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

When the merger does not generate asymmetry across banks’ balance sheets, it affects
aggregate liquidity only through the reserve channel. The aggregate supply of private liq-
uidity changes, whereas the aggregate liquidity demand remains the same. Thus, the merger
reduces both aggregate liquidity risk and expected aggregate liquidity needs when the ag-
gregate supply of private liquidity increases through the higher reserve-deposit ratio of the
merged banks. The opposite happens when the aggregate liquidity supply falls.

When the merger generates the internal money market and modifies bank sizes, both the
asymmetry and the reserve channel are at work. Depending on the size of the relative cost
of refinancing, the two channels can reinforce or offset each other. Therefore, we consider
the cases of high and low relative cost of refinancing separately.

Proposition 5 Ifthe relative cost of refinancing is above ρ (rrD > ρ), the merger increases
both aggregate liquidity risk and expected aggregate liquidity needs.

When the relative cost of refinancing is rather high, the asymmetry channel and the
reserve channel work in the same direction. The asymmetry channel increases the variance
of the aggregate liquidity demand, and the reserve channel reduces the aggregate liquidity
supply through the lower reserve holdings of the merged banks. Both these effects make the
system more vulnerable to liquidity shortages and thus more dependent on public liquidity
provision.

Proposition 6 If the relative cost of refinancing is below ρ (rrD < ρ), then the following
holds:

1. There exists a critical level of the relative cost of refinancing g ∈ (σ, ρ) such that the
merger reduces aggregate liquidity risk if the cost of refinancing is below such critical
level, and increases it otherwise;

2. For any small level of asymmetry induced by the merger, there exists a set G of values of
the relative cost of refinancing, with G ⊂ (1, ρ), for which the merger reduces expected
aggregate liquidity needs.

When the cost of refinancing is relatively low, the reserve and the asymmetry channels
drive aggregate liquidity in opposite directions, and the net effect depends on their relative

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