statistique, théorie et gestion de portefeuille - Docs at ISFA

306 9. Mesure de la dépendance extrême entre deux actifs financiers
instance, let us consider the Pareto distribution ¯ FY (y) = 1/(y/y0) µ defined for y ≥ y0, whose
density is equal to PY (y) = (µ/y0)/(y/y0) 1+µ ; the limit (9) gives f(x) = µ/x 1+µ . In contrast,
for the Poisson law ¯ FY (y) = e −ry defined for y ≥ 0 with density PY (y) = re −ry , the limit (9)
gives f(x) = limt→∞ r t e −rt(x−1) = 0 for x > 1. Thus an estimation of the tail of the factor
distribution is sufficient to infer the limit function f(x). Moreover, equation (7) has a rather simple
interpretation since it shows that a non-vanishing coefficient of tail dependence results from the
combination of two phenomena. First, the limit function f(x), which only depends on the behavior
of the factor distribution, must be non-zero. Second, the constant l must remain finite to ensure
that the integral in (7) does not vanish. Thus, the value of the coefficient of tail dependence is
controlled by f(x) solely function of the factor and a second variable l quantifying the competition
of the tails of the distribution of the factor Y and of the idiosyncratic noise ε.
The fundamental result (7) should be of vivid interest to financial economists because it provides a
general, rigorous and simple method for estimating one of the key variable embodying the occurrence
of and the risks associated with extremes in joint distributions. From a theoretical view point, it
also anchors the derivation and quantification of a key variable on extremes in the general class of
financial factor models, thus extending their use and relevance also to this rather novel domain of
extreme dependence, extreme risks and extreme losses.
Up to now, we have assumed that the factor Y and the idiosyncratic noise ε were independent. In
fact, it is important to stress for the sake of generality that the result (7) holds even when they are
dependent, provided that this dependence is not too strong, as explained and made specific at the
end of Appendix A.1.
We now derive two direct consequences of this result (7) (see corollary 1 and 2 in appendix B),
concerning rapidly varying and regularly varying factors 2 , which clearly illustrate the role the
factor itself and the impact of the trade off between the factor and the idiosyncratic noise.
2.2 Absence of tail dependence for rapidly varying factors
Let us assume that the factor Y and the idiosyncratic noise ε are normally distributed (the second
assumption is made for simplicity and will be relaxed below). As a consequence, the joint distribution
of (X, Y ) is the bivariate Gaussian distribution. Refering to the results stated in section 1.1.2,
we conclude that the copula of (X, Y ) is the Gaussian copula whose coefficient of tail dependence
is zero. In fact, it is easy to show that λ = 0 for any non-degenerated distribution of ε.
More generally, let us assume that the distribution of the factor Y is rapidly varying, which describes
the Gaussian, exponential and any distribution decaying faster than any power-law. Then, the
coefficient of tail dependence is identically zero. This result holds for any arbitrary distribution of
the idiosyncratic noise (see corollary 1 in appendix B). It also holds for mixtures of normals or other
distributions fatter than Gaussians, some of which are thought to be reasonable approximations to
empirical stock return distributions.
These statements are somewhat counter-intuitive since one could expect a priori that the coefficient
of tail dependence does not vanish as soon as the tail of the distribution of factor returns is fatter
than the tail the distribution noise returns. Said differently, when the standard deviation of the
idiosynchratic noise ε is small (but not zero), then the idiosynchratic noise component is small and
X and Y are practically identical, and it seems strange that their tail dependence can be equal
2 see Bigham, Goldie, and Teugel (1987) or Embrechts, Kluppelberg, and Mikosh (1997) for a survey of the
properties of rapidly and regularly varying functions
7