Advances in Risk Management

EMANUELE BORGONOVO AND MARCO PERCOCO 65

SA based on partial derivatives (PD) or elasticity (E) leads to limitations in
testing:

1 the sensitivity of a portfolio property with respect to simultaneous

changes in several parameters;

2 the sensitivity of a portfolio property for a strategy involving relative

weight changes other than uniform or proportional ones.

In particular, we have proven that (i) utilizing PDs to rank weights is equiva-
lenttostatetheassumptionthatalltheweightsarechangedbysameamount;
and (ii) utilizingEwould be equivalent to state that all the weights are
changed by the same proportion.
We have illustrated that these limitations can be overcome if one utilizes
the Differential Importance Measure (D) as SA method. We have shown
thatD: (i) makes the evaluation of the impact of changes in several weights
straightforward thanks to the additivity property; and (ii) allows to take into
account the effect of arbitrary relative changes in portfolio weights.
We have discussed the SA ofσpestimated through GARCH models. We
have found the analytical expression of theDof portfolio weights with
respect toσp. We have examined the differences in the expressions for three
possible strategies: the uniform weight change strategy – equivalent to uti-
lizing PDs, the proportional change case – equivalent to utilizingE, and
the “optimal” strategy. We have also provided the expression for the joint
importance of weights with respect toσpby exploitingDadditivity property.
Empirical results have been obtained by applying the proposed approach
to the SAof the portfolio composed by the 30 stocks of the Dow Jones index –
the same portfolio as in Manganelli (2004). We have analysed the importance
of weights for strategies involving uniform and proportional changes in the
weights, utilizingDin cases H1 [equation (3.22)] and H2 [equation (3.23)].
We then focused on the “optimal” strategy, and estimated quantitatively
the degree of similarity of the three TRS by making use of Savage Score
Correlation Coefficients. The corresponding strategy correlation matrix has
shown that the optimal strategy resulted closer to a proportional strategy
than to a uniform one. Such similarity is confirmed also by the magnitude
of total change inσpprovoked by the two strategies. The numerical findings
confirm the fact that the effect of assets varies according to the adopted strat-
egy and that conclusions obtained applying PDs orEcannot be extended to
the examination of generic strategies.
We then studied the effect of diversification by examining the change
inσpwith respect to the portfolio composition. This required the compu-
tation of the sensitivity ofσpto groups of assets [equation (3.24)]. Each
group reflected the industry of operation (sector) of the firm. We have seen
that Manufacturing assets resulted the most important for the uniform and
optimal strategies, while “Telecommunication and ICT” assets resulted as