Application of Cumulative Entropy Measure and PSO Algorithm in Tehran Stock Exchange Petrochemical Companies Portfolio Optimization

The main object of this study is to use a new measure of risk called Cumulative entropy in the Markowitz portfolio optimization model and solve this model using Particle swarm optimization (PSO) to optimize the portfolio of Petrochemical companies by Applying the data consist of monthly returns of the Fifteen petrochemical companies in Tehran Stock Exchange from 2013 to 2019. The Markowitz model uses the variance as a risk measure by default. in this study, a new measure of risk called Cumulative Entropy is introduced. This measure can be used in many issues without considering the limitations of variance (standard deviation). 

The Markowitz model is one of the most important models for solving portfolio optimization problems, but this model has many disadvantages. The Markowitz optimization problem can be solved by simple mathematical programming models when the number of assets to be invested and the market constraints are small, but when the real-world conditions and constraints are taken into account, the problem becomes complex and difficult. One of the methods that have solved human ambiguities in recent years in solving many optimization problems and has been successful in responding to complex problems is the so-called intelligent methods and algorithms. Intelligent methods that were introduced to eliminate the shortcomings of classical (traditional) optimization methods with a comprehensive and random search, largely guarantees the possibility of achieving better results.Due to the mentioned problems in this research, a new criterion by the name Cumulative entropy is introduced which can be used as an alternative to variance in the Markowitz mean-variance optimization model as a risk criterion. Also, due to the mentioned problems for the Markowitz model, in this research, the meta-innovative particle cumulative motion (PSO) algorithm will be used to optimize the stock portfolio for Petrochemical companies stocks.

As can be seen in the PSO algorithm, the average value of the stock return function is less than the average value of the stock return function in the Markowitz model, while the average value of the portfolio risk function is well minimized to a value less than the average portfolio risk function in the Markowitz model. For the final comparison of these two models, using the values of the table, the Reward to Volatility index (which is defined as the ratio of return to portfolio risk) is calculated that in the PSO algorithm is higher than the Markowitz model; Therefore, it can be seen that the PSO algorithm performs portfolio optimization better than the Markowitz model and produces optimal answers 

According to the research findings the Cumulative Entropy measure can be used in many issues without considering the limitations of variance (standard deviation).In the PSO algorithm, the average value of the stock return function is slightly less than the average value of the stock return function in the Markowitz model, while the average value of the portfolio risk function is much less than the average value of the portfolio risk function in Markowitz programming. Comparing the volatility reward index of the PSO algorithm with the Markowitz model, it was observed that the value of this index is higher in the PSO algorithm, which shows PSO algorithm performs portfolio optimization better and produces optimal answers.