Scandinavian Working Papers in Economics

Working Papers,
Örebro University, School of Business

No 2024:9: Minimum VaR and minimum CVaR optimal portfolios: The case of singular covariance matrix

Mårten Gulliksson (), Stepan Mazur () and Anna Oleynik ()
Additional contact information
Mårten Gulliksson: Örebro University School of Science and Technology, Postal: Örebro University, School of Science and Technology, SE - 701 82 ÖREBRO, Sweden
Stepan Mazur: Örebro University School of Business, Postal: Örebro University, School of Business, SE - 701 82 ÖREBRO, Sweden
Anna Oleynik: Department of Mathematics, University of Bergen, Postal: University of Bergen, Department of Mathematics, 5020 BERGEN, Norway

Abstract: This paper examines optimal portfolio selection using quantile-based risk measures such as Valueat-Risk (VaR) and Conditional Value-at-Risk (CVaR). We address the case of a singular covariance matrix of asset returns, which leads to an optimization problem with infinitely many solutions. An analytical form for a general solution is derived, along with a unique solution that minimizes the L2-norm. We also show that the general solution reduces to the standard optimal portfolio for VaR and CVaR when the covariance matrix is non-singular.

Keywords: Minimum VaR portfolio; Minimum CVaR portfolio; Singular covariance matrix; Linear illposed problems

JEL-codes: C58; G11; G32

Language: English

8 pages, October 31, 2024

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