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Bayesian Learning For The Markowitz Portfolio Selection Problem

Author

Listed:
  • CARMINE DE FRANCO

    (OSSIAM, Paris, France)

  • JOHANN NICOLLE

    (OSSIAM, Paris, France2Laboratoire de Probabilités Statistique et Modélisation (LPSM), Paris, France)

  • HUYÊN PHAM

    (Laboratoire de Probabilités, Statistique et Modélisation (LPSM), Université de Paris, Paris, France)

Abstract

We study the Markowitz portfolio selection problem with unknown drift vector in the multi-dimensional 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 nonlearning, using a constant drift in an uncertain context, and analyze the sensitivity of the value of information with respect to various relevant parameters of our model.

Suggested Citation

  • 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.
    • Handle: RePEc:wsi:ijtafx:v:22:y:2019:i:07:n:s0219024919500377
    DOI: 10.1142/S0219024919500377
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    Cited by:

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    3. Pier Francesco Procacci & Tomaso Aste, 2021. "Portfolio Optimization with Sparse Multivariate Modelling," Papers 2103.15232, arXiv.org.
    4. Pier Francesco Procacci & Tomaso Aste, 2022. "Portfolio optimization with sparse multivariate modeling," Journal of Asset Management, Palgrave Macmillan, vol. 23(6), pages 445-465, October.
    5. Masashi Ieda, 2022. "Continuous-Time Portfolio Optimization for Absolute Return Funds," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 29(4), pages 675-696, December.

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