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



For rI > 3rD it is immediate to see that ωm 2ωsq < 0 if Dm 4. For rI 3rD, ωm 2ωsq
can be rearranged as

ωm sq =


(1ksq)2 Dsq


(1 _ k-, + 2 k3 ʌ D

2   lmn + 3 mn J Dm

(1 - ksq)2    Dsq

Suppose for a moment km = ksq and Dm = 2Dsq. Then, the expression simplifies to
k2Dsq (4ksq 1), which is negative because ksq < 1/2. To see that this holds also for
km >ksq, we use (21) and rewrite ωmsq as

ωm


rD

2ωsq = 'rI Dsq


rI μ 1
rD ∖2


λ Dm

D Dsq


Denote now A = (∣ km + 3 km). Since A is decreasing in km and km > ksq for rI3rD,
it follows
ωm 2ωsq < 0 when Dm = 2Dsq. The same holds for Dm < 2. By plotting the
expression (
DD A Dm 1) for Dm > 2 and DD (1,3], one sees that there is a level h (2,4)
of the ratio
Dm such that ωm 2ωsq if Dm h, and ωm 2ωsq otherwise. The plot is
available from the authors upon request.                                         Q.E.D.

Proof of Proposition 4

This proof is a generalization of that of Lemma 1. Let Dtot denote the total deposits
NDsq = Dm +(N2)Dc, and let Rtot denote the total reserves NRsq = Rm +(N2)Rc.
Applying the general formula for the distribution of a weighted sum of uniformly distributed
random variables in Bradley and Gupta (2002) to our model, we obtain the density functions
of the aggregate liquidity demands in the status quo
fsq (Xsq) and after the merger fm (Xm)
as

N

fsq (Xsq ) = (N 1)!(Dsq )N X=0 Γ1)∖ i

(XsqiDsq)+N-1


PN-2 hf 1 ii (N-^ X X    D (i 1 'I D IN-2 I (N-^ X X    iD N--2i

fm (Xm) =


i=1 (1) i-1 (XmDm(i1)Dc)+   + i   (XmiDc)+

(N 2)!Dm(Dc)N-2

The two density functions are plotted in Figure 3. The density fsq (Xsq) is more concentrated
around the mean than
fm(Xm). To verify that this is always the case, we compare the
variances of
Xsq and Xm , which are given by

N

Var(Xsq)=X  Ds2qV ari),

i=1

35



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