Comparing Optimal Portfolio Performance Based on Skew-Normal Distribution and Skew-Laplace-Normal Distribution: A Mean-Absolute Deviation-Entropy Approach

Document Type : Research Paper

Authors

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

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

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

Abstract

Objective
Investors typically seek to strike the optimal balance between potential returns and associated risks in their trades. Various models have been presented to choose the optimal portfolio using different approaches. one of these methods is based on the statistical distribution of asset return. In these methods, the type of distribution of returns is first identified, and a suitable portfolio selection method is then applied based on this identified distribution type. This study compares the effectiveness of the mean-absolute deviation-entropy model utilizing both Skew-Normal Distribution and Skew-Laplace-Normal Distribution for constructing an optimal portfolio in the Tehran Stock Exchange over 36 months from April 2018 to March 2020.
 
Methods
The data used in this study comprises the monthly returns of 181 companies listed on the Tehran Stock Exchange. These returns were gathered from a statistical population of 338 members utilizing Morgan's table and Cochran’s formula. After fitting density functions for Skew-Normal and Skew-Laplace-Normal distributions to the returns, maximum likelihood estimates were obtained using the Stats package and the optim Function in R software. The reliability of these estimates was then checked using bootstrap sampling with 1,000 repetitions. Subsequently, relationships corresponding to the mathematical expectation of return distribution and the objective function representing the risk of absolute deviation were estimated using numerical methods. Therefore, this paper aimed to propose a multi-objective optimization model, namely a mean-absolute deviation-entropy model for portfolio optimization by using a goal-programming approach based on Skew-Normal Distribution and Skew-Laplace-Normal Distribution. The objective functions of the model were to maximize the mean return, minimize the absolute deviation, and maximize the entropy of the portfolio.
 
Results
It can be inferred from the observed values ​​of the descriptive statistics of the monthly stock returns corresponding to the stock exchange symbols that some stocks have different skewness and kurtosis values ​​compared to the normal distribution. For example, The symbol "Shepna" exhibits negative skewness, indicating a left-skewed distribution. Similarly, the distribution of the "Basama" symbol exceeds the normal distribution. These instances suggest that the normal distribution is inadequate for describing monthly return distributions. Instead, distributions with parameters should be employed to account for skewness and kurtosis. According to the obtained results, the model utilizing the Skew- Laplace- Normal distribution has a higher performance ratio than the model based on the Skew-Normal distribution.
 
Conclusion
The reason for this superiority, where the model utilizing the Skew-Laplace-Normal distribution outperforms the model based on the Skew-Normal distribution, is the incorporation of both skewness and kurtosis criteria within the former. Additionally, upon analyzing the descriptive statistics of the symbols, it's evident that the kurtosis of most stock symbols is substantial. Therefore, integrating a combination of higher-order moments (skewness and kurtosis) along with entropy leads to enhanced performance.
 

Keywords

Main Subjects

References

 
Aksarayli, M. & Pala, O. (2018). A polynomial goal programming model for portfolio optimization based on entropy and higher moments. Expert System Applied, 94, 185-192.
Arditti, F.D. (1971). Another look at mutual fund performance. Journal of Financial and Quantitative Analysis, 6(3), 909-912.
Azzalini, A. (1985). A Class of Distribution which includes the normal ones. Scandinavian Journal of Statistics, 12, 171-178.
Behzadi, A. & Bakhtiari, M. (2013). Presenting a model based on mean- entropy- skewness for stock portfolio optimization in fuzzy environment. Journal of Financial Engineering and Securities Management, 5(19), 39-55. (in Persian)
Bera, A.K. & Park, S.Y. (2008). Optimal Portfolio Diversification Using the Maximum Entropy Principle. Econometric Reviews, 27(15), 484-512.
Choi, B.G., Rujeerapaiboon, N. & Jiang, R. (2016). Multi- period portfolio optimization: Translation of autocorrelation risk to excess variance. Operations Research, 44(6), 801-807.
Dedekhani, H., Abbasi, E., Shiri Qahi, A. & Meshari, M. (2018). Development of mean-absolute deviation (MAD) portfolio optimization model with random- fuzzy mixed uncertainty approach and considering investors attitude to risk. Financial Engineering and Securities Management, 10(40), 84-102. (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.
Gupta, A.K., Chang, F.C. & Huang, W.J. (2003). Some skew-symmetric models. Random Operators Stochastic Equations, 10, 133-140.
Huang, X. (2008). Mean- Entropy Models for Fuzzy Portfolio Selection. IEEE Transactions on Fuzzy Systems, 16(21): 170-176.
Jana, P., Roy, T.K. & Mazumder, S.K. (2009). Multi-objective possibility model for portfolio selection with transaction cost. Journal of Computational and Applied Mathematics, 228, 188-196.
Kasenbacher, G., Lee, J. & Euchukanonchai, K. (2019). Mean-variance vs. mean-absolute deviation: A performance comparison of portfolio optimization models. University of British Columbia, Vancouver BC, Canada, Thesis.
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.
Konno, H. & Suzuki, K.I. (1992). A fast algorithm for solving large scale mean-variance models by compact factorization of covariance matrices. Journal of the operations research society of Japan, 35 (1), 93-104.
Konno, H. & Yamazaki, H. (1991). Mean-Absolute Deviation Portfolio Optimization Model and Its Application to Tokyo Stock Market. Management Science, 37, 519-531.
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.
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.
Li, D. (2016). Optimal Dynamic Portfolio Selection: Multiperiod Meanvariance Formulation. Mathematical Finance, 10(5), 387-406.
Lu, S., Zhang, N. & Jia, L. (2021). A multiobjective multiperiod mean-semientropy-skewness model for uncertain portfolio selection. Applied Intelligence, 51, 5233-5258.
Markowitz, H. (1952). Portfolio Selection, Journal of Finance, 7, 77-91.
Nabizadeh, A. & Behzad, A. (2018). Higher Moments Portfolio Optimization Considering Entropy based on Polynomial Idealistic Programming. Financial Research Journal, 20(2), 191-208. (in Persian)
Philippatos G. & Wilson, C. (1972). Entropy, market risk, and the selection of efficient portfolios. Applied Economics, 4(3), 209-220.
Pindoriya, N. (2014). Multi-Objective Mean- variance- skewness Model for Generation Portfolio Allocation in Electricity Markets. Electric Power Systems Research, 80(10): 1314-1321.
Raei, R., Bajelan, S., Habibi, M., & Nikahd, A. (2017). Optimization of Multi-Objective Portfolios Based on Mean, Variance, Entropy and Particle Swarm Algorithm. Quarterly Journal of Risk Modeling and Financial Engineering, 2(3), 362-379. (in Persian)
Rai, R. & Poyanfar, A. (2018). Advanced investment management. Tehran: Publications of the Organization for the Study and Compilation of Humanities Books in Universities (Samt). (in Persian)
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.
Shannon, C.E. (1948). A Mathematical Theory of Communication. The Bell System Technical Journal, 27, 379-423.
Simkowitz, M.A. & Beedles, W.L. (1978). Diversification in a three-moment world. Journal of Financial and Quantitative Analysis, 13(5), 927-941.
Sun, Y.F., Aw, G., Loxton, R. & Teo, K.L. (2017). Chance-constrained optimization for pension fund portfolios in the presence of default risk. European Journal of Operational Research, 256(1), 205-214.
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.
Viole, F. & Nawrocki, D. (2016). Predicting risk/return performance using upper partial moment/lower partial moment metrics. Journal of Mathematical Finance, 6(5), 900-920.
Zhang, W.G., Liu, Y.j. & Xu, W.J. (2012). A Possibility Mean-Semi Variance-Entropy Model for Multi- Period Portfolio Selection with Transaction Costs. European Journal of Operational Research, 222(5), 341-349.
Zhang, X.L. (2009). Using Genetic Algorithm to Solve a New Multi-period Stochastic Optimization Model. Journal of Computational and Applied Mathematics, 231(13), 114-123.
Industrial Management Journal
Volume 16, Issue 2
2024
Pages 192-214

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