Advances in Risk Management

52 SENSITIVITY ANALYSIS OF PORTFOLIO VOLATILITY

the impact of TRS on portfolio properties. We begin with a simple
example.

Example 1,let

v=a 1 v 1 +a 2 v 2 (3.9)

be the value of a portfolio at a certain point in time. Let alsoa 1 =100,
a 2 =9900,v 1 =10EUR andv 2 =5EUR. The total value of the portfolio is then
v=50500EUR. Let us undertake the SAofvwith respect to the weights, with
reference to two hypothetical trading strategies. The first TRS is to buy one
additional stock of 1 and 2. In this case we have a unitary change ina 1 and
a 2 , for example,da 1 =da 2 =1. Applying equation (3.6), one gets:D 11 =0.667
andD 12 =0.333. This result means that asset 1 is the most influential if a
TRS involving uniform weight changes is considered. Let us consider the
case in which the trader opts for a proportional change in the two assets, for
example, he buys (or sells)% in each of them. Applying equation (3.7),
one gets:D 21 =0.02 andD 22 =0.98. In this case asset 2 would be the most
influential one on the portfolio value.
The above example clearly shows that to evaluate the impact of changes
in portfolio composition one must consider not only the rate of change (PD)
of the portfolio with respect to the weights, but also the relative way in
which the weights are changed [equation (3.5)].^6
We now extend the meaning of “example” showing that evaluating the
impact of portfolio changes by means of the solePDis equivalent to make
the implicit assumption of a TRS involving uniform weight changes.

Proposition 1 Ranking weights based on PDs is equivalent to consider-

ing TRS involving uniform weight changes.

Proof: LetY^0 =f(a^0 ) denote an-asset, differentiable portfolio property as a
function of the current allocationa^0. We use the symbolaiajto indicate
that weightaiis more important thanaj(Borgonovo, 2001b). If one utilizes
PDto rank weights, then one says thataiis more important thanajwhen the
magnitude of the change inY^0 provoked by a change inainamely|fi(a^0 )|,is
greater than the magnitude of the change inY^0 provoked by a change inaj:

aiaj⇔|fi(a^0 )|>|fj(a^0 )| (3.10)

Nothing changes in|fi(a^0 )|>|fj(a^0 )|if one multiplies and divides both sides
for|

∑n

k= 1 fk(a

(^0) )|. One gets:
aiaj⇔|fi(a^0 )|>|fj(a^0 )|⇔
∣
∣fi(a^0 )
∣
∣
∣
∣∑nk= 1 fk(a (^0) )
∣
∣>
∣
∣fj(a^0 )
∣
∣
∣
∣∑nk= 1 fk(a (^0) )
∣
∣ (3.11)
The above is then equivalent to stating:
aiaj⇔|D (^1) i(a^0 )|>|D (^1) j(a^0 )| (3.12)