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9.2. Estimation du coefficient de dépendance de queue 309
Thus, since the minimum of ρ may be very different from the minimum of λ, minimizing ρ almost
surely leads to accept a level of extreme risks which is not optimal.
2.4 Tail dependence between two assets related by a factor model
We now present the second part of our theoretical result. Let X1 and X2 be two random variables
(two assets) of cumulative distributions functions F1, F2 with a common factor Y . Let ε1 and ε2
be the idiosyncratic noises associated with these two assets X1 and X2. We allow the idiosyncratic
noises to be dependent random variables, as occurs for instance if they embody the effect of other
factors Y ′ , Y ′′ , ... which are independent of Y . Our essential assumption is that the distribution of
the factor Y must have a tail not thinner than the tail of the distributions of the other factors Y ′ ,
Y ′′ , ... This hypothesis is crucial in order to detect the existence of tail-dependence. This means
that, for purposes of characterizing tail dependencies in factor models, our model can always be
re-stated as a single factor model where the single factor is the factor with the thickest tail. This
makes our results quite general. Then, the model can be written as
X1 = β1 · Y + ε1 , (17)
X2 = β2 · Y + ε2 . (18)
We prove in appendix A.2, that the coefficient of (upper) tail dependence λ+ = limu→1 Pr{X1 >
F1 −1 (u) | X2 > F2 −1 (u)}, between the assets X1 and X2, is given by the expression
λ+ =
∞
maxl 1
β1 , l 2
β2 dx f(x) , (19)
which is very similar to that found for the tail dependence between an asset and one of its explaining
factor (see equation (7)). As previously, l1,2 denotes the limit, when u → 1, of the ratio
F1,2 −1 (u)/FY −1 (u), and f(x) is the limit, when t → +∞, of t · PY (tx)/ ¯ FY (t).
The result (19) can be cast in a different illuminating way. Let λ(X1, Y ) (resp. λ(X2, Y )) denote
the coefficient of tail dependence between the asset X1 (resp. X2) and their common factor Y . Let
λ(X1, X2) denote the tail dependence between the two assets. Equation (19) allows us to assert
that
λ(X1, X2) = min{λ(X1, Y ), λ(X2, Y )}. (20)
The tail dependence between the two assets X1 and X2 is nothing but the smallest tail dependence
between each asset and the common factor. Therefore, a decrease of the tail dependence between
the assets and the market will also lead automatically to a decrease of the tail dependence between
the two assets. This result also shows that it is sufficient to study the tail dependence between the
assets and their common factor to obtain the tail dependence between any pair of assets.
The result (20) is also useful in the context of portfolio analysis. Not only does it provide a tool
for assessing the probability of large losses of a portfolio composed of assets driven by a common
factor, it also allows us to define novel strategies of portfolio optimization based on the selection
and weighting of stocks chosen so as to balance to risks associated with extreme co-movements.
Such an approach has been tested in (Malevergne and Sornette 2002) with encouraging results.
3 Empirical study
We now apply our theoretical results to the daily returns of a set of stocks traded on the New York
Stock Exchange. In order to estimate the parameters of the factor model (6), the Standard and
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