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



DC       DC        N

Var (Xm) = -mVar(δι) + 4'Var(δm) + ∑-c2Var(δi)

4            4            i=3

DC

Var(δi) -m + ∑Dc2


because Var(δ1) = Var(δ2) = Var(δi).


tains PiC=1


Dm + PN=3 -C] > P


i=3

Since Dm + PiN=3 Dc = PiN=1 Dsq, one ob-


iN=1 Ds2q by Lagrangian maximization. Hence, it is al-


ways V ar (Xm) >Var(Xsq). Since f(Xsq) and f(Xm) are well behaved (they approach


a normal distribution), they intersect only in two points.11 This, along with the sym-
metry of the two density functions around the same mean E[X
m] = E[Xsq] = Dtot and
Var(Xm ) >Var(Xsq ), implies

Φsq = Pr(Xsq > Rtot) > Φm = Pr(Xm > Rtot) for any Rtot < -tot,

and vice versa for Rtot DCot. Using Proposition 1, Rtot = NRsq, and (1), we obtain that
R
tot D2ot if rD < 4 σ. The first statement follows.

Using the definition in (7), we have

Ωm - --sq


Dtot

tot (Xm


Dtot

Rtot)fm (Xm)d(Xm) -      (Xsq

Rtot


Rtot)fsq(Xsq)d(Xsq)


DtotXmfm(Xm)d(Xm) -  DtotXsqfsq(Xsq)d(Xsq)

Rtot                           Rtot

-Rtot(1


Fm(Rtot)) +Rtot(1


Fsq(Rtot)).


Deriving it with respect to Rtot gives

d(Ωm


Ωsq )


dRtot


-Rtotfm (Rtot) + Rtotfsq(Rtot) - (1


Fm(Rtot))


+Rtotfm (Rtot) + (1 - Fsq (Rtot))
Fm(Rtot) - Fsq (Rtot).

Rtotfsq(Rtot)


As showed earlier, Fm(Rtot) - Fsq (Rtot ) > 0 for Rtot Dot and Fm(Rtot)
for Rtot DCot. Also, Fm(0) = Fsq(0) = 0 and Fm(Rtot) = Fsq(Rtot) = 0. This implies

Fsq(Rtot) < 0


Ωm Ωsq > 0 for all Rtot [0, Dtot]. The second statement follows.

Q.E.D.


Proof of Lemma 3

Suppose first -r-D < ρ. In this range, the aggregate reserve/deposit ratio in the status quo
(which coincides with the individual banks’ deposit ratio) is smaller than the one after
merger; i.e.,

k = Rsq = PRsq < K_
k
sq Dsq    NDsq <Km

11A formal proof that this is the case is in Manzanares (2002).

36



More intriguing information

1. Insecure Property Rights and Growth: The Roles of Appropriation Costs, Wealth Effects, and Heterogeneity
2. Strategic Policy Options to Improve Irrigation Water Allocation Efficiency: Analysis on Egypt and Morocco
3. The name is absent
4. The name is absent
5. The Role of Land Retirement Programs for Management of Water Resources
6. Can we design a market for competitive health insurance? CHERE Discussion Paper No 53
7. The name is absent
8. Migrating Football Players, Transfer Fees and Migration Controls
9. Public-private sector pay differentials in a devolved Scotland
10. American trade policy towards Sub Saharan Africa –- a meta analysis of AGOA