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

keep more reserves initially, since this is less costly than obtaining the missing liquidity from
the interbank market at date 1. In this sense, the ratio rD can be defined as the relative
cost of refinancing, which will help us later on to distinguish various scenarios for liquidity
effects. It is a measure of how costly refinancing at date 1 is relative to raising deposits and
reserves at date 0.⁶

Two further implications of Proposition 1 are important for comparing this equilibrium
with the post-merger equilibrium in the next section. First, using the balance sheet equality
(1), we can express equilibrium reserve holdings in terms of an optimal reserve-deposit ratio
as

ksq = Dsq = (¹ - r? !∙                       (¹0)

Note that, whereas the equilibrium reserve holdings in Proposition 1 depend on the loan
market outcome, the reserve-deposit ratio in (10) does not. To exploit this, in what follows
we will mostly focus on this ratio. In practice, the ratios of liquid assets to customers’ sight
deposits or of liquid assets to total assets are among the most frequently used indicators
by banks to assess their own liquidity situation (see, e.g., ECB, 2002, p. 22). Second,
Proposition 1 implies the following corollary.

Corollary 1 In the status quo equilibrium, each bank has liquidity risk φ_sq

_ r^D ŋ _ Lsq Г rD
sq = 2ri Dsq = 2 у ri

The equilibrium liquidity risk φ_sq and the expected liquidity needs ω_sq are increasing
in the deposit rate r^D and decreasing in the refinancing cost r^I . Banks keep low reserves
when r^D is high and r^I is low, because raising more deposits is expensive while borrowing
additional liquidity at date 1 is not. Thus, banks’ demand for reserves decreases with the
ratio ^r∣-, and banks’ liquidity risk and expected liquidity needs increase with it.

4TheEffects of a Merger on Individual Banks’ Behavior

In this section we analyze what happens at the individual bank level when a merger takes
place. The behavior of the merged banks changes in several ways. First, they can exchange

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