AMIYATOSH PURNANANDAM ET AL. 41row rank,PPis positive definite. This property impliesλwPPλw≥0 which
yields (λw(Pη)−)^2 ≤λw(Pη)−and proves that
xw=λw(Pη)−≤ 1 (2.21)Thus, the optimal amount of insurance to purchase is strictly less than one
unit.
The strict inequality in the above corollary reinforces the importance of
diversification. Specifically, we are able to diversify risk more effectively
once the insurance contract becomes available.
To summarize, it is not necessary for firms to purchase the entire insur-
ance contract provided they engage in subsequent portfolio rebalancing. As
indicated in the next corollary, fewer dollars are also required to be spent on
portfolio insurance in this circumstance, a result that is later reinforced by
Proposition 2.5.3.
Corollary 2.5.2 The dollar value of required insurance is less with
portfolio rebalancing,xwICw<ICwo,ifPis of full row rank.Proof: This result follows from equations (2.19) and (2.20),
xwICw=ICwo−qPλwand the fact that the last termqPλw=q(ηw−η) is positive. Indeed,
q(ηw−η)>0 is a consequence of the conditiony=Qδ∗−(Pη)+=Pηw+
(Pη)−xw−(Pη)+≥0 from (2.17) which impliesPηw+(Pη)−−(Pη)+≥0 since
1 >xw≥0. Therefore,Pηw−Pη≥0 with strict inequality in at least one
scenario and by the assumption of no arbitrage,q(ηw−η)>0.
In addition, equation (2.18) implies that neitherδ 1 norηwoare optimal in
the presence of the insurance contract. These statements are formalized in
the following corollary.
Corollary 2.5.3 Ifηis an unacceptable portfolio andPis of full row rank,
then neitherδwo=[
ηwo
0]
nor δ 1 =[
η
1]are optimal in the presence of the insurance contract.Proof:Ifδwois acceptable, then it is also acceptable in the presence of
the insurance contract. But ifδwois optimal, then (2.14) and (2.18) jointly
imply that
P(λwo−λw)=0 and ((Pη)−)λw= 0Hence, with the payoff matrixPbeing of full row rank, it follows that
λwo=λwwith (2.13) implyingλwoPPλwo=0 which contradictsPPbeing
positive definite since (Pη)−is strictly greater than 0 in at least one scenario.