statistique, théorie et gestion de portefeuille - Docs at ISFA

308 9. Mesure de la dépendance extrême entre deux actifs financiers
function, i.e:
Corollary 2 in appendix B.2 shows that
L(ty)
lim = 1, ∀y > 0. (11)
t→∞ L(t)
λ =
1
max
1, l
β
α , (12)
where l denotes the limit, when u → 1, of the ratio FX −1 (u)/FY −1 (u). In the case of particular
interest when the distribution of ε is also regularly varying with tail index α and if, in addition, we
have ¯ FY (y) ∼ Cy ·y −α and ¯ Fε(ε) ∼ Cε ·ε −α , 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
. (13)
When the tail indexes αY and αε of the distribution of the factor and the residue are different,
then λ = 0 for αY < αε and λ = 1 for αY > αε.
The results (12) and (13) are very important both for a financial and and economic perseptive
because they express in the most general and straightforward way the risk of extreme co-movements
quantified by the tail dependence parameter λ within the important class of factor models. That λ
increases with the β factor is intuitively clear. Less obvious is the dependence of λ on the structure
of the marginal distributions of the factor and of the idiosynchratic noise, which is found to be
captured uniquely in terms of the ratio of their scale factors Cε and Cy. The scale factors Cε and
Cy together with the factor β thus replace the variance and covariance in their role as the sole
quantifiers of the extreme risks occurring in co-movements.
Until now, we have only considered a single asset X. Let us now consider a portfolio of assets Xi,
each of the assets following exactly the one factor model (6)
Xi = βi · Y + εi, (14)
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| α · 4
Cεi . Thus, equation (13) shows that the tail dependence between the portfolio and
the factor is
|wi|
λ = 1 +
α · Cεi
( wiβi) α −1 . (15)
· CY
When unlimited short sells are allowed, one can follow a “market neutral” strategy yielding β = 0
and thus λ = 0. But in the more realistic case where only limited short sells are authorized, one
cannot reach β = 0, and the best portfolio, which is the less “correlated” with the large market
moves, has to minimze the tail dependence (15).
This simple example clearly shows that it is very different to minimize the extreme co-movements,
according to (15), and to minimize the (linear) correlation ρ between the portfolio and the market
factor given by
ρ =
1 +
w 2 i · V ar(εi)
( wiβi) 2 V ar(Y )
−1/2
. (16)
4 In the more realistic case where the εi’s are not independent but still embody one or several common factors
Y ′ , Y ′′ , · · ·, the resulting scale factor Cε can be calculated with the method described in Bouchaud, Sornette, Walter
and Aguilar (1998)
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