Application of Clayton Copula in Portfolio Optimization and its Comparison with Markowitz Mean-Variance Analysis

Subject Areas : Financial Accounting

Roya  Darabi ¹ (Department of Accounting, South Tehran Branch, Islamic Azad University, Tehran, Iran.)
Mehdi  Baghban ² (Department of Accounting, South Tehran Branch, Islamic Azad University, Tehran, Iran.)

Received: 2018-03-19 Accepted : 2018-03-19 Published : 2018-03-01

Keywords: Portfolio optimization, Copula Function, utility function, Copula Clayton,

Abstract :

With the aim of portfolio optimization and management, this article utilizes the Clayton-copula along with copula theory measures. Portfolio-Optimization is one of the activities in investment funds. Thus, it is essential to select an appropriate optimization method. In modern financial analyses, there is growing evidence indicating the distribution of proceeds of financial properties is not customary. However, in common risk management methods the main assumption is that the distribution of assets returns is normal. When the distribution of earnings isn’t normal, the linear correlation coefficient isn’t considered to be an appropriate measure to express the dependency structure. The investors are required to make use of methods that concentrate on the aggregated risks, considering the whole positions and the links between risk factors and assets. Therefore, we use copula as an alternative measure to model the dependency structure in this research. In this regard, given the weekly data pertaining to the early 2002 until the late 2013, we use Clayton-copula to generate an optimized portfolio for both copper and gold. Finally, the Sharpe ratio obtained through this method is compared with the one obtained through Markowitz mean-variance analysis to ascertain that Clayton-copula is more efficient in portfolio-optimization.

References:

[1] Alexander C., Market Risk Analysis. West Sussex: John Wiley & Sons, 2008.

[2] Alibaygi H., Optimizing the Portfolio with an Approach of Mean-Half Variance Using Harmonic Search Methods, Master’s Thesis, and Faculty of Administration. (In Persian), 2012.

[3] Aleš Kresta. Application of GARCH-Copula Model in Portfolio Optimization.  Financial Assets and Investing, 2015, 2, P.1-20.

[4] Ang A., Chen, J., Asymmetric Correlations of Equity Portfolios. Journal of Financial Economics, 2002, 63(3), P.443-494.

[5] Artzner P., Delbaen F., Eber, J.M., Heath, D., Coherent Measures of Risk. Mathematical Finance, 1999, 9, P.203-228.

[6] Baumol W.J., An Expected Gain-Confidence Limit Criterion for Portfolio Selection. Management Science, 1963, 10 (1), P.174-182.

[7] Bedford, T., Cooke, R.M., Probability Density Decomposition for Conditionally Dependent Random Variables Modeled by Vines. Annals of Mathematics and Artificial Intelligence, 2001, 32, P.245-268.

[8] Brandimarte P., Numerical Methods in Finance and Economics: A Matlab-Based Introduction, 2^nd Edition, 2006.

[9] Campbell, R. H., Nasdaq.com: Portfolio Optimization. http:./www.nasdaq.com/ investing/ glossary/ p/ portfolio-optimization. (Retrieved 2016-04-03), 2011.

[10] Cherubini, U., Gobbi, F., Mulinacci, S., Romagnoli, S., Dynamic Copula Methods in Finance. John Wiley, NY, 2012.

[11] Cherubini, U., Luciano, E., Vecchiato W., Copula Methods in Finance. John Wiley & Sons, 2004.

[12] Chollete, L., de la Pena, V., Lu, C-C., International Diverse Citation: A Copula Approach. Journal of Banking and Finance, 2011, 35 (2), 403-417.

[13] Costinot A., Roncali T., Teiletche J., Revisiting the Dependence between Financial Markets with Copulas. Working Paper, Credit Lyonnais, 2000.

[14] Dachraoui K., Dionne, G., Stock astic Dominance and Optimal Portfolio. Economics Letters, Elsevier, 2001, 71(3), P.347-354.

[15] Danielsson J., Financial Risk Forecasting: John Wiley & Sons, Ltd, 2011.

[16] De Melo Mendes B.V., Kolev N., How Long Memory in Volatility Affects True Dependence Structure. International Review of Financial Analysis, 2008, 17(5), P.1070-1086.

[17] Embrechts P., McNeil A.J., Straumann D., Correlation: Pitfalls and Alter-Natives a Short. Risk Magazine, 1999, 2, P.69-71.

[18] Fabozzi F. J.,  Kolm P.N., Pachamanova D. A. FocardiS. M., Robust Portfolio Optimization and Management. John Wiley & Sons Inc., 2007.

[19] Grinold C., Kahn N., Active Portfolio Management. McGraw – Hill, 2000.

[20] Hodder, E., Jackwerth, J.C. and Kolokolova, O., Improved Portfolio Choice Using Second-Order Stochastic Dominance. Review of Finance, 2015, 19, P.1623-1647.

[21] Hu, L., Dependence Patterns Across Financial Markets: A mixed Copula Approach. Applied Financial Economics, 2006, 16 (10), P.717-729.

[22] Jahanbakhshi, M., Optimizing the Investment Portfolio of Companies Listed in Tehran Stock Exchange with an approach to Ideal Planning of GP and AHP (Case Study: Omid Investment Management Co.). Master’s Thesis, Faculty of Administration. (In Persian), 2011.

[23] Jan Mossin. Equilibrium in a Capital Asset Market. Econometrical, 1966, 34(4), P.768-783.

[24] Jondeau E., Rockinger M., The Copula-GARCH Model of Conditional Dependencies: An International Stock Market Application. Journal of International Money and Finance, 2006, 25 (5), P.827-853.

[25] Lintner J., The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolio and Capital Budgets. Review of Economics and Statistics, 1965, 47(1), P.13-37.

[26] Liu W., Currencies Portfolio Return: A Copula Methodology. Working Paper, University of Toronto, 2006.

[27] Garcia R., Tsafack G., Dependence Structure and Extreme Co-movements in International Equity and Bond Markets. Journal of Banking and Finance, 2011, 35(8), P.1954-1970.

[28] Gnatievay K., Platen E., Modeling Co-Movements and Tail Dependency in the International Stock Market via Copula. Asia-Pacific Financial Markets, 2010, 17(3), P. 261-302.

[29] Lujie Sun., & Manying Bai., Application of Copula and Copula- CVaR in the Multivariate Portfolio Optimization. Springer Berlin Heidelberg, 2007, 46(14), P.231-242.

[30] Kaplan, S., & Garrick, B. J., On the Quantitative Definition of Risk. Risk Analysis, Wiley Online Library, 2006, 5, P.11-27.

[31] Kakouris I., Rustem B., Robust Portfolio Optimization with Copulas. European Journal of Operational Research, 2014, 235, P.28-37.

[32] Manying Bai, Lujie Sun., Combinatory, Algorithms, Probabilistic and Experimental Methodologies.The Series Lecture Notes in Computer Science , 2007, 46(14), P.231-242.

[33] Manner, H., Estimation and Model Selection of Copulas with an Application to Exchange Rates. Working Paper, University Maastricht, 2007.

[34] Markowitz H., Portfolio Selection. The journal of finance, 1952, 7 (1), P.77-91.

[35] Markowitz, H.M., Portfolio Selection: Efficient Diverse Citation of Investments. Yale University Press, Massachusetts, 1959.

[36] McNeil, A., Frey, R., Embrechts, P., Quantitative risk management: Concepts, techniques and tools. Princeton: Princeton University Press, 223, 2005.

[37] Najafpour H., Optimizing the Portfolio Using Mimetic Algorithm. Master’s Thesis, Faculty of Administration. (In Persian), 2011.

[37] Nelsen R. B., Dependence and Order in Families of Archimedean Copulas. Journal of Multivariate Analysis, 1997, 60(1), P.111-122.

[38] Nelsen R., An introduction to Copulas. 2^th Edition. New York: Springer, 2006.

[39] Ogryczak W., Ruszczy-nski A., From Stochastic Dominance to Mean-Rrisk Models: Semi deviations as Risk Measures. European Journal of Operational Research, 1999, 116, P.33-50.

[40] Ogryczak W., Ruszczy-nski A., On Consistency of Stochastic Dominance and Mean-Semi Deviation Models. Mathematical Programming, 2001, 89, P.217-232.

[41] Ogryczak W., Ruszczy-nski A., Dual Stochastic Dominance and Related Mean-Risk Models. SIAM Journal on Optimization, 2002, 13, P.60-78.

[42] Rachev S. T., Stein M., Copula Concepts in Financial Markets, 2009. 

[43] Raee R., Fallahpour S., Designing a Model for Active Portfolio Management Using VaR and Genetic Algorithm. Accounting and Auditing Investigations, 2011, 18(64), P.19-34. (In Persian).

[44] Raee R., Pouyanfar A., Managing Advanced Investment. Seventh Edition, Tehran: SMT. (In Persian), 2006.

[45] Raee R., Saeedi A., Principles of Financial Engineering and Risk Management. First Edition, Tehran: SMT. (In Persian), 2005.

[46] Resti A., Sironi A., Risk Management and Shareholders’ Value in Banking. West Sussex: John Wiley & Sons, 2007.

[47] Rodriguez J.C., Measuring financial Contagion: A Copula Approach. Journal of Empirical Finance, 2007, 14 (3), P.401-423.

[48] Rockafellar R.T. Uryasev S., Optimization of Conditional Value-at-Risk.  Journal of Risk, 2000, 2, P.21-41.

[49] Roman D., Mitra G., Zverovich V., Enhanced Indexation based on Second-Order Stochastic Dominance. European Journal of Operational Research, 2013, 228, P.273-281.

[50] Sharpe W., Investments. 6^th Edition, 1995.

[51] Shalit H., Yitzhaki S., Marginal Conditional Stochastic Dominance. Management Science, 1994, 40, P.670-684.

[52] Sklar A., Fonctions de r´epartition `an n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, 1959, 8, P.229-231.

[53] Sun W., Rachev S., Fabobozzi F.J., Petko S.K., A new Approach to Modeling Co-Movement of International Equity Markets: Evidence of Unconditional Copula-Based Simulation of Tail Dependence. Empirical Economics, 2009, 36 (1), P.201-229.

[54] Tobin. J., Liquidity Preference as Behavior toward Risk. Review of Economic Studies, 1958, 25, P.65–86.

[55] Wei G., Hu T., Super modular Dependence Ordering on a Class of Multivariate Copulas. Statistics and Probability Letters, 2002, 57, P.375-385.