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. The name is absent
2. The name is absent
3. The name is absent
4. The Variable-Rate Decision for Multiple Inputs with Multiple Management Zones
5. The name is absent
6. Brauchen wir ein Konjunkturprogramm?: Kommentar
7. Multi-Agent System Interaction in Integrated SCM
8. Language discrimination by human newborns and by cotton-top tamarin monkeys
9. The name is absent
10. The name is absent