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 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 (    __ 44(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 ω
sq as in Corollary 1. We obtain:

f (1 — km + 2 km ) Dm (1 — ksq ) Dsq  for rI 3rD

ωm sq =

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

34



More intriguing information

1. Real Exchange Rate Misalignment: Prelude to Crisis?
2. National curriculum assessment: how to make it better
3. EU enlargement and environmental policy
4. Job quality and labour market performance
5. The name is absent
6. The name is absent
7. Party Groups and Policy Positions in the European Parliament
8. Effort and Performance in Public-Policy Contests
9. Solidaristic Wage Bargaining
10. Segmentación en la era de la globalización: ¿Cómo encontrar un segmento nuevo de mercado?