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



reserves and the precision with which banks can estimate the probability of needing liquidity
at date 1. The merger changes the distribution of shocks the merged banks face and makes
them less uncertain about their future liquidity needs. As a consequence, when the relative
cost of refinancing is low and banks keep a low level of reserves, the merged banks increase
their reserves as they are more certain to need them at date 1. The opposite happens in case
of high relative cost of refinancing. In all circumstances, however, the merged banks improve
their liquidity situation, having lower liquidity risk and expected liquidity needs. Moreover,
by lowering refinancing costs, the internal money market generates endogenous financial cost
efficiencies, which reduce, ceteris paribus, the anti-competitive effects of mergers between
banks. These results suggest that merged banks benefit from scope economies in their
liquidity management by raising deposits in two imperfectly correlated deposit markets.
This last result finds empirical support in Hughes et al. (1996), who show that banks
active in imperfectly correlated deposit markets have lower costs of controlling liquidity
risk, especially after consolidation.

Mergers affect market power and therefore change both loan rates and market shares
in our imperfectly competitive loan market. As known from the industrial organization
literature, the overall effect of a merger on loan rates depends on how strong the increase in
market power is relative to potential efficiency gains. Loan rates increase when the market
power effect dominates, and they decrease when the cost efficiency effect prevails. The
novelty here is that the merger may generate efficiency gains through the re-optimization of
their reserve holdings as well as through a potential reduction in lending costs.

The changes the merger induces in banks’ reserve holdings, loan competition and bal-
ance sheets affect also the interbank market and aggregate liquidity. We can disentangle
again two channels. The first one, which we denote as reserve channel, originates directly
from the changes induced in merged banks’ reserve holdings as described above. A merger
leads to higher aggregate liquidity supply and thus lower expected aggregate liquidity needs
when banks increase their reserve holdings; whereas the opposite holds when banks’ re-
serves are reduced. The second channel, the so-called asymmetry channel, relates instead to
the distribution of balance sheet sizes across banks. A merger inducing greater asymmetry
among banks increases the variance of aggregate liquidity demand, thus increasing ceteris
paribus expected aggregate liquidity needs. In contrast, mergers inducing smaller asymme-



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