Application of GARCH-Copula Model in Portfolio Optimization

Aleš Kresta

doi:10.5817/FAI2015-2-1

Application of GARCH-Copula Model in Portfolio Optimization

Vol.6,No.2(2015)

Aleš Kresta

https://doi.org/10.5817/FAI2015-2-1

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Abstract

Although the cornerstone of modern portfolio theory was set by Markowitz in 1952, the portfolio optimization problem is a never-ending research topic for both academics and practitioners. In this problem the future prediction of time series evolution plays an important role. However, it is rarely addressed in research. In the paper we analyze the applicability of the GARCH-copula model. To be more concrete we assume the investor maximizing Sharpe ratio while the future evolution of the time series is simulated by means of the AR(1)-GARCH(1,1) model using the copula modelling approach. The bootstrapping technique is applied as a benchmark. From the empirical results we found out that the GARCH-copula model provides better forecasts of future financial time series evolution than the bootstrapping method. Assuming the investor is maximizing the Sharpe ratio, both the final wealth increases and maximum drawdown decreases when we apply the GARCH-copula model compared to the application of bootstrapping technique.

Keywords:
portfolio optimization; Sharpe ratio; GARCH; copula function

References

Biglova, A. et al. (2004). Different approaches to risk estimation in portfolio theory. The Journal of Portfolio Management, 31(1), pp. 103-112. https://doi.org/10.3905/jpm.2004.443328

Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), pp. 307-327. https://doi.org/10.1016/0304-4076(86)90063-1

Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1(2), pp. 223-236. https://doi.org/10.1080/713665670

Elton, E. J. et al. (2014). Modern Portfolio Theory and Investment Analysis. 9th Ed. Hoboken: Wiley.

Engle, R. F. (1982). Auto-regressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50(4), pp. 987-1007. https://doi.org/10.2307/1912773

Farinelli, S. et al. (2008). Beyond Sharpe ratio: Optimal asset allocation using different performance ratios. Journal of Banking & Finance, 32(10), pp. 2057-2063. https://doi.org/10.1016/j.jbankfin.2007.12.026

Huang, J. J. et al. (2009). Estimating value at risk of portfolio by conditional copula-GARCH method. Insurance: Mathematics and Economics, 45(3), pp. 315-324.

Cherubini, U. et al. (2004). Copula Methods in Finance. Chichester: Wiley.

Cherubini, U. et al. (2011). Dynamic Copula Methods in Finance. Chichester: Wiley.

Konno, H. and Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management Science, 37(5), pp. 519-531. https://doi.org/10.1287/mnsc.37.5.519

Kresta, A. (2015). Financial Engineering in Matlab: Selected Approaches and Algorithms, SAEI, vol. 33. Ostrava: VSB-TU Ostrava.

Lundblad, C. (2007). The risk return tradeoff in the long run: 1836–2003. Journal of Financial Economics, 85(1), pp. 123-150. https://doi.org/10.1016/j.jfineco.2006.06.003

Mandelbrot, B. (1963). The Variation of Certain Speculative Prices. The Journal of Business, 36(4), pp. 394-419. https://doi.org/10.1086/294632

Markowitz, H. M. (1952). Portfolio selection. Journal of Finance, 7(1), pp. 77-91.

Nelsen, R. B. (1997). Dependence and order in families of Archimedean copulas. Journal of Multivariate Analysis, 60(1), pp. 111-122. https://doi.org/10.1006/jmva.1996.1646

Rank, J. (2006). Copulas: From Theory to Application in Finance. London: Risk Books.

Shalit, H. and Yitzhaki, S. (1984). Mean‐Gini, Portfolio Theory, and the Pricing of Risky Assets. The journal of Finance, 39(5), pp. 1449-1468. https://doi.org/10.1111/j.1540-6261.1984.tb04917.x

Sharpe, W. F. (1966). Mutual Fund Performance. The Journal of Business, 39(1), pp. 119-138. https://doi.org/10.1086/294846

Sharpe, W. F. (1994). The Sharpe Ratio. The Journal of Portfolio Management, 21(1), pp. 49-58. https://doi.org/10.3905/jpm.1994.409501

Sklar, A. (1959). Fonctions de repartition à n dimensions et leurs marges. Publications de l'Institut de statistique de l'Université de Paris, 8, pp. 229-231.

Wang, Z.-R. et al. (2010). Estimating risk of foreign exchange portfolio: Using VaR and CVaR based on GARCH–EVT-Copula model. Physica A: Statistical Mechanics and its Applications, 389(21), pp. 4918-4928. https://doi.org/10.1016/j.physa.2010.07.012

Young, M. R. (1998). A minimax portfolio selection rule with linear programming solution. Management Science, 44(5), pp. 673-683. https://doi.org/10.1287/mnsc.44.5.673