Analytic approach to variance optimization under an 1 constraint

  • Imre Kondor
  • Gábor Papp
  • Fabio CaccioliEmail author
Open Access
Regular Article


The optimization of the variance of a portfolio of N independent but not identically distributed assets, supplemented by a budget constraint and an asymmetric 1 regularizer, is carried out analytically by the replica method borrowed from the theory of disordered systems. The asymmetric regularizer allows us to penalize short and long positions differently, so the present treatment includes the no-short-constrained portfolio optimization problem as a special case. Results are presented for the out-of-sample and the in-sample estimator of the regularized variance, the relative estimation error, the density of the assets eliminated from the portfolio by the regularizer, and the distribution of the optimal portfolio weights. We have studied the dependence of these quantities on the ratio r of the portfolio’s dimension N to the sample size T, and on the strength of the regularizer. We have checked the analytic results by numerical simulations, and found general agreement. Regularization extends the interval where the optimization can be carried out, and suppresses the large sample fluctuations, but the performance of 1 regularization is rather disappointing: if the sample size is large relative to the dimension, i.e. r is small, the regularizer does not play any role, while for r’s where the regularizer starts to be felt the estimation error is already so large as to make the whole optimization exercise pointless. We find that the 1 regularization can eliminate at most half the assets from the portfolio (by setting their weights to exactly zero), corresponding to this there is a critical ratio r = 2 beyond which the 1 regularized variance cannot be optimized: the regularized variance becomes constant over the simplex. These facts do not seem to have been noticed in the literature.

Graphical abstract


Statistical and Nonlinear Physics 


  1. 1.
    J.D. Jobson, B. Korkie, Proc. Am. Stat. Assoc. (Bus. Econ. Stat.) 1, 279 (1979) Google Scholar
  2. 2.
    P. Jorion, J. Financ. Quant. Anal. 21, 279 (1986) CrossRefGoogle Scholar
  3. 3.
    O. Ledoit, M. Wolf, J. Empir. Finance 10, 603 (2003) CrossRefGoogle Scholar
  4. 4.
    O. Ledoit, M. Wolf, J. Portfolio Manage. 31, 110 (2004) CrossRefGoogle Scholar
  5. 5.
    O. Ledoit, M. Wolf, J. Multivar. Anal. 88, 365 (2004) CrossRefGoogle Scholar
  6. 6.
    V. Golosnoy, Y. Okhrin, Eur. J. Finance 13, 441 (2007) CrossRefGoogle Scholar
  7. 7.
    T. Shinzato, J. Stat. Mech.: Theor. Exp. 2017, 023301 (2017) MathSciNetCrossRefGoogle Scholar
  8. 8.
    O. Ledoit, M. Wolf, Ann. Stat. 40, 1024 (2012) CrossRefGoogle Scholar
  9. 9.
    J. Bun, J.-P. Bouchaud, M. Potters, My beautiful laundrette: Cleaning correlation matrices for portfolio optimization, 2016, available at (accessed: September 2017)
  10. 10.
    O. Ledoit, M. Wolf, Direct nonlinear shrinkage estimation of large-dimensional covariance matrices, University of Zurich, Department of Economics, 2017, Workingpaper No. 264, p. 46 Google Scholar
  11. 11.
    R. Tibshirani, J. R. Stat. Soc. Ser. B (Methodological) 58, 267 (1996) Google Scholar
  12. 12.
    R. Jagannathan, T. Ma, J. Finance 58, 1651 (2003) CrossRefGoogle Scholar
  13. 13.
    J. Brodie, I. Daubechies, C. De Mol, D. Giannone, I. Loris, Proc. Natl. Acad. Sci. 106, 12267 (2009) ADSCrossRefGoogle Scholar
  14. 14.
    V. DeMiguel, L. Garlappi, F.J. Nogales, R. Uppal, Manage.Sci. 55, 798 (2009) CrossRefGoogle Scholar
  15. 15.
    D. Giomouridis, S. Paterlini, J. Financ. Res. 33, 223 (2010) CrossRefGoogle Scholar
  16. 16.
    M. Carrasco, N. Noumon, Optimal portfolio selection using regularization. University of Montreal, 2012, (accessed: September 2017)
  17. 17.
    J. Fan, J. Zhang, K. Yu, J. Am. Stat. Assoc. 107, 592 (2012) CrossRefGoogle Scholar
  18. 18.
    Y.-M. Yen, T.-Y. Yen, Comput. Stat. Data Anal. 76, 737 (2014) CrossRefGoogle Scholar
  19. 19.
    B. Fastrich, S. Paterlini, P. Winker, Quant. Finance 14, 2019 (2014) MathSciNetCrossRefGoogle Scholar
  20. 20.
    M. Mézard, G. Parisi, M.A. Virasoro, in Spin Glass Theoryand Beyond, World Scientific Lecture Notes in Physics (World Scientific, Singapore, 1987), Vol. 9 Google Scholar
  21. 21.
    F. Caccioli, I. Kondor, M. Marsili, S. Still, Int. J. Theor. Appl. Finance 19, 1650035 (2016) MathSciNetCrossRefGoogle Scholar
  22. 22.
    T. Hastie, R. Tibshirani, J. Friedman, in The Elements of Statistical Learning, Data Mining, Inference, and Prediction, 2nd edn. Springer Series in Statistics (Springer, Berlin, 2008) Google Scholar
  23. 23.
    I. Kondor, G. Papp, F. Caccioli, J. Stat. Mech.: Theor. Exp. 2017, 123402 (2017) CrossRefGoogle Scholar
  24. 24.
    P. Bühlmann, S. Van De Geer, Statistics for High-Dimensional Data: Methods, Theory and Applications (Springer Science Plus Business Media, Berlin, Heidelberg, 2011) Google Scholar
  25. 25.
    I. Varga-Haszonits, F. Caccioli, I. Kondor, J. Stat. Mech.: Theor. Exp. 2016, 123404 (2016) CrossRefGoogle Scholar
  26. 26.
    D. Donoho, J. Tanner, Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 367, 4273 (2009) ADSCrossRefGoogle Scholar
  27. 27.
    D. Amelunxen, M. Lotz, M.B. McCoy, J.A. Tropp, Inform. Inference 3, 224 (2013) CrossRefGoogle Scholar

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© The Author(s) 2019

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Parmenides FoundationPullachGermany
  2. 2.London Mathematical LaboratoryLondonUK
  3. 3.Complexity Science HubViennaAustria
  4. 4.Eötvös Loránd University, Institute for PhysicsBudapestHungary
  5. 5.University College London, Department of Computer ScienceLondonUK
  6. 6.Systemic Risk Centre, London School of Economics and Political SciencesLondonUK

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