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

Using (16), we can express the liquidity risk for the merged banks as

1 - — RRm ⁴Xm dxm   for rI ≤ 3rD

0 D_m
φ_m = Pr(xm >Rm)=

[ /Dm 4(DD-^xm) dxm for rI > 3rD.

Solvingtheintegrals, we obtain φm = 1 — 2Dmm for rI ≤ 3rD and 2 —4Dm + 2Dmm for rI > 3r^D.
D_m                 Dm  D_m

Substituting km = Dm implies

{1 — 2k2    for rI ≤ 3r^D

m

2(1 — k_m)² for r^I > 3r^D .

Substituting km as in (20), we can express the merged banks’ resiliency as

{2[z(rI, r^D)]²    for rI ≤ 3r^D

_

1 — 2( ∖³rD)² for rI > 3rD.

Similarly, from Corollary 1 we can write a bank’s individual resiliency in the status quo as
1 — Φs_q = ksq = 1 — ʌ/rr. Plotting these expressions as a function of the ratio rD, one
immediately sees that 1 — φ_m > 1 — φ_sq always holds, so that φ_m < φ_sq . The plot is available
from the authors upon request.                                                  Q.E.D.

Proof of Corollary 4

Using (16), we can express the expected liquidity needs for the merged banks as

RΓ (_ _ R )4χmd_r + RDm(_  _ R )4(D^m-x^m)d

I R (^xm   ɪɛm) D2 ^txχm + I Dm (^xm   ^2tm)    D2    ^txχm

^m              m_rn           2                  m_rn

ω_m =

C Dm (    __ 4 ∖ ⁴(Dm^-xm) J

I Dm (^xm   m m) )    D2    ^ux'm

2                   ^m

Solving the integrals, we obtain ωm = Dm — Rm + 3 Rmm for rI ≤ 3r^D and 3 (Dm-Rm)
2         3D_m                 3  D_m

for rI > 3r^D. Substituting km = DDm, we obtain

(¹ — km + I km ) Dm for rI ≤ 3r^D

2(1 — km)³Dm       for rI > 3r^D.

To compare ω_m with 2ω_sq , we substitute (20) in the above expression for ω_m and (21) in
the expression for ωsq as in Corollary 1. We obtain:

f (^{1 — k}m ⁺ 2 ^km ) Dm ^{— (1 — k}sq ) Dsq  ^{for r}I ≤ ^3rD

ω_m — 2ω_sq =

I ^rD (D₄m — Dsq)                       for rI > 3rD.

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