statistique, théorie et gestion de portefeuille - Docs at ISFA

460 14. Gestion de Portefeuilles multimoments et équilibre de marché
E Calculation of the moments of the distribution of portfolio returns
Let us start with equation (72) in the 2-asset case :
ˆPS(k)
1
=
2π 1 − ρ2
dy1dy2 exp − 1
2 ytV −1
y1
y + ik χ1w1sgn(y1)
√
2
y2
+χ2w2sgn(y2)
√
2
Expanding the exponential and using the definition (67) of moments, we get
Posing
1
Mn =
2π 1 − ρ2
dy1 dy2
γq1q2 (n, p) = χ1 p χ2 n−p
this leads to
2π 1 − ρ 2
n
p=0
n
χ
p
p
dy1dy2 sgn(y1) p
y1
√
2
Mn =
1χn−p 2 wp 1wn−p 2
×sgn(y2) n−p
q1p
y2
√
2
sgn(y1) p
y1
√
2
q2(n−p)
sgn(y2) n−p
y2
√
2
Let us defined the auxiliary variables α and β such that
α = (V −1 )11 = (V −1 )22 = 1
1−ρ 2 ,
β = −(V −1 )12 = −(V −1 )21 = ρ
1−ρ 2 .
n
p=0
q1
+
q2
q1p
×
. (143)
1
−
e 2 ytV −1y . (144)
q2(n−p)
1
−
e 2 ytV −1y , (145)
n
w
p
p
1wn−pγq1q2 2 (n, p) . (146)
(147)
Performing a simple change of variable in (145), we can transform the integration such that it is defined
solely within the first quadrant (y1 ≥ 0, y2 ≥ 0), namely
γq1q2 (n, p) = χ1 p n−p 1 + (−1)n
χ2
2π 1 − ρ2 +∞ +∞
dy1 dy2
0
0
q1p q2(n−p) y1 y2
α
− √2 √2 e 2 (y2 1 +y2 2 ) ×
× e βy1y2
p −βy1y2
+ (−1) e
. (148)
This equation imposes that the coefficients γ vanish if n is odd. This leads to the vanishing of the moments
of odd orders, as expected for a symmetric distribution. Then, we expand e βy1y2 + (−1) p e −βy1y2 in series.
Permuting the sum sign and the integral allows us to decouple the integrations over the two variables y1 and
y2:
γq1q2 (n, p) = χ1 p n−p 1 + (−1)n
χ2
2π 1 − ρ2 +∞
[1 + (−1)
s=0
p+s ] βs
s!
36
+∞
dy1
0
+∞
× dy2
0
y q1p+s
1
2 q1p 2
y q2(n−p)+s
2
2 q2 (n−p)
2
α
−
e 2 y2 1
α
−
e 2 y2 1
×
. (149)