## Bank Failure Quantitative Models

While a qualitative review of bank failure provides some insight into what causes a bank to fail, these ideas must be subjected to more rigorous testing. Any econometric model of bank failure must incorporate the basic point that insolvency is a discrete outcome at a certain point in time. The outcome is binary: either the bank fails or it does not. The discussion in the previous section shows that banks (or, in Japan, almost the entire banking sector) are often bailed out by the state before they are allowed to fail. For this reason, the standard definition of failure, insolvency (or negative net worth), is still extended to include all unhealthy banks which are bailed out as a result of state intervention, using any of the methods outlined in earlier sections, such as the creation of a 'bad bank' which assumes all the troubled bank's unhealthy assets and becomes the responsibility of the state, and a merger of the remaining parts with a healthy bank.

Some of the methodology employed here is borrowed from the literature on corporate bankruptcy, where a firm is either solvent (with a positive net worth) or not. In situations where the outcome is binary, two econometric methods commonly used are discriminant or logit/probit analysis. Multiple discriminant analysis is based on the assumption that all quantifiable, pertinent data may be placed in two or more statistical populations. Discriminant analysis estimates a function (the 'rule') which can assign an observation to the correct population. Applied to bank failure, a bank is assigned to either an insolvent population (as defined above) or a healthy one. Historical economic data are used to derive the discriminant function that will discriminate against banks by placing them in one of two populations. Early work on corporate bankruptcy made use of this method. However, since Martin (1977) demonstrated that discriminant analysis is just a special case of logit analysis, the multinomial logit model is used here.

The logit model has a binary outcome. Either the bank fails, p = 1, or it does not, p = 0. The right-hand side of the regression contains the explanatory variables, giving the standard equation:

where:

p = 1 if z > 0 p = 0 if z < 0 z = log[p/(1 -p)] P0: a constant term

P': the vector of coefficients on the explanatory variables x: the vector of explanatory variables e: the error term.

It is assumed that var (e) = 1, and the cumulative distribution of the error term is logistic; were it to follow a normal distribution, the model would be known as probit.

Readers are assumed to be familiar with logit analysis; if not they are referred to any good textbook in introductory econometrics. In a simple application of equation (14.1), should x consist of just one explanatory variable (for example capital adequacy), the logit model becomes a two-dimensional sigmoid-shaped curve.

However, there are problems with much of the published literature as it stands. The first is that virtually every econometric study of bank failure is based on US data, because it is only in the US that the sample of bank failures is large enough to allow quantitative tests to be conducted. The problem with interpreting the results relates to the structure of the American banking system, which is very unlike banking structures typical of most other Western countries, with the possible exception of Japan.21

With reference to the US, the most important difference is the absence of a national banking system, dominated by four or five banks, with branches located across the country. US banking regulations have, historically, focused on discouraging the concentration of banks, and it is only in the late 1990s that severe branching restrictions across (and sometimes within) states have been removed. With such a different banking structure, empirical results based on US data should be treated with caution if applied to other Western countries. An added problem is the difficulty in testing for the effects of macroeconomic variables when only one country is studied.

A final, potential problem relates to the use of the multinomial logit function in estimating bank failure. These studies rely on a cross-section of failed banks either in a given year, or over a number of years. They are using panel data, and for this reason, an alternative model could be a panel data logit specification first described by Chamberlain (1980). The 'conditional' logit model for panel data is:

where:

a.: captures individual group effects, and is separate for each group; i = failed or non-failed

Chamberlain shows that if a multinomial logit regression is used on panel data, and the number of observations per group is small (except in the US, where the number of failed banks was very high in the early 1930s and throughout the 1980s), the result is inconsistent estimates, arising from omitted variable bias. Furthermore, the a.. allow a test for group effects, addressing the question of whether there is something unique to the group of bank failures. In equation (14.2), the a.. are not considered independent of x.

None the less, the results of key studies which use a multinomial logit model are reviewed to identify the statistically significant variables causing bank failure.22

Espahbodi (1991), Martin (1977) and Thomson (1992) employ US financial ratio data based on a cross-section of bank failures. Their observations are confined to the US, and, with one exception, these studies did not test for macroeconomic influences on bank failure. Thomson used a two-step logit model, and a sample of FDIC-insured failed and healthy banks. He tried to capture the effects of macroeconomic variables using state gross domestic product, state-level personal income, and county-level employment data. He found a number of variables to be statistically significant with the right sign. These were a proxy for net worth or solvency, non-deposit liabilities/(cash + securities), overheads/total assets, net income after taxes/total assets, loans to insiders/total assets, dummies for unit banking state and bank holding companies, the log of total assets, and the log of average deposits per banking office.

Espahbodi's (1991) data consisted of 48 banks23 that failed in 1983, matched with another 48 healthy banks. The matching was based on the FDIC's membership status, bank size and geographical location. He finds two sources of fund measures (the ratio of loan revenue to total income and interest income on US Treasury securities to total operating income), a use of funds variable (interest paid on deposits/total operating income) and a deposit composition (total time and savings operations/total demand deposits) measure to be significant and right-signed. However, various measures of liquidity, capital adequacy (total loans/total equity capital), efficiency (total operating expenses/total operating income), loan quality (reserve for possible loan losses/total loans) and loan volume were found to be insignificant as explanations for why banks fail.

Martin's (1977) data set consisted of 58 US banks that failed between 1970 and 1976, and different combinations of 25 financial ratios, classified into four groups: asset risk, liquidity, capital adequacy and earnings. In his best regression, using 1974 data, he found measures of profitability (net income/total assets), asset quality (commercial loans/total loans and gross charge-offs/net operating income), and capital adequacy (gross capital to risky assets) to be significant and right-signed.

Avery and Hanweck (1984) report results from one of the largest samples employing a logistic function. They used 100 failed US banks from the 1979-83 period, and for non-failed banks, a 10 per cent random sample of all insured commercial banks, totalling 1,190 banks. They find earnings after taxes on assets, the ratios of capital to assets and loans to assets, and a bank's percentage of industrial and commercial loans to be statistically significant variables in explaining bank failure. The higher the percentage of commercial and industrial loans, the greater the probability of failure. Likewise, Barth et al. (1985) and Benston (1985), employing US data, report various financial ratios as significant variables in explaining bank failure.

Barker and Holdsworth (1994), in an unpublished research paper, conducted an extensive study of 859 failed and 12,364 non-failed institutions for the 1986-91 period using six logit specifications. They divided the possible causes of bank failure into three categories: risk taking, mismanagement and fraud, and external and systemic factors. There was some statistical support for all three categories, but the best explanation of bank failure was risk taking by banks, measured by concentration of loans in specific areas (namely, real estate). Mismanagement and systemic factors (such as local real estate market conditions) contributed to bank failure. The authors attempted to measure the extent to which fraud contributes to bank failure, but their proxies, loans to insiders (significant and positive) and the ratio of non-interest expenses to total assets (significant and negative - 'wrong-signed', according to the authors) are not necessarily indications of fraud. Non-interest expenses could be interpreted as a sign of diversification and, therefore, may have the right sign.

Heffernan (1995) is the first published work to employ an international database and to estimate a conditional logit model and the more standard, multinomial logit model. The fact that several countries are included made it possible to test for macroeconomic variables, in addition to financial ratios and bank-size variables.

This global study of the causes of bank failure was made possible by access to financial ratio data for banks reporting to IBCA, a ratings firm based in New York and London.24 Since this firm is rating banks from around the world, one of its principal aims is to ensure that the financial ratio data reported to its clients are comparable across banks.

The countries included in the sample were Australia, Finland, France, Norway, Sweden and the United States. All had financial ratio data on healthy and failed banks. Other variables tested were various measures of bank size and a number of macroeconomic indicators.

The potential for multicollinearity problems, arising from lack of independence among the variables, made a stepwise (forward and backward) procedure appropriate. Variables were dropped from or added to the model one at a time based on their contribution to the overall fit of the model, as measured by pseudo or McFadden R2 (MR2).25

The period of estimation ran from 1989 to 1992. The sample was pooled across countries and a total of 88 regressions were run, but of these, only 28 were worth reporting, based on the size of the MR2 and the significance of coefficients as measured by ¿-ratios. The number of observations for these 28 regressions ranged from 155 to 20 5.26 Even among the 28 equations, the MR2 ranged from 0.23 to 0.784.

The key findings of the 1995 paper are as follows. The financial ratio, net income/total assets, a measure of profitability, was available for most of the banks in the sample. It is correctly signed (that is, as profitability rises, the probability of failure falls) and statistically significant at the 99 or 95 per cent confidence levels in 11 of the 15 regressions which tested this variable. Other measures of profitability, such as net interest revenue/total assets or the ratio of net income to equity, were correctly signed but either insignificant or weakly significant.

The risk-weighted capital assets ratios were, during this period, reported by only a few banks in the sample, and for this reason, could not be tested. However, the ratio of equity to total assets is widely reported. Any measure of capital adequacy should have a negatively signed coefficient. Of the 26 regressions testing this variable, six were wrong-signed but insignificant, and, of the 21 remaining regressions, eight were statistically significant and correctly signed at the 99 or 95 per cent confidence levels. The other financial ratios tested, including measures of liquidity and internal capital generation (retained profit), yielded inconclusive results.

Four measures of bank size were tested, all involving the banks' assets, measured in US dollars. Only one, the log of (bank i's assets in US\$/US nominal GDP) had any explanatory power. In the nine regressions tested, four had negatively signed coefficients significant at the 99 or 95 per cent confidence levels. One regression had a positively signed coefficient (significant at 0.95) and the other four (two of which had positive signs; two negative) were insignificant. This would suggest that the larger the bank, the lower the probability of failure, which may be due to 'too-big-to-fail' intervention by the authorities, resulting in closer regulatory scrutiny of these banks compared to smaller ones. Or it may indicate that larger banks have a lower probability of failure because their portfolios are more diversified.

Five macroeconomic variables were tested for significance, including the growth rate of real GDP, the nominal and real effective exchange rates, nominal and real interest rates, a price index and the rate of inflation. The coefficients on all the variables had the expected sign,27 and except for real GDP growth rate, were significant, at the very minimum, at the 90 per cent level of confidence. The coefficient on the real interest rate variable was positively signed and significant at the 95 per cent level of confidence, but in only one of the nine regressions testing this variable. Of the eight regressions testing the nominal interest rate, two were positively signed but insignificant; the coefficients on the other five were significantly positive at the 95 per cent level of confidence. From these tests, we can conclude that the state of the macro economy can often contribute to bank failures, and the nominal interest rate in particular is an important explanatory variable: as interest rates rise, the probability of bank failure also increases.

The IBCA individual rating for each bank ranges from 5 (A - excellent) to 1 (E - the bank has a serious problem and is likely to require external support). The firm rates a bank based on financial ratios, size, macro-economic variables (to a lesser extent), and on the knowledge of its field experts. To avoid problems of multicollinearity, the model was run in a separate regression, and the coefficient on the rating variable was significantly negative at the 99 per cent level of confidence, with a very respectable McFadden R2 of 0.56.

Multinomial logit regressions were run on the same pool of banks. However, the overall performance of the model was poor compared to the conditional logit model, with MR2s ranging from 0.16 to 0.32, though the significance of the coefficients on the individual explanatory variables was similar, and the constant term significant at confidence levels of 99 or 95 per cent.

An updated version of the above models was recently tested by this author.28 The number of countries included in the data set was increased to include Spain, an expanded data set for France, Italy and the United Kingdom, with the period of estimation running from 1982 to 1995. Table 14A.1 in the appendix lists the healthy and failed banks used in the sample. Unlike many studies, there was no pairwise grouping of data (matching a healthy bank with a failed one) because of the possibility of sample bias, given the low frequency of bank failure. The criterion for including banks was based on the availability of financial ratio data from Fitch IBCA.

Rather than conducting logit regressions on every combination of variables as was done in the 1995 paper, the variables found to be significant or near-significant in that study were tested. These included internal capital generation, three measures of profitability, capital adequacy, liquidity and bank size. It was also possible to include a loan-loss measure: the ratio of loan-loss reserves to total loans. The macroeconomic variables include effective exchange rates and interest rates, real and nominal. The definitions for these variables may be found in Table 14.1. The Fitch IBCA rating was also tested in a separate regression.

A total of 21 regressions were run for the different combination of variables, using both multinomial and conditional (for panel data) logit models. With an updated version of the software package, it was possible to employ a more powerful test of model superiority. The Hausman29 test was devel-

 Table 14.1 Definitions of explanatory variables tested Variable Definition INT K GEN Internal capital generation, or an increase in bank equity from retained profit PROFIT1 Net income/total assets (average) PROFIT2 Net interest revenue/total assets (average) PROFIT3 Net income/equity (average) CAP ADEQ Equity/total assets (average) LIQUIDITY Liquid assets/(customer + short-term funding) LLOSS Loan-loss reserves/loans SIZE Log of US\$ assets for bank i/US nominal GDP RINT Average annual real interest rate for country i* NINT Average nominal real interest rate for country i REE Average annual real effective exchange rate for country i: as the real exchange rate rises, the real value of the home currency rises NEE Nominal effective exchange rate INF Annual inflation rate for country i IND Annual consumer price index for country i RGDP Annual real GDP growth rate for country i RA IBCA rating for bank i

Note: *Annual average computed from monthly market rate.

Note: *Annual average computed from monthly market rate.

Sources: Financial ratios (all data are year-end, unless otherwise stated), size variable, and RA: Fitch IBCA; macroeconomic variables: IMF, International Financial Statistics, various years.

oped to test for specification error, and can be used to compare a given model with a hypothesized alternative. If there is no misspecification, there exists a consistent and asymptotically efficient estimator, but that estimator is biased and inconsistent if the model is misspecified.

In the conditional panel logit model, recall that the a; are different for the failed and healthy bank groups. The a; cannot be identified but are common to banks that fail, for example, rogue traders or bank managers looting a bank before it fails. The Hausman test, applied here, is asking whether the as for the failed and healthy groups are the same or not. They are homogeneous if the same, heterogeneous if substantially different. The Hausman test discriminates between these two hypotheses.

Of the 21 regressions, there was only one where the conditional logit for panel data model could be accepted as superior with a 99 per cent level of confidence, thereby rejecting the hypothesis of homogeneous as in this one case.30 The results are reported in Table 14.2.

As can be seen from Table 14.2, while the Hausman test confirms the panel logit regression to be superior, all the coefficients are rendered insig-

Table 14.2 Panel logit model accepted (using the Hausman test) as superior to its multinomial logit counterpart

Independent variable

Conditional logit

Multinomial logit

Coefficient (t-ratio)

Coefficient (t-ratio)

Constant

n.a.

2.45 (2.00**)

INT K GEN

-0.0222 (-0.679)

0.0138 (2.21**)

PROFIT1

-4.31 (-1.84*)

-0.960 (4.26)

-0.876 (0.71)

-0.183 (1.44*)

LIQUIDITY

-0.11 (-0.77)

-0.560 (3.50)

LLOSS

0.0335 (0.06)

-0.0403 (-0.53)

SIZE

16.2 (1.71*)

-0.351 (1.60*)

INF

0.400 (0.70)

0.258 (2.54)

Hausman/MR2

Hausman: 49.33 (7d.f.)

MR2: 0.500

Observations

335

p = 1 (bank failure); p = 0 (healthy bank). Abbreviations: See Table 14.1. Levels of significance:

Bold: significant at (at least) 99% level of confidence. "significant at 97.5% level of confidence.

*significant at 90% level of confidence, except profit measure, which is significant at 95% level of confidence.

Hausman = 49.33, which, with 7 degrees of freedom, is significant at the 99.95% level of confidence, using the cumulative x-square distribution.

Notes:

p = 1 (bank failure); p = 0 (healthy bank). Abbreviations: See Table 14.1. Levels of significance:

Bold: significant at (at least) 99% level of confidence. "significant at 97.5% level of confidence.

*significant at 90% level of confidence, except profit measure, which is significant at 95% level of confidence.

Hausman = 49.33, which, with 7 degrees of freedom, is significant at the 99.95% level of confidence, using the cumulative x-square distribution.

nificant, with the exception of the profitability and bank size coefficients, which are significant at the 95 per cent level of confidence. The sign on the coefficient for the size variable changes, from negative to positive, suggesting the probability of failure rises with bank size. Also, the significance level for profitability drops, compared to a 99.95 per cent significance level in the multinomial logit model. The internal capital generation (INT K GEN), capital adequacy, and loan-loss variables are all wrong-signed.

Only one of the 21 regressions passes the Hausman test with a high level of confidence, and, at the same time, most of the coefficients on the explanatory variables are insignificant, sometimes with the wrong signs, with the exception of profitability variable. The implication to be drawn is that the hypothesis of homogeneity of the as cannot be rejected, and therefore the conditional panel logit model is not generally superior to the multinomial logit model.

This finding contradicts previous (1995) results but they were obtained with less sophisticated software (which relied upon a comparison of the MR2 because the Hausman test was not available) and a smaller sample size.

The details of the results of the 21 multinomial regressions are reported in the working paper, but a summary of the findings appears in Table 14.3. It is sensible to concentrate on the regressions run using PROFIT1, defined as the ratio of net income to total assets, with the other financial ratios (capital adequacy, liquidity, loan losses), bank size and the sequence of macroeconomic variables listed in Table 14.1. The results of the first seven regressions are reported in Table 14.3. In terms of MR2 (with one exception), measuring the overall fit of the model and the significance of PROFIT1 (as compared to the other two measures of profit employed in the subsequent 14 regressions), it is quite clear that the regressions employing the PROFIT1 variable were superior.

The coefficient on the ratio of net income to total assets (PROFIT1) is very highly significant (at the 99.95 per cent level of confidence), with the expected negative sign. The same is true of the internal capital generation variable (the use of retained profits to increase bank equity) and liquidity. The coefficient on loan losses was negative and nearly significant at the 95 per cent confidence level in the first seven equations. At first this result may appear counterintuitive, but it may be that banks which set aside reserves are explicitly acknowledging loan-loss problems and take appropriate action, thereby avoiding failure.31 The measure for bank size, the rate of growth of the bank, had a significantly negative coefficient (95 per cent level of confidence) in only two equations, suggesting that as bank size increases, the probability of failure falls.

The inflation and real exchange rate coefficients were, in their respective