Mean Variance Optimization of Non-Linear Systems and Worst-case Analysis

Proof of Proposition 6

Statement 1. From the proof of Proposition 4, Km = ksq implies Φ_m = Φsq when rrrD = σ,
and Φ_m < Φsq when rrD < σ. Since Km > ksq in the range rrD < ρ, it is Φ_m < Φsq when
rrI = σ. The strict inequality and continuity imply that there must exist a neighborhood
where -rr-D > σ and Φ_m < Φsq. For -rr-D > ρ, Φ_m > Φsq (from Proposition 5); hence, there
must exist a critical level g ∈ (σ, ρ) (with σ < ρ from the proofs of Propositions 2 and 4)
such that as Φ_m < Φ_sq if rD < g, and Φ_m > Φ_sq otherwise. The first statement follows.

Statement 2. From Proposition 2, km = ksq for rrD = 1 and rrD = ρ, and km > ksq for
1 < r^rD < ρ. This induces the same relation between Km and ksq, so that Km — ksq is
first increasing and then decreasing in the interval rD ∈ (1, ρ). By Proposition 4, when
Dm = 2D_c there is a neighborhood of rrI = 1 where Ωm — Ωsq > 0. Also, when rrD = ρ
and Dm = 2D_c, Ωm > Ωsq. When rrD = 1, it is always Ωm = Ωsq = D^ot. From Lemma
3, when Dm = 2D_c it is Ωm — Ωsq < 0 for all rrD ∈ (1, ρ) and Ωm = Ωsq when rD = ρ.
By continuity, if one fixes a sufficiently small level of asymmetry in the deposit bases across
banks (Dm — 2D_c sufficiently small), then Ωm — Ωsq > 0 in an immediate neighborhood of
r^rD = 1. Given that Km — ksq is increasing around rrD = 1, there will be a higher ratio rrD,

named g, such that if the merger generates that asymmetry when rrrD = g, then Ωm — Ω,
and Ωm — Ωsq < 0 in the immediate right neighborhood. Again by continuity, Ωm — Ω,

sq

sq

=0

>0

in an immediate neighborhood of --rI = ρ. Given that Km — ksq is decreasing around rrI = ρ,
there will be a smaller ratio rrD, named g, such that, when rrD = g, then Ωm — Ωsq = 0 and
Ωm — Ωsq < 0 in the immediate left neighborhood. The second statement follows. Q.E.D.

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