Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

A general result valid for any regularly varying distribution was provided. Let F Y
follow a regularly varying distribution with tail index α:
F Y (y) = L(y) ∗ y −α
where L(y) is a slowly varying function, i.e.
then:
(3.38)
L(t · y)
lim = 1 for all y > 0, (3.39)
t→∞ L(t)
λ + =
1
�
max 1, l
β
� α
(3.40)
with l given by equation (3.36). To obtain λ − the limit u → 1 − has to be replaced by
u → 0 + .
The Parametric Approach by Sornette & Malevergne
Following the parametric approach according to Sornette & Malevergne we estimate
a parametric form of tail distribution, which for extreme market risks is assumed to
follow a power-law distribution. For ν = νY = νε we have F Y (y) = Ĉy ∗ y −ν and
F ε(ε) = Ĉε ∗ ε −ν for large y and ε, then the coefficient of tail dependence is a simple
function of the ratio Cε/CY of the scale factors:
λ =
1
1 + β −α · Cε
CY
(3.41)
If the tail indexes αY and αε of the distribution of the factors and the residue are
different, then λ = 1 for αY < αε and λ = 0 for αY > αε.
So far we have only considered a single asset with vector of returns X. Let us now
consider a portfolio of assets with vectors of returns Xi, where each asset follows the
linear factor model:
Xi = βi · Y + εi
(3.42)
with independent noises εi, whose scale factors are Cεi . The portfolio X = � wiXi,
with weights wi, also follows the factor model with a parameter β = � wiβi and noise
ε, whose scale factor is Cε = � |wi| ν · Cεi . The tail dependence between the portfolio
and the market index can now be obtained by equation (3.41):
� �
|wi|
λ = 1 +
α · Cεi
( � wiβi) α �−1 (3.43)
· CY
Tail Dependence between two Assets related by a Factor Model
The tail dependence between two assets related by a factor model given by:
X1 = β1 · Y + ε1
X2 = β2 · Y + ε2
22
(3.44)
(3.45)