Advances in Risk Management

EMANUELE BORGONOVO AND MARCO PERCOCO 51

As a consequence,

∑n

s= 1

Ds(x^0 ,dx)= 1 (3.4)

for example, the sum of theDi(i=1,...,n) of all parameters is always

equal to unity.

B Equation (3.2) shows thatDaccounts for the relative parameters changes
through the dependence ondx. In fact, equation (3.2) can be rewritten as:

Ds(x^0 ,dx)=

fs(x^0 )

∑n

j= 1 fj(x

(^0) )dxj
dxs
(3.5)
In the hypothesis of uniform parameter changes (H1) (dxj=dxs∀j,s), one
finds:
D (^1) s(x^0 )=
fs(x^0 )
∑n
j= 1 fj(x
(^0) ) (3.6)
In the hypothesis of proportional changes (H2)
(
dxj
x^0 j =ω∀j
)
, one finds:
D (^2) s(x^0 )=
fs(x^0 )·x^0 s
∑n
j= 1 fj(x
(^0) )·x 0
j
(3.7)
It can be shown thatDgeneralizes other local SA techniques as the
Fussell–Vesely importance measure and Local Importance Measures based
on normalized partial derivatives, also known as Criticality Importance or
E.^5 More specifically, in case H2 it holds that (Borgonovo and Peccati, 2004):
D (^2) s(x^0 )=
Es(x^0 )
∑n
j= 1 Ej(x
(^0) ) (3.8)
whereEs(x^0 ) is the elasticity ofYwith respect toxsatx^0. Equation (3.8) shows
thatEproduces the importance of parameters for proportional changes in
their. In the next section, we examine how these results affect the SA of
portfolio properties.

3.3 EFFECT OF RELATIVE WEIGHT CHANGES

We now show a first portfolio management implication of equations (3.6)

and (3.7): relative weight changes cannot be neglected when evaluating