Portfolio Optimization under Varying Market Risk Conditions: Copula Dependence and Marginal Value Approaches
محورهای موضوعی : Financial EconomicsJila Ahmadi 1 , Hasan Ghodrati Ghezaani 2 , Mehdi Madanchi Zaj 3 , Hossein Jabbari 4 , Aliakbar Farzinfar 5
1 - Department of Management, Kashan Branch, Islamic Azad University, Kashan, Iran
2 - Department of Management, Kashan Branch, Islamic Azad University, Kashan, Iran
3 - Department of Financial Management, Electronic Unit, Islamic Azad University, Tehran, Iran
4 - Department of Accounting and Management, Kashan Branch, Islamic Azad University, Kashan, Iran
5 - Department of Accounting and Management, Kashan Branch, Islamic Azad University, Kashan, Iran
کلید واژه: COPULA, Extreme Value Theory, Asset Portfolio, Market risk,
چکیده مقاله :
This paper aims to investigate the portfolio optimization under various market risk conditions using copula dependence and extreme value approaches. According to the modern portfolio theory, diversifying investments in assets that are less correlated with one another allows investors to assume less risk. In many models, asset returns are assumed to follow a normal distribution. Consequently, the linear correlation coefficient explains the dependence between financial assets, and the Markowitz mean-variance optimization model is used to calculate efficient asset portfolios. In this regard, monthly data-driven information on the top 30 companies from 2011 to 2021 was the subject to consideration. In addition, extreme value theory was utilized to model the asset return distribution. Using Gumbel’s copula model, the dependence structure of returns has been analyzed. Distribution tails were modeled utilizing extreme value theory. If the weights of the investment portfolio are allocated according to Gumbel’s copula model, a risk of 2.8% should be considered to obtain a return of 3.2%, according to the obtained results.
This paper aims to investigate the portfolio optimization under various market risk conditions using copula dependence and extreme value approaches. According to the modern portfolio theory, diversifying investments in assets that are less correlated with one another allows investors to assume less risk. In many models, asset returns are assumed to follow a normal distribution. Consequently, the linear correlation coefficient explains the dependence between financial assets, and the Markowitz mean-variance optimization model is used to calculate efficient asset portfolios. In this regard, monthly data-driven information on the top 30 companies from 2011 to 2021 was the subject to consideration. In addition, extreme value theory was utilized to model the asset return distribution. Using Gumbel’s copula model, the dependence structure of returns has been analyzed. Distribution tails were modeled utilizing extreme value theory. If the weights of the investment portfolio are allocated according to Gumbel’s copula model, a risk of 2.8% should be considered to obtain a return of 3.2%, according to the obtained results.
[1] Tehrani, R., Fallah Tafti, S., Asefi, S., Portfolio Optimization Using Krill Herd Metaheuristic Algorithm Considering Different Measures of Risk in Tehran Stock Exchange, Financial Research Journal, 2018; 20(4): 409-426, (in Persian). doi:10.22059/FRJ.2019.244004.1006538
[2] Haddadi, M. R., Nademi, Y., Farhadi, H., Forecasting the Global Gold Price Movement with Marginal Distribution Modeling Approach: An Application of the Copula GARCH Gaussian, Financial Engineering and Securities Management, 2020; 11(42): 67-88, (in Persian). dor: 20.1001.1.22519165.1399.11.42.4.5
[3] Raghfar, H., Ajorlou, H., Calculation of Value at Risk of Currency Portfolio for a Typical Bank by GARCH-EVT-Copula Method. Iranian Economic Research Quarterly, 2016, 21, P.113-141. (in Persian). doi: 10.22054/ijer.2016.7238
[4] Sina, A., Fallah Shams, M., Comparison of Value Risk Models and Coppola-CVaR in Portfolio Optimi-zation in Tehran Stock Exchange. Journal of Financial Management Perspectives, 2020, 10(29), P.125-146. (in Persian)
doi: 10.52547/jfmp.10.29.125
[5] Sabahi, S., Mokhatab Rafiei, F., Rastegar, M. A., Mixed- Asset Portfolio Optimization. Monetary and Finance Economics, 2020, 27(19), P.49-278. (in Persian), doi: 10.22067/pm.v27i19.84579
[6] Sayadi, M., Karimi, N., Modeling the Dependency Structure between Stocks of Chemical Products Re-turn, Oil Price and Exchange Rate Growth in Iran; an Application of Vine Copula. Journal of Economic Modeling Research, 2019; 10(38):45-94. (in Persian)
[7] Fallahpour, S., Baghban, M., Application of Copula-CVaR in Portfolio Optimization and Comparative with Mean-CVaR. Quarterly Journal of Economic Research and Policies, 2014, 22(72), P.155-172. (in Persian)
[8] Keshavarz Haddad, G. R., Heyrani, M., Estimation of Value at Risk in the Presence of Dependence Structure in Financial Returns: A Copula Based Approach, Journal of Economic Research, 2014, 49(4), P.82-123
[9] Lalegani, E., Zehtabian, M., Application of Copula and Simulated Returns in the Portfolio Optimization with Conditional Value-at-Risk (CVaR) in Tehran Stock Exchange (TSE), Quarterly of Investment Knowledge, 2018; 7(26): 1-16.
[10] Adcock, C.J., Mean–variance–skewness efficient surfaces, Stein’s lemma and the multivariate ex-tended skew-student distribution. Eur. J. Oper. Res. 2014; 234(2):392–401. doi: 10.1016/j.ejor.2013.07.011
[11] Christoffersen, P., Errunza, V., Jacobs, K., Jacobs, K., Jin, X., Correlation dynamics and international diversification benefits, International Journal of Forecasting, 2014, 30(3), P.807-824. doi:10.1016/j.ijforecast. 2014.01.001
[12] Embrechts, P., Lindskog, F., McNeil, A., modelling dependence with copulas and applications to risk management. In S. T. Rachev (Ed.), Handbook of heavy tailed distribution in finance, Elsevier, 2003, 329–384.
doi: 10.1016/B978-044450896-6.50010-8
[13] Gao, C.T., Zhou, X.H., Forecasting VaR and ES using dynamic conditional score models and skew Student distribution. Econ. Model, 2016; 53: 216–223. doi: 10.1016/j.econmod.2015.12.004
[14] Gong, X.L., Liu, X.H., Xiong, X., Measuring tail risk with GAS time varying copula, fat tailed GARCH model and hedging for crude oil futures. Pac.Basin Financ. 2019; 55: P.95–109. doi: 10.1016/j.pacfin.2019.03.010
[15] Goode, J., Kim, Y.S., Fabozzi, F.J., Full versus quasi MLE for ARMA-GARCH models with infinitely divisible innovations. Appl. Econ, 2015; 47(48):5147–5158. doi: 10.1080/00036846.2015.1042203
[16] Han, Y. Li, P., Xia, Y., Dynamic Robust Portfolio Selection with Copulas, Finance Research Letters, 2019; 21:190-200. doi: 10.1016/j.frl.2016.12.008
[17] Hartmann, P., Straeman, S., de Vries, C.G., Asset market linkages in crisis periods, Review of Econom-ics and Statistics, 2004; 86(1):313-326.
[18] Henserson, A., Comparison of Value Risk Models and Coppola-CVaR in Portfolio Optimization in Financial market, Financial Management Perspective, 2020; 10(29):125-146. doi: 10.52547/jfmp.10.29.125
[19] Karma, M., Dependence structure and portfolio risk in Indian foreign exchange market: A GARCH-EVT-Copula approach, Quarterly Review of Economics and Finance, 2017; 64: 275-291. doi: 10.1016/j.qref.2017.01.007
[20] Krzemienowski, A., Szymczyk. S., Portfolio optimization with a copula-based extension of condi-tional value-at-risk. Annals of Operations Research. 2016; 237(1-2): 219-236
[21] Sahamkhadam, M., Andreas S., and Östermark, R., Portfolio optimization based on GARCH-EVT-Copula forecasting models, International Journal of Forecasting, 2018; 34(3):497-506.
[22] Zoia, M.G., Biffi, P., Nicolussi, F., Value at risk and expected shortfall based on Gram-Charlier-Like expansions. J. Bank. Financ. 2018; 93:92–104.
[23] Alotaibi, TS., Dalla Valle, L., Craven, M.J., The Worst Case GARCH-Copula CVaR Approach for Portfolio Optimisation: Evidence from Financial Markets, Journal of Risk and Financial Management, 2022; 15(10):482.
[24] Sahamkhadam, M., Stephan, A., and Östermark, R., Copula-based Black–Litterman portfolio optimi-zation, European Journal of Operational Research, 2022; 297(3):1055-1070.
[25] Andrew W. L., Wu, Lan and Zhang, Ruixun and Zhao, Chaoyi, Optimal Impact Portfolios with Gen-eral Dependence and Marginals, doi:10.2139/ssrn.4177277.
[26] Jacob, M., Neves, C., Vukadinović Greetham, D., Extreme Value Theory. In: Forecasting and As-sessing Risk of Individual Electricity Peaks, Mathematics of Planet Earth, 2020; 18: 34-49.
[27] Atilgan, Y., Turan G., Bali, K., Ozgur Demirtas, A., Gunaydin, D., Left-tail momentum: Underreaction to bad news, costly arbitrage and equity returns, Journal of Financial Economics, 2020;135(3): 725-753. doi: 10.1016/ j.jfineco.2019.07.006
[28] Babalooyan, S., Nikoomaram, H., Vakilifard, H. R., Rahnamay Roodposhty, F., Evaluating Extreme Dependence between Tehran security exchange and international Stock Markets Using Multivariate Ex-treme Value Theory (MEVT), Journal of Investment Knowledge, 2018; 27:241-256. (in Persian)
[29] Markowitz, H., Portfolio Selection, Journal of Finance, American Finance Association, 1952; 7(1): 77-91. doi: 10.2307/2975974
[30] Forbes, K.J., and Rigobon, R., No Contagion, Only Interdependence: Measuring Stock Market Comovements, The Journal of Finance, 2002; 57:2223-2261. doi:10.1111/0022-1082.00494
[31] Rezaei, A., Falahati, A., Sohaili, K., Portfolio Optimization Using Three-Objective Particle Swarm Optimization, Quarterly Journal of Applied Theories of Economics, 2019; 5(4):31-52.
[32] Tahani, T.S., Dalla Valle, L., Craven, M.J., The Worst Case GARCH-Copula CVaR Approach for Port-folio Optimisation: Evidence from Financial Markets, J. Risk Financial Manag, 2022; 15:482. doi:10.3390/jrfm 15100482
[33] Sklar, A., Fonctions de repartition a n dimensions et leurs marges, Publications de Institut de Statis-tique de I Univercite de Paris, 1959; 8: 299-331.
[34] Hadadi, M. R., Nademi, Y., Farhadi, H., Forecasting the Global Gold Price Movement with Marginal Distribution Modeling Approach: An Application of the Copula GARCH Gaussian and t. Financial Engi-neering and Portfolio Management, 2020; 11(42):67-88. (in Persian)
