Advances in Risk Management

40 INCORPORATING DIVERSIFICATION INTO RISK MANAGEMENT

with linear complementarity conditions:
{
y−QQλ=−(Pη)−
y≥0, λ≥0, λy= 0

(2.17)

fory=Qδ∗−(Pη)+. Denote the optimal solution to (2.17) by (ηw,xw,λw)
which yields:
[
ηw
xw

]

=

[

η

0

]

+

[

P

((Pη)−)

]

λw (2.18)

Therefore, the second equation of (2.18) implies the optimal amount of
insurance to purchase equals:

xw=((Pη)−)λw≥ 0 (2.19)

Hence, conditional on additional rebalancing fromηtoηw, the price of the
insurance contract isICw·xw+qηw=qηwowhich is equivalent to

ICw=

qP(λwo−λw)

((Pη)−)λw

(2.20)

by equation (2.19) and the relationshipηwo−ηw=η+Pλwo−η−Pλw=
P(λwo−λw).
The magnitude ofxwin equation (2.19) quantifies the importance of diver-
sification. Additional portfolio rebalancing reduces the required amount of
portfolio insurance contract from 1 toxwwhenPis of full row rank as proved
in the next corollary.
With little loss of generality, the matrixPis of full row rank with the
availableNrisky assets exceeding the number of scenariosM. For example,
consider a collection of futures contracts and options ranging across differ-
ent maturities and strike prices. Although the payoffs of these derivative
securities are correlated, it is important to clarify the distinction between
linear dependencies in the columns ofPversus its rows. In particular, cor-
relation between theNrisky securities influences the column rank of this
payoff matrix but not its row rank. Indeed, diversification implies the more
linearly dependent securities included in the optimization problem, the less
drastic is the necessary portfolio rebalancing to achieve acceptability. More
importantly, the row rank ofPis a function of how “close” theMscenarios
are to one another. However, since the scenarios involve extreme events,
redundancy in the rows ofPis not anticipated since this would imply the
scenarios produce identical payoffs for each asset.

Corollary 2.5.1 The optimal amount of portfolio insurance to purchase,

xw, is strictly less than one unit ifPis of full row rank.

Proof: The inequalityxw≤1 follows fromλwQQλw=λw(Pη)−by (2.17),
which is equivalent toλwPPλw+(λw(Pη)−)^2 =λw(Pη)−. WhenPis of full