MINIMUM TSALLIS PORTFOLIO

Year 2022, Volume: 7 Issue: 1, 90 - 102, 27.06.2022

Erhan Ustaoğlu , Atif Evren

https://doi.org/10.54452/jrb.1030739

Abstract

Mean-variance portfolio optimization model has been shown to have serious drawbacks. The model assumes that assets returns are normally distributed that is not valid for most of the markets and portfolios. It also relies on asset’s covariance matrices for the calculation of portfolio’s risk that is open to estimation errors. Moreover, these optimization errors are maximized by the method that result in poor out-of-sample performances. In this study, we propose a new portfolio optimization method based on minimization of Tsallis entropy, which is valid for any underlying distribution. First, we show that the Tsallis entropy can be employed as a risk measure for portfolio analysis. Then we demonstrate the validity of the model by comparing its performance with those mean-variance and minimum-variance portfolios using BIST 30 data.

Keywords

Portfolio optimization, entropy, minimum Tsallis portfolio

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.
• Batra, L., & Taneja, H. C. (2020). Portfolio optimization based on generalized information theoretic measures. Communications in Statistics-Theory and Methods, 1-15.
• Bera, A.K. & Park, S.Y. (2008). Optimal portfolio diversification using the maximum entropy principle. Econometric Reviews, 27, 484–512.
• Best, M. J., & Grauer, R. R. (1991). On the sensitivity of mean-variance-efficient portfolios to changes in asset means: Some analytical and computational results. The Review of Financial Studies, 4(2), 315–342.
• Black F., & Litterman, R. (1992). Global Portfolio Optimization. Financial Analysts Journal, 48, 28–43.
• Chan L.K.C., Karceski, J., Lakonishok, J. (1999). On portfolio optimization: Forecasting covariances and choosing the risk model. Review of Financial Studies, 12, 937-74.
• DeMiguel, V., Garlappi, L., & Uppal, R. (2009). Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy? Review of Financial Studies, 22, 1915–1953.
• Green R.C., & Hollifield, B. (1992). When Will Mean-Variance Efficient Portfolios Be Well Diversified? Journal of Finance, 47, 1785-1809.
• Jagannathan, R., & Ma, T. (2003). Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps. Journal of Finance, 58(4), 1651-1683
• James, W., & Stein, C. (1961). Estimation with Quadratic Loss. In Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, 1, University of California Press, Berkeley, California, 361-379.
• Jorion, P. (1968). Bayes-Stein estimation for portfolio analysis. Journal of Financial and Quantitative Analysis, 21(3), 279–292. Lassance, N., & Vrins, F. (2019). Minimum Rényi entropy portfolios. Annals of Operations Research, 1-24.
• Ledoit, O., & Wolf, M. (2004). Honey, I Shrunk the Sample Covariance Matrix. Journal of Portfolio Management, 30, 110–119.
• Mercurio, P.J., Wu, Y., Xie, H. (2020). An Entropy-Based Approach to Portfolio Optimization. Entropy, 22(3):332.
• Michaud, R. (1989). The Markowitz Optimization Enigma: Is Optimized Optimal? Financial Analysts Journal, 45(1), 31–42.
• Philippatos, G.C., & Wilson, C.J. (1972). Entropy, market risk, and the selection of efficient portfolios. Applied Economics, 4, 209-220.
• Rotela Junior, P., Rocha, L. C., Aquila, G., Balestrassi, P. P., Peruchi, R. S., & Lacerda, L. S. (2017). Entropic data envelopment analysis: a diversification approach for portfolio optimization. Entropy, 19, 352-362.
• Smimou, K., Bector, C.R., & Jacoby, G. (2007). A subjective assessment of approximate probabilities with a portfolio application. Research in International Business and Finance, 21, 134–160.
• Stein, C. (1956). Inadmissibility of the Usual Estimator for the Mean of a Multivariate Normal Distribution. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Contributions to the Theory of Statistics, University of California Press: California, USA, 197-206.
• Tsallis, C. (1988). Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics, 52, 479-487.
• Usta, I. & Kantar, Y.M. (2011). Mean-Variance-Skewness-Entropy Measures: A Multi-Objective Approach for Portfolio Selection. Entropy, 13, 117-133.
• Xu, Y., Wu, Z., Long, J., & Song, X. (2014). A maximum entropy method for a robust portfolio problem. Entropy, 16, 3401–3415.
• Yahoo finance historical prices. (2021). Retreived from https://finance.yahoo.com/quote/XU030.IS/¬history?p=XU030.IS
• Zeng, X., Wu, J., Wang, D., Zhu, X., & Long, Y. (2016). Assessing Bayesian model averaging uncertainty of groundwater modeling based on information entropy method. Journal of Hydrology, 538, 689–704.
• Zhang, J., & Li, Q. (2019). Credibilistic Mean-Semi-Entropy Model for Multi-Period Portfolio Selection with Background Risk. Entropy, 21, 944.
• Zhang, W.G., Liu, Y.J., & Xu W.J. (2012). A possibilistic mean-semivariance-entropy model for multi-period portfolio selection with transaction costs. European Journal of Operations Research, 222, 341-349.
• Zhou, J., Shen, J., Zhao, Z., Gu, Y., & Zhao, M. (2019). Performance of Different Risk Indicators in a Multi-Period Polynomial Portfolio Selection Problem Based on the Credibility Measure. Entropy, 21, 491.
• Walters, J. (2014). The Black-Litterman Model in Detail. http://dx.doi.org/10.2139/ssrn.1314585
• Woerheide, W., and D. Persson. 1993. An index of portfolio diversification. Econometric Reviews 2 (2):73–85.