IDEAS home Printed from https://ideas.repec.org/p/hal/wpaper/hal-01923917.html
   My bibliography  Save this paper

Bayesian learning for the Markowitz portfolio selection problem

Author

Listed:
  • Carmine de Franco

    (OSSIAM)

  • Johann Nicolle

    (LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique, OSSIAM)

  • Huyên Pham

    (LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique, CREST - Centre de Recherche en Economie et en Statistique - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique)

Abstract

We study the Markowitz portfolio selection problem with unknown drift vector in the multidimensional framework. The prior belief on the uncertain expected rate of return is modeled by an arbitrary probability law, and a Bayesian approach from filtering theory is used to learn the posterior distribution about the drift given the observed market data of the assets. The Bayesian Markowitz problem is then embedded into an auxiliary standard control problem that we characterize by a dynamic programming method and prove the existence and uniqueness of a smooth solution to the related semi-linear partial differential equation (PDE). The optimal Markowitz portfolio strategy is explicitly computed in the case of a Gaussian prior distribution. Finally, we measure the quantitative impact of learning, updating the strategy from observed data, compared to non-learning, using a constant drift in an uncertain context, and analyze the sensitivity of the value of information w.r.t. various relevant parameters of our model.

Suggested Citation

  • Carmine de Franco & Johann Nicolle & Huyên Pham, 2018. "Bayesian learning for the Markowitz portfolio selection problem," Working Papers hal-01923917, HAL.
    • Handle: RePEc:hal:wpaper:hal-01923917
    DOI: 10.1142/S0219024919500377
    Note: View the original document on HAL open archive server: https://hal.science/hal-01923917v1
    as

    Download full text from publisher

    File URL: https://hal.science/hal-01923917v1/document
    Download Restriction: no
    File URL: https://libkey.io/10.1142/S0219024919500377?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Lakner, Peter, 1998. "Optimal trading strategy for an investor: the case of partial information," Stochastic Processes and their Applications, Elsevier, vol. 76(1), pages 77-97, August.
    2. Klein, Roger W. & Bawa, Vijay S., 1976. "The effect of estimation risk on optimal portfolio choice," Journal of Financial Economics, Elsevier, vol. 3(3), pages 215-231, June.
    3. Bodnar, Taras & Mazur, Stepan & Okhrin, Yarema, 2017. "Bayesian estimation of the global minimum variance portfolio," European Journal of Operational Research, Elsevier, vol. 256(1), pages 292-307.
    4. Barry, Christopher B, 1974. "Portfolio Analysis under Uncertain Means, Variances, and Covariances," Journal of Finance, American Finance Association, vol. 29(2), pages 515-522, May.
    5. Aguilar, Omar & West, Mike, 2000. "Bayesian Dynamic Factor Models and Portfolio Allocation," Journal of Business & Economic Statistics, American Statistical Association, vol. 18(3), pages 338-357, July.
    6. L.C.G. Rogers, 2001. "The relaxed investor and parameter uncertainty," Finance and Stochastics, Springer, vol. 5(2), pages 131-154.
    7. Lakner, Peter, 1995. "Utility maximization with partial information," Stochastic Processes and their Applications, Elsevier, vol. 56(2), pages 247-273, April.
    8. Jakša Cvitanić & Ali Lazrak & Lionel Martellini & Fernando Zapatero, 2006. "Dynamic Portfolio Choice with Parameter Uncertainty and the Economic Value of Analysts' Recommendations," The Review of Financial Studies, Society for Financial Studies, vol. 19(4), pages 1113-1156.
    9. Amine Ismail & Huy^en Pham, 2016. "Robust Markowitz mean-variance portfolio selection under ambiguous covariance matrix ," Papers 1610.06805, arXiv.org, revised Mar 2017.
    10. David Bauder & Taras Bodnar & Nestor Parolya & Wolfgang Schmid, 2021. "Bayesian mean–variance analysis: optimal portfolio selection under parameter uncertainty," Quantitative Finance, Taylor & Francis Journals, vol. 21(2), pages 221-242, February.
    11. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    12. Doron Avramov & Guofu Zhou, 2010. "Bayesian Portfolio Analysis," Annual Review of Financial Economics, Annual Reviews, vol. 2(1), pages 25-47, December.
    13. Best, Michael J & Grauer, Robert R, 1991. "On the Sensitivity of Mean-Variance-Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results," The Review of Financial Studies, Society for Financial Studies, vol. 4(2), pages 315-342.
    14. Paul A. Samuelson, 2011. "Lifetime Portfolio Selection by Dynamic Stochastic Programming," World Scientific Book Chapters, in: Leonard C MacLean & Edward O Thorp & William T Ziemba (ed.), THE KELLY CAPITAL GROWTH INVESTMENT CRITERION THEORY and PRACTICE, chapter 31, pages 465-472, World Scientific Publishing Co. Pte. Ltd..
    15. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    16. Frost, Peter A. & Savarino, James E., 1986. "An Empirical Bayes Approach to Efficient Portfolio Selection," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 21(3), pages 293-305, September.
    17. Merton, Robert C., 1980. "On estimating the expected return on the market : An exploratory investigation," Journal of Financial Economics, Elsevier, vol. 8(4), pages 323-361, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Shubhangi Sikaria & Rituparna Sen & Neelesh S. Upadhye, 2019. "Bayesian Filtering for Multi-period Mean-Variance Portfolio Selection," Papers 1911.07526, arXiv.org, revised Aug 2020.
    2. Trichilli, Yousra & Abbes, Mouna Boujelbène & Masmoudi, Afif, 2020. "Islamic and conventional portfolios optimization under investor sentiment states: Bayesian vs Markowitz portfolio analysis," Research in International Business and Finance, Elsevier, vol. 51(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Carmine De Franco & Johann Nicolle & Huy^en Pham, 2018. "Bayesian learning for the Markowitz portfolio selection problem," Papers 1811.06893, arXiv.org.
    2. Carmine De Franco & Johann Nicolle & Huyên Pham, 2019. "Bayesian Learning For The Markowitz Portfolio Selection Problem," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(07), pages 1-40, November.
    3. Carmine De Franco & Johann Nicolle & Huyên Pham, 2019. "Dealing with Drift Uncertainty: A Bayesian Learning Approach," Risks, MDPI, vol. 7(1), pages 1-18, January.
    4. Bauder, David & Bodnar, Taras & Parolya, Nestor & Schmid, Wolfgang, 2020. "Bayesian inference of the multi-period optimal portfolio for an exponential utility," Journal of Multivariate Analysis, Elsevier, vol. 175(C).
    5. Penaranda, Francisco, 2007. "Portfolio choice beyond the traditional approach," LSE Research Online Documents on Economics 24481, London School of Economics and Political Science, LSE Library.
    6. David Bauder & Taras Bodnar & Nestor Parolya & Wolfgang Schmid, 2021. "Bayesian mean–variance analysis: optimal portfolio selection under parameter uncertainty," Quantitative Finance, Taylor & Francis Journals, vol. 21(2), pages 221-242, February.
    7. Taras Bodnar & Mathias Lindholm & Vilhelm Niklasson & Erik Thors'en, 2020. "Bayesian Quantile-Based Portfolio Selection," Papers 2012.01819, arXiv.org.
    8. Bodnar, Taras & Lindholm, Mathias & Niklasson, Vilhelm & Thorsén, Erik, 2022. "Bayesian portfolio selection using VaR and CVaR," Applied Mathematics and Computation, Elsevier, vol. 427(C).
    9. David Feldman, 2007. "Incomplete information equilibria: Separation theorems and other myths," Annals of Operations Research, Springer, vol. 151(1), pages 119-149, April.
    10. Yuanyuan Zhang & Xiang Li & Sini Guo, 2018. "Portfolio selection problems with Markowitz’s mean–variance framework: a review of literature," Fuzzy Optimization and Decision Making, Springer, vol. 17(2), pages 125-158, June.
    11. Bodnar, Taras & Mazur, Stepan & Okhrin, Yarema, 2017. "Bayesian estimation of the global minimum variance portfolio," European Journal of Operational Research, Elsevier, vol. 256(1), pages 292-307.
    12. Kellerer, Belinda, 2019. "Portfolio Optimization and Ambiguity Aversion," Junior Management Science (JUMS), Junior Management Science e. V., vol. 4(3), pages 305-338.
    13. Taras Bodnar & Vilhelm Niklasson & Erik Thors'en, 2022. "Volatility Sensitive Bayesian Estimation of Portfolio VaR and CVaR," Papers 2205.01444, arXiv.org.
    14. Bodnar, Taras & Mazur, Stepan & Nguyen, Hoang, 2022. "Estimation of optimal portfolio compositions for small sampleand singular covariance matrix," Working Papers 2022:15, Örebro University, School of Business.
    15. Lorenzo Garlappi & Raman Uppal & Tan Wang, 2007. "Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach," The Review of Financial Studies, Society for Financial Studies, vol. 20(1), pages 41-81, January.
    16. Vasyl Golosnoy, 2010. "No-transaction bounds and estimation risk," Quantitative Finance, Taylor & Francis Journals, vol. 10(5), pages 487-493.
    17. David Bauder & Taras Bodnar & Stepan Mazur & Yarema Okhrin, 2018. "Bayesian Inference For The Tangent Portfolio," Journal of Enterprising Culture (JEC), World Scientific Publishing Co. Pte. Ltd., vol. 21(08), pages 1-27, December.
    18. Muhinyuza, Stanislas & Bodnar, Taras & Lindholm, Mathias, 2020. "A test on the location of the tangency portfolio on the set of feasible portfolios," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    19. Michele Longo & Alessandra Mainini, 2015. "Learning and Portfolio Decisions for HARA Investors," Papers 1502.02968, arXiv.org.
    20. Francisco Peñaranda & Enrique Sentana, 2024. "Portfolio management with big data," Working Papers wp2024_2411, CEMFI.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:wpaper:hal-01923917. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.