# Sparse precision matrices for minimum variance portfolios

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## Abstract

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.

## Keywords

Minimum variance Precision matrix Graphical lasso Tlasso## Notes

### Acknowledgements

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.

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