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

Figure 4: This figure compares the copula-based estimation of MES with the nonparametric kernel estimation
following Brownless–Engle (2011). In the left panel, data are generated by the student t copula with degree of
freedom df = 5. In the right panel, data are generated by the Rotated Gumbel Copula. Both margins are set to
be student t with df = 5. The parameters of both copula models are selected such that the Kendall correlation of
generated data is 0.5. The X-axis denotes the unconditional cut-off value τ which determines the tail event of the
market return when r mt V aR rm (τ). The simulations are implemented for 5000 times.
Proposition 2.2 implies that the marginal expected shortfall of financial institution i can be written as:
MSE i,t−1 (K) = −E t−1 (r i,t |r m,t K)
(
= −σ i,t E t−1 ɛ i,t |ɛ m,t
K )
σ m,t
= − σ ∫ 1
i,t
τ t
0
ɛ i,t
(u) ∂C(u, τ t)
du
∂u
F −1
where the threshold value K = r m (τ) is the unconditional quantile at τ% of the market return r mt , which defines
the tail event when the market return exceeds K (namely, r m,t K). τ t = F ɛm,t ( K
σ m,t
) is the CDF of the market
return evaluated at the scaled threshold value
K
σ m,t
. C(u, τ) is the copula function between the innovation process
ɛ i,t and ɛ m,t .
It is noteworthy that the value of MES relies not only on the dependence structure determined by copula
function C(u, v; θ), but also on the marginal characteristics such as volatility σ it and F −1
ɛ i,t
. In other words, two firms
with the same dependence structure (C(u, v; θ)) with market but with different volatility (σ it ) would otherwise be
considered as similarly systemically important, but will be treated differently by the measures of MES. This has
been highlighted by Archaya et al. (2012) in the case of the joint Gaussian distribution where the linear correlation
ρ captures the full dependence structure. Therefore, unlike ∆CoV aR or ∆CoV aR , which solely depend
on dependence structure determined by the copula model, MES is a firm-specific value, which is determined by
the dependence structure between firms and market as well as the marginal characteristics of financial firms such
as volatility σ it . Thus, it is possible to observe a situation where a financial institution is more systemically risky
according to the MES measure (higher volatility), but could be less risky based on the ranking of ∆CoV aR
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