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Regulatory optimal bank size.

Abstract This paper presents a study of potential outcomes of bank growth. Banks grow by expanding market presence within the geographic region within which they are domiciled and by expanding presence into other regions via new implantations. Growth leads to improved diversification, but also results in an increase in the risk of catastrophe that a bank's failure may engender. The conclusion is that there will exist a threshold size of bank at which the rate of growth in its systemic risk exceeds the rate of decline in its risk of insolvency. An empirical study of US bank call report data provides results that are consistent with the theory presented in the first part of the paper.

Keywords Banks. Systemic risk. Diversification. Optimal size

JEL Classification D21. E50. G20 L10 R10

Introduction

This paper studies the heterogeneity present across bank structures, as well as the heterogeneity that exists in the interbank markets. Banks are observed to be spatially separated from heterogeneously allocated investment alternatives and to have investment costs that are partially dependent upon distance. Limitations to divisibility of investment alternatives, as well as distance costs, result in heterogeneous portfolios, and this creates heterogeneity in systemic risk.

In the presence of distance costs, spatial separation of investment alternatives requires a bank to grow geographically in order to reduce the distance between it and a new subset of alternative investments. Bank growth thereby leads to improved diversification and a reduction in the risk of insolvency. An additional outcome is that it also increases a bank's linkages to other banks. This increased exposure to other banks, in combination with its own greater weight in the money supply, results in an increasing systemic risk in the event of its failure.

Background

Allen and Saunders (1986) explain the heterogeneity found in the structure of the interbank market for Federal Funds in the U.S.A. as an outcome of a game theoretic ranking of banks in the competition for funding. However, distance costs explain the asymmetrical structure within this market quite handily and, moreover, provide a rationale for banks to grow larger. As demonstrated below, in the presence of distance costs, a heterogeneous, geographic dispersion of asset correlations will constrain banks to geographic expansion for the purposes of achieving better diversification.

The works of Pyle (1971), Hart and Jaffee (1974), and Kim and Santomero (1988) depict banks as portfolio managers. The studies of Rochet and Tirole (1996), Freixas et al. (2000) and Allen and Gale (2000) model risk present in the interlinkages between banks. While the research of Hotelling (1929), Coase (1937), Salop (1979), and Fujita et al. (2001), discuss different aspects of the problem of spatial separation and the strategic implications of growth through geographically separate emplacement. This paper builds upon the fundamentals discussed in this body of literature by demonstrating that the natural search for improved performance results in a regulatory optimal size of bank.

In the literature, game theoretic arguments are frequently used to describe the potential for systemic failure. However, one must not lose sight of the fact that an asset deflation event may cause a bank to become 'insolvent' in the sense that it no longer has a required level of capital. Indeed, system wide catastrophic outcomes can occur in the absence of rational runs on banks. The interbank market for funds serves as a conduit for the transmission of failure and exposure to the market creates conditions for systemic failure.

The Model

In order to simplify the approach it is assumed that banks issue capital and liabilities in the from of deposits and borrowings from other investors (including banks). Capital is set in the initial period and varies depending upon asset valuations and income. New capital injections are not possible and no dividends are paid. Furthermore, asset-liability composition of banks is assumed to be independent a la Klein (1971) and Monti (1972). Managers are constrained to maintaining a minimum, risk-weighted capital ratio, in default of which their bank may be declared insolvent by the regulatory authority and subject to reorganization or liquidation. The risk-return function associated with assets is assumed to be convex.

Distance Costs

Monitoring and other costs associated with the reduction of moral hazard are to some extent related to the relative distance of the asset from the monitor, where the concept of distance includes anything that reduces the bank's ability to monitor an asset (see also Coase 1937). Distance cost includes factors which are essentially geographic in nature (proximity relationships, language barriers, etc.), that engender expense that is associated with search and transactions (including monitoring). Distance costs reduce the expected return of a potential investment from what would otherwise have been obtained.

Justification for such costs is also found in the research of Hotelling (1929) and Salop (1979) who present models where, in the presence of transportation costs, ceteris paribus, clients prefer to deal with suppliers who have lower costs. (1) An outcome of such cost is that, in order to achieve greater diversification, a bank will eventually need to create branches or subsidiaries in locations that are characterized by substantial reduction in the geographical separation from investment alternatives.

Lemma 1 The higher the distance cost [C.sub.d] associated with an investment, the lower its relative weight will be in a bank's portfolio.

Assuming that the cost of distance [C.sub.d] is at least quasi monotonically increasing in geo-spatial separation, the [C.sub.d] associated with a specific asset will be quasi monotonically decreasing in the size of the bank. The expected return on bank [B.sub.i.sup.']s portfolio would thus be equal to:

E{[M.summation over (a = 1)]([r.sub.a] - [C.sub.d,a])[A.sub.a] - [N.summation over (l = 1)][C.sub.l] - [C.sub.0]} (1)

where:

[r.sub.a] return on bank assets

[C.sub.d] distance costs,

A portfolio assets (total number=M),

L liabilities (total number=N),

[C.sub.1] cost of liabilities,

[C.sub.o] cost of overhead.

It is probable that distance also plays a role in the cost of liabilities, but the exclusion of this effect from consideration will not change the conclusions of this paper, given the assumed separability of the decision concerning the structure of the balance sheet.

Accepting the theory of Markowitz (1952), the relative weight of each asset in a bank's portfolio is set to maximize the following:

max[N.summation over (i=1)] E [([r.sub.i]-[C.sub.d,i])[x.sub.i] / [[x.sub.i.sup.t]V[x.sub.i].sub.1/2] (2)

s.t.[N.summation over (i = 1)][x.sub.i] = 1 (3)

where:

V is the variance-covariance matrix of asset returns,

[x.sub.i] is the weight of asset i.

Since the returns of each asset are encumbered by the monitoring costs associated with the asset and assuming these costs will be a function [C.sub.d] that is increasing in the 'distance', the composition of the portfolio will not correspond to the 'market' portfolio of available alternatives. Consider a spectrum of geographically separate assets that are progressively further away from each other. In this case, the return on the ith investment [[zeta].sub.i] would be:

[[zeta].sub.i][equivalent to]([r.sub.i] - [C.sub.d,i])[therefore][[zeta].sub.i][less than or equal to][r.sub.i]if[C.sub.d,i] > 0 (4)

Assuming that [C.sub.d] is unbounded and increasing in distance, then as d [vector] [infinity] Prob ([[zeta].sub.i] < 0) [right arrow] 1

Let [A.sub.i][member of][A.sub.1] ... n then since [[zeta].sub.i]=([r.sub.i]-[C.sub.d,i)][less.than or equal to][r.sub.i] ceteris paribus, the weight, [x.sub.i] of [A.sub.i] held in portfolio will, be quasi-monotonically decreasing in d, i.e., as the costs associated with distance to an asset rise, the weight of the asset in the portfolio will decline.

A small bank, due to the limited availability of suitable alternative investments within its viable distance basin, would most likely have a relatively higher risk profile within its loan portfolio and would seek to balance this with a larger amount of low risk assets such as deposits at other banks, CDs, Fcd Funds or other such 'institutionalized' short term assets. To this extent, smaller banks would be expected to have a relatively larger share of their assets in such reduced risk securities. Consequently, the expectation is that smaller banks will have a higher exposure to the sell-side of the interbank market than larger banks.

As noted by Klein (1973), issues seek to limit the divisibility of their liabilities in order to reduce costs. Banks have a raison d'etre in that they intermediate between small investors and borrowers. This argument is applicable to the banking sector itself, wherein borrowing banks may encounter costs in marketing their liabilities to other banks. If part of the expense has to do with distance costs as described above, this would favor an accretion of interbank relationships with neighboring banks. The actual number of risky assets held by a bank will be increasing in its size since growth results in reduced distance costs to a larger subset of alternative investment assets, as well as the ability to acquire larger, indivisible assets. Larger banks would therefore tend to be net purchasers of funds from interbank markets.

The larger the size of a bank, the greater the number of direct linkages that will exist between it and other banks. This follows immediately from the assumptions that costs are related to distance and that distance is non-increasing in the size of a bank. Let banks h, i and j be equidistant from each other, and then let [B.sub.i] grow larger such that [SIGMA][A.sub.n, i] > [SIGMA][A.sub.m, j] = [SIGMA][A.sub.o, h]. Assume that under the new market configuration [C.sub.d,i][less than or equel to][C.sub.dj] where [C.sub.d,i] and [C.sub.d,i] are the distance costs born by bank h if it invests in the assets of the banks i and j. Assume moreover that the expected returns on the investments in each of the banks exclusive of distance costs are the same, i.e., E[[r.sub.n, h]] = E[[r.sub.n, i]] = E[[r.sub.n, j]. Since the cost associated with investment in [B.sub.i] is less than or equal to that for [B.sub.j] 'then by lemma 1 [B.sub.h] will have a weighting of [B.sub.i] that is greater than or equal to that of [B.sub.j]. Moreover, since [SIGMA][A.sub.n, i] > [SIGMA][A.sub.m, j] = [SIGMA][A.sub.o, h] it will generally be true that [SIGMA][A.sub.n, i] > [SIGMA][L.sub.m, j] [congruent to] [SIGMA][L.sub.o, h]; thus, from the point of view of [B.sub.h,] there will generally exist a greater number of investment opportunities in [B.sub.i] than in [B.sub.j].

Optimal Size

Consider that at time t, bank [B.sub.i] chooses a portfolio such that:

[M.summation over (i = 1)][A.sub.i,t] + [R.sub.t] = [N.summation over (j = 1)][L.sub.j,t] + [K.sub.t] (5)

[K.sub.t] = [K.sub.t - 1] + I (6)

where A, R, L, K and I stand for, respectively, the bank's assets, reserves, liabilities, capital and income. Income is for the period running from t-1 to t and includes revaluation gains or losses. This means that the bank selects its portfolio of assets such that it equals its resources and that its next period capital is a strict function of its income in the current period.

At time t, initial liabilities are engaged in and assets are invested (subsequently, engagements and investments are made in each period). At time t+l, information is received on asset value probability distributions and banks revise their asset values and capital. Bank accounting information is subsequently communicated to other agents. Hence, a bank may be insolvent by the time that depositors are notified.

if [K.sub.t + i][less than or equal to][N.summation over (i = 1)][k*.sub.[i,t + 1]]([A.sub.i,t + l]) then the bank is insolvent (7)

Here, k* represents the minimum rate of capital required to be allocated to a specific asset. The probability that a bank will be insolvent in period t+l can therefore be expressed as follows:

P[[K.sub.t] + [M.summation (i=1)] [r.sub.i][A.sub.i,t] - [N.sunmmation (j=1) [C.sub.j][L.sub.j] - [C.sub.o] < [M.summation (i=1)] [k*.sub.i,t+1] [A.sub.i,t+1]] (8)

It is immediately clear that the probability of insolvency is decreasing in [K.sub.t]. Suppose that bank [B.sub.i][not.equal.to][B.sub.1] holds a vector of assets [A.sub.b] corresponding to a portion of [B.sub.1] liabilities. In the event of the insolvency of [B.sub.1], accounting standards require that [A.sub.b] be written down to their reasonably expected recovery value on [B.sub.i.sup.']s balance sheet. The probability of [B.sub.i] becoming insolvent will therefore depend to some extent upon the solvency of [B.sub.1]. Abstracting from indirect linkages, and game theoretical issues over whether or not [B.sub.1] would actually be closed by the authorities, we can state this latter probability as:

P[[K.sub.t] -E + [(M-b).summation (i=1)] [r.sub.i,t][A.sub.i,t] - [N.sunmmation (j=1) [C.sub.j][L.sub.j] - [C.sub.o] < [M.summation (i=1)] [k*.sub.i,t+2] [A.sub.i,t+2]] (9)

where

E = - [N.sunmmation (b=1)] ([A.sub.b,t+2] - [A.sub.b,t] (10)

This is the probability that the remaining capital of bank [B.sub.i] is lower than that required on the assets in its portfolio. It is to be noted that, generally speaking, the capital requirement on bank deposits is substantially lower than that on loans; however, the capital required would jump to 100% in the event that the bank holding the deposits is declared insolvent. The probability of insolvency is therefore increasing in the exposure that a bank has to a bank that has been declared insolvent.

Extensions to indirect linkages are obvious: the solvency of [B.sub.1] may have an impact on the probability of default of a third company in which both [B.sub.i] and [B.sub.1] hole assets. The eventual default of [B.sub.1] may thus increase the probability of insolvency of [B.sub.i] even in the case where there are no direct linkages between the two banks.

Nevertheless, as it grows, a bank's portfolio better approximates that of the market due to the reduction in costs separating it from alternative investments. This results in a progressive reduction of the probability of its default, ceteris paribus, due to the improvement of the return/risk index on the portfolio. To this extent then, a larger bank, although a greater risk to the system, would be more stable and better able to resist local fluctuations in underlying systematic risk. Unfortunately, extreme events do happen and therefore, even if the probability of failure is very small, it is still possible and, indeed, given sufficient time, it is a nearly certain event.

Decreasing Benefits from Diversification

It is a well known phenomenon that the return/risk characteristics of a portfolio improve at an asymptotically decreasing rate, the rate of improvement is determined by the degree of covariance between new alternative investments and the existing portfolio. Those alternatives with lower correlation coefficients provide larger improvement in return/risk and those high correlation provide negligible improvement. As the number of alternative assets in the portfolio grows very large, its variance approaches the average covariance of the underlying investment universe. The contribution to performance from the idiosyncratic risk of each individual investment becomes generally smaller as the number of alternative assets in the portfolio grows.

This outcome will obtain in the presence of the distance costs hypothesized above; however, the rate at which the decline obtains may be slower. Geographic expansion does not take place continuously, but rather, in leaps and bounds, usually via the acquisition of new, geographically separate units. The return on a subset of portfolio assets logically rises as the bank grows closer to those assets since the distance costs associated with the assets declines.

The improvement to portfolio performance from the reduction in distance costs will not be negligible, but it is easy to see that such reduction will also tail off as the size of the bank grows. A logistic model demonstrates the exposure of the bank to minimal distance costs. As a bank expands, the investment alternatives to which it does not have minimal costs declines asymptotically. A simple version of such a model would be:

A = 1-[e.sup.-rt] (11)

where

A is the percent of the asset universe that has minimal distance costs,

t is time

r is the rate function at which the bank is growing

Assuming a positive rate of growth, as[right arrow][infinity] (i.e., as size approaches maximum) the improvement in the bank's portfolio return derived from reduced distance costs will decline asymptotically to zero.

A threshold size of bank will therefore exist beyond which its utility to an altruistic regulatory authority is dominated by that of a bank of smaller size. An altruistic authority is here considered 'altruistic' in the sense that its decisions give no consideration to the acquisition of private benefits obtained. Moreover, policies are formulated for the specific benefit of the economy as a whole. The systemic risk that an individual bank represents to the authority may be defined as the risk of a contraction of system wide economic wealth subsequent to the idiosyncratic failure of the bank. An altruistic authority must therefore perceive increasing disutility in a bank as its systemic risk rises.

Define the utility function of the altruistic authority as concave and increasing in a bank's probability of not becoming insolvent and decreasing in the systemic risk that obtains from the bank.

U [[B.sub.i])=F([P.sub.i],[J.sub.i],[epsilon].sub.i]) (12)

where:

F([P.sub.i],[J.sub.i],[epsilon]) = [P.sub.i]-[J.sub.i]+[[epsilon].sub.i] (13)

U altruistic regulator's utility function

[B.sub.i] the i th bank

[P.sub.1] Probabilistic wealth function of the ith bank's idiosyncratic risk of not defaulting

[J.sub.i] Probabilistic cost function of a systemic failure subsequent to the ith bank's failure

[[epsilon].sub.i] other externalities herein assumed to be independent of bank size.

Both [P.sub.1] and [J.sub.i] are functions that have been demonstrated above to depend positively upon bank size (extrapolating from moral hazard issues once the bank has become Too Big To Fail). As a bank grows in size, its probability of not defaulting increases at an asymptotically decreasing rate; thereby, increasing utility to the authority at a decreasing rate. It has been demonstrated above that the systemic risk engendered by a bank's growth increases at a rate that is at least linear; thereby, resulting in a decrease in the regulator's utility that is at least a linear rate. It obtains therefore that there will be a size beyond which the rate of decrease in the bank's probability of insolvency will drop below the rate of increase in the disutility of associated systemic risk. The optimal size is, of course, at the point where further growth results in an increase in utility that is just equal to the increase in the disutility of the growth.

Studies by Benston (1965), Berger et al. (1999) amongst others have indicated that most of the benefits to banks from diversification are achieved at a rather early stage in their development. Hence, the precise point at which a bank acquires an optimal size will undoubtedly depend upon the rate of increase in systemic risk. Studies simulating the relationship between size and systemic risk are clear in their conclusion that most risk to the system comes from probability of collapse of the largest banks in the system. These studies are not unanimous with regards to how large a bank needs to be to become a prominent risk (see Upper and Worms (2002) and Wells (2002), as well as other sources mentioned within their studies). However, it would appear from this literature that banks accounting for over 25% of the total banking assets of a nation are a clear threat to the nation, and, depending upon the degree of interrelatedness allowed within the simulation, banks with as little as 15% of total system banking assets could be a significant threat. More research must be done in this area.

Empirical Evidence

It has been proposed above that smaller banks should be net suppliers of Fed Funds and short term deposits, while larger banks should be net demanders of the same. Systematically high exposure to Federal Funds and repossessions on the asset side by small banks coupled with systematically low exposure to similar assets by large banks would provide some support for the relationships proposed in the theoretical model presented hereinabove. Earlier empirical studies of the Federal Funds market (Allen and Saunders 1986; Stigum 1982; Maerowitz 1981; Lucas et, al. 1977) using call report data have documented that such exposures do indeed exist. The present study adds to this research by deriving correlations between distance from banking centers and exposures. This test is an innovation that to this author's knowledge has not previously been published.

Bank call reports provide aggregated amounts of Federal Funds and repossessions, which provide an indication of the structure of interbank relationships across the U.S. bank sector as a whole, at least for those banks that actually engage in activity in these markets. A source of bias in call reports is that the data is not consolidated (e.g., Citizens Bank is represented as approximately 28 separate banks) and this will give rise to distortions resulting from double counting in some cases, for example, in the case where two sister banks hold assets and/or liabilities in one of its siblings. This concerns principally larger banks that would borrow less from smaller banks than would otherwise be the case; hence, the expectation is that this source of bias will result in less favorable statistics for the looked upon relationships than would otherwise be the case.

The study examines year end bank balance sheet data over a 13-year period of 1993-2005. The test of the relationship between distance and exposure to interbank activity consists of correlations between exposure and distance. The distance being measured as that between the centroids of banks' postal code regions and the geographic center of the nearest major banking city center. To reassure that larger banks are indeed headquartered near metropolitan centers (as predicted by the theory developed in Fujita et al. 2001, amongst others), correlations were also calculated between total asset size and distance.

As high exposures are concerned, Table 1 presents the number of offering and purchasing banks that had FREPO (Federal Funds and repossessions offered) or FREPP (Federal Funds and repossessions purchased) outstanding in magnitude greater than 10% of their total assets. The number of such banks is presented in the following table for each of the years as well as the arithmetic mean total assets of the concerned banks (see the Table 5 in the Appendix for a breakdown of the number of banks in each of the years studied).
Table 1 Number of highly exposed banks

           Number of        Mean        Number of       Mean value
          FREPO banks      value       FREPP banks     assets (a)
        [greater than    assets (a)    value expo.
        or equal to]10%              [greater than
                                     or equal to]10%

Dec-93            1,498         248               457        2,481
Dec-94              815         526               591        2,286
Dec-95            1,539         363               439        2,916
Dec-96            1,289         250               438        2,581
Dec-97            1,353         526               389        5,016
Dec-98            1,952         380               367        5,191
Dec-99              973         527               434        4,600
Dec-00            1,061         647               366        5,782
Dec-01            1,346         931               317        6,778
Dec-02            1,156         984               272        7,224
Dec-03              960       1,001               304        7,726
Dec-04              805       1,874               293        7,148
Dec-05              885       1,477               297       11,213

(a) Arithmetic mean of asset values in $ millions.


In each of the time periods covered by the survey, the mean value of the total assets of highly exposed Federal Fund offering banks is substantially below that of highly exposed Fed Fund purchasing banks. Moreover, the number of highly exposed offering banks is considerably larger than that of highly exposed purchasing banks.

Relationship between Exposure and Distance from Money Centers

The FDIC call report data sets include zip codes of bank headquarter locations. Using the centroid as a proxy for the bank's actual location, distances in kilometers were calculated between banks and the geographical center of 15 major metropolitan centers in the U.S.A. Table 2 below presents the correlations between log exposure to either the FREPO or FREPP of each bank, and the distance to the nearest city center, as well as the correlation between asset size and distance.
Table 2 Correlation between bank exposure and distance to nearest
city center

Year  FREPO (%)  FREPP(%)  Total assets (%)

1993       1.59     -8.87            -24.75
1994       1.28     -7.56            -24.04
1995       2.95     -9.39            -23.27
1996       5.41     10.31            -22.81
1997       3.60    -10.04            -22.71
1998       4.14    -10.23            -22.31
1999       2.66     -8.52            -20.91
2000      -0.22     -6.87            -20.37
2001       2.33     -6.44            -20.21
2002       2.78     -6.86            -20.54
2003       7.37      6.92            -20.82
2004       3.72     -6.83            -20.38
2005       1.84     -6.81            -18.71


The statistics provide support for the existence of the relationships proposed hereinabove, i.e., that the farther a bank is from a city center, the more likely it is to be exposed to interbank lending and the contrary for interbank borrowing. The former relationship is not as strong as might have been expected; and, correlations for both sides of the market are sensitive to the actual cities from which the distance is measure. Indeed, the eastern and western seaboards buck the trend present in the rest of the data. The method used to measure distance from the nearest center is a probable cause, i.e., using postal code centroids as the location of banks may not be well adapted to highly concentrated regions of economic activity. Correlations between distance and log asset size of banks that are active in either of the two interbank markets are strongly negative and this is strong evidence that banks headquartered farther from city centers are generally smaller in size.

Least Squares Regressions

To test the relationships between exposure and bank size, an interactive multiple regression was run recursively on the 13 years of panel data. Bank size, as measured by total assets, was regressed against exposure in the interbank markets. The following linear relationship is tested:(2)

Y = [alpha] + [5.summation over (i = 1)][[beta].sub.i][X.sub.i] + [epsilon] (14)

Y is the log asset size, [x.sub.1]is the log exposure to FREPO of banks reporting only FREPO activity, [x.sub.2] is the log exposure to FREPO of banks reporting both FREPO and FREPP activity, [x.sub.3] is the log exposure to FREPP of banks reporting only FREP activity, and [x.sub.4] is the log exposure to FREPP of banks reporting both FREPO and FREPP activity, and [x.sub.5] is a dummy variable that is equal to 1 for banks reporting on both sides of the market. Note that both FREPO1 and FREPP1 are equal to zero when the dummy variable equals 1.

Table 3 presents the mean coefficients from the interactive regression runs on annual data over the thirteen year period of the study (see Table 4 in the Appendix for annual results). As expected the coefficients indicate a negative relationship between bank size and exposure to FREPO and the contrary for FREPP. The mean [R.sup.2] is nearly 18% indicating that about a sixth of the variation in the FREPO and FREPP exposure variables can be accounted for by variation in bank size. The t-stats were all quite large and indicate that the null hypotheses on all beta coefficients can be rejected with a large amount of confidence. The .F-stat also allows great confidence in the rejection of the null hypothesis that all of the true coefficients are zero.
Table 3 Mean multiple regression coefficients (1993-2005)

                    [alpha]   [[beta].sub.1]  [[beta].sub.2]

Coefficient means    11.26            -5.17            -4.58
Coefficient SDs       0.21             1.31             0.32
t-stat means        534.95            -6.57           -16.59
r-stat SDs           27.69             1.83             1.64
                   [R.sup.2]         F-stat
Test stat mean       17.46%          470.38
Test stat SD          0.03            92.29

                   [[beta].sub.3]  [[beta].sub.4]  [[beta].sub.5]

Coefficient means            6.92            9.67            0.61
Coefficient SDs              3.45            2.71            0.08
t-stat means                 8.91           11.98           11.39
r-stat SDs                   4.15            2.63            1.66
Test stat mean
Test stat SD

Table 4 Interactive regression coefficients

Year    [alpha]  [[beeta.sub.1]  [[bera].sub.2]  [[beeta].sub.3]

Dec-93    10.98           -2.58           -4.44             0.46
         558.36           -3.38          -15.26             0.69
Dec-94    10.94           -5.04           -4.48             4.59
         583.89           -5.61          -12.77             7.06
Dec-95    11.08           -3.80           -4.32             8.34
         555.21           -5.27          -16.04            11.07
Dec-96    11.11           -4.54           -4.57            12.08
         567.42           -5.56          -17.19            16.21
Dec-97    11.15           -5.48           -4.30             8.61
         551.51           -7.25          -16.56            11.21
Dec-98    11.27           -6.72           -4.29             8.24
         526.32          -10.10          -19.38            11.26
Dec-99    11.19           -7.66           -5.00             7.52
         544.94           -9.79          -18.43            10.27
Dec-00    11.26           -5.41           -4.40            10.50
         533.54           -7.13          -17.30            12.29
Dec-01    11.37           -4.77           -4.61            11.77
         508.27           -6.49          -17.23            12.55
Dec-02    11.45           -6.35           -5.18             5.11
         493.09           -7.41          -17.11             6.37
Dec-03    11.48           -5.14           -4.98             4.72
         518.46           -5.87          -17.30            5.9(5
Dec-04    11.51           -5.57           -4.87             4.50
         508.10           -6.21          -15.85             6.09
Dec-05    11.57           -4.15           -4.15             3.45
         505.18           -5.38          -15.27             4.81

Year   [[beta].sub.4]  [[beta].sub.5]  [R.sup.2]  F-stat

Dec-93           9.12            0.63       0.13   383.2
                10.73           12.11
Dec-94           6.82            0.63       0.15   387.8
                11.33           12.58
Dec-95           9.77            0.56       0.20   563.2
                11.42           10.39
Dec-96           6.69            0.47       0.20   607.4
                 9.06            5.75
Dec-97           5.01            0.63       0.19   450.8
                 6.55           11.68
Dec-98          13.24            0.70       0.22   606.7
                14.31           12.84
Dec-99           9.44            0.60       0.18   492.0
                13.82           11.58
Dec-00           8.53            0.55       0.18   487.6
                10.41            9.70
Dec-01          11.07            0.46       0.18   536.2
                13.74            5.05
Dec-02          13.47            0.67       0.16   458.5
                14.76           11.86
Dec-03          12.43            0.67       0.17   447.8
                15.53           12.52
Dec-04          12.39            0.70       0.16   397.8
                14.31           13.13
Dec-05           1.11            0.69       0.14   296.0
                 9.53           12.95

t-stats in italics


Summary and Conclusion

The empirical examination of the call report data provides evidence that is favorable to the proposition that smaller banks lend proportionately more to larger banks. It is also favorable to the proposition that smaller banks that are located further from money centers lend proportionately more to other banks. Large banks, as a group, are clearly net purchasers of funds and tend to be headquartered in city centers.

This behavior can be explained by distance costs as such expenses create the incentives for banks to grow geo-spatially. Larger banks thereby have access to a greater universe of feasible investment alternatives and become net borrowers of short-term funds on the interbank market. Smaller banks located further from metropolitan centers seek to lend funds on the interbank market in order to balance off the higher risk of their constrained loan portfolios. Since benefits to diversification tail-off, while systemic risk associated with growth continues to rise, there will be a regulatory (socially) optimal size of bank

Possible extensions of this study would include multinomial analysis of the determinants of utility functions as concerns the establishment of progressively greater thresholds of exposure on either side of the market. Also, further analysis of the factors at play in the relative importance of distance from alternative banking centers, as well as the impact of economic activity in exposure to either side of the interbank markets.

Appendix
Table 5 Numbers of banks

          Total    Reporting both   Reporting   Reporting    Reporting
         in data  FREPO and FREPP  only FREPO  only FREPP   no activity
          base

Dec-93    13,324            2,695       7,649       1,248         1,732
Dec-94    12,643            2,370       6,025       2,417         1,831
Dec-95    12,002            2,516       6,780       1,219         1.487
Dec-96    11,479            2,354       6,104       1,551         1,469
Dec-97    10,946            2,301       5,858       1,421         1,366
Dec-98    10,484            2,271       6,074         942         1,197
Dec-99    10,240            2,067       4,757       2,025         1,391
Dec-00     9,920            2,155       4,854       1,599         1,312
Dec-01     9,630            2,104       4,993       1,304         1,213
Dec-02     9,369            2,054       4,925       1,160         1.215
Dec-03     9,194            1,862       4,496       1,570         1,266
Dec-04     8,988            1,865       4,298       1,604         1,221
Dec-05     8,832            1,934       4.200       1,573         1,125
Mean      10,542            2,196       5,463       1,510         1,371
SD         1,448              252       1,042         383           213

FREPO: Fed funds and repos on the offer side (assets), FREPP: Fed
funds and repos on the purchase side (liabilities)


(1) A survey completed by Degryse and Ongena (2004) explores the increasing body of literature affirming the existence and importance of such costs, see also Fujita et al. (2001).

(2) See Greene (2003) for requirements concerning the inversability of statistical relationships.

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R. L. McFadden

UNY Prague, School of Business Administration Legerova 72, Prague, Czech Republic

e-mail: rlmcf@noos.fr
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