loans L of the bank is r . The bank is able to raise the quality of its loan
portfolio e units above q0 if it devotes resources to this purpose with a cost
of C(e). I assume that dC/de > 0 and d2C/de2 > 0. The level of quality
achieved by the bank is q = q0 + e. The quality of the loan portfolio has
an influence on the distribution of returns G(r|q). Higher levels of q will
be assumed to shift the distribution of returns in the sense of either first-
or second-order stochastic dominance. Analytically, the underlying density
function is assumed to have positive support on [r, r] and:
(FOSD) Gq(r|q) ≤ 0 ∀r ∈ [r, r] with strict inequality for some r; or
(SOSD) fyy Gq(r∖q)dr ≤ 0 ∀ ∈ [r, r] with strict inequality for some y.
At time t1, the cash flows are collected, which consist of the average gross
return rL less the cost γ(L) of processing L and less the owed payment to
depositors D. Limited liability of banks is assumed. Hence, deposits D are
serviced first. Outside shareholders receive a proportion z of the residual.
The rest goes to the owner of the initial share. In case of bankruptcy, the
bank is closed and equity holders receive nothing. The whole cash flows go
to depositors who have to bear the residual loss.
Given the investment and financing opportunities available to the bank,
I define the initial wealth of the bank for a given set of financial decisions as
with
rr
(1-z)
Jr
(rL- γ(L) - D)dG(r|q) - C(q - q0)
(1)
r = D + γ (L)
L
(2)
The bank has to satisfy the following financing constraint
reE = z
rr
J b (rL - γ(L) - p(q)D)dG(r\q)
(3)
and the cash flow constraint
L = D + E.
(4)
Using E = kL with k being the capital-asset ratio, (3), and (4), (1) can be
rewritten as
rr
rb
(rL - γ(L) - (1 - k)L)dG(r\q)
- rekL - C(q - q0)
(5)
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