statistique, théorie et gestion de portefeuille - Docs at ISFA

454 14. Gestion de Portefeuilles multimoments et équilibre de marché
B Generalized efficient frontier and two funds separation theorem
Let us consider a set of N risky assets X1, · · · , XN and a risk-free asset X0. The problem is to find the
optimal allocation of these assets in the following sense:
⎧
⎨
⎩
inf wi∈[0,1] ρα({wi})
i≥0 wi = 1
i≥0 wiµ(i) = µ ,
In other words, we search for the portfolio P with minimum risk as measured by any risk measure ρα
obeying axioms I-IV of section 2 for a given amount of expected return µ and normalized weights wi.
Short-sells are forbidden except for the risk-free asset which can be lent and borrowed at the same interest
rate µ0. Thus, the weights wi’s are assumed positive for all i ≥ 1.
B.1 Case of independent assets when the risk is measured by the cumulants
To start with a simple example, let us assume that the risky assets are independent and that we choose to
measure the risk with the cumulants of their distributions of returns. The case when the assets are dependent
and/or when the risk is measured by any ρα will be considered later. Since the assets are assumed
independent, the cumulant of order n of the pdf of returns of the portfolio is simply given by
Cn =
(90)
N
wi n Cn(i), (91)
i=1
where Cn(i) denotes the marginal nth order cumulant of the pdf of returns of the asset i. In order to solve
this problem, let us introduce the Lagrangian
N
N
L = Cn − λ1 wi µ(i) − µ − λ2 wi − 1 , (92)
i=0
where λ1 and λ2 are two Lagrange multipliers. Differentiating with respect to w0 yields
i=0
λ2 = µ0 λ1, (93)
which by substitution in equation (92) gives
N
L = Cn − λ1 wi (µ(i) − µ0) − (µ − µ0) . (94)
i=1
Let us now differentiate L with respect to wi, i ≥ 1, we obtain
so that
n w ∗ i n−1 Cn(i) − λ1(µ(i) − µ0) = 0, (95)
w ∗ 1
i = λ1
n−1
Applying the normalization constraint yields
1
w0 + λ1
n−1
N
i=1
1
µ(i) − µ0
n−1
. (96)
n Cn(i)
1
µ(i) − µ0
n−1
n Cn(i)
30
= 1, (97)