Advances in Risk Management

AMIYATOSH PURNANANDAM ET AL. 45

using subadditivity. However,ρ(γηc) equals zero since this portfolio is accepted by the
regulator,γηc∈Aη. Hence,ρ(η 2 )≤ρ(η 1 ) and the monotonicity of riskfree capital is proved.
Consider a portfolioη ∈Aηsuch thatPη−i<0 for somei. It must be proved thatρ(η)>0.
Proceed by contradiction by supposing thatρ(η)=0 which implies thatη∈Aηby Defini-
tion 2.2.2. However, Definition 2.2.1 requires thatPη≥0 forη∈Aη, contradictingPη−i< 0
for anyi. Hence, relevance is proved.
Consider a portfolioηthat does not belong to the acceptance set. By the Separating
Hyperplane Theorem (the acceptance setAηis a convex subset ofRN+^1 (according to
Proposition 2.2.1) and non-empty), there exists a pointη∗on the boundary ofAηsuch
that‖η−η∗‖ 2 is the unique minimum distance ofηfrom setAη. Now consider any scalar
γand letu ̃be the unit directional vector in the directionη∗−η. The vectorη+γ·u ̃is a
point along the path of minimum distance and proves the shortest path property:

ρ(η+γ·u ̃)=‖η+γ·u ̃−η∗‖ 2

=

∥

∥∥

∥η+γ·

(

η∗−η

‖η−η∗‖ 2

)

−η∗

∥

∥∥

∥

=

∥∥

∥∥η∗−η−γ·

(

η∗−η

‖η−η∗‖ 2

)∥∥

∥∥

2

=

(

1 −

γ

‖η−η∗‖ 2

)

‖η−η∗‖ 2

=‖η−η∗‖ 2 −γ

=ρ(η)−γ

APPENDIX B: PROOF OF PROPOSITION 2.4.2

It is sufficient to prove that any two solutions to the linear complementary conditions in
(2.10) yield the same optimal portfolioη∗. Therefore, our procedure is optimal. Let (y 1 ,
λ 1 ) and (y 2 ,λ 2 ) denote two solutions to (2.10) with the following conditions:











y 1 −PPλ 1 =Pη

y 2 −PPλ 2 =Pη

λ 1 ≥0,λ 2 ≥0,y 1 ≥0,y 2 ≥ 0

λ 1 y 1 =0,λ 2 y 2 = 0.

(2.23)

We proceed to show:

Pλ 1 =Pλ 2

with both solutions generating the same optimal portfolioη∗=η+Pλifori=1, 2. From
(2.23):

{

λ 1 Pη=−λ 1 PPλ 1

λ 2 Pη=−λ 2 PPλ 2