Markowitz, in his Portfolio selection theory, stated that investors select their portfolios according to two criteria of risk and return. Accordingly, he presented his mathematical model. One of the criticisms of this model is that while investors, practically, consider different criteria in forming their portfolios, it only considers the return mean and return standard deviation. Liquidity is one of the most important criteria in forming portfolios. The present research aims at merging this criterion with Markowitz’s suggested model in Iran’s market using liquidity filtering, liquidity constraints and thus forming a model by using of which investors form a portfolio whose return, risk and liquidity is optimal. The research results show that liquidity in high levels has an effect on investors decisions and their efficient frontiers. 

Following the publication of Markowitz’s Article in 1952, coming up with the best way for optimizing the portfolio has been always one of the concerns ahead of the activists in the investment management industry. In the recent decade, the introduction of the mathematical and operational research models is one of the activities which could affect the portfolio optimization. The present research tries to optimize portfolio making use of the robust optimization and the estimate of the portfolio return and risk and the comparison of the predicted risk and return of this model with those of the classic one. The research studies 115 monthly portfolios within some ten years and estimates the risk and return of each portfolio based on two robust optimization and classic model. At the next stage, the research makes use of average pair test to determine any significant difference between the risk and return predicted for the above model. The present research determined that the return predicted for the robust model is way above that of the classic model; and the risk predicted for the robust model is way below the risk predicted for the classic model. while the real risk of portfolios optimized by the robust method is way below those optimized by the classic method. The results obtained on the estimate of the return coincide with the findings of the foreign studies and does not have any difference with these researches in risk estimation. It goes without saying that none of the foreign and domestic studies have dealt with the performance of the portfolios optimized by these two models.

Risk assessment and the selection of an optimal portfolio selection are critical issues in financial research, investment firms, and among investors. Traditional optimization models like Mean-Variance, Value-at-Risk, Conditional Value-at-Risk, and Omega often rely on historical returns, which can be insufficient in optimizing return and reducing risk. Recently, researches have used predictive return models in the optimization process. This study uses ARIMA model to predict stock returns alongside a mean-variance optimization model (ARIMA_MV) using Iranian exchange market data from 1395 to 1401. First, stock returns have been predicted using ARIMA model and error criteria such as MSE, MAD, HR, HR+ and HR- are estimated. The results show that despite its simplicity, the ARIMA model has a relatively good performance in predicting the return of stock. Then, using the predicted return and the mean variance model, the optimal portfolio has been calculated in the sliding window process. The results show that portfolios calculated by ARIMA_MV model outperform the Tehran Exchange Index (TEDPIX), risk-adjusted criterion like Sharpe and Jensen's alpha ratios and mean_variance model with historical data (HMV). With the model developed in this project (ARIMA_MV), investment companies can offer shareholders higher returns and lower risks, potentially increasing company value in the capital market and boosting shareholder wealth.

Portfolio selection is an important issue for researchers and practitioners. The portfolio must incorporate what the investor believes to be an acceptable balance between risk and reward. The classical Markowitz model uses the variance as the risk measure and is a quadratic programming problem, which is difficult to find the global optimal solution for those problems. Many attempts have been made to linearize the portfolio optimization problem. This paper presents a linear fuzzy portfolio selection model with fuzzy return rates and risks, where the objective is to minimize the downside risk constrained so that a given expected return should be achieved. Furthermore, the proposed approach is demonstrated and validated by a numerical example from real stocks dataset obtainable from Tehran stock exchange market.

In today's world, optimizing investment portfolios has received increasing attention. While predicting the expected returns of investment options and incorporating them into the objective function for profit maximization is a common practice, the most significant innovation in current research is the minimization of prediction error as the objective function. This innovation advises investors to emphasize not only on profit and risk but also on the predictability of investment options when forming an investment portfolio. Integrating return prediction from traditional time series models into portfolio formation can enhance the performance of the primary portfolio optimization model. Since machine learning and deep learning models have demonstrated a significant superiority over time series models, this paper combines return prediction in portfolio formation with machine learning models, namely Random Forest, and deep learning model, Long Short-Term Memory (LSTM). To evaluate the performance of the proposed model, five years of historical data from 2017 to 2021 are used for five industry sectors: banking, automotive, pharmaceutical, metal, and petroleum. The experimental results demonstrate that the mean-variance optimization models perform better when return prediction is done using Random Forest

Portfolio optimization problem is one of the most interesting and practical issues in the financial markets. Portfolio optimization is applied as an applicable and efficiant tool for helping investors in their decision making by allocating wealth to the different asset with controlling the return and risk. The purpose of this paper is to develop a novel portfolio selection and optimization method in portfolio selection problem by considering the return and risk under fuzziness. In the paper, two possibilistic programming model is developed by applying measures of the probabilistic and possibilistic mean and downside risk of fuzzy return. The performance of the proposed models was evaluated by using historical data introduced by Markowitz and data of the Tehran Stock Exchange. The results of the paper show that the proposed models are able to propose an appropriate portfolio for investors with optimizing the return and risk, simultaneously, in terms of different investment strategies.

Choosing a stock portfolio in investment discussions is a difficult and difficult task. Deciding which stock is in a better position compared to other stocks and deserves to be selected and placed in one's investment portfolio, and how to allocate capital between these stocks, is a complex issue. Considering the importance of the issue,The main purpose of this research was to investigate the selection of the portfolio of stock assets based on the method, risk and return of the stock market in the Tehran stock market. This study is post-event in terms of practical and methodological purpose, the data collected annually from 1391 to 1402, which includes twenty active companies in the Iranian stock market. which was selected using the risk minimization model based on the Markotiz model of the optimal portfolio. The results and analysis show that the selection of the efficient portfolio is based on the efficiency frontier diagram of the latest stock price of the companies based on yield and risk, for example,Amir Kabir Kashan steel has less risk and more yield than all the items. Pars Industrial Carbon Black is less risky than Fan Avran Petrochemical, Chemical Daro Pars. However, it is less efficient compared to Techno-Avaran petrochemicals, but it is more efficient compared to Dorofos chemical. Khuzestan steel has the highest risk and zero return of all items and should not be chosen. Chador Melo, Khark Petrochemical, Mobarakeh Steel have more risk and very very little return,And should not be chosen.

This research is aimed to address the optimization of a product portfolio problem under uncertainty using the principles of , nancial portfolios theory. Since the success of a product portfolio is dependent on strategic decision making as well as on future changes of return, the return is best considered when it is deemed an uncertain parameter. The speci, c innovation of this research is the use of a robust optimization approach and providing an exact solution algorithm based on the model of Bertsimas and Sim. Given the uncertainty of the returns, the product portfolio model was developed based on the robust counterpart formulation of Bertsimas and Sim. An exact solution algorithm was also formulated to reduce the solution time. The results obtained by applying the model to a real case study of the dairy industry in Iran showed that increasing the con, dence level would decrease total returns of the portfolio and increase its total risk. A comparison between the proposed algorithm and similar methods showed that, on average, it would make 3% improvement in the solution time.