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

on the frequency with which aggregate liquidity demand exceeds aggregate supply, but also
on the magnitude of each excess. As noted earlier, the merger increases the variance of the
distribution of X_m and thus the probability of events with very low and very high demands.
If banks do not hold reserves, these increases offset each other and the expected aggregate
liquidity needs are the same before and after the merger. By contrast, when banks hold
positive reserves, they can cover the events with low aggregate liquidity demand. Hence, the
higher probability of extreme events with high aggregate liquidity demand is not outweighed
any more by the higher frequency of low demand events, and the expected aggregate liquidity
needs grow.

Note that the results of Proposition 4 crucially depend on the fact that we are focusing on
mergers that lead to more asymmetry in banks’ balance sheets, like those among large banks.
Of course, mergers may also have the opposite effect of making banks more symmetric.
This could happen, for example, when mergers involve smaller banks. In such a case, the
functioning of the asymmetry channel is reversed and mergers reduce expected aggregate
liquidity needs. We need to keep this in mind when discussing our results further below.

5.2 Interaction with the Reserve Channel

In this section we reintroduce the possibility for the merged banks to use the internal money
market. We first analyze how this affects aggregate liquidity through the reserve channel.
Denote as

Rm + Pi=3 Rc = km Dm ⁺ Pi=3 kcDc

Dm + (N - 2)Dc = Dm + (N - 2)Dc

the aggregate reserve-deposit ratio after the merger. Since competitors choose the same ratio
as in the status quo (k_c = k_sq), the change in K_m is solely determined by the change in the
merged banks’ reserve-deposit ratio. Hence, it follows from Proposition 2 that K_m increases
when the relative cost of refinancing is relatively low (because then k_m > k_sq), whereas
it decreases otherwise. The following lemma describes how the change in the aggregate
reserve-deposit ratio alone affects aggregate liquidity.

Lemma 2 Suppose the merger does not cause any asymmetry in banks’ balance sheets
(D_m = 2D_c). Then, it decreases aggregate liquidity risk and expected aggregate liquidity
needs if the relative cost of refinancing is below ρ (rrD < ρ), and it increases them otherwise.

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