University of Isfahan Journal of Asset Management and Financing 2383-1189 11 4 2023 12 22 Expansion of the Markowitz Model in Portfolio Optimization Considering Realistic Constraints Expansion of the Markowitz Model in Portfolio Optimization Considering Realistic Constraints 65 92 28246 10.22108/amf.2024.137329.1793 FA Hossein Alizadeh MSc, Department of Industrial Engineering and Futures Studies, Faculty of Engineering, University of Isfahan, Isfahan, Iran Kamran Kianfar Assistant Professor, Department of Industrial Engineering and Futures Studies, Faculty of Engineering, University of Isfahan, Isfahan, Iran Journal Article 2023 04 12 This research aims to expand the Markowitz model in a way that aligns more closely with real-world conditions, considering various fundamental analysis factors and market constraints. Conducted on the Tehran Stock Exchange, this study utilizes two measures, semi-variance and mean absolute deviations, alongside the variance measure in the Markowitz model to better estimate risk levels. In addition, several constraints such as cardinality constraint, threshold constraint, and segmentation constraint are employed to bring the results of the Markowitz-based model closer to reality. To ensure that the stock return metric is not solely based on stock price changes, this research incorporates nine important fundamental analysis metrics in filtering company stocks and as a return metric in the Markowitz model. Due to the computational complexity of the mathematical programming model, sample problems were also solved using the Harmony Search algorithm. The results indicate that the mathematical model performs better in terms of the distance from the ideal point efficiency metric, while the harmony search algorithm excels in uniformity metrics, exploring diverse solutions, and solution time. Increasing the range of cardinality and threshold constraints results in selecting more stocks in the portfolio, and simultaneously, the risk and return objective functions will improve concurrently. Keywords: Markowitz Model, Stock Portfolio Selection, Fundamental Analysis, Stock Risk Management, Harmony Search Algorithm. Introduction In this study, the semi-variance and mean absolute deviation models are used alongside the mean-variance model for comparison. Developed models for optimizing stock portfolios heavily rely on realistic constraints. Budget constraints, cardinality constraints, threshold constraints, and segmentation constraints are among the most crucial constraints in the stock market, which were utilized in this research for portfolio selection (Mehrjerdi & Rasaei, 2013). Besides return metrics, indicators such as P/E (price to earnings per share), ROE (return on equity), six-month turnover rate, and company total asset ratio are used in the current research for initial stock screening, followed by metrics such as P/S (price per share to sales), ROA (return on assets), quick ratio, and turnover to total market value in the objective functions of the Markowitz model (Asgarnezhad, 2018). This study, conducted on the Tehran Stock Exchange, utilizes nine important and strategic industry groups that have a significant impact on the overall index. The study period covers the second six months of the year 2022, and questionnaires were completed by capital market experts and brokers from the Mashhad Mofid brokerage. The main question of this study in the field of financial optimization is how to expand the Markowitz (1952) model to better align with real-world conditions. Alongside the main question, several subsidiary questions arise, including: 1) How can stocks be selected based on realistic constraints? 2) How is the balance between risk and return established in portfolio selection? 3) Which fundamental analysis indicators are more important in stock portfolio selection on the Tehran Stock Exchange? Materials and Methods The research aims to minimize portfolio investment risk and maximize expected portfolio return. The first objective seeks to select a combination of stocks with the least investment risk, utilizing three different measures: variance, semi-variance, and mean absolute deviation. Due to the abundance of factors and sub-factors, two questionnaires were designed for financial market specialists to prioritize factors through analytic hierarchy process (AHP) analysis. Fifteen experts with over five years of experience in financial, accounting, or managerial fields were selected through purposive sampling. In the proposed mathematical programming model, sub-factors with higher importance within each group are incorporated into the objective function, while less important ones are used as filters before entering the model. For instance, ROA, P/S, quick ratio, market value of the company to industry, and percentage of companies' operating profit are included in the objective function, while ROE, P/E, turnover rate, and percentage of total assets serve as filters. Due to the nonlinearity and NP-hardness of the portfolio optimization problem, traditional mathematical programming models may not yield solutions within a reasonable timeframe, especially for large-scale problem instances. Hence, the harmony search algorithm is employed to reduce solution time and approximate the optimal solution. The algorithm is implemented in MATLAB software, where a solution string representing the investment proportion in each stock is defined, allowing for the quick derivation of other variable values in the mathematical programming model. Research Findings The results of the study indicate that all 15 questionnaires had inconsistency rates lower than 0.1, indicating a good level of consistency. Therefore, the geometric mean was calculated from these 15 questionnaires. In the questionnaires, factors influencing the five selected factors were compared with each other (See Table 1). Table 1. Results of Expert Opinions in the Questionnaire for Selected Factors Factor Sub-factor (average of weight) Profitability ROA (0.392) ROE (0.374) Net Profit Margin on Sales (0.127) Operating Expense Ratio (0.108) Valuation P/S (0.302) P/E (0.301) Ln(A/B) (0.162) Ln(B/P) (0.124) Liquidity Asset Liability Ratio (0.388) Quick Ratio (0.374) Equity to Dept Ratio (0.125) Current Ratio (0.113) Volume Turnover to total market 3-month (0.447) Turnover to total market 1-month (0.260) Turnover rate 3-month (0.178) Turnover rate 1-month (0.114) Growth Total assets (0.394) Operating profit (0.320) Gross profit (0.286) All necessary data have been obtained from www.tsetmc.com and www.codal.ir websites and have been incorporated into the filters. To compare the results of the exact solution method and the harmony search metaheuristic, different sizes of sample problems need to be defined. For this purpose, sample groups N10 to N55, consisting of 10 to 55 stocks, have been defined, including categories such as oil derivatives, basic metals, chemical products, automobiles, metal extraction, food products, investments, pharmaceuticals, and banks. In the N30 group, the GAMS software requires a minimum solution time of 560 seconds, while the harmony search algorithm needs only 33 seconds to obtain local optimal points. A comparison between the two solution methods is illustrated in Figure 1, demonstrating that the SEMICOV objective function performs best for risk calculation in both exact and metaheuristic solution methods. Both figures exhibit robustness, and the solutions are well dispersed in the space. Figure 1. Pareto Charts of Three Objective Functions for Group N30 The study compares the exact method and the harmony search algorithm for optimizing stock portfolios. Results show the harmony search method's efficiency, especially in larger dimensions, despite slight differences with the exact method. The SEMICOV criterion performs well, but the MAD criterion shows instability. Additionally, the harmony search algorithm proves superior in uniformity, scalability, and solution time. Sensitivity analysis highlights the importance of parameter variations in portfolio optimization. Discussion of Results and Conclusion The results revealed that the semi-variance objective yielded better results in both the exact solution method and the harmony search algorithm in most groups of the test instances. The harmony search algorithm provides a good approximation of the optimal solution in a much shorter time compared to the mathematical model. The harmony search algorithm performed better in terms of uniformity and spread metrics of Pareto points, but the exact solution method was superior in calculating the distance from the ideal point metric. The sensitivity analysis conducted on the N30 group highlighted the significant impact of altering constraint selection ranges on the objective function and, consequently, on the stock portfolio. Initially, expanding the ranges of both the cardinality and threshold constraints led to the inclusion of stocks with higher returns and lower risks at the Pareto frontier across all risk assessment metrics. Specifically, in the semi-variance objective function, the optimal stock portfolio was found in subgroup with a return of 0.84 and a corresponding risk of 0.146. As the range decreased in subgroup , returns diminished to 0.526 with a risk of 0.15. In subgroup , where both constraints became more stringent, returns reached 0.284 with a risk value of 0.16. Consequently, the analysis concluded that broader constraint ranges corresponded to lower risks and higher returns, and vice versa. The applications of this research are for individual shareholders, financial institutions, investment funds, and portfolio managers who can use this research to improve their returns and reduce their trading risk. This research aims to expand the Markowitz model in a way that aligns more closely with real-world conditions, considering various fundamental analysis factors and market constraints. Conducted on the Tehran Stock Exchange, this study utilizes two measures, semi-variance and mean absolute deviations, alongside the variance measure in the Markowitz model to better estimate risk levels. In addition, several constraints such as cardinality constraint, threshold constraint, and segmentation constraint are employed to bring the results of the Markowitz-based model closer to reality. To ensure that the stock return metric is not solely based on stock price changes, this research incorporates nine important fundamental analysis metrics in filtering company stocks and as a return metric in the Markowitz model. Due to the computational complexity of the mathematical programming model, sample problems were also solved using the Harmony Search algorithm. The results indicate that the mathematical model performs better in terms of the distance from the ideal point efficiency metric, while the harmony search algorithm excels in uniformity metrics, exploring diverse solutions, and solution time. Increasing the range of cardinality and threshold constraints results in selecting more stocks in the portfolio, and simultaneously, the risk and return objective functions will improve concurrently. Keywords: Markowitz Model, Stock Portfolio Selection, Fundamental Analysis, Stock Risk Management, Harmony Search Algorithm. Introduction In this study, the semi-variance and mean absolute deviation models are used alongside the mean-variance model for comparison. Developed models for optimizing stock portfolios heavily rely on realistic constraints. Budget constraints, cardinality constraints, threshold constraints, and segmentation constraints are among the most crucial constraints in the stock market, which were utilized in this research for portfolio selection (Mehrjerdi & Rasaei, 2013). Besides return metrics, indicators such as P/E (price to earnings per share), ROE (return on equity), six-month turnover rate, and company total asset ratio are used in the current research for initial stock screening, followed by metrics such as P/S (price per share to sales), ROA (return on assets), quick ratio, and turnover to total market value in the objective functions of the Markowitz model (Asgarnezhad, 2018). This study, conducted on the Tehran Stock Exchange, utilizes nine important and strategic industry groups that have a significant impact on the overall index. The study period covers the second six months of the year 2022, and questionnaires were completed by capital market experts and brokers from the Mashhad Mofid brokerage. The main question of this study in the field of financial optimization is how to expand the Markowitz (1952) model to better align with real-world conditions. Alongside the main question, several subsidiary questions arise, including: 1) How can stocks be selected based on realistic constraints? 2) How is the balance between risk and return established in portfolio selection? 3) Which fundamental analysis indicators are more important in stock portfolio selection on the Tehran Stock Exchange? Materials and Methods The research aims to minimize portfolio investment risk and maximize expected portfolio return. The first objective seeks to select a combination of stocks with the least investment risk, utilizing three different measures: variance, semi-variance, and mean absolute deviation. Due to the abundance of factors and sub-factors, two questionnaires were designed for financial market specialists to prioritize factors through analytic hierarchy process (AHP) analysis. Fifteen experts with over five years of experience in financial, accounting, or managerial fields were selected through purposive sampling. In the proposed mathematical programming model, sub-factors with higher importance within each group are incorporated into the objective function, while less important ones are used as filters before entering the model. For instance, ROA, P/S, quick ratio, market value of the company to industry, and percentage of companies' operating profit are included in the objective function, while ROE, P/E, turnover rate, and percentage of total assets serve as filters. Due to the nonlinearity and NP-hardness of the portfolio optimization problem, traditional mathematical programming models may not yield solutions within a reasonable timeframe, especially for large-scale problem instances. Hence, the harmony search algorithm is employed to reduce solution time and approximate the optimal solution. The algorithm is implemented in MATLAB software, where a solution string representing the investment proportion in each stock is defined, allowing for the quick derivation of other variable values in the mathematical programming model. Research Findings The results of the study indicate that all 15 questionnaires had inconsistency rates lower than 0.1, indicating a good level of consistency. Therefore, the geometric mean was calculated from these 15 questionnaires. In the questionnaires, factors influencing the five selected factors were compared with each other (See Table 1). Table 1. Results of Expert Opinions in the Questionnaire for Selected Factors Factor Sub-factor (average of weight) Profitability ROA (0.392) ROE (0.374) Net Profit Margin on Sales (0.127) Operating Expense Ratio (0.108) Valuation P/S (0.302) P/E (0.301) Ln(A/B) (0.162) Ln(B/P) (0.124) Liquidity Asset Liability Ratio (0.388) Quick Ratio (0.374) Equity to Dept Ratio (0.125) Current Ratio (0.113) Volume Turnover to total market 3-month (0.447) Turnover to total market 1-month (0.260) Turnover rate 3-month (0.178) Turnover rate 1-month (0.114) Growth Total assets (0.394) Operating profit (0.320) Gross profit (0.286) All necessary data have been obtained from www.tsetmc.com and www.codal.ir websites and have been incorporated into the filters. To compare the results of the exact solution method and the harmony search metaheuristic, different sizes of sample problems need to be defined. For this purpose, sample groups N10 to N55, consisting of 10 to 55 stocks, have been defined, including categories such as oil derivatives, basic metals, chemical products, automobiles, metal extraction, food products, investments, pharmaceuticals, and banks. In the N30 group, the GAMS software requires a minimum solution time of 560 seconds, while the harmony search algorithm needs only 33 seconds to obtain local optimal points. A comparison between the two solution methods is illustrated in Figure 1, demonstrating that the SEMICOV objective function performs best for risk calculation in both exact and metaheuristic solution methods. Both figures exhibit robustness, and the solutions are well dispersed in the space. Figure 1. Pareto Charts of Three Objective Functions for Group N30 The study compares the exact method and the harmony search algorithm for optimizing stock portfolios. Results show the harmony search method's efficiency, especially in larger dimensions, despite slight differences with the exact method. The SEMICOV criterion performs well, but the MAD criterion shows instability. Additionally, the harmony search algorithm proves superior in uniformity, scalability, and solution time. Sensitivity analysis highlights the importance of parameter variations in portfolio optimization. Discussion of Results and Conclusion The results revealed that the semi-variance objective yielded better results in both the exact solution method and the harmony search algorithm in most groups of the test instances. The harmony search algorithm provides a good approximation of the optimal solution in a much shorter time compared to the mathematical model. The harmony search algorithm performed better in terms of uniformity and spread metrics of Pareto points, but the exact solution method was superior in calculating the distance from the ideal point metric. The sensitivity analysis conducted on the N30 group highlighted the significant impact of altering constraint selection ranges on the objective function and, consequently, on the stock portfolio. Initially, expanding the ranges of both the cardinality and threshold constraints led to the inclusion of stocks with higher returns and lower risks at the Pareto frontier across all risk assessment metrics. Specifically, in the semi-variance objective function, the optimal stock portfolio was found in subgroup with a return of 0.84 and a corresponding risk of 0.146. As the range decreased in subgroup , returns diminished to 0.526 with a risk of 0.15. In subgroup , where both constraints became more stringent, returns reached 0.284 with a risk value of 0.16. Consequently, the analysis concluded that broader constraint ranges corresponded to lower risks and higher returns, and vice versa. The applications of this research are for individual shareholders, financial institutions, investment funds, and portfolio managers who can use this research to improve their returns and reduce their trading risk.

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