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

For rI > 3rD it is immediate to see that ω_m — 2ωs_q < 0 if Dm < 4. For rI ≤ 3rD, ω_m — 2ω_sq
can be rearranged as

(¹ _ k-, ₊ ² k³ ʌ D

2   ^lmn ⁺ 3 mn J Dm

(1 ^- ksq)²    Dsq

Suppose for a moment k_m = k_sq and D_m = 2D_sq. Then, the expression simplifies to
k2Dsq (4ksq — 1), which is negative because ksq < 1/2. To see that this holds also for
km >ksq, we use (21) and rewrite ωm — 2ωsq as

Denote now A = (∣ — km + 3 km). Since A is decreasing in km and km > ksq for rI ≤ 3rD,
it follows ωm — 2ωsq < 0 when Dm = 2Dsq. The same holds for Dm < 2. By plotting the
expression ( DD A Dm — 1) for Dm > 2 and DD ∈ (1,3], one sees that there is a level h ∈ (2,4)
of the ratio Dm such that ωm ≤ 2ωs_q if Dm ≤ h, and ωm > 2ωs_q otherwise. The plot is
available from the authors upon request.                                         Q.E.D.

Proof of Proposition 4

This proof is a generalization of that of Lemma 1. Let D_tot denote the total deposits
NDsq = Dm +(N — 2)Dc, and let Rtot denote the total reserves NRsq = Rm +(N — 2)Rc.
Applying the general formula for the distribution of a weighted sum of uniformly distributed
random variables in Bradley and Gupta (2002) to our model, we obtain the density functions
of the aggregate liquidity demands in the status quo f_sq (X_sq) and after the merger f_m (X_m)
as

N

^fs^q (Xs^q ) = (N — 1)!(Dsq )^N X=0 Γ¹)∖ i

PN^-2 hf 1 iⁱ (N^-^ X X    D (i 1 'I D IN^-2 I (N^-^ X X    iD N-^-2i

_i=1 (—1) _i-1 (Xm — Dm — (i — 1)Dc)₊   + _i   (Xm — iDc)₊

(N — 2)!Dm(Dc)N^-2

The two density functions are plotted in Figure 3. The density f_sq (X_sq) is more concentrated
around the mean than f_m(X_m). To verify that this is always the case, we compare the
variances of Xsq and Xm , which are given by

N

Var(Xsq)=^X  D_s²_qV ar(δi),

i=1

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