Elsevier

International Journal of Forecasting

Volume 34, Issue 3, July–September 2018, Pages 497-506
International Journal of Forecasting

Portfolio optimization based on GARCH-EVT-Copula forecasting models

https://doi.org/10.1016/j.ijforecast.2018.02.004Get rights and content

Abstract

This study uses GARCH-EVT-copula and ARMA-GARCH-EVT-copula models to perform out-of-sample forecasts and simulate one-day-ahead returns for ten stock indexes. We construct optimal portfolios based on the global minimum variance (GMV), minimum conditional value-at-risk (Min-CVaR) and certainty equivalence tangency (CET) criteria, and model the dependence structure between stock market returns by employing elliptical (Student-t and Gaussian) and Archimedean (Clayton, Frank and Gumbel) copulas. We analyze the performances of 288 risk modeling portfolio strategies using out-of-sample back-testing. Our main finding is that the CET portfolio, based on ARMA-GARCH-EVT-copula forecasts, outperforms the benchmark portfolio based on historical returns. The regression analyses show that GARCH-EVT forecasting models, which use Gaussian or Student-t copulas, are best at reducing the portfolio risk.

Introduction

A range of different portfolio optimization methods, generally consisting of two steps (Markowitz, 1952), have been proposed over the last few decades. The first step involves forecasting the future returns of the underlying assets. One of the models that has been used for this step is the generalized autoregressive conditional heteroscedasticity (GARCH) extreme value theory (EVT) copula (Longin, 1996). The second step consists of optimal portfolio allocation, which is achieved by defining each asset’s weight in the corresponding portfolio. There are three main methods that are used for this step: Min-CVaR (minimizing the conditional Value-at-Risk), GMV (minimizing the variance) and CET (maximizing the Sharpe ratio).

GARCH-EVT-copula models are used mainly for minimizing the downside risk, and are considered to be an improvement on the traditional GARCH volatility models. In general, the downside risk can be measured by the Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) (Gençay & Selçuk, 2004). The latter measures the expected loss at a specific confidence level. GARCH models can be employed for forecasting asset returns (Crato & Ruiz, 2012), though it is recommended that forecasters take into account the fact that the financial assets (e.g. stock prices, interest rates and exchange rates) possess the features of fat-tailed distributions Bondt & Thaler (1985), Dacorogna & Pictet (1997), Harmantzis et al. (2006). EVT can be used to estimate the tail behavior of the returns of these financial assets, and the volatility has been shown to be an important factor in extreme value forecasting models Gençay & Selçuk (2006), Gençay et al. (2003). A combination of GARCH and EVT models has been proposed and used in several previous studies Bhattacharyya & Ritolia (2008), Chan & Gray (2006), Deng et al. (2011), McNeil & Frey (2000). The basic idea behind this combination is that EVT is suitable for the independently and identically distributed series that can be derived from the residuals obtained from GARCH models.

Furthermore, one of the characteristics of financial time series is that movements in one series can affect those in other series (Christoffersen, Errunza, Jacobs, & Jin, 2014). This provides researchers with a vast array of choices when modeling the dependency structures between different types of financial assets, such as stock markets, exchange rates, commodities and so forth. Of the various choices, copula models have shown adequate practicality, perhaps because of the technique employed in copula models. In this technique, originally shown by Sklar (1959), the correlations between assets are obtained from the joint distribution and then used in each separate marginal distribution.

The application of GARCH-EVT-copula models to risk management has been the focus of several previous studies. For instance, a study performed by Wang, Chen, Jin, and Zhou (2010) used three copula models to estimate the investment risk of foreign currencies. Of these copulas, both the Clayton and Student-t copulas yield better estimates of the correlation between exchange rates than the Gaussian copula. Berger (2013) forecasts the portfolio risk (VaR) by employing a time-varying dynamic conditional correlation (DCC) copula in combination with EVT, which leads to a better estimation of VaR than a static copula approach. In addition, a semi-parametric GARCH-EVT-copula model was proposed by Koliai (2016), who performed stress tests on corresponding portfolios and found that different models affect stress scenarios in various different ways.

In another study, based on the DCCGARCH-copula (but without EVT), two dynamic robust portfolio optimization methods are considered and compared with non-robust portfolios (Han, Li, & Xia, 2017). Several more complex copula models, including the regular vine (R-vine), canonical vine (C-vine) and drawable vine (D-vine), are examined by Zhang, Wei, Yu, Lai, and Peng (2014), who use these copulas to forecast both VaR and CVaR. The authors conclude that D-vine is better than the other vine copulas in terms of forecasting the CVaR. Bhatti and Nguyen (2012) suggest the use of a conditional EVT and time-varying copula for modeling the tail dependency between stock markets. Wang, Jin, and Zhou (2010) use the GARCH-EVT-copula model to evaluate the risk of foreign currencies. They concentrate on Student-t, Gaussian and Clayton copulas, and, in correspondence with Huang, Lee, Liang, and Lin (2009), who implement GARCH-copula and GJRGARCH-copula models for estimating the VaR of a portfolio of stock indexes, point out that the Student-t provides better estimation of the VaR when compared with other copula models.

Most studies of the GARCH-EVT-copula model have focused on its application to forecasting and the examination of the resulting downside risk. However, some studies have used these models not only for risk modeling, but also for portfolio allocation and back-testing. For instance, Huang and Hsu (2015) consider two GARCH-EVT-copula models (based on Student-t and Gaussian copulas) for simulating future returns of stock markets, and use a rolling window to compute the optimal weights based on the Min-CVaR allocation for the out-of-sample period. In addition, they also conduct four re-balancing strategies (daily, weekly, biweekly and monthly) and evaluate the performance of each model during the financial crisis and post-crisis periods. Their results indicate that daily and weekly strategies based on GARCH-EVT-copula models outperform others with respect to both the annual average return and the Sharpe ratio. Another study performed by Low, Alcock, Faff, and Brailsford (2013) employed a Clayton C-vine copula for testing the portfolio performance (accumulation wealth) based on minimizing the conditional value-at-risk. Two portfolio sizes are considered, namely 12 and 3 dimensions. They find that the use of this model is more appropriate for the portfolio with more assets.

Following Huang and Hsu (2015) and Wang, Chen et al. (2010), we use the GARCH-EVT-copula and ARMA-GARCH-EVT-copula models for performing out-of-sample forecasting and portfolio allocation. We begin by applying rolling window estimation and using univariate GARCH(1,1) and ARMA(1,1)-GARCH(1,1) models separately to obtain the parameters for one-day-ahead forecasts, then use EVT for tail modeling and for obtaining the uniforms. As for the dependency structure, we consider five d-dimensional copula models, namely Student-t, Gaussian, Clayton, Frank and Gumbel. Finally, we use the parameters from ARMA-GARCH (or GARCH) models and the dependency structures from copula models to simulate one-day-ahead returns. We capture the performances of these models fully by applying them to stock markets, and also evaluate different re-balancing frequencies.

Our work contributes to the existing literature in several ways. First, the addition of CET and GMV portfolios allows us to present novel results. As was mentioned earlier, the GARCH-EVT-copula and ARMA-GARCH-EVT-copula models are used in the first step of portfolio optimization by forecasting future returns and volatilities. In contrast to previous studies, we use the simulated returns from the copula model not only in the Min-CVaR, but also for the CET and GMV portfolios. Moreover, the aim of our study is to answer the question of how much of a reduction in risk (or gain in returns) can be achieved by combining the different forecasting models with EVT and the above-mentioned portfolio optimization techniques. To the best of our knowledge, this is the only study to do so by estimating the contribution of each part in risk modeling using a regression analysis based on the sample of portfolio returns obtained from back-testing. We find that almost all of the risk models decrease the portfolio risk significantly. Overall, GARCH forecasting models in combination with EVT and copula models appear to be particularly suitable for the CET optimization framework.

The rest of the paper is organized as follows. Section 2 presents the methodology, including the ARMA-GARCH model, extreme value theory, copula models and optimization methods. The data set is described in Section 3. Section 4 provides details of our empirical results. We describe the robustness analysis in Section 5. Finally, concluding remarks are provided in Section 6.

Section snippets

Methodology

The GARCH-EVT-copula approach assumes that the returns are ergodic processes (Boltzmann, 1896) and the residuals are independently identically distributed (i.i.d.) random variables.

Data

Our data set includes daily adjusted prices of ten stock indexes (S&P 500, FTSE 100, DAX 30, EURO STOXX, MSCI World, CAC 40, OMXC 20, OMXH, OMXS 30 and TOPIX) obtained from Thomson Reuters’ Datastream. We define logarithmic returns. The sample period starts in August 1996 and ends in August 2016, giving a total of 5218 observations. Similarly to Huang and Hsu (2015) and Low et al. (2013), we implement a rolling window of 1260 observations.1

Results

We employ GARCH-EVT and ARMA-GARCH-EVT models to define the marginal distribution for the innovations. We assume that the conditional distribution for residuals in the GARCH(1,1) is Student-t. However, for the ARMA(1,1)-GARCH(1,1), we use a normal distribution as the conditional distribution.2

Impact of risk modeling on the portfolio risk and performance

We determine and evaluate the impact of model choice on portfolio risk by summarizing the portfolio returns generated from each strategy, calculating the standard deviation (SD), first percentile (P1) and 99th percentile (P99) of the 3958 out-of-sample predicted portfolio returns. The first two statistical measures SD and P1 describe the portfolio risk, with the latter describing the downside risk in particular, while P99 describes the positive (wanted) potential in terms of the upper tail of

Conclusions

This study uses GARCH-EVT-copula and ARMA-GARCH-EVT-copula models to forecast and simulate the one-day-ahead returns of ten stock indexes. Using the forecasts, we employ three portfolio optimization techniques (Min-CVaR, GMV and CET) to compute the optimal weights and perform portfolio back-testing for the out-of-sample period based on different rebalancing strategies. Furthermore, we perform regression analyses to examine how, on average, the various risk modeling strategies affect three

Acknowledgments

Some of the computations for this study were done using the R packages “rugarch” and “spd” Ghalanos (2015), Ghalanos (2018). We want to thank Bernhard Pfaff for his guidance in the use of GARCH-copula models (Pfaff, 2016). In addition, we are grateful to seminar participants at the 10th International Conference on Computational and Financial Econometrics (CFE 2016) in Seville. Finally, we thank Scott Hacker, Kristofer Månsson, Gazi Salah Uddin and Pär Sjölander for their helpful comments and

Maziar Sahamkhadam is a Ph.D. student at the School of Business and Economics, Linnaeus University, Sweden. His research is in the area of portfolio optimization and risk modeling. He holds a master degree in International Financial Analysis from Jönköping International Business School, Sweden.

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Maziar Sahamkhadam is a Ph.D. student at the School of Business and Economics, Linnaeus University, Sweden. His research is in the area of portfolio optimization and risk modeling. He holds a master degree in International Financial Analysis from Jönköping International Business School, Sweden.

Andreas Stephan studied industrial engineering, statistics and economics at TU Berlin and received a Ph.D. in economics at the Humboldt University Berlin. He previously held positions at the Social Science Research Center in Berlin (WZB), at the German Institute for Economic Research (DIW Berlin) and as an Assistant Professor of economics at the European University Viadrina in Frankfurt/Oder. His main areas of interest are financial economics including financial engineering. Previous work includes the estimation of implicit betas from option prices. He is affiliated with CESIS - Centre of Excellence for Science and Innovation Studies at the Royal Institute of Technology, Stockholm, and with CeFEO - Centre for Family Enterprise and Ownership, Jönköping.

Ralf Östermark studied accounting and economics at Turku School of Buiness Economics and Åbo Akademi University (ÅAU). He has held positions in business economics and accounting at ÅAU. He is currently Professor in accounting and optimization systems at ÅAU. His main areas of interest are high performance computing and financial engineering. Previous work includes multiperiod portfolio management systems and scalability testing of challenging numerical problems.

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