statistique, théorie et gestion de portefeuille - Docs at ISFA

446 14. Gestion de Portefeuilles multimoments et équilibre de marché
The first point to note is the difference between currencies and stocks. For small as well as for large returns,
the exponents c− and c+ for currencies (excepted Poland and Thailand) are all close to each other. Additional
tests are required to establish whether their relatively small differences are statistically significant. Similarly,
the scale factors are also comparable. In contrast, many stocks exhibit a large asymmetric behavior for large
returns with c+ −c− 0.5 in about one-half of the investigated stocks. This means that the tails of the large
negative returns (“crashes”) are often much fatter than those of the large positive returns (“rallies”).
The second important point is that, for small returns, many stocks have an exponent 〈c+〉 ≈ 〈c−〉 2 and
thus have a behavior not far from a pure Gaussian in the bulk of the distribution, while the average exponent
for currencies is about 1.5 in the same “small return” regime. Therefore, even for small returns, currencies
exhibit a strong departure from Gaussian behavior.
In conclusion, this empirical study shows that the modified Weibull parameterization, although not exact on
the entire range of variation of the returns X, remains consistent within each of the two regimes of small
versus large returns, with a sharp transition between them. It seems especially relevant in the tails of the
return distributions, on which we shall focus our attention next.
8 Cumulant expansion of the portfolio return distribution
8.1 link between moments and cumulants
Before deriving the main result of this section, we recall a standard relation between moments and cumulants
that we need below.
The moments Mn of the distribution P are defined by
ˆP (k) =
+∞
n=0
(ik) n
where ˆ P is the characteristic function, i.e., the Fourier transform of P :
Similarly, the cumulants Cn are given by
Differentiating n times the equation
ln
ˆP (k) =
ˆP (k) = exp
+∞
n=0
(ik) n
+∞
−∞
n! Mn
+∞
n! Mn , (61)
dS P (S)e ikS . (62)
n=1
=
(ik) n
n! Cn
+∞
n=1
(ik) n
. (63)
n! Cn , (64)
we obtain the following recurrence relations between the moments and the cumulants :
Mn =
n−1
n − 1
MpCn−p ,
p
(65)
p=0
n−1
n − 1
Cn = Mn −
CpMn−p . (66)
n − p
p=1
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