Banking Supervision in Integrated Financial Markets: Implications for the EU

banks to have a higher capital-asset ratio. As the marginal cost of increasing
the capital-asset ratio is the same for all banks, the regulator will do so until
the bank has the same probability of failure as a high-quality bank.

Corollary 2 Under [FOSD] or [SOSD], the equilibrium probability of bank
failure increases with the cost of equity r^e, i.e.,

de{^{G(r^b|q⁽qo^))} = 1 > 0 ∀qo ∈ [q^,q]               (14)

dr^e                  b              —

at the solution to [RP].

When the cost of equity rises relative to the cost of debt, it becomes more
costly to avoid bankruptcy. Consequently, bankruptcy is permitted to occur
with greater frequency in equilibrium.

Corollary 3 Given [FOSD] or [SOSD], the equilibrium probability of bank
failure is decreasing with b, at the solution to [RP]:

d              r^e  1

db{G(r‰))} = ^- — < 0.

As the excess losses from bankruptcy increase, the regulator will optimally
ensure a reduced incidence of bankruptcy.

To derive additional results in the case of [SOSD], I assume that the single
crossing property (SCP) holds:

(SCP): There exists b ∈ (r,r) such that G_q(r|q) Q 0 as r Q b.

Corollary 4 Suppose [FOSD] holds, or [SOSD] and (SCP) hold with r^b > rb.
Then the higher the quality of the bank’s portfolio, the lower the required
capital-asset ratio, i.e., dk/dq < 0 at the solution to [RP].

Proof. From equations (2) and (13), G((^1-k)L^+γ(L) |q) = re^-1. Therefore

dk₌GB < 0

dq      g(∙)       .

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