statistique, théorie et gestion de portefeuille - Docs at ISFA

456 14. Gestion de Portefeuilles multimoments et équilibre de marché
Since ρα(w1, · · · , wN) is a homogeneous function of order α, its first-order derivative with respect to wi is
also a homogeneous function of order α − 1. Using this homogeneity property allows us to write
∂ρα
∂wi
−1 ∂ρα
λ1 (w
∂wi
∗ 1, · · · , w ∗ N) = (µ(i) − µ0), i ∈ {1, · · · , N}, (106)
1
−
λ1
α−1 w ∗ 1
−
1, · · · , λ1
α−1 w ∗
N = (µ(i) − µ0), i ∈ {1, · · · , N} . (107)
Denoting by { ˆw1, · · · , ˆwN} the solution of
this shows that the optimal weights are
∂ρα
( ˆw1, · · · , ˆwN) = (µ(i) − µ0), i ∈ {1, · · · , N}, (108)
∂wi
w ∗ i = λ1
1
α−1 ˆwi. (109)
Now, performing the same calculation as in the case of independent risky assets, the efficient portfolio P
can be realized by investing a weight w0 of the initial wealth in the risk-free asset and the weight (1 − w0)
in the risky fund Π, whose weights are given by
˜wi =
Therefore, the expected return of every efficient portfolio is
ˆwi
N i=1 ˆwi
. (110)
µ = w0 · µ0 + (1 − w0) · µΠ, (111)
where µΠ denotes the expected return of the market portfolio Π, while the risk, measured by ρα is
so that
ρα = (1 − w0) α ρα(Π) , (112)
µ = µ0 + µΠ − µ0
ρα(Π) 1/α ρα 1/α . (113)
This expression is the natural generalization of the relation obtained by (Markovitz 1959) for mean-variance
efficient portfolios.
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