Portfolio Optimization using the Optimized \(\alpha\) in Renyi Entropy


  • Alireza Sajedi Department of Statistics, Islamic Azad University, Science and Research Branch, Tehran
  • Gholamhossein Yari Department of Statistics, Islamic Azad University, Science and Research Branch, Tehran




Entropy, Portfolio optimization, Utility function, Intelligent optimization algorithm


In this paper, a new approach of portfolio optimization using Renyi entropy is presented. In the proposed method, we use Renyi Entropy as a measurement of risk by determining the minimum return. We also attempt to examine the relationship between Renyi Entropy's \(\alpha\)-level and risk. Then, a single-objective function with penalty function approach based on Renyi Entropy-mean-semi-variance models is developed. As a result, we find the optimized \(\alpha\) denoting the most appropriate portfolio related to the risk level denoted by the investor. Finally by providing an illustrative example, the validity of this method is checked and the conclusion is drawn.


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A.K. Bera and S.Y. Park, Optimal portfolio diversification using the maximum entropy principle, Econometric Reviews 27(4-6) (2008), 484 – 512, DOI: 10.1080/07474930801960394.

G. Bugár and M. Uzsoki, Portfolio optimization strategies: Performance evaluation with respect to different objectives, Journal of Transnational Management 16(3) (2011), 135 – 148.

C.C. Coello, Treating objectives as constraints for single objective optimization, Engineering Optimization 32(3) (2000), 275 – 308.

R. Dobbins, W.F. Witt and J. Fielding, Portfolio theory and investment management, Blackwell Business (1994).

R.C. Green and B. Hollifield, When will mean-variance efficient portfolios be well diversified?, The Journal of Finance 47(4), (1992), 1785 – 809, DOI: 10.1111/j.1540-6261.1992.tb04683.x.

J.N. Kapur and H.K. Kesavan, Entropy Optimization Principles With Applications, Academic Press, New York (1992).

J. Ke and C. Zhang, Study on the optimization of portfolio based on entropy theory and mean-variance model, in: Service Operations and Logistics and Informatics, 2008, IEEE/SOLI 2008, IEEE International Conference on 2008 October 12 (Vol. 2, pp. 2668-2672), DOI: 10.1109/SOLI.2008.4682988.

J. Kennedy, Particle swarm optimization, in: Encyclopedia of Machine Learning, pp. 760–766, Springer, US (2011).

H. Markowitz, Portfolio selection, The Journal of Finance 7(1) (1952), 77 – 91, DOI: 10.1111/j.1540-6261.1952.tb01525.x.

R.C. Merton, An analytic derivation of the efficient portfolio frontier, Journal of Financial and Quantitative Analysis 7(4) (1972), 1851 – 1872, DOI: 10.2307/2329621.

G.C. Philippatos and C.J. Wilson, Entropy, market risk, and the selection of efficient portfolios, Applied Economics 4(3) (1972), 209 – 220, DOI: 10.1080/00036847200000017.

A. Rényi, On Measures of Entropy and Information, Hungarian Academy of Sciences Budapest Hungary (1961).

C.E. Shannon, A mathematical theory of communication, ACM SIGMOBILE Mobile Computing and Communications Review 5(1) (2001), 3 – 55, DOI: 10.1002/j.1538-7305.1948tb01338.x.

B. Tessema and G.G. Yen, A self adaptive penalty function based algorithm for constrained optimization, in Evolutionary Computation, 2006, CEC 2006, IEEE Congress on 2006 July 16 (pp. 246-253), DOI: 10.1109/CEC.2006.1688315.

I. Usta and Y.M. Kantar, Mean-variance-skewness-entropy measures: a multi-objective approach for portfolio selection, Entropy 13(1) (2011), 117 – 133, DOI: 10.3390/e13010117.

Y. Xu, Z. Wu, L. Jiang and X. Song, A maximum entropy method for a robust portfolio problem, Entropy 16(6) (2014), 3401 – 3415, DOI: 10.3390/e16063401.

G. Yari, M. Rahimi and P. Kumar, Multi-period multi-criteria (MPMC) valuation of American options based on entropy optimization principles, Iranian Journal of Science and Technology, Transactions A: Science 41(1) (2017), 81– 86, DOI: 10.1007/s4099017-0206-0.




How to Cite

Sajedi, A., & Yari, G. (2019). Portfolio Optimization using the Optimized \(\alpha\) in Renyi Entropy. Communications in Mathematics and Applications, 10(1), 111–122. https://doi.org/10.26713/cma.v10i1.899



Research Article