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

Theorem 1 (Sklar)(1959) Let G be a joint distribution function with n marginals F i .
exists an n dimension copula C such that
(
)
G(x 1 , x 2 , · · · , x n ) = C F 1 (x 1 ), F 2 (x 2 ), · · · , F n (x n )
Then there
If all margins F i are continuous, then the copula C is unique.
In contrast to the traditional dependence measure like Pearson’s correlation ρ and Kendall’s τ, the copula model
is capable of measuring the whole dependence structure including tail dependence between random variates.
Definition 1: Tail Dependence
Let X 1 and X 2 be two random variables with CDF F 1 and F 2 . Then the lower tail dependence (LTD)
coefficient of two random variables (X 1 , X 2 ) is defined as
LT D(X 1 , X 2 ) =
(
)
lim Pr X 2 < F −1
u−→0 + 2 (u)|X 1 < F1 −1
C(u, u)
(u) = lim
u−→0 + u
Analogously, the upper tail dependence (UTD) coefficient of two random variables (X 1 , X 2 ) is defined
as
(
)
UT D(X 1 , X 2 ) = lim Pr X 2 > F −1
u−→1 − 2 (u)|X 1 > F1 −1
1 − 2u + C(u, u)
(u) = lim
u−→1 − 1 − u
In the following, a wide range of copula models are to be used to estimate the systemic risk measures ∆CoV aR
and MES, and we will study their relationship with the tail dependence between the financial institutions and
market.
2.2.1 ∆ CoVaR Estimation
If the joint distribution F (x, y) has a continuous marginal distribution, Sklar’s Theorem (1959) stated that
there exists a unique copula function such that any bivariate distribution can be represented by the combination
of their marginal distributions in a certain form of copula function:
(
)
F (x, y) = C F x (x), F y (y); θ = C(u, v; θ)
where u and v are the marginal distributions F x (x) and F y (y) of x and y respectively. Differentiating C(u, v; θ)
with respect to u, we can obtain the conditional distribution of y given x (See the proof in the Appendix):
F (y|x) =
∂C(u, v; θ)
∂u
= C 1 (u, v; θ) = C 1 (F x (x), F y (y); θ)
where F (y|x) denotes the conditional distribution of y given x and C 1 (u, v; θ) represents the partial derivative of
the copula with respect to the first argument.
By defining τ = F (y|x) and inverting C 1 with respect to the second argument can derive the marginal distribution
of y: F y (y) as
F y (y) = C −1
1 (F x(x), τ; θ)
Let x = V aR x (τ), and y = CoV aRτ
y|x . The solution for CoVaR in the context of the copula framework can be
explicitly expressed as:
( (
CoVaR y|x
τ
= F −1
y
C −1
1
9
F x (V aR x (τ)), τ, θ) ) (1)