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Comparing the performance of Median or Mean and other risk indicators in Portfolio Optimization

Document Type : Research Paper

Author

Assistant professor, Faculty of Economics, University of Kharazmi, Tehran, Iran.

Abstract

EXTENDED ABSTRACT
INTRODUCTION
Markowitz model of asset portfolio optimization has some deficits, one of the most important of which is the normality assumption of stock market returns. Normality of returns has been rejected in numerous studies and has been shown that mean is not a good maximization objective anymore. Mean of returns is quite sensitive to outliers. There are three different ways of facing this problem in the literature. The first approach is generally to abandon the modern portfolio theory and turn to using meta-heuristic algorithms in portfolio optimization. The second approach still considers the modern portfolio theory important and valuable and uses it with adjustments. Some studies under the ultra-modern portfolio theory have focused on the inappropriateness of variance and use other measures of risk. Some others studies focus on drastic changes in the optimal portfolio as a result of changes in input values and use robust statistics. The third approach, on the other hand, tries to avoid the mentioned problems simply by using different parameters in optimization instead of average and variance. This paper based on the third approach seeks to use the median instead of the mean in stock portfolio optimization. The purpose is to compare the performance of mean and median in optimization. Moreover, the variance is not enough to control the risk because of heavy tails of return distribution and, thus, this paper incorporates various risk measures into models to test which one performs better beside median as an alternative to mean-variance models.
 
METHODOLOGY
Five median maximization models are presented with different risk measures of mean absolute deviations (MAD), value at risk (VaR), average value at risk (CVaR), and maximum loss (ML). Models are solved using GAMS software package and daily real data of twenty stocks from Tehran Stock Exchange from the beginning of 2016 to the end of 2019. For this purpose, the models are first employed for portfolio optimization in a certain period of 50 days. After computation, then the optimal weights are used for the next period of 50 days. This procedure is then repeated for the next 100 days to the end of 2019. Finally, the average and the distribution of returns of the optimal portfolio obtained from different models are compared with three other models: the mean optimization method without any control for risk, the mean optimization method subject to value at risk (VaR) constraint, and the portfolio with equal weights (EqW).
 
FINDINGS
Findings show that the median has a better performance in portfolio optimization. The model of median maximization gains higher returns in seventy percent of cases and a higher return on average. This means that a higher value of portfolio would be obtained using median in optimization. As a second conclusion, it was also shown that the median optimization method results in a portfolio with higher degree of diversification. The result remains true adding various risk measures to the model showing that median optimization instead of mean obtains a more diversified portfolio. In comparison among different median optimization models, it was also shown that CVaR and MAD risk measures controls the risk better than VaR and Maximum loss and obtains even further diversification.
 
CONCLUSION
The results generally show that portfolio optimization models based on the mean instead of the median and subject to risk measures of the average value at risk (VaR) and the mean absolute deviations (MAD) have a better performance in return maximization, control of risk and portfolio diversification.

Keywords

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References
Ansari Mahabadi, S.; Massah Bavani, A.R. & Bagheri, A. (2018). Improving adaptive capacity of social-ecological system of Tashk-Bakhtegan Lake basin to climate change effects – A methodology based on Post-Modern Portfolio Theory. Ecohydrology & Hydrobiology, 18(4). 365-378.
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. https://www.sid.ir/paper/883624/fa [in Persian]
Benati, S. (2011). Heuristic methods for the optimal statistic medians problem. Computers & Operations Research 38(1), 379–386.
Benati, S. (2015). Using medians in portfolio optimization. Journal of the Operational Research Society 66, 720 –731.
Benati, S & Rizzi R. (2009). The optimal statistical median of a convex set of arrays. Journal of Global Optimization. 44 (1), 79–9. 
Ben Salah, H.; Chaouch, M.; Gannoun, A.; De Peretti, C. & Trabelsi, A. (2018). Mean and median-based nonparametric estimation of returns in mean-downside risk portfolio frontier. Annals of Operations Research, 262. 653–681
Boyle, P. P., Siu, T. K., & Yang, H. (2002). Risk and probability measures. Risk, 15(7). 53–57
Branda, M.; Bucher, M.; Cervinka, M. & Schwartz, A. (2018). Convergence of a Scholtes-type regularization method for cardinality-constrained optimization problems with an application in sparse robust portfolio optimization. Computational Optimization and Applications, 70. 503-530.
Broadie, M. (1993). Computing efficient frontiers using estimated parameters. Annals of Operations Research 45(1), 21–58.
Chen, X.; Song, P.; Gao, K. & Qiao, Y. (2017). The Application in the Portfolio of China's A-share Market with Fama-French Five-Factor Model and the Robust Median Covariance Matrix. International Journal of Economics, Finance and Management Sciences, 5(4). 222-228.
Chen, J.M. (2016). A Four-Moment Capital Asset Pricing Model. In: Postmodern Portfolio Theory. Quantitative Perspectives on Behavioral Economics and Finance. Palgrave Macmillan, New York. https://doi.org/10.1057/978-1-137-54464-3_10
Cont, R. (2001). Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance 1(2), 223–236.
Cooper, L.; Evnine, J.; Finkelman, J.; Huntington, K. & Lynch, D. (2016). Social Finance and the Postmodern Portfolio: Theory and Practice. The Journal of Wealth Management, 18(4). 9-21.
Dai, W. (2018). Mean-Entropy Models for Uncertainty Portfolio Selection. In: Multi-Objective Optimization; Springer: Singapore.
DeMiguel, V. & Nogales F.J. (2009). Portfolio selection with robust estimation. Operations Research 57(3), 560–577.
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.
Erfani, A. & Safari S. (2014). A study of return cyclical pattern monthly in Tehran stock (by using moving block bootstrap). Financial Knowledge of Securities Analysis, Vol. 7, No.22, PP. 47-59. https://jfksa.srbiau.ac.ir/article_2925.html?lang=en [in Persian]
Fabozzi, F. J.; Kolm, P. N.; Pachamanova, D. A. & Focardi, S. M. (2007). Robust Portfolio Optimization and Management. John Wiley & Sons, Inc., Hoboken, New Jersey.
Geambasu, C.; Sova, R.; Jianu, I. & Geambasu, L. (2013). Risk measurement in post-modern portfolio theory: differences from modern portfolio theory. Economic Computation and Economic Cybernetics studies and Research, 47. 486-508.
Gerber, S.; Markowitz, H. M & Pujara, P. (2015). Enhancing multi-asset portfolio construction under Modern Portfolio Theory with a robust co-movement measure. SSRN Electronic Journal. DOI: 10.2139/ssrn.2627803
Grossi, L. & Laurini F. (2011). Robust estimation of efficient mean-variance frontiers. Advances in Data Analysis and Classification 5(1), 3–22.
Gupta, P., Mehlawat M. K. & Mittal G. (2012). Asset portfolio optimization using support vector machines and real-coded genetic algorithm. Journal of Global Optimization. 53, 297–315.
Hu, J.; Harmsen, R.; Crijns-Graus, W. & Worrel, E. (2019). Geographical optimization of variable renewable energy capacity in China using modern portfolio theory. Applied Energy, 253.
Huang, D.; Zhou, J.; Li, B.; Hoi, S. C. H. & Zhou, S. (2016). Robust Median Reversion Strategy for Online Portfolio Selection. IEEE Transactions on Knowledge and Data Engineering, 28(9).
Huo, L., Kim T. H. & Kim Y. (2012). Robust estimation of covariance and its application to portfolio optimization. Finance Research Letters. 9(3) 121-134.
Jagannathan, R. & Ma T. (2003). Risk reduction in large portfolios: Why imposing the wrong constraints helps. Journal of Finance 58(4): 1651–1684.
Kamali, S. (2014). Portfolio Optimization Using Particle Swarm Optimization and Genetic Algorithm. Journal of Mathematics and Computer Science, 10(2). 85-90
Kara, G., Ozmen A. & Weber G. W. (2019). Stability advances in robust portfolio optimization under parallelepiped uncertainty. Central European Journal of Operations Research. 27, 241-261.
Karandikar, R. (2012). Modelling in the Spirit of Markowitz Portfolio Theory in a Non-Gaussian World. Current Science, 100(6). 666-672.
Katterbauer, K., Oguz C. & Salman S. (2012). Hybrid adaptive large neighborhood search for the optimal statistic median problem. Computers & Operations Research 39(11), 2679–2687.
Li, J.Y.M. (2018). Technical Note—Closed-Form Solutions for Worst-Case Law Invariant Risk Measures with Application to Robust Portfolio Optimization. Operational Research, 66(6).
Markowitz, H. M. (1952). Portfolio selection. Journal of Finance 7(1): 77–91.
Mercurio, P. J.; Wu, Y. & Xie, H. (2020). An Entropy-Based Approach to Portfolio Optimization. Entropy, 22(3).
Ortobelli, S., Rachev, S. T., Stoyanov, S., Fabozzi, F. J., & Biglova, A. (2005). The proper use of risk measures in portfolio theory. International Journal of Theoretical and Applied Finance, 8(8). 1107–1133.
Puerto, J.; Rodriguez-Madrena, M. & Scozzari, A. (2020).  An application of the p-median problem in optimal portfolio selection. IX Workshop on Locational Analysis and Related Problems.
Qiu, H., Han F., Liu H. & Caffo B. (2015). Robust portfolio optimization, in: Advances in Neural Information Processing Systems (NIPS), 28, 46–54.
Raei, R. & Nabizadeh A. (2013). Testing Stock Return Distribution in the Tehran Stock Exchange. Journal of Financial Management Strategy. Vol.1, No.1, pp. 1-15. 10.22051/JFM.2014.952 [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).
Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2. 21–41.
Rom, B. M. & Ferguson, K. W. (1993). Post-Modern Portfolio Theory Comes of Age. The Journal of Investing, 2(4). 27.33.
Rotela, P. (2017). Entropic Data Envelopment Analysis: A Diversification Approach for Portfolio Optimization. Entropy, 19. 352.
Schulmerich, M.; Leporcher, Y.M.; & Eu, C.H. (2015). Modern Portfolio Theory and Its Problems. In: Applied Asset and Risk Management. Management for Professionals. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55444-5_2
Shams, S. & Esfandiari Moghaddam, A. T. (2017). The impact of herding behavior on the performance of investment companies based on modern and post modern portfolio theory. Journal of Financial Research, 19(1). 97-118.
Shannon, C. (1948). A Mathematical Theory of Communication: Part 1. The Bell System Technical Journal, 27(3). 379–423.
Sefiane, S. & Benbouziane M. (2012). Portfolio Selection Using Genetic Algorithm. Journal of Applied Finance & Banking, 2(4). 143-154.
Sornette, D. (2004). Why Stock Market Crash: Critical Events Is Complex Financial Systems. Princeton University Press: Princeton.
Sortino, F. & Price, L. N. (1994). Performance Measurement in a Downside Risk Framework. The Journal of Investing, 3(3). 59-64.
Swisher, P. & Kasten, G.W. (2005). Post-modern portfolio theory. Journal of Financial Planning, 18(9).
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. 10.22059/JFR.2017.62580 [in Persian]
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. 10.22059/FRJ.2019.244004.1006538 [in Persian]
Torki, L.; Esmaeli, N. & Haghparast, M. (2023). Comparison of GARCH Family Models in Estimating Value at Risk and Conditional Value at Risk on the Tehran Stock Exchange. Quarterly Journal of Quantitative Economics, 19 (4), 43-78. 10.22055/jqe.2021.33186.2240 [in Persian]
Trzpiot, G. & Majewska J. (2008). Investment decisions and portfolio classification based on robust methods of estimation. Operations Research and Decisions 1, 83–96.
Tukey, J. W. (1960). A survey of sampling from contaminated distributions. In: I. Olkin (ed). Contributions to Probability and Statistics. Stanford University Press: Stanford, 448–485.
Viswanathan, L. & Maheswaran S. (2017). An Investigation into non-normality of stock returns. Asian Journal of Empirical Research, Asian Economic and Social Society, Vol.7 (2), 19-27.
Yang, L., Couillet R. & McKay M. R. (2015). A Robust Statistics Approach to Minimum Variance Portfolio Optimization. IEEE Transactions on Signal Processing 63(24).
Yanou, G. (2013). Extension of the random matrix theory to the L-moments for robust portfolio selection. Quantitative Finance 13(10), 518–531.
Zhou, R. (2017). Properties of Risk Measures of Generalized Entropy in Portfolio Selection. Entropy, 19. 657.
Zhu, H., Wang Y., Wang K. & Chen Y. (2011). Particle Swarm Optimization (PSO) for the constrained portfolio optimization problem. Expert Systems with Applications, 38(8). 10161-10169.
 
 
 
Quarterly Journal of Quantitative Economics
Volume 20, Issue 1 - Serial Number 76
Vol. 20, No. 1, Spring 2023
June 2023
Pages 99-138
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History
  • Receive Date: 27 February 2021
  • Revise Date: 16 September 2021
  • Accept Date: 25 October 2021
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