Using (16), we can express the liquidity risk for the merged banks as
1 - — RRm 4Xm dxm for rI ≤ 3rD
0 Dm
φm = 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 > 3rD.
Dm Dm Dm
Substituting km = Dm implies
{1 — 2k2 for rI ≤ 3rD
m
2(1 — km)2 for rI > 3rD .
Substituting km as in (20), we can express the merged banks’ resiliency as
{2[z(rI, rD)]2 for rI ≤ 3rD
_
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 — Φsq = ksq = 1 — ʌ/rr. Plotting these expressions as a function of the ratio rD, one
immediately sees that 1 — φm > 1 — φsq always holds, so that φm < φsq . 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 )4χmdr + RDm(_ _ R )4(Dm-xm)d
for rI ≤ 3rD
for rI > 3rD .
I R (xm ɪɛm) D2 txχm + I Dm (xm 2tm) D2 txχm
m mrn 2 mrn
ωm =
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 ≤ 3rD and 3 (Dm-Rm)
2 3Dm 3 Dm
for rI > 3rD. Substituting km = DDm, we obtain
(1 — km + I km ) Dm for rI ≤ 3rD
ωm
2(1 — km)3Dm for rI > 3rD.
To compare ωm with 2ωsq , we substitute (20) in the above expression for ωm and (21) in
the expression for ωsq as in Corollary 1. We obtain:
f (1 — km + 2 km ) Dm — (1 — ksq ) Dsq for rI ≤ 3rD
ωm — 2ωsq =
I rD (D4m — Dsq) for rI > 3rD.
34
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