-2324, May.
    6. Jingtang Ma & Zhengyang Lu & Zhenyu Cui, 2022. "Delta family approach for the stochastic control problems of utility maximization," Papers 2202.12745, arXiv.org.
    7. Ren Liu & Johannes Muhle-Karbe, 2013. "Portfolio Choice with Stochastic Investment Opportunities: a User's Guide," Papers 1311.1715, arXiv.org.
    8. Baojun Bian & Xinfu Chen & Min Dai & Shuaijie Qian, 2021. "Penalty method for portfolio selection with capital gains tax," Mathematical Finance, Wiley Blackwell, vol. 31(3), pages 1013-1055, July.
    9. Martin Haugh & Garud Iyengar & Chun Wang, 2016. "Tax-Aware Dynamic Asset Allocation," Operations Research, INFORMS, vol. 64(4), pages 849-866, August.

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