Optimal minimum variance portfolios can be analytically computed and require only the estimate of the inverse of the covariance matrix, commonly referred to as the precision matrix. Graphical models, which have demonstrated exceptional performance in uncovering the conditional dependence structure among a given set of variables, can provide reliable estimates of the precision matrix. This paper introduces two novel graphical modeling techniques: Gslope and Tslope, which use the Sorted \(\ell _1\) -Penalized Estimator (Slope) to directly estimate the precision matrix. We develop ad hoc algorithms to efficiently solve the underlying optimization problems: the Alternating Direction Method of Multipliers for Gslope, and the Expectation-Maximization algorithm for Tslope. Our methods are suitable for both Gaussian and non-Gaussian distributed data and take into account the empirically observed distributional characteristics of asset returns. Through extensive simulation analysis, we demonstrate the superiority of our new methods over state-of-the-art estimation techniques, particularly regarding clustering and stability characteristics. The empirical results on real-world data support the validity of our new approaches, which often outperform state-of-the-art methods in terms of volatility, extreme risk, and risk-adjusted returns. Notably, they prove to be effective tools for dealing with high-dimensional problems and heavy-tailed distributions, two critical issues in the literature.
