Advances in Risk Management

66 SENSITIVITY ANALYSIS OF PORTFOLIO VOLATILITY

the most influential in a proportional strategy. In the optimal strategy, most
of the diversification would be due to the Manufacturing and Energy cate-
gories, since the corresponding assets are responsible for 66 percent of the
change inσp.
Future research by the authors will involve the application of the pro-
posed method to the SA of other portfolio properties (for example, VaR),
and the examination of its role in asset allocation with dynamic portfolio
optimization.

APPENDIX: PROOF OF PROPOSITION 3

The PDs ofσtwith respect to the portfolio weights are (Theorem 1 of Manganelli, 2004):

∂ht

∂ai

=

∂zt

∂ai

·θ+zt·

∂θ

∂ai

(3.27)

where∂zt/∂aiis immediately derived due to the linear dependence, and∂θ/∂aidenotes
theith row of the matrix:
[
¶q
¶a

]

=−LTaq×L−qq^1 (3.28)

where ∂θ/∂a=[∂θj/∂ai](i=1,...,n, j=1,...,m), Laθ = ∂^2 L/∂aiθj andLθθ=∂^2 L/∂θsθr andLis given by equation (3.18). Equa-
tion (3.28) is found by implicit differentiation from the solution to the set of conditions
determining the parameters:

∂LT

∂θi

∣∣

∣∣

θ

= 0 i=1, 2,...,m (3.29)

namely,θ={θ 1 ,θ 2 ,...,θm}, which can be regarded as an implicit function of the portfolio
weights:

θ=g(a^0 ) (3.30)

wheregis anm-dimensional vector ofn-dimensional functions ata^0. Combining equation
(3.28) with equations (3.27) and (3.2), one finds the result.

NOTES

1. See the monography Shephard (2005) for an overview of the literature on volatility
   estimation.


2. By TRS we mean any change in portfolio composition.


3. After the seminal works of Bollerslev (1986) and Engle (1982), GARCH models
   have become widespread tools in investment management (Duan, 1995; Manganelli,
   2004).


4. We consider the VSA strategy proposed by Manganelli (2004).