Interbank Lending and Systemic Risk: An Empirical Analysis for Switzerland

GEORGE SHELDON and MARTIN MAURER*

1. INTRODUCTION

In a recent survey of the literature, STAUB (1997) notes a rapidly increasing number of theoretical papers on various aspects of systemic risks. The rising interest1 of theorists is matched by an increasing preoccupation with the practical consequences of systemic risk by the supervisory authorities. The president of the Federal Banking Commission of Switzerland, for example, recently justified the direct supervision of large banks by the regulatory authorities with systemic risk (HAURI, 1998), and not with the protection of small depositors.

Systemic risks are for financial market participants what Nessie, the monster of Loch Ness, is for the Scots (and not only for them): Everyone knows and is aware of the dan­ger. Everyone can accurately describe the threat. Nessie, like systemic risk, is om­nipresent, but nobody knows when and where it might strike. There is no proof that any­one has really encountered it, but there is no doubt that it exists.

The literature on systemic risk comprises theoretical models that analyze specific as­pects of systemic risk, on the one hand, and empirical analyses of historical events con­sidered to be a financial crisis, on the other. The following paper combines the theore­tical literature's focus on one particular aspect of systemic risk with the empirical literature's adherence to concrete historical events. The study focuses exclusively on the potential threat stemming from interbank lending relationships, the so-called domino ef­fect. It assesses the probability that a single bank failure could spread through a bank sys­tem via existing interbank lending and borrowing relationships. The investigation is based on accounting data drawn from banks that operated in Switzerland during the period 1987-95.

The paper unfolds as follows. The next section provides a brief survey of the difficul­ties encountered in defining systemic risk and determining the conditions under which a

* Labor Market and Industrial Organization Research Unit (FAI), University of Basle, Postfach, CH-4003 Basel and Swiss Bankers' Association, respectively. Without wishing to implicate them, we thank our discussant EVA TERBERGER as well as HANS GERSBACH, KLAUS PÖTZELBERGER and MARKUS STAUB for

helpful discussions.

1. FREIXAS and ROCHET ( 1997) provide a survey of formal models of systemic risk. A broader overview, in­cluding conceptual issues, empirical evidence and political aspects, appears in KAUFMAN (1995). DAVIS (1995) offers a wealth of factual knowledge and attempts to clear up confusion on several conceptual points.

Swiss Journal of Economics and Statistics 1998, Vol. 134(4.2)685-704

686 GEORGE SHELDON AND MARTIN MAURER

financial crisis can occur. It also reviews the empirical literature, highlighting the role of interbank credit relations in financial crises during the past 25 years. The third section develops the analytical approach, which is based on entropy maximization. Section 4 discusses the data base, and section 5 presents our results. The final section summarizes our findings and interprets the results.

2. BACKGROUND

Economists agree that systemic risk constitutes a market failure, but this unanimity is clouded by one decisive factor: there is no agreement as to what systemic risk encom­passes exactly. At a recent conference on «Banking, Financial Markets, and Systemic Risk» (KAUFMAN, 1995), a number of prominent researchers offered definitions that dif­fer substantially. Most radically, SCHWARTZ (1995) suggests dispensing with the term al­together. She proposes using instead the term «financial crisis», defined as an event which is «precipitated by desperate actions of the public in a flight to currency that sud­denly squeezes the reserves of the banking system. In a futile attempt to restore reserves, the banks may call loans, refuse to roll-over existing loans, or resort to selling assets.» (SCHWARTZ, 1995, p. 25). Other economists prefer a broader use of the term. MISHKIN

(1995) defines systemic risk as the probability that the informational function of finan­cial markets breaks down. BARTHOLOMEW and WHARDEN (1995) regard systemic risk as the likelihood of a sudden, unexpected collapse of confidence. The choice of definition is important2, since the exact nature of systemic risk determines whether (and when) au­thorities should intervene and what regulatory instruments are appropriate.

Most authors would include in their definition of systemic risk macro-shocks (balance of payments difficulties, speculative attacks on currencies, debt problems, deregulation) and inherent systemic instabilities (rational bubbles, uncertainty, credit rationing)3. How­ever, for SCHWARTZ, such occurrences may just lead to a pseudo crisis, in which assets lose their value and investors earn a negative return, but the economy as a whole remains otherwise unaffected. Therefore, such events cannot - in her opinion - be viewed as a fi­nancial crisis. According to her view, the simple fact that many institutes are hit by an ex­ogenous disturbance does not constitute a specific problem of the system as such, but is the sum of individual problems. However, based on the definition of MISHKIN or BARTHOLOMEW and WHARDEN such an increase in the sum of individual problems seri­ously affects the informational function of the financial markets and/or leads to a col­lapse in confidence.

2. See BARTHOLOMEW and WHARDEN (1995) on the difficulties of reaching a consensus and on the impli­cations of diverging views on the issue itself.

3. See DAVIS ( 1995) for an attempt to classify the sources of systemic risk. SCHWARTZ ( 1995) discusses the literature very briefly. STAUB (1995) also includes a brief overview.

INTERBANK LENDING AND SYSTEMIC RISK 687

A broad definition considers any disturbance that is sufficiently strong and works it­self through the system as constituting systemic risk4. As such, it does not address spe­cific causes nor refer to specific transmission mechanisms. Given the complexity of the issue, BARTHOLOMEW and WHARDEN (1995) prefer listing the crucial factors of a crisis, which somehow interact, rather than limiting the discussion to some specific issue in an attempt to achieve conceptual rigor. Factors to be included are: - Type and incidence of any shock. Either a large institution or a number of small firms

face a disruption of their activity affecting a significant number of other institutes. - Intensity and strength of interbank relations. The stronger the links between the insti­

tutes, the larger the secondary effects from an initial disturbance. Interbank loans are a classic example. The failure of a bank to fulfill its contractual obligations imposes losses on correspondent banks. Failed settlements in payment systems are another threat. Linkages exist between institutions (banks and non-banks), markets and coun­tries. The growth of markets has increased, thus complicating firm, market and coun­try linkages, and has led to larger, more complex and less transparent counterparty risks. Three mechanisms are particularly relevant:

- Information effects. Real or apparent difficulties in a bank induce depositors and in­vestors to suspect difficulties in other banks. If they withdraw their funds, solvent banks confront liquidity problems and face a crisis.

- Interbank relations. Financial relationships between banks, of which interbank loans are probably the most evident, favor a rapid propagation of disturbances. The trans­mission effect here is based on contracts and not, as above, on information.

- Payment and settlement risks. A disfunctionality in the system can lead to the illiq-uidity of solvent banks.

- Structure and diversification of the institutions ' portfolios. An unfavorable shock is more likely to lead to failure if asset concentration in an institute is high. Similarly, where the asset and liability portfolios of several institutes are alike, the susceptibili­ty to shocks increases, particularly when this homogeneity is strengthened by portfo­lio concentration effects.

- Informational efficiency of the market. Quantity, quality and distribution of informa­tion to market participants and regulators can influence the probability of contagion, be it through bank panics, funding crunches or inadequate policy measures.

4. This discussion also has an impact on how we classify the banking crisis of the 30s in the US. SCHWARTZ considers this episode the last financial crisis. However, CALOMIRIS and MASON (1997) interpret the hi­storical evidence differently, stating that «...failures during the panic reflected the relative weakness of failing banks in the face of a common asset value shock rather than contagion...» (p. 881). In other words, in the terminology of SCHWARTZ, they view that banking crisis as a pseudo financial crisis.

688 GEORGE SHELDON AND MARTIN MAURER

Two other factors may be added: - Situation of the real economy. The risk of a collapse is larger during a recession, which

weakens profits and hence the ability to withstand additional disturbances, and at the same time increases the probability and severity of disturbances and shocks.

- Instruments and policy of the supervisory authority and the central bank. Apart from supervision and regulation to limit the probability of a systemic risk, the central banks' use of their instruments and timing is a decisive factor.

A short look at previous financial crises - based on the cases studies in DAVIS (1995) and in GAO (1997) and on a review of the Swiss Regional Banking Crisis in the early 90's -helps shed some light on the actual factors leading to such episodes, although such an overview admittedly has to be incomplete and subjective: - Some financial crises have been caused by a problem arising in a large institute (Her-

statt, Continental Illinois, Hunt). - A common portfolio structure across banks has been an important catalyst for crisis

(Sweden, Swiss Regional Bank Crisis). Similarly, concentration of loans in certain sectors has caused several distress episodes (Latin American debt crisis, Canadian, Norwegian, Swiss Banking Crisis).

- Interbank relations have acted as transmission mechanisms in different episodes as­sociated with systemic problems, e.g., the settlement system in Herstatt and the fund­ing of investment houses by banks in Sweden's financial crisis. Interbank lending per se only had an important effect in the Continental Illinois crisis, however, when the lending facility of Continental Illinois to smaller local banks was seriously endan­gered after foreign banks withdrew funds and problems arose in raising funds on the national market.

- Information effects or the preoccupation with them appear to have been much more important than interbank relations.

- Most episodes have taken place in a time of weak economic performance of the na­tional economy, often accompanied by strict monetary policy. During a transition pe­riod, deregulation too has contributed to systemic risk.

- Central banks have reacted very early. It is impossible to deduce what would have happened had there been no intervention.

In contrast to the long list of possible causes of systemic risk, the following paper focus­es exclusively on the transmission mechanism interbank lending relationships provide, ignoring all informational aspects. In other words, with the exception of the vagaries of market returns, this study assumes certainty.

INTERBANK LENDING AND SYSTEMIC RISK 689

3. METHODOLOGICAL APPROACH

The empirical analysis in this study views systemic risk as the likelihood that the failure of one bank will trigger a chain reaction causing other banks linked to that bank through interbank loans to fail, the so-called domino effect. The study focuses on three aspects of the problem: - the probability that a bank will fail in a given period, - the path, based on the interbank loan structure, that such a shock will take, and - the effect of this shock on the solvency of those banks linked through loans to the de­

faulting bank. In short, the study concentrates on the first link in a potential chain reaction, looking to see whether a single bank default is indeed likely to cause other banks to fail or, instead, to simply dissipate in the first round.

The analysis is based on two important assumptions: - A single factor causes a bank to default on its interbank loans: insolvency, defined

here as a drop in a bank's revenues that wipes out its equity. Other possible initiating forces are ignored.

- Default is complete. Insolvency implies that the lending bank loses the entire book value of its loans to the defaulting bank.

To begin, consider the likelihood of a bank defaulting in a given period. Denote the prob­ability that a given bank will fail as P, where 0 < P < 1, and assume that P is uniform across all I banks in the banking system. Furthermore, assume that the individual proba­bilities of failure are initially independent across banks. Then the probability that no banks fail within the given period equals

(1-P)1 . (1)

Hence, the probability that at least one will fail is5

l - ( l - P ) 1 . (2)

Under the assumption that at least one bank must collapse to trigger a chain reaction of bank defaults, equation (2) gives the upper bound on the level of systemic risk in a bank­ing system containing I banks with uniform default probability P.

5. To our knowledge, HALL ( 1977) first employed this simple law of probability to develop a model of job matching. PAROUSH (1988) uses it in a banking context.

690 GEORGE SHELDON AND MARTIN MAURER

The question then arises as to what determines P. In answering this question we draw on SHELDON (1995). His approach rests on the simple observation that a bank is insolvent by definition when its losses exceed its level of capital, i.e., when

net income < -capital (3)

Capital carries a minus sign since net income is negative when a loss occurs. Adding «overhead», here defined as taxes plus expenses on personnel, materials and office space, to both sides of (3) yields

«revenues» < - (capital - «overhead») (4)

Based on our definition of «overhead», «revenues» correspond to total revenues stemming from all aspects of a bank's business (services as well as loans and investments), minus interest expenses, paid commissions and fees, and loan loss provisions. The addition of «overhead» to net income acknowledges that revenue generated to cover these expenses is also subject to risk, while deducting these expenses from capital takes account of the fact that these costs must be met first before the claims of a bank's debtholders can be honored.

Finally, dividing (4) through by the value of total assets yields

ROA < a - C A R , (5)

where ROA represents the return on assets and a corresponds to the ratio of «overhead» to total assets. The expression on the right hand side of the inequality defines the default threshold of a bank. Should ROA fall below this level, the bank is considered to be in­solvent in this study.

a, CAR and total assets are taken as being given at the beginning of a period, while ROA is viewed as being unknown until the end of the period. To capture this uncertainty, ROA is treated as a random variable with mean E(ROA) and standard deviation CT(ROA).

Under these conditions the probability P that a bank i will be insolvent at the end of the period is

P (bank i fails) = P(ROAj < a{ - CAR;)

[ROA; - E(ROAj)- CARj) a; - CAR; - E(ROAj) | , [ a(ROAi) < a(ROAj) ] ( }

= F(Zi),

INTERBANK LENDING AND SYSTEMIC RISK 691

where Z; symbolizes the default threshold of bank i in standardized form (appearing in full to the right of the inequality sign in the expression enclosed in square brackets) and F represents an arbitrary probability distribution which transforms the range (_oo, +00) of Zj into associated probabilities varying from 0 to 1.

From (6) it should be apparent that the probability that a given bank will be insolvent at the end of the period depends on (i) the relative size a of its overhead to total assets, (ii) its capital-to-assets ratio CAR, (iii) the mean E(ROA) and (iv) standard deviation a(ROA) of its rate of return on assets ROA, and on (v) the probability distribution F of its standardized ROA. Given F, banks with high overhead, a volatile and low rate of re­turn, and a low capital-to-assets ratio are more likely to fail, everything else equal.

The choice of F is a critical issue since different probability distributions can yield dif­ferent probabilities for any given default threshold z. A common practice in risk man­agement is to assume that F corresponds to the standard normal distribution. A more ro­bust approach, however, is to assume no more than that a bank's ROA is distributed symmetrically about its mean E(ROA). In this case, the Chebychev inequality implies that

= F(Zi)<^V. (7)

The Chebychev inequality is a well-known theorem of probability theory6 that places an upper bound on the probability that an arbitrary random variable will diverge from its mean by a given number of standard deviations. Since the Chebychev inequality does not depend on a particular parametric form of F, the results based on it are more robust.

Given the determinants of P, the next issue is the path that a default shock will take within the banking system. This is assumed to depend on the structure of interbank lend­ing relationships. The structure of these relationships is probably most easily viewed from the perspective of an interbank-lending matrix as pictured in Figure 3.1. The ma­trix contains M rows and N columns. Each row represents one of M banks that have loans outstanding with one or more of the N borrowing banks within the same banking system. A bank need not be both lender and borrower, which is why N need not equal M. Nor need a bank be either one or the other, so that M, N < I. py represents the share of total interbank loans within the system that consist of loans from bank i as lender to bank j as borrower. Summing across the columns in row i produces xx, the share of all interbank loans within the system provided by bank i, while summing down column j yields Cj, the share of all interbank loans received by bank j . The outer column and row containing these sums represent the marginal distributions of loans across lenders and borrowers, re­spectively. Hence

E r i = S c j = l . (8)

6. See SNELL ( 1988) for example.

692 GEORGE SHELDON AND MARTIN MAURER

Figure 3.1: Interbank-Lending Matrix

N Borrowing Banks

J3 fl Ä

M

.2 "O a a> J S

Pli

•

•

Pil

• •

•

PMI

. . .

• • •

• • •

Pij

•

•

•

Pu

• •

•

PMJ

. . .

• • •

• • •

PIN

•

•

•

PiN

• •

•

PMN I'M

C N

In general, the only information a researcher possesses with regard to the interbank-lend­ing matrix pictured in Figure 3.1 consists of the absolute levels associated with rj and Cj. Information of this sort is readily available from bank balance sheets. Missing are data on the internal values p^ of the matrix.

The missing data problem can be neatly summarized by the following equation sys­

tem:'

where

P'

Ap = B , (9)

B' = A =

[pn,..., P|N,..., Pij,..., pMh —» PMNI' ^„ . . „^^„ . . „CNJand (M+N) x (M-N)-dimensional matrix containing ones and zeros in the ap­propriate cells so that pre-multiplying the (M-N) x 1 column vector p with A yields the row and column sums contained in the (M+N) x 1 column vector B.

7. Bold type denotes matrices and vectors.

INTERBANK LENDING AND SYSTEMIC RISK 693

A and B are known, the former by definition [see (8)] and the latter from bank account­ing statements. What is unknown are the elements of p. Solving for these is not a viable alternative, however, since (9) contains more unknowns than equations. In other words, the equation system is under-identified. This is the essence of the missing data problem.

Under-identification implies that more than one solution to (9) exists. The problem is which solution to choose. Since (9) is the only information at our disposal it seems nat­ural to seek that solution which injects the minimum amount of additional and, by defi­nition, non-verifiable information into the data. This insight thus suggests choosing that solution which maximizes the entropy of the interbank-lending matrix. Entropy, as used in the mathematical theory of communication (see SHANNON/WEAVER, 1949), measures the amount of information that a message contains. In this context, the informational content of a message (i.e., its entropy) is greatest when the message recipient is the most uncertain as to the outcome of a given event, for example, when the perceived odds are 50:50. In this case, a message containing the outcome of the event provides the greatest possible increase in knowledge to the recipient. Hence, informational content and ex ante uncertainty are positively linked.

Entropy in an informational setting is defined as

- p ' l n p , (10)

where the column vector p contains the perceived probabilities that the events in ques­tion will occur. As is easily seen, in the two-state event of our example the entropy is indeed maximized when the odds pertaining to the possible outcome stand at 50:50. Applied to the problem at hand, p represents the missing cell values of the interbank-lending matrix. In this context, maximizing (10) subject to the linear restrictions con­tained in (9)8 chooses that solution which leads to the most even («uncertain») distribu­tion of loans across the cells in the interbank-lending matrix9, given the known structure of the marginal distributions contained in B10. Without these linear restrictions the max­imization of (10) would yield a uniform distribution in which all p^'s were equal. The only thing preventing this solution from occurring is the structure of the marginal distri­butions. Hence, it is this information alone which forms the solution. No other informa­tion enters in.

8. 0 < Pij < 1 is a further restriction. 9. This smoothing or averaging effect of entropy maximization has led to its use, particularly in computer

tomography, to restore images from which only fragmentary information exists (see FRIEDEN, 1980). The usefulness of entropy maximization in such applications is intuitively clear if one views the interbank-lending matrix as a grid in which a given number of squares are empty and in need of filling.

10. Entropy maximization has a long tradition in economics. IRELAND and KULLBACK (1968) show that the RAS procedure developed by Nobel laureate RICHARD STONE (1962) and others to update input-output tables with the help of marginal distributions is a variant of entropy maximization.

694 GEORGE SHELDON AND MARTIN MAURER

Although maximizing entropy minimizes the amount of external information im­posed on the solution, it nevertheless projects a great deal of behavioral structure on the data. For one, maximizing entropy maximizes the degree of loan diversification for given marginal distributions within the banking system. Hence, maximizing entropy minimizes the amount of idiosyncratic risk in the system, which may or may not provide a good approximation of reality. Furthermore, maximizing entropy implies to a certain extent11 that a bank's choices of which banks to lend to and from which to borrow are stochastically independent. That means that the relative distribution of a bank's loans across banks has no bearing on the relative intensity with which that bank borrows funds from other banks. Maximizing entropy thus minimizes the possible presence of house-bank relationships. In view of these behavioral implications of entropy maximization, the entropy-maximizing solution to (9) probably presents the lower bound on the true amount of systemic risk resting in any given banking system. Note though, that the be­havioral implications of entropy maximization is less a fault of the methodology than of the shortage of information on interbank lending relationships at our disposal. Addition­al knowledge of the true values of any cells in the interbank-lending matrix can easily be incorporated in the system of restrictions (9) in order to improve the realism of the esti­mates12. This basic openness of the methodology proposed here is one of its major strengths.

In principle, maximizing (10) subject to (9) is a simple matter. The second derivative of (10) is a diagonal matrix with diagonal elements -py"1, indicating that (10) is strictly concave from below. Thus, any commonly used gradient procedure such as the Newton-Raphson method for finding an optimum should have no problem converging to a solu­tion. The main obstacle is the possible size of the problem. Assuming 450 banks that all lend and borrow with one another leads to a system of 900 equations with over 200000 unknowns. The A matrix alone would take up about 1.5 gigabytes of memory in this case. Aggregating the banks into groups can help to make the problem more manageable, however, especially if no bank is left ungrouped, since in this case maximizing entropy subject to (9) yields the solution

Pij = rj-Cj, (11)

implying stochastic independence between a bank's source and target for interbank loans and thus obviating the need to store A.

11. In this case, one must assume that banks can lend and borrow with themselves, which is not a crucial as­sumption if the number of banks is large (say 500), as the assumption then only really affects the cell entries in the main diagonal of the interbank-lending matrix. We thank KLAUS PÖTZELBERGER from the University of Economics in Vienna for proving to us that the maximization of ( 10) subject to (9) implies stochastic independence in this special case.

12. For example, we know that an individual bank cannot borrow and lend with itself so that p̂ = 0 for i = j .

INTERBANK LENDING AND SYSTEMIC RISK 695

4. DATA

Interbank transactions are restricted here to short-term (0-3 month) interbank time de­posits (Bankenkreditoren auf Zeit) and loans (Bankendebitoren auf Zeit). Money market papers and derivatives, which also count as interbank financial transactions, are ignored for lack of data. However, STAUB (1997) reports that short-term interbank loans and de­posits are the principal instruments for the closing of short-run liquidity gaps among Swiss banks (as opposed to derivatives, which are used for risk management purposes). So although we do not include all sources of systemic risk arising from interbank finan­cial ties, we do focus on a major source.

The data used in this study stem from two sources. The figures on the marginal distri­butions of the interbank-lending matrix contained in the vector B are drawn from the an­nual reports (Das schweizerische Bankwesen) of the Swiss National Bank from 1987 to 1995. To obtain consistent estimates of the interbank linkages necessitates including all banks, and the annual reports are the only generally accessible source containing all banks. The annual report only publishes aggregate figures, however, grouped into 12 bank categories (see Table 4.1), consisting of cantonal banks, commercial banks (Han­delsbanken), investment banks (Börsenbanken), consumer credit institutes, foreign-con­trolled banks, financial companies (Finanzgesellschaften), credit cooperatives, private banks, branches of foreign-owned banks and other, special-purpose banks13.

Table 4.1: Banks in Switzerland, 1987-95

Bank Group 1987 1988

Cantonal 29 29 Large 5 5 Regional 214 213 Commercial 27 26 Investment 49 48 Consumer 11 11 Special 4 4 Foreign 111 116 Coop 2 2 Financial 130 133 Branches 17 17 Private 23 22

Total 622 626

Source: Das schweizerische Bankwesen,

1989

29 5

210 25 51 11 4

118 2

137 17 22

631

various

1990

29 4

204 25 51 11 5

126 2

130 16 22

625

years

1991

28 4

189 24 54 10 4

130 2

112 16 19

592

1992

28 4

174 23 57 9 4

134 2

101 14 19

569

1993

28 4

155 20 56

7 4

143 2

79 13 18

529

1994

27 4

135 19 58 5 4

140 1

71 13 17

494

1995

25 4

127 20 54

5 5

141 1 -

14 17

413

Avg.

28 4

180 23 53 9 4

129 2

112 15 20

567

13. For a general description of these bank groups see the annual report of the Swiss National Bank or SHELDON (1995)

696 GEORGE SHELDON AND MARTIN MAURER

To calculate default thresholds in accordance to equation (6) requires individual bank data. These stem from the annual published balance sheets and income statements of the first eight bank categories listed in Table 4.1 and cover the same sample period 1987-9514. The construction of the variables appearing in (6) were discussed above. It remains to add that bank capital (CAR) also encompasses published reserves and roughly corresponds to Tier 1 funds as defined in the Basle Capital Accord. Total assets only include on-balance-sheet items, but this poses no restriction since assets merely serve as a standardization device in (6). Any revenues off-balance-sheet items might generate appear in a bank's income statement and thus enter our calculations.

5. RESULTS

Table 4.1 indicates that the number of banks that operated in Switzerland in the sample period averages out to 576. Setting this value equal to I in equation (2) produces the curve in Figure 5.1, which - assuming that at least one bank must fail to trigger a chain reaction - gives the upper bound on the level of systemic risk associated with various values of P, the probability that a particular bank will fail in a given observation period. As the figure indicates, a relatively low individual probability of default (horizontal axis) suffices to make the chance of any bank failing in a given period rather likely. For ex­ample, even when the average probability that a particular bank will fail in a given time span equals 0.1 percent, the likelihood that at least one of 576 banks will collapse is al­most 45 percent. In fact, at a default probability of 1 percent, such an event is almost cer­tain.

90 -

? 00 5.

i TO

I * 40

! A 3°-

1 » I 2 0

0

Figure 5.1: Systemic Risk as a Function of Bank Failure Probabilities

0.0 0.3 0.4 0J 0.6 0.7

Default Probability for a Sfalle Bank (in %) 0.8 0.9

14. The data were kindly supplied to us by the Swiss National Bank with the consent of the banks concer­ned, to whom we extend our thanks.

INTERBANK LENDING AND SYSTEMIC RISK 697

Table 5.1 presents the average value of P for the groups of banks for which individual ac­counting data stood at our disposal. The results are based on equation (6) and cover the period 1987-95. The calculated probabilities represent annual risks and give an average bank's risk of being bankrupt at the end of a year, given its financial position at the be­ginning of the year and the variability of its ROA. Corresponding figures for the period 1987-93 appear in SHELDON (1995). The earlier study produces very similar results. For example, regional banks still carry the greatest risk of insolvency, whether one assumes normally distributed («Norm.») rates of return or relies merely on the Chebychev in­equality («Cheb.»). Under the assumption of normally distributed rates of return, the av­erage overall probability of default for a bank is 0.8 percent according to Table 5.1, which when viewed from the perspective of Figure 5.1 makes it appear quite likely that at least one bank in Switzerland will fail in any given year.

Table 5.1: Probability of Bank Failure, Switzerland, 1987-95

P(default) in % BANKS

Cantonal Large Regional Commercial Investment Consumer Special Foreign

All

CASES

29 4.

213 23 60 11 4

136

480

a

0.009 0.018 0.010 0.036 0.093 0.046 0.018 0.061

0.037

CAR

0.041 0.061 0.052 0.170 0.280 0.094 0.171 0.241

0.141

E(ROA)

0.012 0.023 0.009 0.043 0.125 0.057 0.033 0.077

0.046

(T(ROA)

0.002 0.002 0.010 0.011 0.032 0.013 0.019 0.019

0.015

Cheb.

0.9 0.1 6.9 2.6 1.0 2.3 0.7 0.4

3.5

Norm.

0.2 0.0 1.6 0.7 0.0 0.5 0.0 0.0

0.8

The large likelihood of a bank failure underscores the importance of knowing how such a shock might impact on other banks linked to the failing bank through interbank loans. Table 5.2 presents the interbank lending structure based on applying the entropy-maxi­mization approach to data taken from the annual reports of the Swiss National Bank. The marginal distributions («Domestic», «Foreign», «Total») stem directly from these re­ports, whereas the values appearing in the matrix cells result from maximizing (11) sub­ject to the marginal distributions appearing in the column and row labeled «Domestic». The values represent annual averages for the period 1987-95. The group «Non-Banks» is a residual category necessitated by the fact that the calculated sum of funds borrowed by banks located in Switzerland from other banks in Switzerland (column «Domestic») exceeds the sum of domestic interbank loans (row «Domestic») by almost 17.5 billion Swiss francs.15 The sources of this discrepancy are somewhat unclear. For one, Swiss ac­counting rules do not define lending and borrowing in an exactly symmetrical fashion.

15. See the cell at the intersection of the row «Non-Banks» with the column «Domestic».

698 GEORGE SHELDON AND MARTIN MAURER

Furthermore, large banks apparently treat interbank loans to small financial institutes as loans to non-banks (STAUB, 1997, p. 13). If smaller banks regard such loans as interbank loans then the sums will not be consistent. Finally, the Swiss National Bank, clearing houses (Pfandbriefzentralen) and foreign brokers, which accounting rules view as banks with regard to interbank loans, are missing in Table 5.2. If this residual category is omit­ted, the solution of the entropy maximization problem breaks down.

Table 5.2: Interbank-Lending Matrix, Switzerland, 1987-95 (1000 Swiss Francs)

Cantonal

1345530

998419

372396

264113

256923

13519

3326

705007

181859

67089

131878

217840

2717701

7275599

746500

8022099

Large

4427160

3407571

1217833

835165

824124

42274

9755

2242002

550684

212506

468340

711157

8733488

23682059

71212875

94894934

RegionalCommercial

359492

300152

98881

76451

69382

3550

724

195411

44533

20297

42265

59975

744186

2015298

36125

2051423

599525

460401

163852

119684

113927

5892

1375

315239

77644

31006

61201

97011

1208340

3255098

2667625

5922723

Investment

221669

153078

61295

41478

42183

2239

597

113368

31165

10323

19321

35199

442962

1174877

1380750

2555627

Consumer

27093

18591

7812

5046

4997

323

75

14099

4016

1309

2547

4414

52313

142634

115875

258509

Special

2083

1239

620

353

421

16

8

1037

338

55

145

340

4356

11011

0

non

Foreign

1345039

1080477

364439

256278

245509

12925

2674

688905

159680

67719

150877

214903

2630493

7219920

19328375

26548295

Coop

8575

6174

2367

1620

1610

88

22

4425

1172

406

791

1359

17018

45625

0

45625

Financial

252125

207200

67877

49835

46585

2381

469

129993

29029

13445

29117

40778

497041

1365875

5286750

6652625

Branches

206904

175013

55100

37705

36275

1817

311

99923

20597

10399

25910

32997

385925

1088875

8393000

9481875

Private

15948

12639

4341

3069

2897

169

34

8452

2034

822

1735

2526

31459

86115

82375

168500

Domestic

8811144

6820955

2416814

1690797

1644832

85192

19368

4517861

1102750

435375

934125

1418500

17465283

0

109250150

156613246

Foreign

3189749

69651076

14692

4713985

3986358

2609

26

20219533

0

6273250

5294750

190500

0

Total

12000893

76472031

2431507

6404782

5631189

87801

19394

24737394

1102750

6708625

6228875

1609000

17465283

113536527 160899523

4286277

4286277

The shaded areas in Table 5.2 contain the differences between the column sums (total loans provided) and the row sums (total loans received) that appear directly above or to the left of the differences. As the figures indicate, the sum of interbank loans provided to banks outside Switzerland exceeded the sum of those received from abroad by about 4.3 billion Swiss francs in the sample period. Moreover, foreign interbank lending relation­ships clearly dominated domestic linkages in volume, the former exceeding the later by more than double. This holds particularly true for large banks, foreign-controlled banks, financial companies and Swiss branches of foreign banks where foreign interbank trans­actions surpasses domestic ones by a ratio of at least 3:1. Conspicuous exceptions to this rule are cantonal banks, regional banks, special-interest banks, credit cooperatives and private banks, where the opposite holds. The strong international bias of the interbank lending relationships of banks in Switzerland clearly limits the degree to which domes­tic interbank loans can contribute to systemic risk in the Swiss banking system.

Table 5.3 divides the cell entries in Table 5.2 by the sum of all domestic interbank loans. Shaded areas denote values equal or greater than 1 percent. As the shaded areas clearly show, the large universal banks are the major borrowers of interbank funds from

INTERBANK LENDING AND SYSTEMIC RISK 699

all bank groups, and the non-banks the greatest provider. This is a direct result of the fact that large banks receive 50 percent of all domestic interbank loans, while non-banks pro­vide almost 37 percent of them. Under the rules of entropy maximization this structure must also hold for each individual bank category, i.e., each bank category must direct 50 percent of its loans to large banks and receive 37 percent of its funds from non-banks. Only as yet unavailable information can verify whether this implication of entropy max­imization is roughly true in reality. Note that the proportionality structure our results ex­hibit is not a general result of entropy maximization, but rather applies only when mar­ginal distributions are the sole information available.

Table 53: Interbank-Lending Matrix, Switzerland, 1987-95 (relative values)

Cantonal

Large

Regional

Commercial

Investment

Consumer

Special

Foreign

Coop

Financial

Branches

Private

Non-Banks

Domestic

Cantonal

<fòfàa:?l Large

-<M#3 '

^uß^^m^ 0.008

0.006

0.005

0.000

0.000

IWJffik ÌQÌ0J8

0017

0.001

0.000

^ ä ^ p p 0.004

0.001

0.003

0.005

'•WJS:' 0.154

OL012

0.004

l$w

0.500

Regional

0.008

0.006

0.002

0.002

0.001

0.000

0.000

0.004

0.001

0.000

0.001

0.001

0X>16

0.043

Commercial

0013

6.010

0.003

0.003

0.002

0.000

0.000

0.007

0.002

0.001

0.001

0.002

0.026

0.069

Investment

0.005

0.003

0.001

0.001

0.001

0.000

0.000

0.002

0.001

0.000

0.000

0.001

0.009

0.025

Consumer

0.001

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.001

0.003

Special

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

Foreign

0.028

0.023

0.008

0.005

0.005

0.000

0.000

0.015

0.003

0.001

0.003

0.005

0.056

0.152

Coop

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.001

Financial

0.005

0.004

0.001

0.001

0.001

0.000

0.000

0.003

0.001

0.000

0.001

0.001

0.010

0.029

Branches

0.004

0.004

0.001

0.001

0.001

0.000

0.000

0.002

0.000

0.000

0.001

0.001

0.008

0.023

Private

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.001

0.002

Dome;

0.18

0.14

0.05

0.03

0.03

0.00

0.00

0.09:

0.02:

O.OO

0.021

o!03<

0.36«

1.0«

On the basis of the figures appearing in Table 5.2, we can go a step further and investi­gate to what extent the default of a borrowing bank will push the lending banks linked to it into bankruptcy (domino effect). To do this, we must first estimate the typical size of a loan default shock emanating from a bank group. We arrive at our estimates by dividing the values in each column in Table 5.2 by the average number of banks (see Table 4.1) in the corresponding bank group. This procedure assumes that borrowed funds are distrib­uted evenly across the banks making up a bank group. In addition, we must assess how many banks within a bank group are hit by the default shock. In this paper we compare two extreme cases. The one assumes that just one bank within a bank category receives the shock, and the other that the shock is distributed evenly across all banks within the bank group, save the one defaulting if of the same group. The calculated size of the shocks are then compared to the default thresholds of the banks within a bank group to assess whether a domino effect will ensue.

The results pertaining to the first case, in which just one bank takes the full impact of a loan default, appear in Table 5.4. The figures to the right of the vertical line in the table

700 GEORGE SHELDON AND MARTIN MAURER

represent the ROA that an average-size bank of the bank group appearing at the left can expect to achieve after suffering the full force of a default shock emanating from an av­erage-size bank in the bank category shown at the head of the column. The values ap­pearing in the first column of figures give the minimum, mean and maximum default thresholds (a-CAR) of the banks in the bank group appearing at the left. The shaded areas mark those cases where a loan default shock would surpass at least the highest de­fault threshold of the group of banks receiving the shock. For example, an average-size cantonal bank could expect to realize an ROA of -11.4 percent if an average-size large bank were to default on its loans and the loans of this bank provided by the group of can­tonal groups stemmed from a single average-size cantonal bank. As the threshold values indicate, a shock of this sort would bankrupt even the cantonal bank with the lowest de­fault threshold of -4.4 percent. As is to be expected, almost no average-size bank would be able to withstand the full force of a loan default from a large bank. The only excep­tions are other large banks, consumer-credit banks and special-purpose banks. The latter two groups of banks appear so resilient because they supply virtually no interbank loans (see Table 5.3).

Table 5.4: Risk of Contagion When One Bank Alone Absorbs a Default Shock

Lending Bank's E(ROA) after an Average Default Shock from Bank Group Threshold Cantonal Large Regional Commercial Investment Consumer Other Foreign Coop Financial Branches Private

min -0.044

mean -0.032 0.006 -0.114 0.012 0.009 0.012 0.012 0.012 0.011 0.012 0.012 0.011 0.012 max -0.020

min -0.050

mean -0.043 0.023 0.018 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 max -0.033

min -0.488 mean -0.043 -0.018 -0.555 0.008 -0.005 0.007 0.007 0.009 0.003 0.006 0.008 0.002 0.009 max -0.004

il min -0.416 mean -0.135 0.039 -0.039 0.043 0.041 0.043 0.043 0.043 0.043 0.043 0.043 0.042 0.043 max 0.018

min -0.586

mean -0.187 0.104 -0322 0.124 0.114 0.123 0.124 0.125 0.121 0.123 0.124 0.120 0.125 max -0.020

min -0.080 mean -0.048 0.056 0.042 0.057 0.056 0.056 0.056 0.057 0.056 0.056 0.057 0.056 0.057 max 0.016

min -0.289

mean -0.153 0.033 0.025 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 max -0.029

min -0.695 mean -0.180 0.041 -0.659 0.076 0.058 0.074 0.075 0.077 0.070 0.074 0.076 0.068 0.077 max -0.031

INTERBANK LENDING AND SYSTEMIC RISK 701

If we assume instead and more realistically that the borrowings of a defaulting bank from a specific bank group are uniformly distributed within a bank group so that the shock hits all banks in the group equally hard, then no lending bank should expect to fail. This is shown clearly in Table 5.5 that is based on this assumption. As a comparison with Table 5.1 indicates, the expected ROA are barely affected by a single default shock in this case.

Table 5.5: Risk of Contagion When All Banks Absorb a Default Shock

Lending Bank's E(ROA) after an Average Default Shock from Bank Group Lender Threshold Cantonal Large Regional Commercial Investment Consumer Other Foreign Coop Financial Branches Private

Cantonal min -0.044

mean -0.032 0.012 0.008 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.012 max -0.020

Large min -0.050

mean -0.043 0.023 0.022 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023 0.023

max -0.033

Regional min -0.488 mean -0.043 0.009 0.006 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009

max -0.004

Commercial min -0.416 mean -0.135 0.043 0.040 0.043 0.043 0.043 0.043 0.043 0.043 0.043 0.043 0.043 0.043 max 0.018

Investment min -0.586 mean -0.187 0.125 0.117 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 max -0.020

Consumer min -0.080 mean -0.048 0.056 0.055 0.057 0.056 0.057 0.057 0.057 0.057 0.057 0.057 0.057 0.057 max 0.016

Special min -0.289 mean -0.153 0.033 0.031 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 0.033 max -0.029

Foreign min -0.695 mean -0.180 0.077 0.072 0.077 0.077 0.077 0.077 0.077 0.077 0.077 0.077 0.077 0.077

max -0.031

In summary, since large banks according to Table 5.1 are least likely to fail and as no bank within a bank group is likely to supply all the loans of that bank group to a large bank, the chances that a defaulting bank will pull down other banks linked to it through interbank loans seem rather slim in Switzerland.

702 GEORGE SHELDON AND MARTIN MAURER

6. CONCLUSIONS

Our analysis of the systemic risk in Switzerland stemming from the structure of inter­bank loans has shown that although the likelihood of a bank insolvency in any given year is quite high, the chances of a bank failure propagating through the banking system via the network of interbank loans are quite low. This is a comforting result and conforms with the findings of CALOMIRIS and MASON (1997) who fail to find evidence that conta­gion effects contributed to bank failures during the 1932 Chicago bank panic. Yet before one takes too much comfort in our results, one should consider the limitations of our study.

For one, it should be noted that the domestic interbank loans studied here are quanti­tatively less important in Switzerland than cross-border interbank loans, which outsize the former by more than double. Furthermore, our study concentrates on a single idio­syncratic shock. A series of simultaneous default shocks would undoubtedly place a greater strain on the network of interbank loans. In addition, our analysis ignores any ef­fects a tightly knit network of interbank loan relationships may have on the role of infor­mation asymmetries. Closer relationships could lead to greater market transparency, on the one hand, but to a faster propagation of rumors, on the other. Moreover, the network of interbank loans studied is based on maximum loan portfolio diversification (entropy), which rules out any clumped risks above and beyond those implied by the marginal dis­tributions of interbank loans. But this is less a fault of our methodology than a short­coming of the data at hand. Knowledge of the true values of any cells in the interbank-lending matrix can easily be incorporated into our approach in order to add to the accuracy of our results. Finally, one should bear in mind that a low risk of contagion needs to be weighted against the costs of its consequences before a final judgment is felled. For example, DZIOBEK and PAZARBASIOGLU (1997) report the cumulative costs of restructuring the banking sector after a systemic crisis to have been 4.3,9.9 and 15.0 per­cent of annual GDP in Sweden, Finland and Spain, respectively. Thus, low risks do not necessarily mean low costs, especially in Switzerland with its large banking sector.

What our results do suggest, however, is that domestic interbank lending is not the foremost threat to financial market stability in Switzerland. In that respect we are in a similar situation as the Scots with Nessie: although Nessie might strike, more dangerous threats to their security exist.

INTERBANK LENDING AND SYSTEMIC RISK 703

LITERATURE

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G.KAUFMAN (1995), p. 3-17.

CALOMIRIS, C , J. MASON (1997), «Contagion and Bank Failures During the Great Depression: The June 1932 Chicago Banking Panic,» in: American Economic Review, 87(5), pp. 863-883.

DAVIS, E. (1995), Debt, Financial Fragility and Systemic Risk, Oxford. DZIOBECK C , C. PAZARBASIOGLU (1997), «Lessons from Systemic Bank Restructuring:

A Survey of 24 Countries,» IMF Working Paper, WP/97/161, Washington. FREIXAS, X., J. ROCHAT (1997), Microeconomics of Banking, Cambridge, Mass. FRIEDEN, B. (1980), «Statistical Models for the Image Restoration Problem,» in:

Computer Graphics and Image Processing, 134 ( 1 ), pp. 40-58. GAO (United States General Accounting Office) (1997), «Financial Crisis Management:

Four Financial Crises in the 1980s,» GAO/GGD-97-96, Washington DC, May. HALL, R. (1977), «An Aspect of the Economic Role of Unemployment,» in:

G. HARCOURT (ed.), Microeconomic Foundations of Macroeconomics, London, pp. 354-372. .

HAURI, K. (1998), «Grossbankenaufsicht - die EBK nimmt die Herausforderung an,» Press Conference of the Federal Banking Commission from April, 21 1998, Bern.

IRELAND, C , S. KULLBACK (1968), «Contingency Tables with Given Marginals,» in: Biometrika, 55, pp. 179-86.

KAUFMAN, G. (ed.) (1995), Banking, Financial Markets, and Systemic Risk, Research in Financial Services, Vol.7, Greenwich/London, 1995.

MISHKIN, F (1995), «Comment on Systemic Risk,» in: G. KAUFMAN (1995), pp. 31^45. PAROUSCH, J. (1988), «The Domino Effect and the Supervision of the Banking System,»

in: Journal of Finance, 43(5), pp. 1207-1218. SCHWARTZ, A. (1975), «Systemic Risk and the Macroeconomy,» in: G. KAUFMAN (1995),

pp. 19-30. SHANNON, C , W. WEAVER (1949), The Mathematical Theory of Communication, Urbana. SHELDON, G. (1995), «A Limit-Risk Capital Adequacy Rule: An Alternative Approach

to Capital Adequacy Regulation for Banks with an Empirical Application to Switzer­land,» in: Swiss Journal of Economics and Statistics, 131(5), pp. 773-805.

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berichte 5/97, WWZ, University of Basle, Basle. STONE, R. (1962), «Multiple Classifications in Social Accounting,» in: Bulletin of the

International Statistical Institute, 39(3), pp. 215-33.

704 GEORGE SHELDON AND MARTIN MAURER

SUMMARY

Systemic risk in banking has gained renewed prominence in the literature in recent years. To date, empirical studies aimed at assessing the quantitative importance of systemic risk have analyzed the outcomes of historical banking crises. This paper takes a new tack by attempting to assess the level of systemic risk currently in a banking system on the basis of interbank loan structures. We construct a matrix of interbank loans for Switzerland based on known marginal loan distributions and the principle of entropy maximization. Our results suggest that the latent systemic risk associated with the interbank loan struc­ture existing among Swiss banks in 1987-95 posed little threat to the stability of the Swiss banking system.

ZUSAMMENFASSUNG

Das Problem des systemischen Risikos ist in letzter Zeit auf vermehrtes Interesse in der Bankliteratur gestossen. Bislang stützten sich Versuche, die empirische Tragweite des Problems zu bestimmen, auf historische Vorfälle. Der vorliegende Beitrag wählt einen anderen Weg, indem er das aktuelle Risikopotential auf der Basis bestehender Interban-kenkreditbeziehungen abzuschätzen versucht. Dabei wird auf der Grundlage gegebener Randverteilungen und des Prinzips der Entropiemaximierung eine Matrix von Interban-kenkreditbeziehungen erstellt. Die Ergebnisse deuten darauf hin, dass das Risikopoten­tial, das sich aus dem Netzwerk an Interbankenkreditbeziehungen ergibt, das 1987-95 in der Schweiz bestand, kaum eine Gefahr für das Schweizer Bankensystem stellte.

RESUME

Le problème du risque systémique a pris ces derniers temps de plus en plus d'importan­ce dans la littérature bancaire. Les événements historiques ont jusqu'à présent servi de base aux tentatives d'appréhender empiriquement l'ampleur du problème. La présente contribution s'inscrit dans une autre démarche, en ce sens qu'elle cherche à estimer le risque potentiel actuel en se fondant sur les relations de crédits interbancaires en cours. A cet effet, une matrice des relations de crédits interbancaires a été établie sur le princi­pe des répartitions marginales et sur celui de la maximisation de l'entropie. Il résulte de ces recherches que le potentiel de risques émanant du réseau de relations de crédits in­terbancaires existant en Suisse de 1987 à 1995 n'a guère constitué un danger pour le sys­tème bancaire suisse.