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A lender of last resort vs. a securities market as a means to reduce liquidity risk: the
case of the Anglo-Palestine Bank
Raphaël Franck, Miriam Krausz
Bar Ilan University, Department of Economics, 52900 Ramat Gan, Israel
E-mails: [email protected] and [email protected]
Under a stylized model, this paper investigates the relationships between commercial
banks' asset management and the financial institutional structure. More specifically, we
compare the effects of the existence of a market for short-term securities with those of a
lender of last resort on the structure of banks' assets. We use historical data of the Anglo-
Palestine Bank to simulate the results of our model. The conclusion is that the development
of a financial market matters more for emerging economies than a lender of last resort since
it eases the issuance of long-term loans that are necessary for stable growth.
Keywords: Financial markets, Lender of last resort, Liquidity risk.
JEL Code: G21, O16, N25
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1. INTRODUCTION
There is a large body of literature on the relationship between financial development and
economic growth. Studies by La Porta et al. (1997, 1998) have notably emphasized the relation
between financial development and investors’ protection. Beck et al. (2000) find that financial
intermediary development is connected to both real per capita GDP growth and total factor
productivity growth.
A parallel may be drawn between these researches dealing with the effects of institutions on
economic development and those related to the impact of institutions and regulations on banks’
asset allocation. As surveyed by Dewatripont and Tirole (1994) and Freixas and Rochet (1997),
banking regulations are said to provide greater financial stability to banks, which may be
translated into ambitious long-term investments.
This paper focuses on the case of the Israeli Anglo-Palestine Bank (APB), which has been
operating for a long time, and has at the same time witnessed a rapid transition of the economy.
Therefore, the evolution of its changing balance sheet may be analyzed as an immediate result of
a changing institutional framework. The APB was founded in 1902 and adopted its current name,
the “Bank Leumi LeIsrael” (BLL), in 1950. In the days of the Ottoman Empire and the British
mandate, the APB operated in what was then a developing market. This was a time when the
infrastructure of the local economy was set up, including buildings, industries and roads. A vast
amount of funding was required. However, the APB was limited in the amount of loans it could
extend. One of the reasons was the lack of a supportive system in the case of a liquidity crisis.
While the APB could turn to British banks and the British Stock Exchange, the costs of
borrowing were clearly very high. Thus, the lack of a lender of last resort (LLR) and securities
market might have impeded the ability of the APB to generate growth through long-term loans.
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This paper suggests a model relating changes in commercial banks’ asset allocation to
changes in the financial institutional structure. These include the existence of a market for short-
term assets and a lender of last resort. We argue that banks alone cannot sustain growth without
the support of other financial institutions because of liquidity risk. Unlike Arestis et al. (2001),
who treat banks and stock market development as two parallel systems, we focus on the
interaction between the two. As such, our perspective also differs from that of Amable and
Chatelain (2001), who analyze the effects of a stock market on the structure of the banking sector
and how it amends imperfect competition.
Our framework builds upon elements of two strands of research in banking. The first one,
characterized by Kareken and Wallace (1978), Koehn and Santomero (1980), Kim and
Santomero (1988) and Rochet (1992) amongst others, focuses on the changes in commercial
banks’ portfolio choices as a result of regulation. The second one stems from the analysis of
bank runs in line with Diamond and Dybvig (1983), and expanded by Hellwig (1994) and Von
Thadden (1997) amongst others. In their seminal paper, Diamond and Dybvig (1983) provide a
rationale for the existence of banks by showing that they improve the risk sharing of simple
competitive markets by transforming illiquid assets. However, because of asymmetric
information, and a sequential service constraint a la Wallace (1988), banks are vulnerable to runs
since they do not know the consumption timing of their depositors.
In our model, the bank is a profit-maximizing risk-neutral institution, which operates in an
environment a la Diamond and Dybvig (1983) with three assets: short- and long-term investment
technologies are available and represent different levels of liquidity, while cash is a storage
technology. Because depositors may withdraw early, the bank is compelled to keep some of its
deposits in the form of interest-free liquid reserves. It is even possible that, at a given point in
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time, the bank’s reserves do not cover the depositors’ desired withdrawals. Under such
circumstances, the bank has to liquidate part or all of its investment technologies at a cost.
Depending on the existence of institutions like a securities market and/or a lender of last resort
(LLR), the bank may find it less costly not to invest a certain part of its deposits and keep them
liquid. Moreover, Diamond and Dybvig (1983) show that agents need a bank because they lack a
financial market for investing. In our model, the market is for the bank, facilitating the
investments it performs on behalf of its depositors. The assumption we make is that depositors
have no access to the market.
The remainder of the article is organized as follows. Section 2 provides some historical
background on the APB. Section 3 provides a model of the bank’s asset allocation in the
presence of a securities market. Section 4 analyzes the bank’s asset allocation in the presence of
a LLR. Section 5 focuses on the bank’s asset allocation under the joint existence of a securities
market and a LLR. Section 6 provides simulations of the model calibrated on APB data. Section
7 summarizes the result and concludes.
2. SOME HISTORICAL BACKGROUND ON THE ANGLO-PALESTINE BANK
At the time the APB was founded, the future state of Israel was a province of the Ottoman
Empire, which was then in delinquency. As documented by Pamuk (2000), there were no
banking regulations, no monetary policy, and not even a single currency. Several foreign
currencies were able to compete in the same province. Halevi et al. (1981) note that all the
branches of the APB in Palestine held their balance sheets in French Franc, which was the most
widespread currency in the province. However, the consolidated balance sheet was laid out in
Pound Sterling, because the bank was registered in Great Britain.
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This lack of regulation came to an end in 1917, when the British army invaded Palestine
during World War I, which ultimately led the Society of Nations to give Great Britain a colonial
mandate over this region three years later. The economic policy of the British mandate had a
major effect on the development of the Palestinian economy, as Gross (1982), Metzer (2001) and
Nathan et al. (1946) demonstrated. The British instituted in 1917 a single currency, the Egyptian
Pound, whose exchange rate with the British Pound was connected through a currency board. A
regional currency, called the Palestine Pound, was only instituted ten years later, under the same
terms. In the meantime, a regional stock exchange was established in 1920. Banking regulations
were established in 1935, following a bank run which had caused several bankruptcies. To avoid
additional damage to the Palestinian economy, the APB, which had then become the largest bank
in the province, acted as a private lender of last resort to other banks, under the supervision of the
British colonial authority.
In 1948, the independent state of Israel was established. The APB, which by then was called
the "Bank Leumi Leisrael" (BLL), then acted as the Israeli Central Bank, though it was still a
private bank. It helped foster the development of the newborn state, as documented by Patinkin
(1959). Six years later, the Bank of Israel was created to be the state-run Israeli Central Bank,
and became de facto the national lender of last resort. The BLL then became once again another
private bank. It survived the eras of high inflation in Israel in the 1970s and the 1980s, the 1983
banking crisis, and the various changes in regulation and monetary policy that were analyzed by
Razin and Sadka (1993) and Ben Bassat (2002). The BLL is at present the second largest bank in
Israel.
The 100-year old APB thus allows the effects of changes in the institutional environment on
a bank’s asset allocation to be assessed. Figure 1 shows the relative share of securities, loans and
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cash on deposits held by the bank over almost a century. Note that for some years, the sum of
relative shares may be superior to 1. This is because deposits are only a part of the bank's
liabilities, and some of the bank's assets may be funded by the bank's equity. This is especially
the case for the bank's first years of existence. In addition, in 1903, a very high percentage of
assets are held in cash and securities. When the stock market was set up in 1920, the APB still
held a high percentage of cash. But by 1935, when banking regulations were instituted, the share
of loans increased. It is also worth noting that at this time, the APB mainly financed its activities
from deposits. Also, from 1935 onwards, there is a steady increase in the share of loans. This rise
was interrupted by the 1983 Israeli banking crisis. This crisis explains the abnormally high
percentage of cash in the APB's balance sheet. If it were not for this crisis, Figure 1 would show
a steady increase in the share of loans from 1935 to 2001.
[Insert Figure 1]
3. THE MODEL
3.1 Basic specification
Consider a discrete-time framework with a stock market where securities may be exchanged
for cash, no LLR, and three time periods t=0, 1, 2, 3. There are two types of technologies: short-
term technologies S, which mature at t=2, and long-term technologies L, which mature at t=3.
The source of uncertainty in this model does not stem from the returns on the available
technologies, but from the amount of withdrawals that the bank faces at t=1. We define β as the
random variable describing the proportion of withdrawals at t=1, that ranges over the interval
[ ]1;0 . We however assume that the bank knows the cumulative distribution function F of β at
t=0, with f its density function. This uncertainty on β is not ad hoc but stems from the behavior
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of depositors. Indeed, none of them would intentionally deposit until t=1, because this is
tantamount to holding cash, which is denoted C. A priori, depositors intend to withdraw at t=2 or
t=3 in proportions α and 1-α respectively, which are known to the bank. This is because the bank
shares part of the returns on the technologies with the depositors. Hence, the latter have
incentives not to withdraw at least before t=2. Some depositors may however have an
unexpected need for cash at t=1, which the bank must provide.
At t=0, the bank receives deposits from its customers, who do not have access to any
investment technology, in the total amount of D. The bank, which is deemed a profit-orientated
risk-neutral financial institution with limited liability, then decides how to allocate D so as to
maximize its expected profit Π at t=3
Π= ( )βα ,,,,,, Ls rrLSCf (1)
subject to
DLSC =++ (2)
where Sr is the riskless gross return on short-term technologies to the bank, with 1>Sr , and Lr
is the riskless gross return on long-term technologies to the bank, with 1>Lr and LS rr < .
In this framework, there is an unlimited elastic demand of L and S so that their respective
returns are taken as given by the bank. Note that L and S may be viewed as long-term loans and
short-term securities. The cash C is a storage good, each purchased unit of which at time t=0
gives one unit in return at all subsequent points in time.
We now focus on the sequence of events in the model. At time t=1, Dβ of deposits are
withdrawn. If the cash C is sufficient to pay depositors, the bank pays from cash. If not, the bank
sells some or all of its short-term assets S on the market at a loss.
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At t=2, a share ( )αβ−1 of the depositors withdraw. At the same time, the bank’s short-term
assets mature. The bank may pay the depositors with its remaining cash C and/or remaining
short-term assets S. If the sum of remaining cash and securities is not sufficient, the bank goes
bankrupt. But if it is sufficient, all securities that have not been consumed are stored in cash. This
is because the alternative is to buy new securities and convert them into cash at t=3. But the bank
will never choose to do so since securities can only be sold at a lower value before they come to
maturity.
At t=3, the remaining share ( )( )αβ −− 11 of depositors withdraw. At the same time, the
bank’s loans L mature. It uses part or all of L, as well as all the remaining cash C and short-term
assets S, to pay depositors. All remaining returns belong to the bank, which then closes.
3.2 The price-taking bank’s asset allocation in the presence of a securities market
In this section, we have assumed that long-term loans cannot be traded on the securities
market. Besides, there is no LLR in this economy.
In the securities market, the smaller the supply of cash is, the higher the return the bank will
have to pay for each unit of cash1. Therefore, in a small market, cash holders may behave like
price-makers, who set a mark-up of the fair price of cash, while the bank is a price-taker.
Denoting Q the mark-up of the fair price, we write that
Q=1
11−
−
Cη,
1 In other words, the price of cash in asset terms is high if the market is small.
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where ( )( )hhC ψψη ∂= is the elasticity of demand faced by suppliers of cash and reflects the
level of competition in the market, h is the demand function for cash and ψ the price of one unit
of cash in terms of securities so that C= )(ψh .
If the securities market is fully liquid, i.e., there is an endless supply of cash in exchange for
assets, then ∞=Cη , and there is no mark-up. Denoting δ the inverse of the mark-up of the fair
price, that is 1−= δQ , we find that there is an endless supply of cash if
1=δ .
Hence, if δ <1, the supply of cash in the market is limited. In such a case, the fair price of
cash in the securities market may be expressed as a function of Sr . Agents who are willing to
give up x units of cash at t=1 want at least x units of cash at t=2, which correspond to an amount
of securities worth ( )Srx . Therefore, the price of one unit of cash in terms of securities, denoted
ψ , is equal to:
( )1
1 11−
−
−=
CSr η
ψ ,
or equivalently
( ) 1−= Srδψ .
Therefore, if the bank needs to liquidate securities to pay depositors at time t=1, the amount
of assets it must sell is equal to ( ) SrCD δβ − . The smaller δ , i.e., the less liquid the market, the
more securities that must be sold to obtain the necessary units of cash. The loss to the bank that
is felt at t=2 is ( ) δβDCSrS −− .
Taking into account these assumptions, we rewrite the bank’s expected profit Π in equation
(1) as follows
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[ ] ( )∫−
−+
−++=ΠDD
DCSr
SL
S
dfDCSrLrα
α
ββ0
( )
( ) ββδ
βδ
δαδα
dfDCSrLrD
DCSr
DCSL
S
∫−−+
−−−+++
1 111 (3)
The first term on the right-hand side (RHS) of equation (3) is defined over the interval
containing all values for which the bank has sufficient cash at t=1 and sufficient securities at t=2
to serve withdrawals. The second term on the RHS of equation (3) is defined over the interval
containing all values of β for which cash is not sufficient to repay withdrawals at t=1 and the
bank sells securities early. However, securities are sufficient to serve withdrawals at t=1 and t=2.
Furthermore, it must be noted that the sum of the two integrals in equation (3) represents the
bank’s probability of solvency. But the boundaries of the integrals are determined by the bank’s
asset allocation decision. Therefore, when the bank chooses the optimal values of C, L and S, it
also determines its own probability of solvency. We define aF1 + aF2 as the probability that the
bank remains solvent, with ( )∫−
−+
=DD
DCrS
a
aS
a
dfFα
α
ββ0
1 and ( )( )
∫−
−+
=δα
δαδ
ββ1
2
DDCrS
DC
a
aS
a
a
dfF , where aC , aL and aS
denote the optimal levels of cash, loans and securities in the presence of a securities market.
We are then able to determine the bank’s optimal allocation of cash, securities and loans as
functions of D by optimizing equation (3) under the constraint of equation (2). To simplify the
presentation of the results for optimal allocation it is assumed that β has a uniform distribution,
which implies that all values of β can occur with equal probability. Hence the value of ( )βf is a
positive constant which is denoted af . We thus arrive at the following result.
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RESULT 1. The bank’s optimal asset allocation is
Φ−=
Sa
a
rfDC 11 ( )
−
−−−+αδ 11
221
aa
aaa
LSfF
FFFrr ( ) ( )
( )−
−−
−−−
− 2111
11
δααδ
αα
SS rr
( )( )
−
−+− S
aaa
SL rfFFrrα121 ( ) ( )
( )
−−
−−−
− 2111
11
δααδ
αα
SS rr
( )( ) ( ) ( )( )
−−−−
−
−+−− 2121 111
1 δααδ
α LLS
aaa
SL rArrfFFrr
( ) ( ) ( )( )
−−−−
−
−−−+− 2212
21 111
1 δαδαδ
αδ LL
aa
a
LS rArfFF
FFrr (4)
Φ=
Sa
a
rfDL ( )
−
−−−+αδ 11
221
aa
aaa
LSfF
FFFrr ( ) ( )
( )−
−−
−−−
− 2111
11
δααδ
αα
SS rr
( )( )
−
−+− S
aaa
SL rfFFrrα121 ( ) ( )
( )
−−
−−−
− 2111
11
δααδ
αα
SS rr (5)
( )( ) ( ) ( )( )
−−−−
−
−+−Φ
= 2121 111
1 δααδ
α LLS
aaa
SLS
aa rArrfFFrr
rfDS
( ) ( ) ( )( )
−−−−
−
−−−+− 2212
21 111
1 δαδαδ
αδ LL
aa
a
LS rArfFF
FFrr (6)
with
( )( )( )
( ) ( )( )
( ) ( )( )( )
( ) ( )( )
−
−−−
−
−−+
−−
−
−
−−−
−
−−+
−−
=Φ
222
212
11
11
111
1
11
11
111
1
δαδαδ
δααδ
αα
δαδ
δαδαδ
α
LL
SS
LL
SS
rAr
rr
rAr
rr
( )1
11
11
21 −−
+−
=δαα
A , ( )22 11
1δαδ
α −+
−=A .
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PROOF. Optimizing equation (3) under equation (2) leads to the following first-order conditions:
[ ] 0F211 =−+Φ−+++=∂Π∂ λLSLL rfDCSrLrFrL
[ ] 211 FSS
SLS rDD
rfDCSrLrFr
S+
−−+++=
∂Π∂
α
( ) ( )δαδδαδαδ
δ −
−
−−+
−−+++1
111
1 3
Df
DDCSr
DCSrLr SSL
0111 2 =−Φ
−−−+++ λ
δδfDCSrLr SL
[ ]DD
fDCSrLrFC SL α−
−+++=∂Π∂ 1
1
( ) ( )δαδδαδαδ
δδ −
−
−−+
−−+++1
111
1 32
Df
DDCSr
DCSrLrF S
SL
0111 2 =−
−−−++− λ
δδ Df
DCDCSrLr SL
with ( )
−
+−==
αα
β11 D
rSDCff S
aa
,
==
DCff
a
β2 , ( )
−
−+==
δαδαδ
β13 D
DCrSff
aS
a
.
Rearranging these three conditions leads to the optimal amounts of cash, loans and
securities as given by equations (4), (5) and (6). ■
We find that all three assets depend on Lr and Sr . It must be noted that, when choosing S, the
return on loans Lr is emphasized; when choosing L, the return on securities Sr is emphasized.
Furthermore, we find that the probability of early liquidation aF2 affects the bank's asset
allocation. So does the degree of liquidity δ in the market. The manner in which δ influences the
bank's asset allocation will be assessed in the simulations carried out later in this paper.
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4. THE PRICE-TAKING BANK’S ASSET ALLOCATION IN THE PRESENCE OF A
LENDER OF LAST RESORT
Consider now what happens if a lender of last resort (LLR) is introduced in the model of
Section 3.1 instead of a market. We assume that the LLR always has the means to lend to the
bank and bail it out if need be.
Indeed, the bank may be compelled to call on the LLR to borrow in order to serve the
depositors’ withdrawals at either time period t=1 and t=2. The amount of borrowing depends at
each time period on the difference between the quantity of withdrawals and the sum of cash and
liquidated assets.
At t=1, if cash is not sufficient to cover withdrawals, the bank may borrow the amount
CD −β at the gross rate of ρ, with ρ> Sr .
At t=2, there are two cases in which the bank has to borrow. In the first instance, the bank
borrowed at t=1 and does not have sufficient cash and liquidated assets at t=2 to repay both the
depositors and the loan so that
( ) ( ) SSrDCD >−+− αβρβ 1
In the second instance, the bank did not borrow at t=1 but does not have sufficient cash and
liquidated assets at t=2 to repay depositors,
( ) ( )CDSrD S −−>− βαβ1
At t=3, the bank repays the loans to the LLR and refunds the depositors.
We rewrite the bank’s expected profit
( ) ( )∫−
−+
++−=ΠDD
DCSr
SL
S
dfCSrDLrα
α
ββ0
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( ) ( )( )[ ][ ] ( ) ββαρβραρρ
αα
dfDCSrDC
DDDCSr
SS
∫−
−+
−−+−−++ 111
( )[ ] ( )∫−
−+
−−++DD
DCSr
DC
S
S
dfDCSrαρ
αρ
ββρβρ 1
( ) [ ]( )[ ] ( ) ββααρβρρρ
αραρ
dfDCSr
DDDCSr
SS
∫−
−+
+−+−−++1
2 11 (7)
The first integral on the RHS of equation (7) is defined over the interval containing all values of
β for which the bank has sufficient cash and securities to serve depositors at t=1 and t=2. The
second integral is defined over the interval containing all values of β for which the bank has
sufficient cash to repay depositors at t=1 but must borrow to serve them at t=2 because the value
of its securities are not sufficient. The third integral is defined over the interval containing all
values of β for which the cash is not sufficient to serve depositors at t=1 so that the bank has to
borrow at t=1, but where the value of securities is sufficient to serve depositors at t=2 and repay
the loan from the previous period. The fourth integral is defined over the interval containing all
values of β for which the cash is not sufficient to serve depositors at t=1 so that the bank has to
borrow at t=1, and for which the value of securities is also not sufficient to serve depositors and
repay the loan at t=2 so that the bank also has to borrow at this time period.
We denote bF1 the probability of the bank having sufficient cash and securities at t=1 and
t=2, bF2 the probability of the bank having to borrow only at t=2, bF3 the probability of having to
borrow only at t=1, bF4 the probability of having to borrow at t=1 and t=2. These probabilities
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are worth ( )∫−
−+
=DD
DCrS
b
bS
b
dfFα
α
ββ0
1 , ( )∫−
−+
=D
C
DDDCrS
b
b
bS
b
dfF
αα
ββ2 , ( )∫−
−+
=DD
DCrS
DC
b
bS
b
b
dfFαρ
αρ
ββ3 , and
( )∫−
−+
=1
4
DDDCrS
b
bS
b
dfF
αραρ
ββ , where bC , bL and bS denote the optimal amounts of cash, loans and
securities for the bank in the presence of a LLR.
In this case, we are able to determine optimality conditions that the bank must fulfill in
order to obtain the maximal profit. We arrive at the following result.
RESULT 2. The optimality conditions the bank must fulfill to obtain maximum profit are
( )( ) SLbb rrFF −=−+ 142 ρ (8)
( ) ( )( )
−
+−−−=
−+1
11 342 ρ
ρραα bbS
bS
bb FFr
DDCrS
f (9)
with
==
DCff
bb β2 .
PROOF. See appendix.
The condition in equation (8) provides the optimal amount of loans and securities. On the
left-hand side (LHS) of equation (8), the expected marginal cost of borrowing at optimum must
equal the marginal cost of investing in securities instead of loans on the right hand side (RHS).
Equation (9) is the condition on the optimal amount of securities and cash to be held. bf 2 is the
probability that bCD =β , which is the condition that states the upper limit before the bank has to
borrow from the LLR at t=1. The expression in brackets on the LHS of equation (9) is the
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amount of excess securities at t=22. It is a ratio whose numerator gives the difference between
the expected values of securities and the expected amount of withdrawals at t=2 while the
denominator is the amount of deposits. Thus, the LHS of equation (9) is the marginal excess
amount of securities held by the bank. The RHS of equation (9) provides the expected return on
securities relative to the cost of borrowing from the LLR. Hence, for the optimal allocation, the
amount of securities and cash held are such that at the margin, the proportion of excess securities
is proportional to the return on securities.
5. THE PRICE-TAKING BANK’S ASSET ALLOCATION IN THE PRESENCE OF A
SECURITIES MARKET AND A LENDER OF LAST RESORT
As the next step, we combine the cases dealt with in Sections 3 and 4. We hence analyze the
effects on the bank’s asset allocation of the joint existence of a securities market and a LLR.
In this case, if cash is not sufficient to satisfy the withdrawals at t=1, then the bank may
either sell part or all of its securities on the market, or take a loan from the LLR. In the former
case, the bank sells an amount denoted S worth
( ) SrCDS δβ −= ,
which entails at time period t=2 a loss that amounts to SrS , that is to ( ) δβ CD − .
Instead, if the bank takes a loan from the LLR, the amount it borrows is ( )CD −β , which
entails a loss at time period t=2 worth ( )ρβ CD − . Thus, the bank will always take a loan at t=1
if
2 At bf 2 , that is at the point where bCD =β , ( ) ( ) DDrSDDCrS S
bbS
b )1( βααα −−=−+ . Here, D)1( βα −
represents the actual withdrawals at t=2. Therefore, DrS Sb )1( βα −− is the excess amount of securities at bCD =β .
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δβ CD − ( )ρβ CD −> ,
or equivalently
( ) 1−< δρ . (10)
This is the same result as in the case analyzed in Section 4, where there is a LLR but no
market. In addition, the bank also calls the LLR if it needs new loans at time period t=2.
If inequality (10) does not hold, the bank will sell its securities at t=1 on the market. If it is
again short of cash at t=2 and cannot cover withdrawals, it will have to borrow from the LLR.
This is the case which we now deal with3.
We rewrite the bank’s expected profit:
( ) ( ) +−++=Π ∫−
−+DD
DCSr
SL
S
dfDCSrLrα
α
ββ0
( ) ( )( )[ ] ( ) ++−−−−++∫−
−+
DC
DDDCSr
SLS
dfDDCSrLr
αα
ββααβρρρ 11
( )[ ] ( ) +−−−++∫−
+−
− ββδβδαδ
αδ
dfDDCSrLrDDSrDC
DC
SL
S
11
( ) ( )( )( )[ ] ( ) ββραδρβραδρρ
αδαδ
dfDDCSrLr
DDDCSr
SLS
∫−
−+
−−−+−−−++1
11/1 (11)
The first term on the RHS of equation (11) is defined over the interval containing all values
of β for which the bank has sufficient cash and sufficient securities to serve depositors at t=1 and
t=2. The second term is the integral that is defined over the interval containing all values of β for
which the bank has sufficient cash to serve withdrawals at t=1 but must borrow from the LLR at
t=2. The third integral is defined over the interval containing all values of β for which the bank
3 We leave out the case where ρ=(δ)−1. Since the bank has been assumed to be risk neutral, it may be assumed for the sake
of the argument that it is equally likely to call upon the LLR or seek cash on the securities market at t=1.
Page 18
18
does not have sufficient cash at t=1 so that it sells securities early, but where the value of the
remaining securities is sufficient to serve depositors at t=2. The fourth integral is defined over
the interval containing all values of β for which the bank must sell securities early at t=1 and
must borrow at t=2 because the value of remaining securities is not sufficient to serve
withdrawals.
We denote dF1 the probability of the bank having sufficient cash and securities for
withdrawals at t=1 and t=2, dF2 the probability of the bank having to borrow from the LLR at
t=2, dF3 the probability of the bank having to sell securities early at t=1, and dF4 is the
probability of the bank having to sell securities early at t=1 and borrow from the LLR at t=2.
These probabilities are worth ( )∫−
−+
=DD
DCrS
d
dS
d
dfFα
α
ββ0
1 , ( )∫−
−+
=D
C
DDDCrS
d
d
dS
d
dfF
αα
ββ2 , ( )∫−
−+
=DD
DCrS
DC
d
dS
d
d
dfFαδ
αδ
ββ3
and ( )∫−
−+
=1
4
DDDCrS
d
dS
d
dfF
αδαδ
ββ , where dC , dL and dS denote the optimal amounts of cash, loans and
securities for the bank in the presence of a LLR. We arrive at the following result.
RESULT 3. The optimality conditions the bank must fulfill to obtain maximum profit are
( ) ( )S
ddSLdd
rFFrr
FF 3142
+−=+ρ , (12)
( )( ) ( )( )1
1 341
122 −
+−++−=
−+ −
ρρδραα dd
Sdd
Sd
Sd
d FFrFFrD
DCrSf , (13)
with
==
DCff
dd β2 .
PROOF. See appendix.
Page 19
19
Equation (12) is the condition on the optimal amount of loans and securities. On its LHS, we
find the expected cost of borrowing4. At the optimal allocation, this cost must equal the RHS of
equation (12) which displays the expected loss to the bank from investing in securities instead of
loans, that is the expected loss from not holding enough loans.
Equation (13) is the condition on the optimal amount of securities and cash to be held. df 2 is
the probability that dCD =β , which is the condition that defines the upper limit before the bank
has to borrow from the LLR or sell securities at t=1. The expression in brackets on the LHS of
equation (13) is similar to that of equation (9) and reflects the amount of excess securities at t=2.
The RHS of equation (13) provides the expected return on securities, given that the bank may
have to borrow from the LLR, relative to the cost of borrowing. At the optimal allocation, the
amount of securities and cash held are such that at the margin, the proportion of excess securities
is proportional to the return on securities.
6. NUMERICAL SIMULATIONS
In this section, we study how the bank’s asset allocation evolves under different institutional
frameworks by simulating the models of the previous sections. We choose to calibrate the
simulation on data from the APB in 1920, just at the time when the Palestine stock market was
created. Starting from this initial situation, we are able to assess the differences in asset
allocation when there is just a securities market in the economy, just a LLR, or both. We also
compare our simulation results with the APB’s actual asset allocation.
4 The left-hand side of equation (12) may also be viewed as the expected penalty for not holding sufficient liquid assets
and too much loans.
Page 20
20
We use the data compiled by Nathan et al. (1946) on the Palestinian economy during the
British mandate and set 12.0=Sr , 2.0=Lr . The amount of deposits held by the APB was worth
D=₤209554. We investigate the changes in asset allocation when the share of early withdrawals
at time t=2 α, the degree of liquidity δ and the cost of borrowing from the LLR ρ vary5. We
assume a uniform distribution of β . Tables 1, 2 and 3 respectively display the results for an
economy with a securities market and no LLR, no securities market and a LLR, and both a
securities market and a LLR according to the maximum expected profit.
[Insert Table 1]
[Insert Table 2]
[Insert Table 3]
In an economy with a securities market and no LLR, Table 1 shows that the greater α, the
smaller L and C and the greater S. This result suggests that the bank's ability to invest in long
term loans is reduced when it expects a large proportion of depositors to withdraw early. Since
the bulk of withdrawals in the case of a high α are at t=2, the bank invests heavily in securities
while forsaking cash because it is able to liquidate the securities early due to the existence of a
market. In addition, when a large proportion of withdrawals is expected at t=3, then more funds
are invested in loans. The bank must however hold more cash in this case because loans cannot
be liquidated early. The greater α , the higher the bank’s expected profit because less deposits
are devoted to cash.
In addition, Table 1 shows that for each α, a decrease in δ entails a rise in S and a reduction
in C and L. This is because the greater the return the bank is able to obtain from early liquidation
of securities, the less securities needed to serve early withdrawals at t=1. It may thus allocate
5 Let it be reminded that, for the simulation to be consistent, ρ>rS, ρ<rL and ρ>(δ)−1.
Page 21
21
more funds to the profitable long-term loans. However, the bank holding less securities in more
liquid markets is partly offset by its holding more cash.
Under the assumption that there is a LLR but no securities market, Table 2 shows that the
greater α, the greater S and the smaller L and C are. In addition, as the cost of borrowing ρ rises,
the share of loans L declines except for very high and low values of ρ where the trend is
opposite. Moreover, when the value of α is low, it appears that the bank invests more in L when
ρ rises. This means that the cost which the bank incurs when it borrows from the LLR is offset
by the profit it makes by investing in the profitable long-term technologies. Such an observation
has a significant implication for the conduct of monetary policy. It indeed suggests that monetary
policy may only be implemented efficiently through the lending channel if a large percentage of
depositors are expected to withdraw their deposits early. Opposite results would have been
expected according to the bank lending channel theory of monetary policy as exemplified by
Bernanke and Blinder (1992) and Kashyap and Stein (2000). The latter show that an increase in
the cost of borrowing from the Central Bank is associated with a reduction in the volume of
lending, but that banks with more liquid balance sheets are less affected by monetary policy. We
explain our results by our adding the dimension of expected withdrawals.
In addition, the comparison between Tables 1 and 2 suggests that, on the whole, the bank
investments in loans are lower in the case of a LLR than in the case of a securities market. This
implies that securities markets are more important for economic development than a LLR.
Furthermore, the highest amount of cash is held in the case of a LLR, again suggesting a waste of
resources and greater efficiency of the market.
In the presence of a securities market and a LLR, a rise in α entails a decline in L and a hike
in S and C, as shown in Table 3. In addition, α is the only variable influencing asset allocation. It
Page 22
22
would thus appear that changes in the level of liquidity δ are offset by the borrowing undertaken
by bank from the LLR, for any level of δ. Our results imply that, in the presence of a LLR, there
is no public incentive to develop markets because lending is not affected. The banks, however,
do have an incentive to develop markets because profits are higher when markets are more
liquid. Clearly, this particular analysis lacks the tradeoff faced by the bank of greater competition
when markets are more developed. However, the level of liquidity in markets does not impede
growth if a LLR exists.
Finally, we compare the simulation results with the actual balance sheets of the APB between
1918 and 1925, that is, before and after the establishment of the Palestine stock exchange in
1920. Figure 2 displays the evolution in the relative share of cash, securities and loans on total
assets over this period. It appears that loans increased relative to cash and securities after 1920.
In addition, the level of cash relative to loans and securities is also lower. These observations are
in line with the simulation results in Table 2. They thus emphasize the effects of a stock market
on a bank's asset allocation6.
[Insert Figure 2]
7. CONCLUSION
The results presented in this paper show the manner in which the level of development of
financial markets affects the banks' optimal allocation and, in particular, the level of liquidity of
their asset allocation and the amount of long-term lending they are willing to take upon
6 We are unable to compare simulation results in the case of an environment with a LLR without a securities market
because the stock exchange was created before the 1954 establishment of the Bank of Israel. Before that, the APB itself
acted as a LLR, on the behalf of the State. Hence, events do not provide the appropriate balance sheet date for simulation
comparison in the situation where both a LLR and a stock exchange exist.
Page 23
23
themselves. It is shown that banks are provided with incentives to engage more in long-term
lending as securities markets become more liquid.
The existence of a LLR in the absence of a securities market leads to the lowest level of long-
term lending. It also appears that, in most cases, the higher the cost of borrowing from the LLR
is, the lower the level of long-term lending. Besides, when both a LLR and a securities market
exist, the level of loans is neither affected by the level of liquidity on the securities market nor by
the cost of borrowing from the LLR, but by the expected levels of withdrawals at each time
period.
Our results suggest that the development of financial markets complements the banking
system. It also appears that financial markets matter more for emerging economies than a lender
of last resort.
Page 24
24
APPENDIX
Throughout this proof, we denote λ the Lagrange multiplier.
Proof of Result 2
Optimizing equation (7) under equation (2) leads to the following first-order conditions:
0=−=∂Π∂ λLrL
( ) 211 FrDD
rfCSrFr
S SS
SS ρα
+−
++=∂Π∂
( ) ( )( )DD
rfDD
DCSrDCSr SS
S αρρα
αα
ραρρ−
−−+
−−+
−−−+− 1111
( ) ( ) 233 1 FrDD
rf
DDDCSr
DCSrFr SSS
SS ραρ
ραρ
αρρ +
−
−
−−+
−++
( ) ( )( ) ( )DD
rf
DDDCSr
DCSrFr SSSS αρ
ραραραρ
αρρρρ
−
−+−−−
−−+
−+− 322
4 111
0=− λ
( ) ( ) ( )( )Df
DCDCSrF
DDfCSrF
C SS2
21
1 111
−−+−−−+++
−++=
∂Π∂ ρραραρρρ
α
( ) ( )( )DD
fDD
DCSrDCSr S
S αρρα
αα
ραρρ−
−−+
−−+
−−−+− 1111
( ) ( )DD
rf
DDDCSr
DCSrFr SSSS αρ
ραρ
αρρ
−
−
−−+
−++ 33 1
( ) 4221 F
Df
DCDCSrS ρρρ +
−−+−
( ) ( )( )DD
fDD
DCSrDCSr S
S αρρ
ραραρ
αρραρρ
−
−−−
−−+
+−−+− 322 111 0=− λ
with =1f ( )
−
−+=
αα
β1D
DCrSf
bS
b
,
==
DCff
b
β2 ( )
−
−+==
αραρ
βD
DCrSff
bS
b
3 .
Rearranging these three conditions leads to the optimality conditions in equations (8) and (9). ■
Proof of Result 3
Optimizing equation (11) under equation (2) leads to the following first-order conditions:
Page 25
25
04321 =−+++=∂Π∂ λFrFrFrFrL LLLL
( ) 211 FrDD
rfDCSrLrFr
S SS
SLS ρα
+−
−+++=∂Π∂
( ) ( )DD
rf
DDSrDC
DDCSrLr SSSL α
ααα
αρρρ
−
+−
−+−
−−−++− 111 3FrS+
( )DD
rf
DDSrDC
DDCSrLr SSSL αδ
αααδ
αδδδ
ρ−
+−
−
+−
−−−+++ 3111
−+ 4FrS ρ ( ) ( ) ( )[( ) −−−+−−−++ 111 ραδρρα
δρρ DDCSrLr SL
( )DD
rf
DDSrDC SS
αδαα
αδαδ
−
+−
−
+−× 31 0=− λ
=∂Π∂C
1 ( ) ( )Df
DCDDCSrLr SL
211
+−−−−+++ ααρρρ
( ) ( )DD
fDD
SrDCDDCSrLr S
SL ααα
αα
ρρρ−
+−
−+−
−−−++ 111
( )DDf
DDSrDC
DDCSrLrF SSL αδδδαδ
αδδδ −
−
−+−
−−++++ 33 111
DfCDDCSrLr SL
211
−−−++−
δδ
( ) ( ) ( )DDf
DDSrDC
DDCSrLrF SSL αδδ
ραδρ
αδαδ
ραδρρ
δρ
−
−−
−+−
+−−−++++ 34 11
0=− λ
with
=1f ( )
−
+−=
αα
β1D
rSDCf S
dd
,
==
DCff
d
β2 ,( )( )
−+−
== − αδαδβ 13 D
rSDCffSdd
.
Rearranging these three conditions leads to the optimality conditions in equations (12)
and (13). ■
Page 26
26
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29
Table 1. Simulation results under the existence of a securities market
A. High level of expected withdrawals at t=2 (α=0.9)
L S C Π E() δ=0.95 27063.50 176957.20 5533.28 14726 δ=0.9 25676.04 180239.10 3638.90 14818.23 δ=0.85 24004.87 183907 1642.12 14847.81
B. Intermediary level of expected withdrawals at t=2 (α=0.8)
L S C Π
δ=0.95 33016.80 163859 12678.21 14610.50
δ=0.9 31827.26 167140.80 10585.91 14823.36
δ=0.85 30364.72 170808.80 8380.513 14980.17
C. Low level of expected withdrawals at t=2 (α=0.7)
L S C Π
δ=0.95 38970.11 150760.8019823.14 14494.87
δ=0.9 37978.48 154042.6017532.91 14828.48
δ=0.85 36724.56 157710.5015118.90 15112.52
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30
Table 2. Simulation results under the existence of a LLR
A. High level of expected withdrawals at t=2 (α=0.9)
L S C Π ρ=1.19 10.17 176855.53 32688.28 8750.15 ρ=1.18 0.71 176623.94 32929.34 10546.36 ρ=1.17 10.03 176385.27 33158.73 10546.36 ρ=1.16 14.15 176144.17 33395.66 11414.46 ρ=1.15 21.06 175902.83 33650.00 12264.57 ρ=1.14 41.41 175648.25 33864.33 13095.92 ρ=1.13 0.59 175404.35 34149.05 13913.48
B. Intermediary level of expected withdrawals at t=2 (α=0.8)
L S C Π ρ=1.19 17.57 165140.66 44355.76 8276.92 ρ=1.18 12.48 164845.92 44695.52 9264.60 ρ=1.17 11.70 164546.20 44996.08 10244.28 ρ=1.16 26.71 164238.52 45288.75 11219.01 ρ=1.15 47.87 163925.40 45580.72 12192.79 ρ=1.14 8.76 163622.31 45922.91 13170.78 ρ=1.13 10.96 163305.10 46237.97 14159.33
C. Low level of expected withdrawals at t=2 (α=0.7)
L S C Π ρ=1.19 11752.06 151311.02 46490.90 7129.93 ρ=1.18 11568.99 151075.37 46909.62 8326.61 ρ=1.17 10940.00 150966.27 47647.72 9527.29 ρ=1.16 9441.77 151100.90 49011.33 10739.42 ρ=1.15 6329.14 151680.88 51543.97 11997.29 ρ=1.14 34.73 153125.81 56393.44 13275.34 ρ=1.13 22.55 152765.36 56766.07 14654.83
Page 31
31
Table 3. Simulation results under the joint existence of a securities market and a LLR
A. High level of expected withdrawals at t=2 (α=0.9)
L S C Π δ=0.95 ρ=1.19 41162.06 168391.60 0.34 19886.10 ρ=1.13 41162.06 168391.60 0.34 2084.57 δ=0.9 ρ=1.19 41162.08 168391.75 0.17 12594.46 ρ=1.13 41162.08 168391.75 0.17 13921.60 δ=0.85 ρ=1.19 41162.09 168391.81 0.10 4444.92 ρ=1.18 41162.09 168391.81 0.10 4734.60
B. Intermediary level of expected withdrawals at t=2 (α=0.8)
L S C Π δ=0.95 ρ=1.19 59871.82 149680.63 1.55 19392.11 ρ=1.13 59871.82 149680.63 1.55 20980.29 δ=0.9 ρ=1.19 59871.91 149681.36 0.73 12100.51 ρ=1.13 59871.91 149681.36 0.73 14056.34 δ=0.85 ρ=1.19 59871.94 149681.60 0.46 3950.99 ρ=1.18 59871.94 149681.60 0.46 6317.72
C. Low level of expected withdrawals at t=2 (α=0.7)
L S C Π δ=0.95 ρ=1.19 785481.45 130968.56 3.99 18897.99 ρ=1.13 785481.45 130968.56 3.99 21114.83 δ=0.9 ρ=1.19 78581.67 130970.45 1.88 11606.55 ρ=1.13 78581.67 130970.45 1.88 14191.04 δ=0.85 ρ=1.19 78581.75 130971.06 1.19 3457.09 ρ=1.18 78581.75 130971.06 1.19 6452.47
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32
Figure 1. The APB’s relative share of securities, loans and cash on deposits
0
0.5
1
1.5
1903 1913 1920 1935 1948 1954 1983 1990 2001
Cash/Deposits Securities/Deposits Loans/Deposits
Source: Bank Leumi LeIsrael.
Page 33
33
Figure 2. The relative share of cash, securities and loans on total assets
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1918 1919 1920 1921 1922 1923 1924 1925
Cash/Assets Securities/Assets Loans/Assets
Compiled by the authors from Bank Leumi LeIsrael data.
