Dependence risk analysis in energy, agricultural and precious metals commodities: a pair vine copula approach

ABSTRACT

We apply pair vine copulas, specifically the C-vine and R-vine copulas, to examine the conditional multivariate dependence pattern/structure and R-vine copula-based value-at-risk (VaR) to assess financial portfolio risk. We examine the co-dependencies of 13 major commodity markets (which include three energy commodities, six agricultural commodities and four precious metals prices) from 2 January 2003 to 19 December 2016. Dividing our sample into three sub-periods, namely pre-GFC, GFC and post-GFC, we find that the dependencies among commodities undergo changes in a complex manner, changing in different financial conditions, and that the Student-t copula appears on the maximum number of occasions, especially during the GFC period, signifying the existence of fatter tails in the distributions of returns. We further show that the co-dependencies computed using R-vine copulas are best suited to compute the portfolio VaR during the considered time period.

KEYWORDS:

• R-vine
• VaR
• dependence structure
• tree structure
• commodity markets

JEL CLASSIFICATION:

• C61
• G11
• C51

Acknowledgments

The corresponding author thanks for supports from the National Natural Science Foundation of China under Grant No. 71974181, 71774152, No. 91546109 and Youth Innovation Promotion Association of Chinese Academy of Sciences (Grant: Y7X0231505) are acknowledged.

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplementary material

Supplemental data for this artcile can be accessed here.

Notes

¹ For the analysis R Core Team (2019) software was used and copula analysis was conducted using Vine Copula and portfolio analysis was conducted using fPortfolio (2017) packages of R.

² To model the multivariate dependence structure/tree there are particularly three types of models: 1. C vine, 2. D-vine and 3. R-vine. In case of C-vine tree, a root node is chosen for each tree and all pair wise dependencies with respect to this node are modelled conditioned on all previous root nodes. It follows that C-vine trees have a star structure. In case of D-vines, a similar process of construction is followed wherein a specific order for the variables is selected. The first tree models the dependence of the first and second variables, of the second and third, and so on, using pair copulas whereas in the second tree, the co-dependence analysis can proceed by modelling the conditional dependence of the first and the third variables, given the second variable; the pair (2, 4 × 3), and so forth. However, in real life, we neither know the tree structure and the bivariate copula families nor the bivariate copula parameters (bivariate copula families are a little exception in this regard as we restrict ourselves to the seven copula families).Such situation can be overcome by using regular (R) vines which have proven to be a flexible tool in high-dimensional dependence modelling. Since dependence pattern and copula fitness to the series under consideration are unknown, use of R vine seems to be a good choice.

³ We are thankful to the anonymous referee for suggesting to include the Joe BB1 copula.

⁴ In the paper, we report the results for the full sample only. Please refer to the Appendix for the sub-sample results. In the equally weighted portfolios, all commodities received a weight of 7.69% and the portfolio expected return and standard deviation were found to be 2.2% and 17.94%, respectively. The results have not been presented to save space; however, they are available upon request.

⁵ To analyse the sensitivity of the results with respect to the rolling window size we also chose a 500-day rolling window and found that the results were robust to the window size selected. Though these results are not presented, they are available upon request.

⁶ To analyse the sensitivity of the results with respect to the rolling window size we also chose 500-day rolling window and found that the results were robust to the window size selected. Though these results are not presented, they are available upon request.

⁷ We would like the readers to note that while performing the analysis of efficient portfolios with and without short-sales, the portfolios are not equally weighted.

⁸ The analysis of efficient portfolios is based on in-sample results only as we estimate the mean and covariance matrices based on the entire sample, which is infeasible for real investments. Further, we have attempted to do the same analysis on simulated data from the GARCH and copula models, but we faced serious convergence issues.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [71974181,71774152,91546109]; Youth Innovation Promotion Association of the Chinese Academy of Sciences [Y7X0231505].