Comparison analysis of copula-based and Markowitz portfolio methods

Authors

DOI:

https://doi.org/10.15587/2312-8372.2016.75572

Keywords:

portfolio, financial risk, copula, hierarchical copula, Archimedean copula

Abstract

In this paper, the objects of study are securities (stocks) and portfolio.

The main problem of the study is portfolio optimization. One of the first portfolio methods was presented by Henry Markowitz with his Modern Portfolio Theory (MPT), which is considered as a classic and the most popular one in modern investing. MPT provides the following assumptions: variance is used as a measure of risk, portfolio stock returns distribution is considered as a normal one. However, these assumptions do not represent real processes in the modern economy. First, in terms of modern volatile economy portfolio stock returns distribution curve has heavy tails, which is not typical for normal distribution. Secondly, in case of variance as a measure of risk probability of extreme events, such as a simultaneous increase or decrease of stock prices, are not taken into account.

So Markowitz method no longer meets the requirements of the modern financial market and there is a need to study alternative and more valid portfolio methods.

In this paper, copula-based approach is considered in contrast to the classical one. In the method assumption about the normality of stock returns is rejected and Value-at-Risk (VaR) is considered as a valid risk measure. VaR assessment is based on an information about random distribution. Since the normality assumption was rejected, to assess portfolio stock returns distribution need to be defined. To do these copula-functions was used.

Stochastic optimization problem using VaR was solved with a modified Nelder-Mead method.

As a result of the dynamic optimization return of copula-based portfolio for 2015 was 12,1 % of the initial investment sum, while the portfolio, constructed with the classical method, showed losses of 4,1 %.

Since in copula-based approach incorrect normality assumption is rejected and a valid risk measure is chosen, copula-based portfolio is much more effective than the Markowitz one.

Author Biographies

Olha Tupko, National Technical University of Ukraine «Kyiv Polytechnic Institute», Bldg. 14, 14-b, Politekhnichna st., Kyiv, Ukraine, 03056

Educational-Scientific Complex «Institute for Applied System Analysis»

Natalia Tupko, National Aviation University, Kosmonavta Komarova 1, Kyiv, Ukraine, 03058

Candidate of Physical and Mathematical Sciences, Associate Professor

Department of higher and computational mathematics

Nataliia Vasil'eva, Odessa State Academy of Civil Engineering and Architecture, Str. Didrihsone, 4, Odessa, Ukraine, 65029

Candidate of Physical and Mathematical Sciences, Associate Professor

Department of higher mathematics

References

  1. Investment. (2003). Investopedia, LLC. Available: http://www.investopedia.com/terms/i/investment.asp#ixzz4EhUH0JRl
  2. Stock. (2003). Investopedia, LLC. Available: http://www.investopedia.com/terms/s/stock.asp#ixzz4EhZKi2Hd
  3. Markowitz, H. (1952). Portfolio selection. The Journal of Finance, Vol. 7, № 1, 77–91.
  4. Penikas, G. I. (2014). Ierarhicheskie kopuly v formirovanii riskov investitsionnogo portfelia. Journal of Applied Econometrics, Vol. 35, № 3, 18–38.
  5. J. P. Morgan/Reuters. (1996). RiskMetrics – Technical Document. Ed. 4. New York: Morgan Guaranty Trust Company. Available: http://pascal.iseg.utl.pt/~aafonso/eif/rm/TD4ePt_2.pdf
  6. Basel Committee on Banking Supervision. (2004). International Convergence of Capital Measurement and Capital Standards. A Revised Framework. Bank for International Settlements. Available: http://www.bis.org/publ/bcbs107.pdf
  7. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. (1999, July). Coherent Measures of Risk. Mathematical Finance, Vol. 9, № 3, 203–228. doi:10.1111/1467-9965.00068
  8. Karadag, D. T. (2008). Portfolio risk calculation and stochastic portfolio optimization by a copula based approach. Graduate Program in Industrial Engineering, Bogazici University. Available: http://www.ie.boun.edu.tr/~hormannw/BounQuantitiveFinance/Thesis/karadag.pdf
  9. Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris, 8, 229–231.
  10. Daul, S., De Giorgi, E. G., Lindskog, F., McNeil, A. (2003, March 13). The Grouped t-Copula with an Application to Credit Risk. SSRN Electronic Journal. Available at: www/URL: http://doi.org/10.2139/ssrn.1358956
  11. Luo, X., Shevchenko, P. V. (2010, November). The t copula with multiple parameters of degrees of freedom: bivariate characteristics and application to risk management. Quantitative Finance, Vol. 10, № 9, 1039–1054. doi:10.1080/14697680903085544
  12. Hofert, M. (2011, January). Efficiently sampling nested Archimedean copulas. Computational Statistics & Data Analysis, Vol. 55, № 1, 57–70. doi:10.1016/j.csda.2010.04.025
  13. Garcia, R., Tsafack, G. (2011, August). Dependence structure and extreme comovements in international equity and bond markets. Journal of Banking & Finance, Vol. 35, № 8, 1954–1970. doi:10.1016/j.jbankfin.2011.01.003
  14. Patton, A. J. (2012, September). A review of copula models for economic time series. Journal of Multivariate Analysis, Vol. 110, 4–18. doi:10.1016/j.jmva.2012.02.021
  15. Savu, C., Trede. M. (2006). Hierarchical Archimedean Copulas. University of Muenster. Available: httm://www.uni-konstanz.de/micfinma/nference/Files/papers/Savu_Trede.pdf
  16. Okhrin, O., Ristig, A. (2014). Hierarchical Archimedean Copulae: The HAC Package. Journal of Statistical Software, Vol. 58, № 4, 1–20. doi:10.18637/jss.v058.i04
  17. Okhrin, O., Okhrin, Y., Schmid, W. (2013, April). On the structure and estimation of hierarchical Archimedean copulas. Journal of Econometrics, Vol. 173, № 2, 189–204. doi:10.1016/j.jeconom.2012.12.001
  18. McNeil, A. J. (2008, May 21). Sampling nested Archimedean copulas. Journal of Statistical Computation and Simulation, Vol. 78, № 6, 567–581. doi:10.1080/00949650701255834
  19. Nelsen, R. B. (2006). An introduction to copulas. Ed. 2. Portland: Springer, 262. doi:10.1007/0-387-28678-0
  20. HAC: Estimation, Simulation and Visualization of Hierarchical Archimedean Copulae (HAC). (12.10.2015). CRAN. Available: https://cran.r-project.org/web/packages/HAC/index.html
  21. Nelder – Mead method. (2016, July 16). Wikipedia. Available: https://en.wikipedia.org/wiki/Nelder%E2%80%93Mead_method
  22. Neldermead: R port of the Scilab neldermead module. (11.01.2015). CRAN. Available: https://cran.r-project.org/web/packages/neldermead/index.html
  23. Copula: Multivariate Dependence with Copulas. (25.07.2016). CRAN. Available: https://cran.r-project.org/web/packages/copula/index.html
  24. Quadprog: Functions to solve Quadratic Programming Problems. (17.04.2013). CRAN. Available: https://cran.r-project.org/web/packages/quadprog/index.html

Downloads

Published

2016-07-26

How to Cite

Tupko, O., Tupko, N., & Vasil’eva, N. (2016). Comparison analysis of copula-based and Markowitz portfolio methods. Technology Audit and Production Reserves, 4(2(30), 65–72. https://doi.org/10.15587/2312-8372.2016.75572

Issue

Vol. 4 No. 2(30) (2016): Information technologies. Mathematical modeling

Section

Mathematical Modeling: Original Research

License

Copyright (c) 2016 Olha Tupko, Natalia Tupko, Nataliia Vasil'eva

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.