statistique, théorie et gestion de portefeuille - Docs at ISFA

This property is verified for all cumulants while is not true for centered moments. In addition, as seen from
their definition in terms of the characteristic function (63), cumulants of order larger than 2 quantify deviation
from the Gaussian law, and thus large risks beyond the variance (equal to the second-order cumulant).
Thus, centered moments of even orders possess all the minimal properties required for a suitable portfolio
risk measure. Cumulants fulfill these requirement only for well behaved distributions, but have an additional
advantage compared to the centered moments, that is, they fulfill the condition (9). For these reasons, we
shall consider below both the centered moments and the cumulants.
In fact, we can be more general. Indeed, as we have written, the centered moments or the cumulants of order
n are homogeneous functions of order n, and due to the positivity requirement, we have to restrict ourselves
to even order centered moments and cumulants. Thus, only homogeneous functions of order 2n can be
considered. Actually, this restrictive constraint can be relaxed by recalling that, given any homogeneous
function f(·) of order p, the function f(·) q is also homogeneous of order p · q. This allows us to decouple
the order of the moments to consider, which quantifies the impact of the large fluctuations, from the influence
of the size of the positions held, measured by the degres of homogeneity of ρ. Thus, considering any even
order centered moments, we can build a risk measure ρ(X) = E (X − E[X]) 2n α/2n which account for
the fluctuations measured by the centered moment of order 2n but with a degree of homogeneity equal to α.
A further generalization is possible to odd-order moments. Indeed, the absolute centered moments satisfy
our three axioms for any odd or even order. We can go one step further and use non-integer order absolute
centered moments, and define the more general risk measure
where γ denotes any positve real number.
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ρ(X) = E [|X − E[X]| γ ] α/γ , (10)
These set of risk measures are very interesting since, due to the Minkowsky inegality, they are convex for
any α and γ larger than 1 :
ρ(u · X + (1 − u) · Y ) ≤ u · ρ(X) + (1 − u) · ρ(Y ), (11)
which ensures that aggregating two risky assets lead to diversify their risk. In fact, in the special case γ = 1,
these measures enjoy the stronger sub-additivity property.
Finally, we should stress that any discrete or continuous (positive) sum of these risk measures, with the same
degree of homogeneity is again a risk measure. This allows us to define “spectral measures of fluctuations”
in the same spirit as in (Acerbi 2002):
ρ(X) = dγ φ(γ) E [(X − E[X]) γ ] α/γ , (12)
where φ is a positive real valued function defined on any subinterval of [1, ∞) such that the integral in
(12) remains finite. It is interesting to restrict oneself to the functions φ whose integral sums up to one:
dγ φ(γ) = 1, which is always possible, up to a renormalization. Indeed, in such a case, φ(γ) represents
the relative weight attributed to the fluctuations measured by a given moment order. Thus, the function φ
can be considered as a measure of the risk aversion of the risk manager with respect to the large fluctuations.
Let us stress that the variance, originally used in (Markovitz 1959)’s portfolio theory, is nothing but the
second centered moment, also equal to the second order cumulant (the three first cumulants and centered
moments are equal). Therefore, a portfolio theory based on the centered moments or on the cumulants
automatically contain (Markovitz 1959)’s theory as a special case, and thus offers a natural generalization
emcompassing large risks of this masterpiece of the financial science. It also embodies several other generalizations
where homogeneous measures of risks are considered, a for instance in (Hwang and Satchell 1999).
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