# Bank Failure: Quantitative Models - Modern Banking

Qualitative reviews of bank failure provide some insight into what causes a bank to fail, but these ideas need to 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 (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.

Much 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 discriminantor 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, most of the studies reported in Table use the multinomial logit model.

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:

z = β0 + β'x +ε -------(XVIII)

where
p = 1 if z >0
p = 0 if z ≤0
z = log[p/(1 −p)]
β0 : a constant term
β': the vector of coefficients on the explanatory variables
x : the vector of explanatory variables
ε : the error term

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

Readers who are unfamiliar with logit analysis will find it explained in any good textbook on introductory econometrics. However, an intuitive idea can be obtained by referring to Figure. In a simple application of equation (XVIII), if x consists of just one explanatory variable (e.g. capital adequacy), the logit model becomes a two-dimensional sigmoid shaped curve, as shown. The probability of failure is on the vertical axis and the explanatory variable, in this case capital adequacy, is on the horizontal axis. Recall, in the logit model, that a bank either fails (p = 1) or it does not (p = 0). As the bank’s capital adequacy (which could be measured in a number of ways) falls (approaching 0 in Figure), the probability of bank failure rises. Note the difference between the logistic curve and the straight line of a standard least squares regression.

A potential problem arises from 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:

z = αi + β'x +ε

The logit model Modern Banking Topics