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

D^C       D^{C        N}

Var (Xm) = -^mVar(δι) + 4'Var(δ_m) + ∑-_c²Var(δi)

4            4            _i=3

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] = Dto^t and
Var(X_m ) >Var(X_sq ), implies

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

and vice versa for R_tot > DCo^t. Using Proposition 1, R_tot = NRsq, and (1), we obtain that
R_tot < D2o^t if rD < 4 ≡ σ. The first statement follows.

Using the definition in (7), we have

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

Rtot                           Rtot

Deriving it with respect to R_tot gives

+R_totf_m (R_tot) + (1 - F_sq (R_tot))
Fm(Rtot) - Fsq (Rtot).

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

Ω_m — Ω_sq > 0 for all R_tot ∈ [0, D_tot]. The second statement follows.

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 = P=ι Rsq < K_
ks^q Dsq    NDsq <K^m

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

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