Advances in Risk Management

EMANUELE BORGONOVO AND MARCO PERCOCO 55

Proposition 3 The differential importance of weightaiwith respect to

σpGARCHfor any change in portfolio composition is given by:

Di(a^0 ,da)=

(

∂zt

∂ai

θ+zt

∂θ

∂ai

)

dai

∑n

j= 1

(

∂zt

∂aj

θ+zt

∂θ

∂aj

)

daj

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

a^0

(3.21)

Proof: The proof is in the Appendix.
Equation (3.21) determines the analytical expression of the importance of
portfolio weights with respect toσpestimated via a GARCH model for the
generic TRS. From equation (3.21), it is then straightforward to estimate the
importance of weights for strategies that foresee a uniform or a proportional
change in weights.

Proposition 4 The importance of individual weights with respect to

σpGARCHfor a TRS that assumes of uniform weight changes is:

D (^1) i(a^0 ,da)=
(
∂zt
∂ai
θ+zt
∂θ
∂ai
)
∑n
j= 1
(
∂zt
∂aj
θ+zt
∂θ
∂aj
)
∣
∣
∣
∣
∣∣
∣
∣
∣
a^0
(3.22)
Proof: Combine equation (3.21) with equation (3.6).
Recalling Proposition 1, equation (3.22) shows that utilizing PDs [equa-
tion (3.27)] to evaluate the impact of weights onσGARCHp , one would not
evaluate the impact of any transaction, but only of TRS involving a uniform
change in the portfolio weights.
Proposition 5 The importance of individual weights with respect to
σpGARCHfor a TRS that assumes proportional weight changes is:
D (^2) i(a^0 )=
(
∂zt
∂ai
θ+zt
∂θ
∂ai
)
a^0 i
∑n
j= 1
(
∂zt
∂aj
θ+zt
∂θ
∂aj
)
a^0 j
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
a^0
(3.23)
Proof: Combine equation (3.21) with equation (3.7).
Recalling Propositions 3.23, equation (3.23) states that utilizing elastic-
ity to evaluate the importance of weights with respect toσGARCHp would be
equivalent to consider only the TRS involving proportional relative weight
changes.
Suppose now that the analyst wants to evaluate the impact of chang-
ing group A vs. group B of portfolio weights. Groups A and B could
be, for instance, the set of assets belonging to two selected Sectors. Let