Advances in Risk Management

EMANUELE BORGONOVO AND MARCO PERCOCO 53

proving that ranking based onPDsis equivalent to ranking based onD (^1) s(a^0 ),
for example, to stating an assumption of uniform parameter changes. Q.E.D.
We now demonstrate that, if instead ofPDs, theEof the weights were
considered, this would be equivalent to make the implicit assumption of a
TRS involving proportional weight changes.
Proposition 2 Ranking weights based on “elasticities” is equivalent to
consider TRS involving proportional weight changes.
Proof: In the same settings as in the above proof, let us assume that one
ranks weights utilizing Elasticity; for example:
aiaj⇔|Ei(a^0 )|>|Ej(a^0 )| (3.13)
Nothing changes in|Ei(a^0 )|>|Ej(a^0 )|if one multiplies and divides both sides
for
∣
∣∑nk= 1 Ek(a^0 )
∣
∣; one gets:
aiaj⇔|Ei(a^0 )|>|Ej(a^0 )|⇔
|Ei(a^0 )|
∣
∣∑nk= 1 Ek(a^0 )
∣
∣>
|Ej(a^0 )|
∣
∣∑nk= 1 Ek(a^0 )
∣
∣ (3.14)
Utilizing equation (3.8), the above is then equivalent to stating:
aiaj⇔|D (^2) i(a^0 )|>|D (^2) j(a^0 )| (3.15)
for example, analysing the influence of weights based onEis equivalent to
stating an assumption of proportional parameter changes. Q.E.D.
As mentioned in section 3.1, the implication on the analysis of TRS of
Propositions is thatPDs/Eare appropriate SA measures on the subset of
strategies involving a uniform/proportional change in the portfolio com-
position. As discussed in section 3.2, these limitations can be overcome by
applyingD. In fact, equation (3.2) accommodates any TRS. Furthermore it
includesPDsandEas particular cases, as equations (3.6) and (3.7) show. In
the next section, we illustrate the application of these concepts to the SA of
σpestimated via GARCH models.

3.4 IMPORTANCE OF PORTFOLIO WEIGHTS IN GARCH

VOLATILITY ESTIMATION MODELS

Models of time-varying volatility have been popular since the early 1990s
in empirical research area of finance, following the influential papers by
Engle (1982). GARCH models are well-known in the time series Econo-
metrics literature. From the initial concern with an economic phenomenon,
for example, time-varying and autoregressive variance of inflation (Engle,