Efficient lending and a new aspect ofgovernment deposit insurance agency♣

Toshiyuki Soma
Graduate School of Economics, Osaka University
Machikaneyama, Toyonaka, Osaka 560, JAPANJuly, 1999

abstract
This paper considers a model where an insolvent bank raises the funds andlends regardless of project return. It is said that one of the purposes ofpromptly corrective action in Japan which was put in action in April, 1998 is
to prevent such a lending. In this paper, we examine what kind of agents
must loan funds to the insolvent bank with projects to lend in order toprevent it from inefficiently lending. We show that government deposit
insurance agency (GDIA)’s loan can make the bank lending efficientalthough a large number of studies has emphasized the point that existenceof GDIA gives a bank an incentive to take too much risk. We also show that
efficient lending can be achieved irrespective of the total deposits to be
insured.
JEL classification : G21, G33, E53.
Keywords : Insolvent bank, Efficient lending, Limited liability, Nationalization,
Caps on insured deposit.

♣

Correspondence: Toshiyuki Soma, 8-9, Uzumasa-Midorigaoka, Neyagawa-shi Osaka,
572-0844 Japan; Phone: (072)821-2472, E-mail: tsouma@hi-ho.ne.jp.
The author is grateful for helpful comments and discussions by Yoshiro Tsutsui, Yuzo
Honda, Takeo Hoshi, Hirofumi Uchida, and seminar participants at Osaka University.
This article also greatly benefited from the suggestions and helpful comments of Hajime
Miyazaki. I also thank Kojun Hamada, Atsushi Shiiba and Yoichi Hizen for valuablecomments. Of course, remaining errors are my own.

1. Introduction
The purpose of this paper is to show how efficient lending by aninsolvent bank can be achieved and to apply this theory to bank closure
policy. The number of insolvent banks has dramatically increased in recent
years in Japan. So how we should dispose of an insolvent bank becomes a hot
topic in Japan. The most popular wisdom is that insolvent banks should beliquidated immediately, possibly with the selective assistance bygovernment’s deposit insurance agency (GDIA). This paper questions such a
wisdom because this policy makes the lending project passed up not only withnegative net present value ( NPV ) but also with positive NPV .
The major result of this paper is that GDIA can enforce the insolventbank’s efficient lending1 if GDIA stands ready to loan funds to the bank inreturn for the bank’s stockholders giving up their equity claims. So, thispaper analyzes the new aspect of GDIA that GDIA’s loan can make the banklending efficient although it has long been recognized that existence of GDIA
gives a bank an incentive to take too much risk. 2 We also discuss animplication and interpretation of GDIA confiscating the bank’s equity claims.
And this policy differs from Acharya and Dreyfus (1989)’s claim which a bankshould be closed whenever its asset-to-deposits ratio falls below a threshold
which is set to be greater than 1 using the option framework.
Gertner and Scharfstein (1991) consider a situation in which a banklends fund to an insolvent firm that has an investment project. They showthat depending on parameters there are two cases considered: one is the casewherein the project is adopted even though the net present value of the
project is negative and the other is the case wherein the project is rejectedeven though the net present value is positive3. Our model is related to thoseof Myers (1977)’s and Gertner and Scharfstein (1991)’s; They have analyzed
problems of investment inefficiency that occurs when the investing firm is
insolvent. Unlike Myers (1977), we assume that investment returns are
unknown at the time of investment. We modify the Gertner and Scharfstein(1991)’s model so that we can interpret it as a lending firm rather than aborrowing firm. Gertner and Scharfstein (1991) show that efficiency of
To avoid confusion of expression, we use the terminology “lend” whenever theinsolvent bank provides funds to the firm and the terminology “loan” whenever someagent provides funds to the insolvent bank.
2 Berlin, Saunders and Udell (1991) survey the literature on this moral hazard problem.
3 Myers (1977) first analyzed this case.
1

1

investment depends on parameters and that it is by sheer coincidence that
efficient investment is achieved. But they do not show how efficientinvestment is achieved. In this paper, we examine what kind of agents mustloan funds to the insolvent bank with project to lend in order to prevent theinsolvent bank from inefficiently lending. The notion of efficiency of banklending using in this paper requires that all and only positive NPV lending
project is carried out regardless of eventual solvency of bank.
Unlike early papers on deposit insurance, we abstract away from thepricing of fair risk-adjusted insurance premium.4 Instead, as in Dreyfus,
Saunders and Allen (1994), we investigate the caps on insured deposit
coverage.
The remainder of the paper is organized as follows. In the next section
we set out the structure of our model. Section 3 analyzes the situation where
the only role of GDIA is to compensate for depositors. We show that under
this situation it is purely by chance that efficient bank lending is achieved.
Section 4 analyzes how the efficient bank lending can be achieved. Section 5
concludes the paper.

2. Basic framework of the analysis
Throughout a paper, we consider a two-period model in which a bank has
become insolvent. At the beginning of Date 1, this bank has liquid assets of
Y and an opportunity of lending I dollars in the project to its client firm. In
Date 2, the bank obtains the return x from this lending. This x is randomvariables distributed over the support [ 0, ∞ ) with the density function f ( x ) ,the cumulative function F ( x ) , and the mean x .5 We assume that the bankbegins Date 1 with the total deposits in the amount of D . Fraction q of thedeposits, however, will be withdrawn by depositors during Date1; theremaining (1 − q) of D will become due in Date 2. Fraction q is taken as
exogenous in this paper. The interest rate on the existent deposit D isnormalized to be zero and all parties are risk neutral. We shall ignore “bankrun” in the analysis by assuming that GDIA can credibly insure all deposits.

4 See, for instance, Merton (1977) and Ronn and Verma (1986).

5 Here the functional form may be arbitrary. We can show that the main results in thispaper hold when x is specified as the return to the insolvent bank from the standard
debt contract.
2

That the bank is already insolvent at the beginning of Date 1 can be
expressed as follows:
A1. Y < D .
Figure 1 described the balance sheet of the bank at the beginning of the Date
1. Thus if the bank is resolved, stockholders receive nothing because ofabsolute priority rule. So, stockholders have the incentive to finance aninvestment dollar I whenever the bank can borrow the invest amount.
Since the bank’s stockholders receive nothing in case of the declarationof bankruptcy to investing project, they are willing to borrow and financeregardless of its returns. In other words, we assume situations wherein thebank must raise ( I + qD − Y > 0 ) in Date 1 to implement the investment,that is, lending to the firm. This is described as
A2. Y < I + qD .
Under A1 and A2, whether the lending to the firm is adopted or not
depends on the decision by the agent who loans funds to the insolvent bank.
We call this agent as a financing agent. The sequence of events and decisions,described in figure 2, is as follows. In Date 1, the insolvent bank lends fundsof I to the firm and pays off the deposit maturing at Date 1 of qD . Under
A1 and A2, it must raise funds ( I + qD − Y ) . The financing agent decides
whether to loan or not. If the bank cannot raise the needed funds in Date 1, it
simply defaults either at the end of Date 1 (the case of Y ≤ qD ) or during
Date 2 (the case of Y ≥ qD ).
In Date 2, if the insolvent bank can raise funds and lend to the firm, itgets returns on lending x and pays off the deposits maturing in Date 2,(1 − q ) D and funds with interest of r to the financing agent,(1 + r ) ( I + qD − Y ). So, debt obligations of the insolvent bank at Date 2
amount to z ≡ (1 − q ) D + (1 + r )( I + qD − Y ) . Note that the insolvent bank pays
the new depositors at r percent interest rate. This is because the insolventbank needs to raise deposits at higher interest rate.
Of course, the ability of the bank to finance the firm’s investment project
depends on availability of the financing agent willing to loan to the defact
insolvent bank. In this paper, potential financing agents who loan funds tothe insolvent bank are GDIA, new depositors, a financing bank, oldstockholders, and new stockholders.6 We assume that the insolvent bank
We call a bank that loans funds to the insolvent bank as a financing bank to
distinguish between these two banks. And we call existent depositors (stockholders)63raises all its needed funds, I + qD − Y , from one agent only.
We define efficient lending that the lending is achieved, if and only if
the net present value ( NPV ) of the lending is non-negative, where NPV is∞

defined as NPV ≡ ∫ xf ( x ) dx − I ≡ x − I . The problem is that the financing
0

agents do not use this criterion as a bench mark of loaning to the insolvent
bank. They offer funds as far as the expected return to them is nonnegative.
And these two criteria usually differ. In what follows, we examine what kindof agents should loan to the insolvent bank in order to achieve the efficientlending.

3. Bank lending without GDIA’s loan
In this section, the only role of GDIA is to compensate for loss ofdeposits.7 We analyze the case where the insolvent bank raises funds withinterest from new depositors, a financing bank, old stockholders, or new
stockholders and show that the bank lending to a firm becomes inefficient inany case.

3.1. Financing with new deposits
In this subsection, we analyze the model in which the insolvent bankraises all its lending funds from new depositors only, namely amount of new
deposit is equal to I + qD − Y . We assume that unlike old depositors, newdepositors stay bank till the end of Date 2. As deposits are completely insured
by GDIA, the expected net returns of the new depositors are given by(1)∫ ( I + qD − Y ) f (x )dx + ∫ [ Z − (1 − q)D ] f ( x)dx − (I + qD − Y ) .
Z

∞

0

Z

The first term is the gross present value of new deposits in the case ofbank’s liquidation. This means that even if the insolvent bank cannot meet
debt obligations Z in Date 2, new depositors recover principal I + qD − Yfrom GDIA. The second term is the gross present value of new deposits whenowning deposits (equities) at the beginning of Date 1 as old depositors (stockholders).
7 The case of GDIA directly loaning to the insolvent bank is analyzed in the next
section.
4

the bank realizes returns exceeding Z . This means that when returns on
lending of the insolvent bank are more than Z , new depositors get principal
and interest.
Equation (1) can be transformed into(2)∞

∫ r ( I + qD − Y ) f ( x)dx ,Zwhich is positive under A2.
The insolvent bank can raise funds from new deposits and therefore lend
regardless of returns on lending because people can profit from the deposits.
It is convenient for the insolvent bank’s stockholders to raise funds from
deposits because stockholders have the incentive to lend regardless of net
present value of lending. This means that GDIA makes completely insured
depositors willing to supply an unlimited amount of deposits regardless of thebank’s returns on lending. We call this effect moral hazard problem ofdepositors caused by GDIA. 8 Let us call that under-lending problem isoccurred if a lending project with positive NPV is passed up andover-lending problem if a lending project with negative NPV is carried out.
Then, we get the following lemma.

Lemma1: When the insolvent bank raises funds from deposit, it nevercauses under-lending problem, but can cause over-lending problem.

3.2. Raising funds from the financing bank9
The financing bank loans to the insolvent bank if net returns of
8 Our result holds in case where the insolvent bank pays a premium to GDIA in raisingfund from new deposits. We define p as flat insurance premium per deposit. So thebank must raise funds [ 1 / ( 1 − p ) ] ( I + q D − Y ) in Date 1. But as long as deposits arecompletely insured by GDIA, new depositor’s action does not change. That is,irrespective of N P V of lending project, new depositor deposits money in this bank.
9 We assume that the financing bank’s debt is junior to deposits. And we are assuming
implicitly that no other bank can have access to this project investment on the groundthat there is the ongoing monitoring and screening investment requisite to theidiosyncratic investment for other lenders, which cannot be easily transferable to other
lenders without having invested in the long term relationship. So, if the insolvent bankcannot raise funds, even the project with positive N P V may be passed up. These
forgone benefits are called the charter value.
5

financing bank with financing are more than those without financing. Thiscondition is described as(3)∞[ x − (1 − q ) D ] f ( x )dx + ∫Z [ Z − (1 − q ) D ] f ( x)dx − ( I + qD − Y ) ≥ 0.
(1 − q ) D∫Z

The left-hand side of inequality (3) is net returns of the financing bankwhen it loans to the insolvent bank. The right-hand side of inequality (3) isreturns that the financing bank receives when it does not loan to theinsolvent bank. This is because that the financing bank has only projects tolend with zero interest rate, so that the financing bank’s opportunity cost is
zero.10
The first term in equation (3) has the integral sign from (1 − q ) D to Z ,reflecting the fact that the bank’s depositors have higher debt seniority on
the insolvent bank than the financing bank. The second term in equation (3)is gross present value of the financing bank when returns on lending of theinsolvent bank are more than Z . Since Z − (1 − q) D = (1 + r )( I + qD − Y ) , thesecond integration is simply the financing bank’s principal and interest. Thismeans that the financing bank gets principal and interest when returns on
the insolvent bank’s lending are more than the insolvent bank’s amount ofdebt obligations in Date 2. The third term is amount of loan by the financing
bank to the insolvent bank. We transform equation (3) into the following
equation, which enables us to judge about efficiency of bank lending.
(4) NPV ≥  qD + ∫0(1 − q ) D(1 − q ) Df ( x ) dx − Y  + ∫ ( x − Z ) f ( x ) dx
(1− q ) D
 z

xf ( x ) dx + ∫∞∞≡V −Y +S,
where V ≡ qD + ∫(1 − q ) D0

xf ( x ) dx + ∫∞( 1− q ) D∞(1 − q ) Df ( x) dx and S ≡ ∫ ( x − Z ) f ( x )dx .
Z

The first term on the right-hand side is what the depositors withdraw in
Date 1. The second term is the expected amount that the insolvent bank pays
out to the depositors in Date 2 in case it actually fails and cannot meet thefull deposit obligations in Date 2. The third term describes the case in which
the insolvent bank can meet the full deposit obligations in Date 2, (1 − q ) D .

10 All our results hold even in general case where opportunity cost is non zero.

6

Note in this case that the insolvent bank may or may not be bankrupt.11 So,
the first three terms on the right-hand side, which we sum up as V , describe
what the insolvent bank pays out to depositors if it obtains loans from thefinancing bank. 12 The fourth term of Y describes what the bankrupt
insolvent bank pays out to depositors in Date 1 if the invest project cannot befinanced. The fifth term of∞∫ ( x − Z ) f ( x)dx , which we shall call S , is theZ

expected returns that stockholders get when the insolvent bank raises funds
and therefore lends to the firm.
V − Y is the transfer from the financing bank to depositors conditionalon bank lending because depositors get Y if the insolvent bank cannot lend
to the firm. S is the transfer from the financing bank to stockholders
conditional on bank lending because equity value is zero if the insolventbank cannot lend to the firm. The interpretation of inequality (4) is that if the
NPV exceeds the sum of these transfers, the financing bank loans to theinsolvent bank and the bank lending can be carried out.
The notion of efficiency of bank lending using in this paper requires that
all and only positive NPV lending project is carried out regardless of
eventual solvency of bank. Efficient lending means that only if NPV ≧0, thebank lending is carried out and if not, it is not carried out. The problem isthat the financing agents do not necessarily use its criterion as a bench markof loaning to the insolvent bank. The financing bank loans whenever NPV
exceeds V − Y + S which may be positive or negative. In other words, thisloan by a financing bank results in either over- or under-lending dependingon the parameters Y , q , D and f ( x ) .
For example, when almost all of depositors draw out their deposits in
Date 1 (q ≒ 1) , the right-hand side of inequality (4) must be positive(under-lending).13 When an excess of liabilities over asset is small ( D − Y ≒0) and the financing bank gets all of the return by setting r sufficiently
high, the right-hand side of inequality (4) can be negative (over-lending).
These show that over-lending and under-lending can occur depending on

11 The insolvent bank defaults if

x is less than Z and does not default if x is morethan Z .
12 As deposits are completely insured by GDIA,
D − V is net expected payments of
GDIA to depositors.
13 In this case,
V = D , so that we can write (4) as N P V ≧ D − Y + S . As both D − Y
and S are positive, respectively, efficient lending is not achieved, even if r → ∞ .
7

parameters.
To summarize this finding,Lemma 2: When the insolvent bank raises funds from the financing bank,it can cause both under-lending problem and over-lending problem.

3.3. Raising funds from old stockholders14
In this subsection, we investigate the situation wherein the insolventbank issues stocks and all these stocks are purchased by incumbent
stockholders. To simplify analysis, we assume that there is no agency
problem among old stockholders.15 And we assume that each old stockholder
purchases newly issued stocks proportionally to his holding equities. Let theratio of number of equities owned by the j-th old stockholder before the newissue be θ j . Then, because the j-th old stockholder purchases the ratio θ j of
newly issued equities, his holding ratio remains the same level of θ j . So, thej-th old stockholder purchases the equities if θ j { ∫∞(1− q ) D

[ x − (1 − q ) D ] f ( x ) dx

− ( I + qD − Y ) } ≥ 0. Summing up this inequality over the stockholders, we get

(5)

∫

∞

(1 − q ) D[ x − (1 − q ) D ] f ( x )dx − ( I + qD − Y ) ≥ 0.

The left-hand side of inequality (5) is what old stockholders receive when
they purchase newly issued equities and the lending of the insolvent bankcan be carried out. The first term shows that the stockholders get returns
only when returns on lending are more than obligations of deposits maturing
in Date 2. The second term is the total amount of newly issued stocks that thestockholders purchase. The right-hand side is what the stockholders receive
when they do not purchase newly issued stocks and the lending of theinsolvent bank cannot be carried out because the bank is insolvent.
Inequality (5) can be transformed into

14 We call stockholders owning equities at the beginning of Date 1 as old stockholders.

This means that the old stockholders behave as if they were one agent. This
assumption is not crucial to our analysis.
15

8

(6)

NPV ≥ V − Y .

V − Y can be positive or negative, so this loan results in either underlending or over-lending. Inequality (6) is equal to inequality (4) when S = 0 .
Thus, the loan condition under which the insolvent bank raises funds fromthe old stockholders is equal to that under which the insolvent bank raises
funds from the financing bank when S = 0 .
To summarize this finding,Lemma 3: When the insolvent bank raises funds from old stockholders, it
can cause both under-lending problem and over-lending problem. But raisingfunds from old stockholders results in under-lending problem less frequently
than raising funds from the financing bank.

3.4. Raising funds from new stockholders
In this subsection, we assume that the newly issued stocks arepurchased by the people except old stockholders.16 Let the number of equities
owned by the old stockholders be M at the beginning of Date 1 and thenumber of newly issued equities be N , so that the total number of equities
becomes M + N . To simplify analysis, we assume that there is no agency
problem among the new stockholders.
The condition under which the new stockholders purchase the stocks isgiven by(7)∞N
[ x − (1 − q ) D ] f ( x) dx − ( I + qD − Y ) ≥ 0 ,M + N ∫(1 −q ) D

which can be transformed into
(8)

NPV ≥ V − Y +

M
S.
M+N

The right-hand side of (8) is bigger than that of (6) byM
S which is
M +N

16 This means that old stockholders do not purchase the new issue. Myers and Majluf(1984) call such stockholders the “passive” old stockholders.

9the equity value of old stockholders.17 This means that the new stockholders
have less incentive to purchase the new issues than the old stockholders
because the new stockholders get only part of equity value.
Thus, we obtain the following lemma.

Lemma 4: When the insolvent bank raises funds from the new stockholders,
it can cause both under-lending problem and over-lending problem. It is
likely to cause under-lending problem compared with the case of raisingfunds from the old stockholders.
The following result formalizes the above discussion:P r o p o s i t i o n １: If deposits are all insured by GDIA, then efficient banklending is achieved only by coincidence of parameters when financing agentsare depositors, a financing bank, old stockholders, and new stockholders.
In the literature dealing with regulatory bank closure policy, the policythat government keeps an insolvent bank in operation until asset maturity iscalled as forbearance policy. Some papers point out that there are benefits forforbearance policy. For example, Nagarajan and Sealey (1995) show thatforbearance can make bank select better assets ex ante in the presence ofmoral hazard. Dreyfus, Saunders and Allen (1994) show that forbearance
policy may be an optimal policy ex post. On the other hand, we examined aneffect of forbearance policy on efficiency of bank lending in this section.
Proposition 1 shows that without GDIA’s direct loan, once the bank is
insolvent, forbearance policy makes bank lending inefficient.

4.

Bank lending with GDIA’s loan

The analysis so far has taken it as given that the only role of GDIA is tocompensate for loss of deposits. In this section, we assume that GDIA can alsoloan to the insolvent bank directly.

17 We can also interpret that

M
S
M + N

is the transfer from the new stockholders to

the old stockholders conditional on bank lending because the equity value of the oldstockholders is zero if the insolvent bank cannot lend to the firm.
104.1. Raising funds from GDIAAs GDIA compensates depositors for all their losses, GDIA has to beburdened with Y − D if it does not loan to the insolvent bank. And even if itloans to the insolvent bank, it has to be burdened with depositors’ losses whenreturns on lending to the firm by the insolvent bank do not suffice forliabilities of deposits. Thus, GDIA loans funds of I + qD − Y to the insolventbank if(9)(1 −q ) D∞

[ x − (1 − q) D ] f ( x)dx + ∫(1−q) D[ x − (1− q) D ] f ( x)dx + ∫Z [ Z − (1 − q) D ] f ( x) dx
0− ( I + qD − Y ) ≥ (Y − D) .

∫Z

The left-hand side of inequality (9) is the net present value of GDIA
when it loans to the insolvent bank. The first term is what GDIA bears for thedepositors who withdraw their deposits in Date 2 when returns on lending of
the bank are less than obligations of deposits maturing in Date 2. The second
term is returns that GDIA gets when returns on lending are more thanobligations of deposit maturing in Date 2 but less than total debt obligationsof Z . The third term shows that GDIA gets a principal and interest whenreturns on lending are more than total obligations in Date 2. The fourth term
is funds that GDIA loans to the bank. The right-hand side is the net present
value of GDIA when it does not loan to the insolvent bank.
Inequality (9) can be transformed into
(10)

NPV ≥ S .

S is by no means negative, so this shows that under-lending problem
can occur, although over-lending problem never occurs. However, S
converges 0 as r → ∞ . That r → ∞ means that GDIA gets not only theprincipal of the loan but also all the equity value, while the existent
stockholders get nothing. Infinite interest rate is unrealistic. However, thesame result can be obtained if GDIA obtains the whole equities at Date 1 andbecomes the only stockholder of the insolvent bank. Note that the equity
value is zero at Date 1, so that GDIA need not pay to stockholders for theacquisition. This is exactly the nationalization of Long-Term Credit Bank of11

Japan and Nippon Credit Bank in 1998 in Japan.
The following result formalizes the above discussion:P r o p o s i t i o n 2 ：When GDIA loans to the insolvent bank and gets all
equity value of the insolvent bank, efficient lending is achieved without
depending on parameters.
That GDIA becomes the only stockholder of the insolvent bank implies
that it assumes unlimited liability because of its obligation of compensation
for deposit loss. And it is why GDIA loan achieves the efficiency of lending.
We will show that once bank stockholders accept an unlimited liability
for their deposits,18 they advance the funds only when the bank lending isefficient. At Date 1, they loan (or they purchase the issues) I + qD − Y to theinsolvent bank and at Date 2 they get x − (1 − q ) D because when x is lessthan (1 − q ) D they must compensate the deposits matured at Date 2. Sotheir net returns are x − I + Y − D , if they loan to the insolvent bank. If they
do not loan to the insolvent bank, they get Y − D , which is negative. So, only
if NPV ≡ x − I ≥ 0 , they prefer to loan to the insolvent bank rather than not
to loan although their net returns are negative unless x − I ≥ D − Y .
Although stockholders’ liability is limited in the real world, only GDIA thatobtained the whole equities functions as the stockholders in the unlimited
liability.

4.2.

Caps on insured deposits19
Thus far, we showed that efficient lending was achieved on theassumption that deposits were all insured. In Japan, GDIA has insured all
the deposits including interest, so that this assumption applies in Japan until
2001. However, it is scheduled to change the system that only deposits less
18 John, John, and Senbet (1991) show that the moral hazard problem of banks taking

on excessive risk arises from existence of the limited liability and the associated convex
payoff to stockholders.
19 Dreyfus, Saunders and Allen (1994) first analyze the optimality of caps on insured
deposits. They show that raising a cap on insurance coverage and allowing the bank tokeep running for an additional period may actually serve to lower GDIA’s failure
resolution cost.
12than ten million yen are insured by GDIA. In this subsection, we will
examine whether our results hold in the system where only a part of deposits
is insured. Because actual provisions are too complex to formalize, we
analyze the following two cases: one is the case wherein GDIA insures acertain ratio of the total deposits of the insolvent bank and the other is thecase wherein GDIA insures a certain amount of deposits that is independentof the total outstanding deposits.
When GDIA insures a certain ratio of the deposits, γD , 0 < γ < 1 , GDIA
loans funds to the insolvent bank, if

∫

(11)

γ (1− q ) D0∞[ x − γ (1 − q) D ] f ( x)dx + ∫γ (1−q) D [ x − γ (1 − q) D ] f ( x) dx− ( I + qγD − Y ) ≥ Y − γD,
which we can transform into NPV ≥ 0.
The left-hand side of inequality (11) is the net present value of GDIA
when it loans to the insolvent bank. The first term is what GDIA bears for thedepositors who withdraw their deposits in Date 2, in the case that returns on
lending of the insolvent bank are less than obligations of deposits maturing
in Date 2. The second term is returns that GDIA gets when returns on
lending of the insolvent bank are more than obligations of deposit maturing
in Date 2. The third term is funds that GDIA loans to the bank. The
right-hand side is the net present value of GDIA when it does not loan to theinsolvent bank.
When GDIA insures a certain amount of deposits, D ( D < D ), GDIA has
to be burdened with Y − D , if it does not loan to the insolvent bank. And if itloans to the insolvent bank, it has to be burdened with D − x when returns
on lending do not suffice for liabilities of deposits and gets x − D whenreturns on lending suffice for liabilities of deposits. Thus, GDIA loans funds of
I + q D − Y to the insolvent bank, if(12) ∫(1− q ) D0

[ x − (1 − q) D ] f ( x)dx + ∫∞(1− q ) D

[ x − (1 − q) D ] f (x)dx − (I + qD − Y ) ≥ Y − D ,which we can transform into NPV ≥ 0.
The following result formalizes the above discussion:

13

P r o p o s i t i o n 3 ：Even if only a part of the deposits is insured, efficient banklending can be achieved, if GDIA loans to the insolvent bank, and gets all
equity value of stockholders.

5. Conclusion
Myth that Japanese banks do not bankrupt is collapsing. So, now inJapan we need to consider how to resolve problems stemming from the bankbankruptcy. So far, only the safety of the deposits has been focused on andthe influence on the firms and bank lending has not been considered.
However, moral hazard problem that stockholders (or managers) of the bankin financial difficulties tend to execute inefficient lending is also serious andshould be resolved. This paper analyzes the closure policy of the insolventbank from a viewpoint of efficiency of lending and shows that in order toachieve efficient bank lending, the financing agent should be GDIA who isgiven the right to acquire all the surplus, if it emerges. This finding
highlights a new role of GDIA.
The policy implication of our findings is that once the bank is insolvent
government should take not the forbearance policy but the prompt
nationalization of the bank by GDIA in order to achieve efficient banklending. Actually, in Japan, in accordance with Article 36 of
the Law
Concerning Emergency Measures for the Revitalization of the Functions of
the Financial System , which was legislated in 1998, GDIA acquired allshares of the Long-Term Credit Bank of Japan and the Nippon Credit Bankand provided necessary support for loans. This newly adopted financial
regulation just corresponds to the procedure proposed in this paper to achieve
efficient lending by the insolvent banks.

14

Figure１: Balance sheet of the insolvent bank at the
beginning of Date 1

asset Y

D value of deposit
S value of equity

Figure 2: Time occurrence of the events（Date1）（Date 2）

• Depositors withdraw qD.
• Insolvent bank raises
funds of (I+qD-Y) andlends I.

• Project pays out x.
• Depositors withdraw (1-q)D.
• Insolvent bank pays his debt.

15

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