Appendix: Robustness analysis
In this paper, the correlation matrix used in the experiment is not estimated from the original dataset, so a reasonable reader would expect significant change of the total VaR by Var-Covar approach if the correlation between risks changes.In order to compare the total VaR and diversification benefit among approaches, a robustness analysis is conducted and shown as follows.
In harmony with the paper, the correlation coefficients of credit risk and market risk, credit risk and operational risk and market risk and operational risk are denoted as ρcm,ρco and ρmo. In order to check how the changes of correlation coefficients will affect the total VaRs and diversification benefits from different approaches, Fig. 1 to Fig. 6 are given. The ρcm,ρcoandρmo are set as 0.66, 0.3 and 0.3 in the paper, so here we changeρcm from 0.46 to 0.86 and change ρco andρmo from 0.1 to 0.5. At each time, one coefficient changes and the other two coefficients keep invariant with the employed coefficients in the experiment of the paper. For each copula with certain correlation assumption, Monte Carlo Simulation runs 100000 times. We set confidence level as 99.9% here, a most frequently-used confidence level of banks and regulatory committees. Fig. 1 and Fig. 4 show the total VaRs and diversification benefits from different approaches with the changes of ρcm. Similarity, Fig. 2 and Fig. 5 show the total VaRs and diversification benefits from different approaches with the changes of ρco. Fig. 3 and Fig. 6 show the total VaRs and diversification benefits from different approaches with the changes of ρmo.
Fig. 1 Total VaRwith the change of correlation coefficient between credit risk and market risk
Fig. 2 Total VaRwith the change of correlation coefficient between credit risk and operational risk
Fig. 3 Total VaRwith the change of correlation coefficient between market risk and operational risk
Fig. 4 Diversification benefit with the change of correlation coefficient between credit risk and market risk
Fig. 5 Diversification benefit with the change of correlation coefficient between credit risk and operational risk
Fig. 6 Diversification benefit with the change of correlation coefficient between market risk and operational risk
From Fig. 1 to Fig. 3 we can see that: (1) the total VaRs from Var-covar constitute the straight line while total VaRs from other approaches fluctuate slightly and constitute broken lines because their results are from Monte Carlo simulation; (2) as the coefficients increase, all total VaRs from different approaches upward trends; (3) the broken lines of copulas are all above the straight lines of Var-covar approach. Besides, among copulas, in general, line of mixture copula is the highest, next are t copula (1df), t copula (10 df) and Gaussian copula successively.
The three findings from Fig. 1 to Fig. 3 reveal that although the total VaR integrated by different approaches will change with the changes of correlation coefficient, the relative order of these approaches is generally changeless. Specifically, mixture copula t copula (1 df) > t copula (10 df) > Gaussian copula > Var-covar approach. Similarly, Fig. 4 to Fig. 6 reveal that the relative order of diversification benefit from different approaches is also changeless in general, namely mixture copula < t copula (1 df) < t copula (10 df) < Gaussian copula < Var-covar approach.
In summary, the robustness analysis results show that the relative order of either total VaRs or diversification benefits from these approaches generally keeps invariant as the correlation changes. Therefore, it is verified that the conclusions of this paper are robust.