statistique, théorie et gestion de portefeuille - Docs at ISFA

402 13. Gestion de portefeuille sous contraintes de capital économique
4.1.1 Case of independent assets
“Super-exponential” portfolio (c > 1)
Consider assets distributed according to modified Weibull distributions with the same exponent c > 1.
The Value-at-Risk is given by
VaRα = ξ(α) 2/c W (0) ·
N
i=1
|wiχi| c
c−1
c−1
2
Introducing the Lagrange multiplier λ, the first order condition yields
∂ ˆχ
∂wi
and the composition of the minimal risk portfolio is
=
λ
ξ(α) W (0)
wi ∗ = χ−c
i
j χ−c
j
which satistifies the positivity of the Hessian matrix Hjk = ∂2 ˆχ
∂wj∂wk {w∗ i }
The minimal risk portfolio is such that
VaR ∗ α = ξ(α)2/c W (0)
i χ−c
1
c
j
, (60)
∀i, (61)
, µ ∗
=
i χ−c
i µi
j χ−c
j
where µi is the return of asset i and µ ∗ is the return of the minimum risk portfolio.
(second order condition).
(62)
, (63)
sub-exponential portfolio (c ≤ 1)
Consider assets distributed according to modified Weibull distributions with the same exponent c < 1.
The Value-at-Risk is now given by
VaRα = ξ(α) c/2 W (0) · max{|w1χ1|, · · · , |wNχN|}. (64)
Since the weights wi are positive, the modulus appearing in the argument of the max() function can be
removed. It is easy to see that the minimum of VaRα is obtained when all the wiχi’s are equal, provided
that the constraint wi = 1 can be satisfied. Indeed, let us start with the situation where
w1χ1 = w2χ2 = · · · = wNχN . (65)
Let us decrease the weight w1. Then, w1χ1 decreases with respect to the initial maximum situation (65) but,
in order to satisfy the constraint
i wi = 1, at least one of the other weights wj, j ≥ 2 has to increase, so
that wjχj increases, leading to a maximum for the set of the wiχi’s greater than in the initial situation where
(65) holds. Therefore,
and the constraint
i wi = 1 yields
w ∗ i = A
A =
χi
, ∀i, (66)
1
i χ−1
i
14
, (67)