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

have
(
)
MES(τ) = −E X|Y V aR y (τ)
= −
= −
= −
= − 1 τ
N∑
i=1
N∑
i=1
(
)
x i P r X = x i |Y V aR y (τ)
(
)
P r X = x i , Y V aR y (τ)
x i (
)
P r Y V aR y (τ)
)
(Y V aR y (τ)|X = x i
N∑ P r
x i
i=1
N∑
i=1
(
) P r(X = x i )
P r Y V aR y (τ)
)
x i P r
(Y V aR y (τ)|X = x i P r(X = x i )
(5)
Let P r(Y = y i ) = v and P r(X = x i ) = u, we have already proved in the beginning of appendix
that the conditional distribution F (y i |x i ) is equal to the first order derivative of copula function
C(u, v) with respect to u
Therefore
)
P r
(Y y i |X = x i =
)
P r
(Y V aR y (τ)|X = x i =
∂C(u, v)
∂u
∂C(u, τ)
∂u
Substituting equation (6) into the equation (5) yields
(
)
MES(τ) = −E X|Y V aR y (τ)
(6)
= − 1 τ
N∑
i=1
∂C(u, τ)
x i P r(X = x i )
∂u
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