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Statistical properties of estimators for the log-optimal portfolio

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  • Gabriel Frahm

    (Helmut Schmidt University)

Abstract

The best constant re-balanced portfolio represents the standard estimator for the log-optimal portfolio. It is shown that a quadratic approximation of log-returns works very well on a daily basis and a mean-variance estimator is proposed as an alternative to the best constant re-balanced portfolio. It can easily be computed and the numerical algorithm is very fast even if the number of dimensions is high. Some small-sample and the basic large-sample properties of the estimators are derived. The asymptotic results can be used for constructing hypothesis tests and for computing confidence regions. For this purpose, one should apply a finite-sample correction, which substantially improves the large-sample approximation. However, it is shown that the impact of estimation errors concerning the expected asset returns is serious. The given results confirm a general rule, which has become folklore during the last decades, namely that portfolio optimization typically fails on estimating expected asset returns.

Suggested Citation

  • Gabriel Frahm, 2020. "Statistical properties of estimators for the log-optimal portfolio," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(1), pages 1-32, August.
    • Handle: RePEc:spr:mathme:v:92:y:2020:i:1:d:10.1007_s00186-020-00701-1
    DOI: 10.1007/s00186-020-00701-1
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    References listed on IDEAS

    as
    1. Frahm, Gabriel & Memmel, Christoph, 2010. "Dominating estimators for minimum-variance portfolios," Journal of Econometrics, Elsevier, vol. 159(2), pages 289-302, December.
    2. repec:hal:journl:peer-00741629 is not listed on IDEAS
    3. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 131-151, January.
    4. Davidson, James, 1994. "Stochastic Limit Theory: An Introduction for Econometricians," OUP Catalogue, Oxford University Press, number 9780198774037.
    5. Merton, Robert C. & Samuelson, Paul A., 1974. "Fallacy of the log-normal approximation to optimal portfolio decision-making over many periods," Journal of Financial Economics, Elsevier, vol. 1(1), pages 67-94, May.
    6. Gabriel Frahm, 2013. "Pricing and Valuation under the Real-World Measure," Papers 1304.3824, arXiv.org, revised Jan 2016.
    7. Kan, Raymond & Zhou, Guofu, 2007. "Optimal Portfolio Choice with Parameter Uncertainty," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 42(3), pages 621-656, September.
    8. Ioannis Karatzas & Constantinos Kardaras, 2007. "The numéraire portfolio in semimartingale financial models," Finance and Stochastics, Springer, vol. 11(4), pages 447-493, October.
    9. Robert M. Bell & Thomas M. Cover, 1980. "Competitive Optimality of Logarithmic Investment," Mathematics of Operations Research, INFORMS, vol. 5(2), pages 161-166, May.
    10. Gabriel Frahm, 2015. "A theoretical foundation of portfolio resampling," Theory and Decision, Springer, vol. 79(1), pages 107-132, July.
    11. Gabriel Frahm, 2016. "Pricing And Valuation Under The Real-World Measure," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(01), pages 1-39, February.
    12. Magnus, J.R. & Neudecker, H., 1979. "The commutation matrix : Some properties and applications," Other publications TiSEM d0b1e779-7795-4676-ac98-1, Tilburg University, School of Economics and Management.
    13. Hakansson, Nils H., 1971. "Capital Growth and the Mean-Variance Approach to Portfolio Selection," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 6(1), pages 517-557, January.
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    Cited by:

    1. Chung-Han Hsieh, 2022. "On Solving Robust Log-Optimal Portfolio: A Supporting Hyperplane Approximation Approach," Papers 2202.03858, arXiv.org.
    2. Dmitry B. Rokhlin, 2021. "Relative utility bounds for empirically optimal portfolios," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(3), pages 437-462, June.

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    More about this item

    Keywords

    Best constant re-balanced portfolio; Estimation risk; Growth-optimal portfolio; Log-optimal portfolio; Mean-variance optimization;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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