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



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