[go: up one dir, main page]
IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1606.08679.html
   My bibliography  Save this paper

Replica approach to mean-variance portfolio optimization

Author

Listed:
  • Istvan Varga-Haszonits
  • Fabio Caccioli
  • Imre Kondor
Abstract
We consider the problem of mean-variance portfolio optimization for a generic covariance matrix subject to the budget constraint and the constraint for the expected return, with the application of the replica method borrowed from the statistical physics of disordered systems. We find that the replica symmetry of the solution does not need to be assumed, but emerges as the unique solution of the optimization problem. We also check the stability of this solution and find that the eigenvalues of the Hessian are positive for $r=N/T

Suggested Citation

  • Istvan Varga-Haszonits & Fabio Caccioli & Imre Kondor, 2016. "Replica approach to mean-variance portfolio optimization," Papers 1606.08679, arXiv.org.
    • Handle: RePEc:arx:papers:1606.08679
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1606.08679
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
    2. Fabio Caccioli & Susanne Still & Matteo Marsili & Imre Kondor, 2013. "Optimal liquidation strategies regularize portfolio selection," The European Journal of Finance, Taylor & Francis Journals, vol. 19(6), pages 554-571, July.
    3. Victor DeMiguel & Lorenzo Garlappi & Raman Uppal, 2009. "Optimal Versus Naive Diversification: How Inefficient is the 1-N Portfolio Strategy?," The Review of Financial Studies, Society for Financial Studies, vol. 22(5), pages 1915-1953, May.
    4. Ravi Jagannathan & Tongshu Ma, 2003. "Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps," Journal of Finance, American Finance Association, vol. 58(4), pages 1651-1683, August.
    5. Fabio Caccioli & Imre Kondor & G'abor Papp, 2015. "Portfolio Optimization under Expected Shortfall: Contour Maps of Estimation Error," Papers 1510.04943, arXiv.org.
    6. 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.
    7. Victor DeMiguel & Lorenzo Garlappi & Francisco J. Nogales & Raman Uppal, 2009. "A Generalized Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms," Management Science, INFORMS, vol. 55(5), pages 798-812, May.
    8. Stefano Ciliberti & Imre Kondor & Marc Mezard, 2007. "On the feasibility of portfolio optimization under expected shortfall," Quantitative Finance, Taylor & Francis Journals, vol. 7(4), pages 389-396.
    9. Pafka, Szilárd & Kondor, Imre, 2003. "Noisy covariance matrices and portfolio optimization II," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 319(C), pages 487-494.
    10. Frahm, Gabriel & Memmel, Christoph, 2010. "Dominating estimators for minimum-variance portfolios," Journal of Econometrics, Elsevier, vol. 159(2), pages 289-302, December.
    11. Imre Kondor & Fabio Caccioli & G'abor Papp & Matteo Marsili, 2015. "Contour map of estimation error for Expected Shortfall," Papers 1502.06217, arXiv.org.
    12. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    13. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
    14. Alexander Kempf & Christoph Memmel, 2006. "Estimating the global Minimum Variance Portfolio," Schmalenbach Business Review (sbr), LMU Munich School of Management, vol. 58(4), pages 332-348, October.
    15. Carlo Acerbi & Dirk Tasche, 2002. "Expected Shortfall: A Natural Coherent Alternative to Value at Risk," Economic Notes, Banca Monte dei Paschi di Siena SpA, vol. 31(2), pages 379-388, July.
    16. I. Kondor & I. Varga-Haszonits, 2008. "Divergent estimation error in portfolio optimization and in linear regression," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 64(3), pages 601-605, August.
    17. 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.
    18. Ledoit, Olivier & Wolf, Michael, 2003. "Improved estimation of the covariance matrix of stock returns with an application to portfolio selection," Journal of Empirical Finance, Elsevier, vol. 10(5), pages 603-621, December.
    19. repec:bla:jfinan:v:58:y:2003:i:4:p:1651-1684 is not listed on IDEAS
    20. Merton, Robert C., 1972. "An Analytic Derivation of the Efficient Portfolio Frontier," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 7(4), pages 1851-1872, September.
    21. 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.
    22. Fabio Caccioli & Imre Kondor & G'abor Papp, 2015. "Portfolio Optimization under Expected Shortfall: Contour Maps of Estimation Error," Papers 1510.04943, arXiv.org.
    23. Martin R. Young, 1998. "A Minimax Portfolio Selection Rule with Linear Programming Solution," Management Science, INFORMS, vol. 44(5), pages 673-683, May.
    24. repec:hal:journl:peer-00741629 is not listed on IDEAS
    25. Plerou, V & Gopikrishnan, P & Rosenow, B & Amaral, L.A.N & Stanley, H.E, 2000. "A random matrix theory approach to financial cross-correlations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 374-382.
    26. Vasiliki Plerou & Parameswaran Gopikrishnan & Bernd Rosenow & Luis A. Nunes Amaral & H. Eugene Stanley, 1999. "Universal and non-universal properties of cross-correlations in financial time series," Papers cond-mat/9902283, arXiv.org.
    27. Pafka, Szilárd & Kondor, Imre, 2004. "Estimated correlation matrices and portfolio optimization," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 343(C), pages 623-634.
    28. Varga-Haszonits, I. & Kondor, I., 2007. "Noise sensitivity of portfolio selection in constant conditional correlation GARCH models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 385(1), pages 307-318.
    29. Gopal K. Basak & Ravi Jagannathan & Tongshu Ma, 2009. "Jackknife Estimator for Tracking Error Variance of Optimal Portfolios," Management Science, INFORMS, vol. 55(6), pages 990-1002, June.
    30. S. Ciliberti & M. Mézard, 2007. "Risk minimization through portfolio replication," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 57(2), pages 175-180, May.
    31. Dickinson, J. P., 1974. "The Reliability of Estimation Procedures in Portfolio Analysis," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 9(3), pages 447-462, June.
    32. Jorion, Philippe, 1986. "Bayes-Stein Estimation for Portfolio Analysis," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 21(3), pages 279-292, September.
    33. Gábor, Adrienn & Kondor, I, 1999. "Portfolios with nonlinear constraints and spin glasses," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 274(1), pages 222-228.
    34. Okhrin, Yarema & Schmid, Wolfgang, 2006. "Distributional properties of portfolio weights," Journal of Econometrics, Elsevier, vol. 134(1), pages 235-256, September.
    35. Vasyl Golosnoy & Yarema Okhrin, 2007. "Multivariate Shrinkage for Optimal Portfolio Weights," The European Journal of Finance, Taylor & Francis Journals, vol. 13(5), pages 441-458.
    36. Imre Kondor & István Varga-Haszonits, 2010. "Instability Of Portfolio Optimization Under Coherent Risk Measures," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 13(03), pages 425-437.
    37. Hiroshi Konno & Hiroaki Yamazaki, 1991. "Mean-Absolute Deviation Portfolio Optimization Model and Its Applications to Tokyo Stock Market," Management Science, INFORMS, vol. 37(5), pages 519-531, May.
    38. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    39. Kondor, Imre & Pafka, Szilard & Nagy, Gabor, 2007. "Noise sensitivity of portfolio selection under various risk measures," Journal of Banking & Finance, Elsevier, vol. 31(5), pages 1545-1573, May.
    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. Nava, Noemi & Di Matteo, T. & Aste, Tomaso, 2018. "Dynamic correlations at different time-scales with empirical mode decomposition," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 502(C), pages 534-544.
    2. Imre Kondor & G'abor Papp & Fabio Caccioli, 2017. "Analytic approach to variance optimization under an $\ell_1$ constraint," Papers 1709.08755, arXiv.org, revised Jul 2018.
    3. Papp, Gábor & Kondor, Imre & Caccioli, Fabio, 2021. "Optimizing expected shortfall under an ℓ1 constraint—an analytic approach," LSE Research Online Documents on Economics 111051, London School of Economics and Political Science, LSE Library.
    4. Shinzato, Takashi, 2018. "Maximizing and minimizing investment concentration with constraints of budget and investment risk," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 986-993.
    5. Li, Yan & Jiang, Xiong-Fei & Tian, Yue & Li, Sai-Ping & Zheng, Bo, 2019. "Portfolio optimization based on network topology," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 515(C), pages 671-681.
    6. Nava, Noemi & Di Matteo, Tiziana & Aste, Tomaso, 2018. "Financial time series forecasting using empirical mode decomposition and support vector regression," LSE Research Online Documents on Economics 91028, London School of Economics and Political Science, LSE Library.
    7. Noemi Nava & Tiziana Di Matteo & Tomaso Aste, 2018. "Financial Time Series Forecasting Using Empirical Mode Decomposition and Support Vector Regression," Risks, MDPI, vol. 6(1), pages 1-21, February.
    8. Tomaso Aste & T. Di Matteo, 2017. "Sparse Causality Network Retrieval from Short Time Series," Complexity, Hindawi, vol. 2017, pages 1-13, November.
    9. Jerome Garnier-Brun & Michael Benzaquen & Stefano Ciliberti & Jean-Philippe Bouchaud, 2021. "A new spin on optimal portfolios and ecological equilibria," Post-Print hal-03378915, HAL.
    10. Imre Kondor & G'abor Papp & Fabio Caccioli, 2016. "Analytic solution to variance optimization with no short-selling," Papers 1612.07067, arXiv.org, revised Jan 2017.

    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. Varga-Haszonits, Istvan & Caccioli, Fabio & Kondor, Imre, 2016. "Replica approach to mean-variance portfolio optimization," LSE Research Online Documents on Economics 68955, London School of Economics and Political Science, LSE Library.
    2. Fabio Caccioli & Imre Kondor & G'abor Papp, 2015. "Portfolio Optimization under Expected Shortfall: Contour Maps of Estimation Error," Papers 1510.04943, arXiv.org.
    3. Imre Kondor & G'abor Papp & Fabio Caccioli, 2016. "Analytic solution to variance optimization with no short-selling," Papers 1612.07067, arXiv.org, revised Jan 2017.
    4. Fabio Caccioli & Imre Kondor & G'abor Papp, 2015. "Portfolio Optimization under Expected Shortfall: Contour Maps of Estimation Error," Papers 1510.04943, arXiv.org.
    5. Caccioli, Fabio & Kondor, Imre & Papp, Gábor, 2015. "Portfolio optimization under expected shortfall: contour maps of estimation error," LSE Research Online Documents on Economics 119463, London School of Economics and Political Science, LSE Library.
    6. Fabio Caccioli & Imre Kondor & Matteo Marsili & Susanne Still, 2014. "$L_p$ regularized portfolio optimization," Papers 1404.4040, arXiv.org.
    7. Fabio Caccioli & Imre Kondor & Matteo Marsili & Susanne Still, 2016. "Liquidity Risk And Instabilities In Portfolio Optimization," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(05), pages 1-28, August.
    8. G'abor Papp & Fabio Caccioli & Imre Kondor, 2016. "Bias-variance trade-off in portfolio optimization under Expected Shortfall with $\ell_2$ regularization," Papers 1602.08297, arXiv.org, revised Jul 2018.
    9. Kourtis, Apostolos & Dotsis, George & Markellos, Raphael N., 2012. "Parameter uncertainty in portfolio selection: Shrinking the inverse covariance matrix," Journal of Banking & Finance, Elsevier, vol. 36(9), pages 2522-2531.
    10. Imre Kondor, 2014. "Estimation Error of Expected Shortfall," Papers 1402.5534, arXiv.org.
    11. Papp, Gábor & Caccioli, Fabio & Kondor, Imre, 2019. "Bias-variance trade-off in portfolio optimization under expected shortfall with ℓ 2 regularization," LSE Research Online Documents on Economics 100294, London School of Economics and Political Science, LSE Library.
    12. DeMiguel, Victor & Martin-Utrera, Alberto & Nogales, Francisco J., 2013. "Size matters: Optimal calibration of shrinkage estimators for portfolio selection," Journal of Banking & Finance, Elsevier, vol. 37(8), pages 3018-3034.
    13. Axel Pruser & Imre Kondor & Andreas Engel, 2021. "Aspects of a phase transition in high-dimensional random geometry," Papers 2105.04395, arXiv.org, revised Jun 2021.
    14. Papp, Gábor & Kondor, Imre & Caccioli, Fabio, 2021. "Optimizing expected shortfall under an ℓ1 constraint—an analytic approach," LSE Research Online Documents on Economics 111051, London School of Economics and Political Science, LSE Library.
    15. Chavez-Bedoya, Luis & Rosales, Francisco, 2022. "Orthogonal portfolios to assess estimation risk," International Review of Economics & Finance, Elsevier, vol. 80(C), pages 906-937.
    16. Chavez-Bedoya, Luis & Rosales, Francisco, 2021. "Reduction of estimation risk in optimal portfolio choice using redundant constraints," International Review of Financial Analysis, Elsevier, vol. 78(C).
    17. Kondor, Imre & Pafka, Szilard & Nagy, Gabor, 2007. "Noise sensitivity of portfolio selection under various risk measures," Journal of Banking & Finance, Elsevier, vol. 31(5), pages 1545-1573, May.
    18. Kircher, Felix & Rösch, Daniel, 2021. "A shrinkage approach for Sharpe ratio optimal portfolios with estimation risks," Journal of Banking & Finance, Elsevier, vol. 133(C).
    19. Frahm, Gabriel & Memmel, Christoph, 2010. "Dominating estimators for minimum-variance portfolios," Journal of Econometrics, Elsevier, vol. 159(2), pages 289-302, December.
    20. G'abor Papp & Imre Kondor & Fabio Caccioli, 2021. "Optimizing Expected Shortfall under an $\ell_1$ constraint -- an analytic approach," Papers 2103.04375, arXiv.org.

    More about this item

    Statistics

    Access and download statistics

    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:arx:papers:1606.08679. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.