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252 9. Mesure de la dépendance extrême entre deux actifs financiers
major equity markets. This was apparent in figures 2, 4 and 6 for ρ +,−
v , and is strongly confirmed on the
conditional Spearman’s ρ.
Interestingly, there is also an asymmetry or directivity in the mutual influence between markets. For instance,
the Chilean and Mexican markets have an influence on the Argentina and Brazilian markets, but the later do
not have any impact on the Chile and Mexican markets. Chile and Mexico have no contagion effect on each
other while Argentina and Brazil have.
These empirical results on the conditional Spearman’s ρ are different from and often opposite to the conclu-
sion derived from the conditional correlation coefficients ρ +,−
v . This puts in light the difficulty in obtaining
reliable, unambiguous and sensitive estimations of conditional correlation measures. In particular, the Pearson’s
coefficient usually employed to estimate the correlation coefficient between two variables is known
to be not very efficient when the variables are fat-tailed and when the estimation is performed on a small
sample. Indeed, with small samples, the Pearson’s coefficient is very sensitive to the largest value, which
can lead to an important bias in the estimation. Moreover, even with large sample sizes, (Meerschaert and
Scheffler 2001) have shown that the nature of convergence as the sample size T tends to infinity of the Pearson’s
coefficient of two times series with tail index µ towards the theoretical correlation is sensitive to the
existence and strength of the theoretical correlation. If there is no theoretical correlation between the two
times series, the sample correlation tends to zero with Gaussian fluctuations. If the theoretical correlation is
non-zero, the difference between the sample correlation and the theoretical correlation times T 1−2/µ converges
in distribution to a stable law with index µ/2. These large statistical fluctuations are responsible for
the lack of accuracy of the estimated conditional correlation coefficient encountered in the previous section.
Thus, we think that the conditional Spearman’s ρ provides a good alternative both from a theoretical and an
empirical viewpoint.
3 Tail dependence
For the sake of completeness, and since it is directly related to the multivariate extreme values theory, we
study the so-called coefficient of tail dependence λ. To our knowledge, its interest for financial applications
has been first underlined by (Embrechts et al. 2001).
The coefficient of tail dependence characterizes an important property of the extreme dependence between
X and Y , using the (original or unconditional) copula of X and Y . In constrast, the conditional spearman’s
rho is defined in terms of a conditional copula, and can be seen as the “unconditional Spearman’s rho” of the
copula of X and Y conditioned on Y larger than the threshold v. This copula of X and Y conditioned on
Y larger than the threshold v is not the true copula of X and Y because it is modified by the conditioning.
In this sense, the tail dependence parameter λ is a more natural property directly related to the copula of X
and Y .
To begin with, we recall the definition of the coefficient λ as well as of ¯ λ (see below) which allows one to
quantify the amount of dependence in the tail. Then, we present several results concerning the coefficient λ
of tail dependence for various distributions and models, and finally, we discuss the problems encountered in
the estimation of these quantities.
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