{"title":"Minimum VaR and minimum CVaR optimal portfolios: The case of singular covariance matrix","authors":"Mårten Gulliksson , Stepan Mazur , Anna Oleynik","doi":"10.1016/j.rinam.2025.100557","DOIUrl":null,"url":null,"abstract":"<div><div>This paper examines optimal portfolio selection using quantile-based risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). We address the case of a singular covariance matrix of asset returns, which may arise due to potential multicollinearity and strong correlations. This 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 <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-norm. We show that the general solution reduces to the standard optimal portfolio for VaR and CVaR when the covariance matrix is non-singular. We also provide a brief discussion of the efficient frontier in this context. Finally, we present a real-data example based on the weekly log returns of assets included in the S&P 500 index.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"26 ","pages":"Article 100557"},"PeriodicalIF":1.4000,"publicationDate":"2025-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Results in Applied Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2590037425000214","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
This paper examines optimal portfolio selection using quantile-based risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). We address the case of a singular covariance matrix of asset returns, which may arise due to potential multicollinearity and strong correlations. This 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 -norm. We show that the general solution reduces to the standard optimal portfolio for VaR and CVaR when the covariance matrix is non-singular. We also provide a brief discussion of the efficient frontier in this context. Finally, we present a real-data example based on the weekly log returns of assets included in the S&P 500 index.