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



different distribution of market shares resulting from loan competition.

As all banks continue to choose reserves according to (10) and as the aggregate demand
for loans is inelastic, the merger does not affect the total amounts of reserves and deposits,
thus leaving the aggregate supply of private liquidity unchanged. The asymmetry of banks’
balance sheets, however, modifies the aggregate liquidity demand, which changes from
Xsq =
PN=1 δiDsq in the status quo to Xm = δι Dm + δ2 Dm + PND δiDc after the merger. Both
Xsq and Xm are weighted sums of N uniform random variables, but in the first case weights
are equal and in the second case they differ (according to deposit sizes
Dm and Dc ). This
brings us to the main result about the asymmetry channel.

Proposition 4 Suppose the merged banks do not exchange reserves internally. Then, the
aggregate liquidity effects of the merger are as follows:

1. The merger decreases aggregate liquidity risk if the relative cost of refinancing is below
a
threshold σ (rD < σ < ρ), and increases it otherwise;

2. The merger always increases expected aggregate liquidity needs.

The intuition behind Proposition 4 is similar to that one behind Lemma 1. Moving
from a uniformly weighted sum of random variables (in the status quo) to a heterogeneously
weighted sum of random variables (after merger) increases the variance of the total sum.
Thus, as Figure 3 illustrates, the distribution of
Xsq gives lower probability to extreme
events —very low and very high realizations of the aggregate liquidity demand— than that of
Xm.

This change in the distribution of Xm reduces the aggregate liquidity risk if the relative
cost of refinancing is low (below the threshold
σ), because it increases the probability that
the aggregate liquidity demand is below the total private supply. This is illustrated in Figure
3, where total reserves —indicated by the vertical line
PiN=1 Ri —arelowandthearea1- Φm
is larger than the diagonally striped area 1 - Φsq . The opposite happens when the relative
cost of refinancing is high.

[FIGURE 3 ABOUT HERE]

Proposition 4 also states that the merger always increases the expected amount of public
liquidity needed. The reason is that the expected aggregate liquidity needs depend not only

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