Sparse precision matrices for minimum variance portfolios

  • Gabriele Torri
  • Rosella GiacomettiEmail author
  • Sandra Paterlini
Original Paper


Financial crises are typically characterized by highly positively correlated asset returns due to the simultaneous distress on almost all securities, high volatilities and the presence of extreme returns. In the aftermath of the 2008 crisis, investors were prompted even further to look for portfolios that minimize risk and can better deal with estimation error in the inputs of the asset allocation models. The minimum variance portfolio à la Markowitz is considered the reference model for risk minimization in equity markets, due to its simplicity in the optimization as well as its need for just one input estimate: the inverse of the covariance estimate, or the so-called precision matrix. In this paper, we propose a data-driven portfolio framework based on two regularization methods, glasso and tlasso, that provide sparse estimates of the precision matrix by penalizing its \(L_1\)-norm. Glasso and tlasso rely on asset returns Gaussianity or t-Student assumptions, respectively. Simulation and real-world data results support the proposed methods compared to state-of-art approaches, such as random matrix and Ledoit–Wolf shrinkage.


Minimum variance Precision matrix Graphical lasso Tlasso 



Sandra Paterlini acknowledges ICT COST Action IC1408 from CRoNoS. Gabriele Torri acknowledges the support of the Czech Science Foundation (GACR) under project 17-19981S, 19-11965S and SP2018/34, an SGS research project of VSB-TU Ostrava. Rosella Giacometti and Gabriele Torri acknowledge the support given by University of Bergamo research funds 2016 2017.


  1. Baba K, Shibata R, Sibuya M (2004) Partial correlation and conditional correlation as measures of conditional independence. Aust N Z J Stat 46(4):657–664CrossRefGoogle Scholar
  2. Banerjee O, Ghaoui LE, d’Aspremont A (2008) Model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data. J Mach Learn Res 9:485–516Google Scholar
  3. Black F, Litterman R (1992) Global portfolio optimization. Finance Anal J 48(5):28–43CrossRefGoogle Scholar
  4. Bouchaud JP, Potters M (2009) Financial applications of random matrix theory: a short review. arXiv preprint arXiv:0910.1205
  5. Brodie J, Daubechies I, De Mol C, Giannone D, Loris I (2009) Sparse and stable Markowitz portfolios. Proc Natl Acad Sci 106(30):12267–12272CrossRefGoogle Scholar
  6. Brownlees CT, Nualart E, Sun Y (2015) Realized networks. Working Paper, SSRNGoogle Scholar
  7. Bruder B, Gaussel N, Richard JC, Roncalli T (2013) Regularization of portfolio allocation. Working Paper, SSRNGoogle Scholar
  8. Cont R (2001) Empirical properties of asset returns: stylized facts and statistical issues. Quant Finance 1:223–236CrossRefGoogle Scholar
  9. DeMiguel V, Nogales FJ (2009) Portfolio selection with robust estimation. Oper Res 57:560–577CrossRefGoogle Scholar
  10. DeMiguel V, Garlappi L, Nogales F, Uppal R (2009a) A generalized approach to portfolio optimization: improving performance by constraining portfolio norm. Manag Sci 55:798–812CrossRefGoogle Scholar
  11. DeMiguel V, Garlappi L, Uppal R (2009b) Optimal versus naive diversification: how inefficient is the 1/N portfolio strategy? Rev Financ Stud 22(5):1915–1953CrossRefGoogle Scholar
  12. Dempster AP (1972) Covariance selection. Biometrics 28(1):157–175CrossRefGoogle Scholar
  13. Engle R (2002) Dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models. J Bus Econ Stat 20(3):339–350CrossRefGoogle Scholar
  14. Fan J, Zhang J, Yu K (2012) Vast portfolio selection with gross-exposure constraints. J Am Stat Assoc 107(498):592–606CrossRefGoogle Scholar
  15. Finegold M, Drton M (2011) Robust graphical modeling of gene networks using classical and alternative t-distributions. Ann Appl Stat 5(2A):1057–1080CrossRefGoogle Scholar
  16. Friedman J, Hastie T, Tibshirani R (2008) Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9(3):432–441CrossRefGoogle Scholar
  17. Friedman J, Hastie T, Tibshirani R (2014) Glasso: graphical lasso-estimation of gaussian graphical models. R packageGoogle Scholar
  18. Goto S, Xu Y (2015) Improving mean variance optimization through sparse hedging restrictions. J Financ Quant Anal 50(6):1415–1441CrossRefGoogle Scholar
  19. Højsgaard S, Edwards D, Lauritzen S (2012) Graphical models with R. Springer, BerlinCrossRefGoogle Scholar
  20. Kan R, Zhou G (2007) Optimal portfolio choice with parameter uncertainty. J Financ Quant Anal 42(3):621–656CrossRefGoogle Scholar
  21. Kolm PN, Tütüncü R, Fabozzi F (2014) 60 years following Harry Markowitz’s contribution to portfolio theory and operations research. Eur J Oper Res 234(2):343–582CrossRefGoogle Scholar
  22. Kotz S, Nadarajah S (2004) Multivariate t-distributions and their applications. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  23. Kremer PJ, Talmaciu A, Paterlini S (2018) Risk minimization in multi-factor portfolios: What is the best strategy? Ann Oper Res 266(1–2):255–291CrossRefGoogle Scholar
  24. Laloux L, Cizeau P, Bouchaud JP, Potters M (1999) Noise dressing of financial correlation matrices. Phys Rev Lett 83(7):1467–1469CrossRefGoogle Scholar
  25. Lam C, Fan J (2009) Sparsistency and rates of convergence in large covariance matrix estimation. Ann Stat 37(6B):4254CrossRefGoogle Scholar
  26. Lange KL, Little RJ, Taylor JM (1989) Robust statistical modeling using the t distribution. J Am Stat Assoc 84(408):881–896Google Scholar
  27. Lauritzen SL (1996) Graph models, vol 17. Clarendon Press, OxfordGoogle Scholar
  28. Ledoit O, Wolf M (2004a) Honey, i shrunk the sample covariance matrix. J Portf Manag 30(4):110–119CrossRefGoogle Scholar
  29. Ledoit O, Wolf M (2004b) A well-conditioned estimator for large-dimensional covariance matrices. J multivar anal 88(2):365–411CrossRefGoogle Scholar
  30. Ledoit O, Wolf M (2011) Robust performances hypothesis testing with the variance. Wilmott 55:86–89CrossRefGoogle Scholar
  31. Markowitz H (1952) Portfolio selection. J Finance 7(1):77–91Google Scholar
  32. McLachlan G, Krishnan T (2007) The EM algorithm and extensions, vol 382. Wiley, HobokenGoogle Scholar
  33. Meucci A (2009) Risk and asset allocation. Springer, BerlinGoogle Scholar
  34. Michaud RO (1989) The Markowitz optimization enigma: is optimized optimal? ICFA Contin Educ Ser 1989(4):43–54CrossRefGoogle Scholar
  35. Murphy KP (2012) Machine learning: a probabilistic perspective. The MIT Press, LondonGoogle Scholar
  36. Rothman AJ, Bickel PJ, Levina E, Zhu J et al (2008) Sparse permutation invariant covariance estimation. Electron J Stat 2:494–515CrossRefGoogle Scholar
  37. Stevens GV (1998) On the inverse of the covariance matrix in portfolio analysis. J Finance 53(5):1821–1827CrossRefGoogle Scholar
  38. Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc Ser B (Methodological) 58(1):267–288Google Scholar
  39. Witten DM, Friedman JH, Simon N (2011) New insights and faster computations for the graphical lasso. J Comput Graph Stat 20(4):892–900CrossRefGoogle Scholar
  40. Won JH, Lim J, Kim SJ, Rajaratnam B (2013) Condition-number-regularized covariance estimation. J R Stat Soc Ser B (Statistical Methodology) 75(3):427–450CrossRefGoogle Scholar
  41. Yuan M, Lin Y (2007) Model selection and estimation in the Gaussian graphical model. Biometrika 94(1):19–35CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Management, Economics and Quantitative MethodsUniversity of BergamoBergamoItaly
  2. 2.Department of Finance, Faculty of EconomicsVŠB-TU OstravaOstravaCzech Republic
  3. 3.Department of Economics and ManagementUniversity of TrentoTrentoItaly
  4. 4.Department of Finance and AccountingEBS Universität für Wirtschaft und RechtWiesbadenGermany

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