Does Tail Dependence Make A Difference In the ... - Boston College

higher value of systemic risk.
Figure 5 compares the ∆CoV aR = , ∆CoV aR and MES estimated by a Normal Copula (without tail dependence)
with those estimated by Student t Copula (with symmetric tail dependence). The marginal distributions for
all copula models are set to be Student t 16 with df = 5. Having eliminated the effect of the marginal distribution,
the discrepancy of systemic risk measures, therefore, can only be attributed to the effect of dependence structures.
It is surprising to find that the behaviors of ∆CoV aR become quite strange. First, ∆CoV aR does not monotonically
increase with the correlation ρ. It declines in the end when the strength of dependence, the correlation ρ,
approaches a high value. Second, the lower tail dependence implied by the Student t copula does not necessarily
result in a higher value of systemic risk than the Normal copula without tail dependence. When the correlation ρ
is sufficiently large, ∆CoV aR estimated by the Normal copula becomes significantly larger than that estimated by
student t copula.
Figure 6: This figure displays the ∆CoV aR = , ∆CoV aR and MES (y − axis) estimated by Normal Copula
(without tail dependence), Rotated Gumbel Copula (with lower tail dependence) and Gumbel Copula (with Upper
tail dependence). The marginal distributions are set to be Student t with df = 5. The x-axis displays kendall
τ, which measures the co-movement strength of data. The parameters of copula model are chosen such that the
kendall τ of generated random variates is equal to the value shown in the x − axis.
Figure 6 provides further evidence that ∆CoV aR is not consistent with dependence measures. As the
average co-movement of data becomes stronger (Kendall τ increases), the ∆CoV aR estimated by Normal copula
(without tail dependence) or Gumbel Copula (with only upper tail dependence) approaches the highest value, which
contradicts the common view that stronger lower tail dependence implied by Clayton or Rotated Gumbel copula
should yield a higher value of systemic risk ∆CoVaR. Furthermore, the value of ∆CoV aR begins to decline when the
co-movement of data becomes even stronger (Kendall τ increases), which again conflicts with the commonly-held
notion that higher interconnectedness between the firm and the market should indicate a larger value of systemic
risk ∆CoV aR.
Figure 7 displays various systemic risk measures against tail dependence (either lower or upper). Again it shows
that ∆CoV aR fails to provide a consistent measure of systemic risk. When tail dependence (X-axis) is sufficiently
16 When I changed the degree of freedom, or control all marginal distributions to be skewed t distribution instead, the main result
remain unchanged.
17