Портфельна оптимiзацiя для реальних даних: пiдходи та виклики

Portfolio optimization continues to be a dynamic field within finance, integrating new theories and technologies to better meet investor needs. As financial markets evolve, so too will the methodologies used to optimize portfolios, making it an area ripe for ongoing research and innovation.Classical Markowitz approach is based on the mean-variance optimization, which quantifies the tradeoff between risk (variance) and return (expected return). This approach had some limitations. It assumes investors are rational, markets are efficient, and asset returns are normally distributed. As a response to the some limitations of Markowitz theory minimum-VaR approach was appeared. This theory recognizes some assymetry, that investors are more concerned about potential losses than gains and incorporates downside risk measures like Value-at-Risk.Despite advancements of the classical Markowitz theory and minimum VaR approach, challenges remain in accurately estimating parameters, singularity of the covariance matrix and managing risks in volatile markets.In this paper we consider the mean-variance and mean-Var optimal portfolios and take into account the case when the covariance estimated matrix is singular. We use the Moore-Penrose pseudoinverse and Singular Value Decomposition (SVD) to find solutions. We apply these approaches and methodics to real financial data, construct mean-variance and mean-Var optimal portfolios and compare the dynamics of expected returns (mean), volatility and VaR for it.Thanks to the proposed approaches, the investor gets a tool that allows him to make decisions about choosing an approach to building an optimal portfolio, as well as taking into account the singularity of the covariance matrix.