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

x_ma = δ1αD_m + δ2(1 - α)D_m ,

when α = 2, and as

Xms = δι Dm + δ₂ Dm

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

for αD_m <X_ma ≤ (1 - α)D_m
for X_ma > (1 - α)D_m ,

Since α < 2, f_roβ(⅛) is steeper than fms(xms) both for x_ma ≤ αD_m and for x_ma >
(1 - α)D_m . 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 <X_ma ≤ (1 - α)D_m . Given that they
are symmetric around the same mean Dm/2 with V ar (Xma) > V ar (Xms), it is:

where F_ma = Pr(X_ma <R_m ) and F_ms = Pr(X_ms <R_m ).

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

X_maf_ma (X_ma )d(X_ma ) -      X_ms f_ms(X_ms )d(X_ms )

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

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