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



Differentiating it with respect to α, we obtain

d        = 2D2Var(δι)(2α - 1) = 0,

∂α

which has a minimum at α = 2.

Step 2. Define now the liquidity demand of the merged banks as

xma = δ1αDm + δ2(1 - α)Dm ,

when α = 2, and as

Xms = δι Dm + δ2 Dm

xma

fma(Xma )=

α(1)Dm

1

(1—α)Dm

Dm xma

α(1-α)Dm


when α = 2. Applying the general formula in Bradley and Gupta (2002) to our case,
the density functions of
xma and xms can be written as (assume α < 2 without loss of
generality):
for X
ma αDm

for αDm <Xma (1 - α)Dm
for Xma > (1 - α)Dm ,

4xm


D2
m


fms(Xms) =


4(Dm—xms)


D2
m


for Xms Dm/2

for Xms > Dm/2.


(16)


Since α < 2, froβ(⅛) is steeper than fms(xms) both for xma αDm and for xma >
(1
- α)Dm . This implies that the two density functions do not cross in these intervals,
whereas they do it in two points in the interval αD
m <Xma (1 - α)Dm . Given that they
are symmetric around the same mean D
m/2 with V ar (Xma) > V ar (Xms), it is:

F

ma


F

ma


> Fms for Rm < —,
< F
ms for Rm > Dm,


(17)


where Fma = Pr(Xma <Rm ) and Fms = Pr(Xms <Rm ).

Denote now as ωma and ωms the expected liquidity needs of the merged banks with asym-
metric deposits and symmetric deposits respectively. We have

ωma - ωms


Z m (Xma
R
m


Rm )fma (Xma)d(Xma) -     (Xms

Rm


Rm)fms(Xms)d(Xms )


Xmafma (Xma )d(Xma ) -      Xms fms(Xms )d(Xms )

(18)


-Rm(1 - Fma(Rm)) + Rm(1 - Fms(Rm)).

30



More intriguing information

1. SOME ISSUES CONCERNING SPECIFICATION AND INTERPRETATION OF OUTDOOR RECREATION DEMAND MODELS
2. The name is absent
3. Estimation of marginal abatement costs for undesirable outputs in India's power generation sector: An output distance function approach.
4. The name is absent
5. The name is absent
6. Neural Network Modelling of Constrained Spatial Interaction Flows
7. Language discrimination by human newborns and by cotton-top tamarin monkeys
8. The Effects of Reforming the Chinese Dual-Track Price System
9. Voluntary Teaming and Effort
10. The name is absent
11. The name is absent
12. A Regional Core, Adjacent, Periphery Model for National Economic Geography Analysis
13. Globalization, Redistribution, and the Composition of Public Education Expenditures
14. The name is absent
15. A Rare Case Of Fallopian Tube Cancer
16. The name is absent
17. The name is absent
18. Real Exchange Rate Misalignment: Prelude to Crisis?
19. THE ECONOMICS OF COMPETITION IN HEALTH INSURANCE- THE IRISH CASE STUDY.
20. IMPLICATIONS OF CHANGING AID PROGRAMS TO U.S. AGRICULTURE