Portfolio Optimization Model based on Median Absolute Deviation-Shannon Entropy by Goal Programming

Document Type : Research Paper

Authors

1 Department of Industrial Management, Faculty of Economics, Management and Administrative Affairs, Semnan University, Semnan, Iran.

2 Department of Business Management, Faculty of Economics, Management and Administrative Affairs, Semnan University, Semnan, Iran.

3 Department of Statistics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.

10.22103/jdc.2024.22442.1435

Abstract

Objective: The main goal of portfolio optimization models is to help investors to achieve the highest rate of return at a certain level of risk. Various models and methods have been presented by researchers to achieve the optimal portfolio with different approaches. The current research is based on portfolio optimization by focusing on assets return distribution and median absolute deviation risk and Shannon entropy diversification criteria. According to this approach, at first returns distribution is identified and then appropriate model is used to select the optimal portfolio and finally performance of the median absolute deviation-Shannon entropy model based on Skew-Normal and Skew-Laplace-Normal statistical distributions are compared.
Method: The data used in this research are monthly returns of 181 stock exchange symbols during a period of 36 months from April 2019 to March 2022, which were randomly obtained from Morgan's table. The portfolio optimization method in this research is three-objective optimization, maximization of average return, minimization of median absolute deviation and maximization of Shannon entropy by using goal programming technique based on Skew-Normal and Skew-Laplace-Normal statistical distributions.
Results: The findings of this research show that median absolute deviation- Shannon entropy model based on the Skew-Laplace-Normal distribution has a higher performance ratio for choosing optimal portfolio in comparison with it’s corresponding model based on the Skew-Normal distribution.
Conclusion: The reason for median absolute deviation-Shannon entropy model based on Skew-Laplace-Normal distribution is considered as preferred model is to pay attention to the descriptive statistics of the skewness and kurtosis of the returns distribution, by considering descriptive statistics of the symbols, the skewness of most of the stock symbols are noticeable. For this reason, simultaneously paying attention to the statistical criteria explaining the characteristics of the skewness and kurtosis of the returns distribution and Shannon entropy diversification criterion lead to it’s better performance.

Keywords

References

آذر، عادل؛ یزدانیان، احمدرضا و قندهاری، مریم (1395). بهینه سازی سبد سهام با استفاده از الگوریتم ژنتیک و روش K-means بهبودیافته بر پایه الگوریتم ژنتیک. همایش ریاضیات و علوم انسانی. [SID].
اسکندری سبزی، سیما و حاجی آقاجانی، اعظم (1402). تأثیر همه‌گیری کووید-19 بر بازار سهام: مطالعه موردی کشورهای منتخب عضو اوپک، مجله توسعه و سرمایه، 8(2)، 168-151. https://jdc.uk.ac.ir/article_3431.html.
بهزادی، عادل و بختیاری، مصطفی (1393). ارائه مدلی بر مبنای میانگین- آنتروپی- چولگی برای بهینه‌سازی سبد سهام در محیط فازی. فصلنامه مهندسی مالی و مدیریت اوراق بهادار، 5(19)، 55-39. https://fej.ctb.iau.ir/article_511497.html.
تقی‌زاده یزدی، محمدرضا؛ فلاح‌پور، سعید و احمدی مقدم، محمد (1395). انتخاب پرتفوی بهینه با استفاده از برنامه‎ریزی فرا آرمانی و برنامه‎ریزی آرمانی ترتیبی توسعه‎یافته. تحقیقات مالی، 18(4)، 591-612. https://jfr.ut.ac.ir/article_62580.html.
تهرانی، رضا؛ فلاح تفتی، سیما و آصفی، سپهر (1397). بهینه‏‌سازی سبد سهام به کمک الگوریتم فراابتکاری دسته‌های میگو با استفاده از معیارهای مختلف از ریسک در بورس اوراق بهادار تهران. تحقیقات مالی، 20(4)، 409-426. https://jfr.ut.ac.ir/article_70018.html.
خندان، عباس (1402). مقایسه عملکرد میانگین با میانه و دیگر شاخص‌های ریسک در بهینه‌سازی سبد سهام، فصلنامه علمی پژوهشی اقتصاد مقداری، 20(1)، 138-99. https://doi.org/10.22055/jqe.2021.36778.2349.
دیده‌خانی، حسین؛ عباسی، ابراهیم؛ شیری قهی، امیر و مشاری، محمد (1398). توسعه مدل بهینه‌سازی پرتفوی میانگین- انحراف مطلق (MAD) با رویکرد عدم قطعیت ترکیبی تصادفی-فازی و در نظر گرفتن نگرش سرمایه‌گذاران به ریسک، فصلنامه مهندسی مالی و مدیریت اوراق بهادار، (10)40، 102-84. https://fej.ctb.iau.ir.
رستمی، محمدرضا؛ کلانتری بنجار، محمود و بهزادی، عادل (1394). گشتاورهای مراتب بالاتر در بهینه‌سازی سبد سهام در محیط فازی، مجله مهندسی مالی و مدیریت اوراق بهادار، 6(24)، 62-41. https://fej.ctb.iau.ir/article_514756.html.
راعی، رضا؛ باجلان، سعید؛ حبیبی، مصطفی و نیک عهد، علی (1396). بهینه‌سازی پرتفوی چند هدفه بر اساس میانگین، واریانس، آنتروپی و الگوریتم ازدحام ذرات، فصلنامة مدلسازی ریسک و مهندسی مالی، 2(3)، 379-362. https://jferm.khatam.ac.ir/article_66584.html?lang=fa.
راعی، رضا و پویان‌فر، احمد (1396). مدیریت سرمایه‌گذاری پیشرفته، تهران، انتشارات سمت. https://samt.ac.ir/fa/book/1582.
محبی، نگین و نجفی، امیرعباس (1397). بهینه‌سازی سبد سرمایه‌گذاری چند دوره‌ای با رویکرد برنامه‌ریزی پویا، فصلنامه مطالعات مدیریت صنعتی، 16(50)، 26-1. https://doi.org/10.22054/jims.2018.9104.
محمدی باغملائی، حسین؛ پارسا، حجت؛ طهماسبی، سعید؛ حاجیانی، پرویز (1400). کاربرد معیار آنتروپی تجمعی و الگوریتم PSO در بهینه‌سازی سبد سهام شرکت‌های پتروشیمی بازار بورس اوراق بهادار. مجله توسعه و سرمایه، 6(2)، 55-41. https://jdc.uk.ac.ir/article_3124.html?lang=fa.
نبی‌زاده، احمد و بهزادی، عادل (1397). گشتاور مراتب بالاتر دربهینه‌سازی پرتفوی با درنظر گـرفتن آنتروپی و استفاده از برنامه‌ریزی آرمانی چندجمله‌ای، فصلنامه تحقیقات مالی، 20(2)، 210-193. https://doi.org/10.22059/frj.2018.255731.1006645.
References
Aksarayli, M., & Pala, O. (2018). A polynomial goal programming model for portfolio optimization based on entropy and higher moments. Expert Systems with Applications, 94, 185-192. https://doi.org/10.1016/j.eswa.2017.10.056.
Azar, A., Yazdanian A. & Ghandehari M. (2019). Stock portfolio optimization using genetic algorithm and adaptive k-means method based on genetic algorithm. Presented in the 4th Seminar of Mathematics and Humanities, Tehran, Iran. [SID] [In Persian].
Azzalini, A. (1985). A Class of Distribution which includes the normal ones. Scandinavian Journal of Statistics, 12(2), 171-178. https://www.jstor.org/stable/4615982.
Behzadi, A., & Bakhtiari, M. (2013). Presenting a model based on mean-entropy-skewness for optimization. Financial Engineering and Portfolio Management, 5(19), 39-55. https://fej.ctb.iau.ir [In Persian].
Bera, A.K., & Park, S.Y. (2008). Optimal Portfolio Diversification Using the Maximum Entropy Principle. Econometric Reviews, 27(4-6), 484-512. https://doi.org/10.1080/07474930801960394.
Cont, R. (2001). Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance 1(2), 223–236. https://doi.org/10.1080/713665670.
Dai, W. (2018). Mean-entropy models for uncertainty portfolio selection. In: Multi-Objective Optimization; Springer: Singapore. DOI: 10.1007/978-981-13-1471-1_2.
DeMiguel, V., Garlappi L., & Uppal R. (2009). Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy? Review of Financial Studies, 22(5), 1915–1953. DOI: 10.1093/rfs/hhm075.
Didekhani, H., Abbasi, E., Shiri Qahi, A., & Mishari, M. (2018). Development of fuzzy random optimization model mean-developing a fuzzy random portfolio optimization model considering investor's risk attitude. Financial Engineering and Portfolio Management, 10(40), 84-102. https://fej.ctb.iau.ir [In Persian].
Erdas, M.L. (2020). Developing a portfolio optimization model based on linear programming under certain constraints: An application on Borsa Istanbul 30 Index. TESAM Akademi Dergisi-Turkish Journal of TESAM Academy, 7(1), 115-141. https://doi.org/10.30626/tesamakademi.696299.
Eskandari Sabzi, S., & Hajiaghajani, A. (2023). The impact of Covid-19 epidemic on stock market; case study of selected OPEC member countries. Journal of Development and Capital, 8(2), 151-168. https://jdc.uk.ac.ir [In Persian].
Gupta, A.K., Chang, F.C., & Huang, W.J. (2003). Some skew-symmetric models. Random Operators Stochastic Equations, 10, 133-140. https://doi.org/10.1515/rose.2002.10.2.133.
Huang, X. (2008). Mean-Entropy Models for Fuzzy Portfolio Selection. IEEE Transactions on Fuzzy Systems, 16(4), 1096-1101. https://doi.org/10.1109/TFUZZ.2008.924200.
Jagannathan, R., & Ma, T. (2003). Risk reduction in large portfolios: Why imposing the wrong constraints helps. Journal of Finance, 58(4), 1651–1684. https://ssrn.com/abstract=310469.
Jana, P., Roy, T.K., & Mazumder, S.K. (2009). Multi-objective possibilistic model for portfolio selection with transaction cost. Journal of Computational and Applied Mathematics, 228(1), 188-196. https://doi.org/10.1016.
Kamali, S. (2014). Portfolio optimization using particle swarm optimization and genetic algorithm. Journal of Mathematics and Computer Science, 10(2), 85-90. http://dx.doi.org/10.22436/jmcs.010.02.01.
Kasenbacher, G., Lee, J., & Euchukanonchai, K. (2017). Mean-variance vs. mean-absolute deviation: A performance comparison of portfolio optimization models. University of British Columbia, Vancouver BC, Canada, Thesis, 1(22), 1-11. DOI: 10.13140/RG.2.2.14459.36644.
Khandan, A. (2023). Comparing the performance of median or mean and other risk indicators in portfolio optimization. Quarterly Journal of Quantitative Economics, 20(1), 99-138. DOI: 10.22055/jqe.2021.36778.2349 [In Persian].
Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its application to Tokyo stock market. Management Science, 37, 519-531. https://doi.org/10.1287/mnsc.37.5.519.
Lam, W.S., Lam, W.H., & Jaaman, S.H. (2021). Portfolio optimization with a mean-absolute deviation-entropy multi-objective model. Entropy, 23(10), 1266. https://doi.org/10.3390/e23101266.
Li, B., & Zhang, R. (2021). A new mean-variance-entropy model for uncertain portfolio optimization with liquidity and diversification. Chaos Solitons Fractals, 146, 1-6. https://doi.org/10.1016/j.chaos.2021.110842.
Lu, S., Zhang, N., & Jia, L. (2021). A multiobjective multiperiod mean-semientropy-skewness model for uncertain portfolio selection. Applied Intelligence, 51, 5233-5258. https://doi.org/10.1007/s10489-020-02079-3.
Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7, 77-91. https://doi.org/10.1111.
Mercurio, P.J.; Wu, Y., & Xie, H. (2020). An entropy-based approach to portfolio optimization. Entropy, 22(3), 332. https://doi.org/10.3390/e22030332.
Mohammadi Baghmolaei, H., Parsa, H., Tahmasebi, S., & Hajiani, P. (2021). application of cumulative entropy measure and PSO algorithm in Tehran Stock Exchange petrochemical companies portfolio optimization. Journal of Development and Capital, 6(2), 41-55. DOI: 10.22103/jdc.2022.17949.1142 [In Persian].
Mohebbi, N., & Najafi, A A. (2018). Multi-period portfolio optimization using dynamic programming approach. Industrial Management Studies Quarterly, 16(50), 1-26. https://doi.org/10.22054/jims.2018.9104 [In Persian].
Nabizade, A., & Behzadi, A. (2017). Higher moments portfolio optimization with entropy based polynomial goal programming. Financial Research Journal, 20(2), 193-210. https://doi.org/10.22059 [In Persian].
Philippatos G., & Wilson C. (1972). Entropy, market risk, and the selection of efficient portfolios. Applied Economics, 4(3), 209-220. https://doi.org/10.1080/00036847200000017.
Raei, R., Bajalan, S., Habibi, M., & Nikahd, A. (2016). Optimization of multi-objective portfolios based on mean, variance, entropy and particle swarm algorithm. Journal of Risk Modeling and Financial Engineering, 2(3), 362-379. https://jferm.khatam.ac.ir/article_66584.html?lang=en [In Persian].
Raei, R., & Pouyanfar, A. (2016). Advanced investment management, Tehran, Samt publications, https://samt.ac.ir [In Persian].
Rasiah, D. (2012). Post-modern portfolio theory supports diversification in an investment portfolio to measure investment's performance. Journal of Finance and Investment Analysis, 1(1), 1-3. https://ideas.repec.org.
Rom, B.M. & Ferguson, K.W. (1993). Post-modern portfolio theory comes of age. The Journal of Investing, 2(4), 27-33. https://www.actuaries.org/AFIR/colloquia/Orlando/Ferguson_Rom.pdf.
Rostami, M., Benjar, M., & Behzadi, A. (2014). Higher order moments in stock portfolio optimization in fuzzy environment. Financial Engineering and Portfolio Management, 6(24), 41-62. https://fej.ctb.iau.ir [In Persian].
Rotela Junior, P., Rocha, L.C.S., Aquila, G., Balestrassi, P.P., Peruchi, R.S., & Lacerda, L.S. (2017). Entropic data envelopment analysis: A diversification approach for portfolio optimization. Entropy, 19(9), 352. https://doi.org.
Samuelson, P.A. (1970). The fundamental approximation theorem of portfolio analysis in terms of means, variances and higher moments. The Review of Economic Studies, 37(4), 537-542. http://hdl.handle.net.
Shannon, C.E. (1948). A mathematical theory of communication. The Bell System Technical Journal, 27, 379-423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x.
Taghizadeh Yazdi, M., Fallahpour, S. & Ahmadi Moghaddam, M. (2017). Portfolio selection by means of Meta-goal programming and extended lexicograph goal programming approaches. Financial Research Journal, 18(4), 591-612. https://doi.org/10.22059/JFR.2017.62580 [In Persian].
Tayi, G.K., & Leonard, P.A. (1988). Bank balance-sheet management: An alternative multiobjective model. Journal of the Operational Research Society, 39(4), 401-410. https://doi.org/10.1057/jors.1988.68.
Tehrani, R., Fallah, S.T., & Asefi, S. (2018). Portfolio optimization using krill herd metaheuristic algorithm considering different measures of risk in Tehran Stock Exchange. Financial Research Journal, 20(4), 409-426. https://doi.org/10.22059/FRJ.2019.244004.1006538 [In Persian].
Zhang, W.G., Liu, Y.J., & Xu, W.J. (2012). A possibilistic mean-semi variance-entropy model for multi-period portfolio selection with transaction costs. European Journal of Operational Research, 222(2), 341-349. DOI: 10.1016/j.ejor.2012.04.023.
Zhou, R. (2017). Properties of risk measures of generalized entropy in portfolio selection. Entropy, 19, 657. DOI:10.3390/e19120657.

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History

  • Receive Date: 02 November 2023
  • Revise Date: 29 June 2024
  • Accept Date: 24 July 2024

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