Dynamical System Modeling and Stability Investigation�DSMSI-2023

Dedicated to the 77th anniversary of the outstanding Ukrainian scientist

professor Denys Khusainov

December 19-21, 2023, Kyiv, Ukraine

Diversification of stock portfolio structure under market restrictions

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Oleksiy Oletsky, National University of Kyiv-Mohyla Akademy

Victor Kulian, Taras Schevchenko National University of Kyiv

Maryna Korobova, Taras Shevchenko National University of Kyiv

Olena Yunkova, Kyiv National Economic University named after Vadym Getman

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Introduction

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Basic principles of decision-making when investing in securities:

• - H. Markowitz approach, using a valid and effective set, investor indifference curves; The theory of H. Markowitz is classic and its principles are the basis of many decision-making strategies in the stock market.
• - Splitting two-criterion investment portfolio optimization problem into two one-criterion; The mathematical two-criterion problem of optimizing a portfolio of risky securities involves maximizing the expected return and minimizing the risk of the investment portfolio.
• - Methods of technical analysis; The most common technology for making practical decisions involves the use of known statistical information about the dynamics of the market value of one share.
• - Methods of fundamental analysis. The theory of fundamental analysis is actively developing and includes new mathematical models and methods for describing the dynamics and procedures for making decisions about the optimal structure of an investment portfolio. In the researches by K. Ito, F. Black, M. Scholes, and R. Merton, new fundamental results are formulated. The results are actively developed and implemented in the practice of decision-making on the stock market.

Dynamical System Modeling and Stability Investigation, DSMSI-2023

Development of Portfolio Investment Model

• The portfolio model was first proposed by H. Markowitz (1952) . Based on the relationship between income and risk, H. Markowitz constructs the mean-variance model. He put forward the risk measurement and the modern principles of portfolio investment. This has laid a theoretical foundation for the researches of securities investment portfolio.
• Sharpe’s (1964) capital asset pricing model have been known and are used by many investors when solving the problems of investing in shares.
• Porter, Huang (2008) have studied the mean-variance model.
• H. Levy and M. Levy (2009), Chiu and Li (2010) have studied the safety-first model.
• Malkiel (2003) have studied the negative effective investment strategy in the efficient market.
• Dierkes (2010) uses the Cumulative Prospect Theory to describe the investor preference.
• Zhang (2014) proposes the multi-period fuzzy portfolio model, and then uses genetic algorithm, hybrid intelligent algorithm and differential approximation algorithm to solve the model.

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Mathematical modeling in portfolio analysis

• Construction of a mathematical model of a stock portfolio

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Choosing of optimal portfolio structure�

Identification of parameters in the task of the structure optimizing of portfolio investments�

The most well-known and common methods of identifying parameters of discrete and continuous models include:

• - correlation-dispersion analysis;
• - approaches based on the principles of the theory of stability;
• - approaches based on methods of analyzing the sensitivity of solutions.

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Optimal portfolio diversification�

E

The risk optimization procedure for the optimal expected return portfolio consists in choosing at each step admissible portfolios that lie on the EF line. This line connects point E, which corresponds to the optimal market value of the portfolio, and point F, which belongs to the efficient set.

Conclusion

• In this study new mathematical formulations of the optimization problems of the stock portfolio structure are given and methods of their solution are developed.
• Mathematical problems formulated on the basis of models of the dynamics of the market value of one share and a portfolio of shares make it possible to solve the problem of optimal diversification of the investment portfolio, taking into account quantitative and qualitative market restrictions on the portfolio structure.
• The approach of H. Markowitz was applied to solve the problem of optimal diversification of the stock portfolio in the presence of market restrictions on the portfolio structure.
• The developed method can be effectively applied to computer programs ("trading robots") that perform automated selection of the optimal portfolio structure.

Thank you for your attention