statistique, théorie et gestion de portefeuille - Docs at ISFA

F.2 General case
We now consider a portfolio with N independent assets. Assuming that the cumulants Cn(i) have the same
sign for all i (according to axiom 1), we are going to show that the minimum of Cn is obtained for a portfolio
whose weights are given by
and we have
µ ∗ n =
wi =
N
i=1
Indeed, the cumulant of the portfolio is given by
subject to the constraint
Cn =
1
N
j=i Cn(j) n−1
1
N
j=1 Cn(j) n−1
µ(i) N
j=i
1
N
j=1 Cn(j) n−1
N
i=1
Cn(i) w n i
463
, (162)
1
Cn(j) n−1
Introducing a Lagrange multiplier λ, the first order conditions yields
so that
Cn(i) w n−1
i
. (163)
(164)
N
wi = 1. (165)
i=1
− λ = 0, ∀i ∈ {1, · · · , N}, (166)
w n−1
i
λ
= . (167)
Cn(i)
Since all the Cn(i) are positive, we can find a λ such that all the wi are real and positive, which yields the
announced result (162). From here, there is no simple condition that ensures µ ∗ n < µ ∗ n+k . The simplest way
is to calculate diretly these quantities using the formula (163).
to compare µ ∗ n and µ ∗ n+k
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