Advances in Risk Management

56 SENSITIVITY ANALYSIS OF PORTFOLIO VOLATILITY

SA={a 1 A,a 2 A,...,akA} andSB={a 1 B,a 2 B,...,amB}, withkAandmBlower than
nbe the assets belonging to group A and B respectively. Then:

Proposition 6 The influence of a change in set A of weights with respect

toσGARCHp is determined by:

DSA(a^0 ,da)=

∑

i=1,...,kA

(

∂zt

∂aiA

θ+zt

∂θ

∂aiA

)

da^0 Ai

∑n

j= 1

(

∂zt

∂aj

θ+zt

∂θ

∂aj

)

a^0 j

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

a^0

(3.24)

for example, it is the sum of the importance of the weights in setA.

Proof: Combine equation (3.21) with equation (3.3).
The above result is a consequence ofDadditivity property and cannot be
obtained utilizing the PDs orEas a means for computing the sensitivity of
σpon the portfolio weights.
Similarly, the influence of a change in set B weights is determined by:

DSB(a^0 ,da)=

∑

i=1,...,mB

(

∂zt

∂aiB

θ+zt

∂θ

∂aiB

)

da^0 iB

∑n

j= 1

(

∂zt

∂aj

θ+zt

∂θ

∂aj

)

a^0 j

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

a^0

(3.25)

Thus, if|DSB(a^0 ,da)|>|DSA(a^0 ,da)|then setBis more influential or as
influentialassetAonσp. Proposition6enablestoperformthejointsensitivity
ofσpon sets of portfolio weights in a straightforward way.
The next section is devoted to the illustration of empirical results and
insights found by application of the results in Propositions 1–6 and to a
portfolio composed by the 30 stocks of the Dow Jones index.

3.5 EMPIRICAL RESULTS: TRADING STRATEGIES

THROUGH SENSITIVITY ANALYSIS

In this section we present the implications of Propositions 1–6 in the analysis
of trading/reallocation strategies. We consider a portfolio of 30 stocks com-
posing the Dow Jones Industrial Average index (Table 3.1), as of 11 March

2002. Daily returns cover the period ranging from 2 January 1992 through
      11 March 2002.
      The classical portfolio choice problem (Campbell, Lo and McKinley, 1997;
      Taggart, 1996) in its dual form is written as:

a∗=argamin[Var(a′y)]

s.t. (3.26)

E(a′y)=μ