The dependence structure between equity and foreign exchange markets and tail risk forecasts of foreign investments

Abstract

Motivated by the importance of the dependence structure between equity and foreign exchange rates in international financial markets, we investigate whether modelling the dependence structure can help forecast the tail risk of foreign investments. We propose a new time-varying asymmetric copula for modelling the dependence structure and forecasting the tail risk. We conduct backtesting on our tail risk forecasts for 12 major developed and emerging markets. We find that modelling the dependence structure can improve the tail risk forecast and make risk management of foreign investments more robust.

Keywords:

• Foreign investments
• Dependence structure
• TVAC model
• Value-at-Risk
• Expected shortfall

JEL classification:

• G15
• F21
• F37

Acknowledgements

The authors would like to thank Andrew Patton and Drew Creal for sharing their copula codes. We are grateful to Carol Alexander, Mario Cerrato, Christian Ewald, Jim Gatheral (the Editor), Neil Kellard, Georgios Sermpinis, Yukun Shi, Sadie Thrift (the Publishing Editor), Sjur Westgaard, two anonymous referees and seminar participants at University of Glasgow, University of Liverpool and Central University of Finance and Economics, and conference participants at the Young Finance Scholars' conference, Macro, Money and Finance Research Group Annual Meeting, International Conference of the Financial Engineering and Banking Society, and Australasian Finance and Banking Conference for their helpful discussions and comments. All authors acknowledge research support from their institutions, and Zhao specifically acknowledges financial support from the National Natural Science Foundation of China (Grant No. 71801117, 71973162). All errors remain their own.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 For example, suppose that the global economic recession has led to a stock market downturn and resulted in losses on both domestic and foreign markets. However, if the foreign currency is negatively correlated with stock prices for some reasons (e.g. “Safe Haven” currency), we can partially hedge the risk of foreign investments (in US dollars).

2 It is econometrically coherent (and practically implementable) to adopt a score-driven update scheme for both univariate and multivariate specifications. However, it is a common practice to use GARCH-type models, especially the GJR-GARCH, with the skewed t distribution to model univariate marginal distributions in the GAS literature, see for instance, Oh and Patton (2018), Eckernkemper (2018) and Bernardi and Catania (2019).

3 We use a rolling window to estimate time-varying tail dependence following Eckernkemper (2018). It is worth noting that the tail dependence significantly increased in all countries after the financial crisis in 2008, which highlights the importance of tail dependence modelling for foreign investments in recent years.

4 Several EU countries were affected by the European sovereign debt crisis after 2009. Thus, we arbitrarily assume that 1 January 2010 is a breakpoint.

5 Christoffersen et al. (2012) use a skewed t copula based on the multivariate skewed t distribution of Demarta and McNeil (2005) and Lucas et al. (2014) use one based on the multivariate generalized hyperbolic t distribution. By contrast, Wang et al. (2013) and Fei et al. (2017) use a Markov switching model to capture the asymmetric dependence structure.

6 We have observed extremely large estimates of the skewness parameter or degrees of freedom parameter when dot.com bubble, mortgage crisis, and sovereign debt crisis occur in the global market. With these outliers, it is not easy to fit the evolution of these parameters by a standard autoregressive process.

7 The model is rejected if it is rejected by either the KS test or the CvM test.

8 The time-varying model seems to have more influence on fitting the bivariate probability distribution than the asymmetric model.

9 We choose this model because it is the most successful univariate VaR models.

10 We thank for reviewer's careful comment on the use of this term. It is true that the correlations we are modelling in multivariate DCC-GARCH only capture linear dependencies, but it is also true that their evolution is nonlinear. Indeed, the multivariate DCC-GARCH are nonlinear time-series models, in the sense that the innovations in their Wald decomposition are not i.i.d (Tsay 2010). Multivariate DCC-GARCH allows a different dependence at the time dimension, e.g. normal time vs. extreme time, but not at location dimension, e.g. centre of probability distribution vs. tail of probability distribution. On the other hand, time-varying copula allows both of these. Therefore, if a model does not allow the different dependence across the location, we use the expression ‘linear dependence’ in our paper.

11 We also evaluate other multivariate GARCH models, such as BEEK (Engle and Kroner 1995) and CCC (He and Teräsvirta 2004). We find that DCC shows more stable estimation results and better forecasting performance than others.

12 Many different copulas exist for modelling the dependence structure. However, because our study is interested in modelling dependence structure between equity and forex, the most commonly used t-copula family in financial time series modelling should be an appropriate choice for our study. The horse-race of various copulas is outside the scope of our study.

Additional information

Funding

Zhao specifically acknowledges financial support from the National Natural Science Foundation of China [grant numbers 71801117, 71973162].