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304 9. Mesure de la dépendance extrême entre deux actifs financiers
not provide a useful measure of the dependence for extreme events, since they are constructed over
the whole distributions.
Another natural idea, widely used in the contagion literature, is to work with the conditional correlation
coefficient, conditioned only on the largest events. But, as stressed by Boyer, Gibson,
and Lauretan (1997), such conditional correlation coefficient suffers from a bias: even for a constant
unconditional correlation coefficient, the conditional correlation coefficient changes with the
conditioning set. Therefore, changes in the conditional correlation do not provide a characteristic
signature of a change in the true correlations. The conditional concordance measures suffer from
the same problem.
In view of these deficiencies, it is natural to come back to a fundamental definition of dependence
through the use of probabilities. We thus study the conditional probability that the first variable is
large conditioned on the second variable being large too: ¯ F (x|y) = Pr{X > x|Y > y}, when x and
y goes to infinity. Since the convergence of ¯ F (x|y) may depend on the manner with which x and y
go to infinity (the convergence is not uniform), we need to specify the path taken by the variables
to reach the infinity. Recalling that it would be preferable to have a measure which is independent
of the marginal distributions of X and Y , it is natural to reason in the quantile space. This leads
to choose x = FX −1 (u) and y = FY −1 (u) and replace the conditions x, y → ∞ by u → 1. Doing so,
we define the so-called coefficient of upper tail dependence (see Coles, Heffernan, and Tawn (1999),
Lindskog (2000), or Embrechts, McNeil, and Straumann (2001)):
λ+ = lim
u→1 − Pr{X > FX −1 (u) | Y > FY −1 (u)} . (2)
As required, this measure of dependence is independent of the marginals, since it can be expressed
in term of the copula of X and Y as
λ+ = lim
u→1− 1 − 2u + C(u, u)
1 − u
. (3)
This representation shows that λ+ is symmetric in X and Y , as it should for a reasonable measure
of dependence.
In a similar way, we define the coefficient of lower tail dependence as the probability that X incurs
a large loss assuming that Y incurs a large loss at the same probability level
λ− = lim
u→0 + Pr{X < FX −1 (u) | Y < FY −1 (u)} = lim
u→0 +
C(u, u)
u
. (4)
This last expression has a simple interpretation in term of Value-at-Risk. Indeed, the quantiles
F −1
−1
X (u) and FY (u) are nothing but the Values-at-Risk of assets (or portfolios) X and Y at the
confidence level 1 − u. Thus, the coefficient λ− simply provides the probability that X exceeds the
VaR at level 1 − u, assuming that Y has exceeded the VaR at the same level confidence level 1 − u,
when this level goes to one. As a consequence, the probability that both X and Y exceed their VaR
at the level 1 − u is asymptotically given by λ− · (1 − u) as u → 0. As an example, consider a daily
VaR calculated at the 99% confidence level. Then, the probability that both X and Y undergo a
loss larger than their VaR at the 99% level is approximately given by λ−/100. Thus, when λ− is
about 0.1, the typical recurrence time between such concomitant large losses is about four years,
while for λ− ≈ 0.5 it is less than ten months.
The values of the coefficients of tail dependence are known explicitely for a large number of different
copulas. For instance, the Gaussian copula, which is the copula derived from de Gaussian multivariate
distribution, has a zero coefficient of tail dependence. In contrast, the Gumbel’s copula used
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