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The principle of scoring is to use a metric for dividing good and bad credits into distinct distributions. The statistical technique is the standard discriminant analysis.

Fisher Discriminant Analysis

Discriminant analysis distinguishes statistically between two or more groups of unit observations (firms or individuals). In the case of credit risk, the technique serves to discriminate between firms likely to default and those not likely to default, up to a given horizon, using current and past values of observable attributes such as financial ratios. Discriminant analysis attempts to do this by using characteristics that differ between groups. For credit risk, such variables include profitability, leverage, size and others. For individuals, income, age and professional activities relate to their credit standing.

The technique forms a linear combination of all discriminating variables to obtain a discriminant function. The discriminant function values are the scores. Discriminant analysis works by fitting the function to data, with observations in all classes that we wish to distinguish, such as defaulters versus non-defaulters. Fitting the function to the data also defines cut-off values of the score functions. The model works by calculating the score, comparing its value to cut-off values, and assigning a class based on these cut-off values provided by the model. For instance, firms with high scores could be classified as non-defaulters and conversely. Classification proceeds with functions that generate the probability of being in a given group based on the score value. To illustrate the principle, we limit the presentation to the two-group case, defaulting and non-defaulting firms, and to Fischers linear discriminant function.

The discriminant function produces a standardized Z score, with mean 0 and standard deviation 1. Averaging each score for one group (such as non-defaulters) provides a mean called the group centroid. The coefficient of the variables of the discriminant function measures the relative contribution of the coefficient to the function. Plotting the number of cases having scores within predefined ranges of values generates the histogram of scores. By plotting scores separately for each group (for instance, defaulters and non-defaulters) we visualize how well the scores differentiate the groups and to what extent they overlap. The wider the gap between the groups means and the lower the overlap, the better the model performs.

A classification function serves to assign, or classify, a case to a group. A case is predicted to be a member of the group for which the value of its classification function is the highest. For only two groups, the discriminant function is simply the difference between the classification functions for each of the two groups. The score of the Fisher discriminant function relates the score to observable attributes:

For every individual, a numerical value of the score results from observed attributes and scores compared to a cut-off value allowing us to assign the firm or individual to one of the two groups. If the value is high, the credit is good, and the higher the score, the higher the quality for the credit. A bad score indicates a lower quality credit.

The outputs of the procedure are the coefficients of the classification function and cut-off values. The coefficients help in interpreting which coefficients best explain the classification, and the posterior probabilities of belonging to a given group. The cut-off values serve to assign a new individual to a group, based on the score value. Posterior probabilities are conditional probabilities of defaulting conditional on the score value. Prior probabilities are default probabilities without any information. For instance, when using a data set of defaulters and non-defaulters, the ratio of defaulters over the total is the prior probability. Since the score value provides information on likelihood of default, the posterior probabilities embed this additional information. Lets assume that high scores relate to bad credit standing. When the company attributes result in a high score value, the posterior probability is higher and vice versa.

Scoring does not use a conceptual framework, such as KMV Credit Monitor, which implements the Merton model. It simply fits a function that best discriminates between high risk and low risk populations. The fit might necessitate repeated calibration to account for changing conditions, although the attributes might also capture the changing conditions.

Therefore, scoring potentially discriminates firms according to their credit standing, and can serve both as a default predictor and as a rating device. For firms, scoring models based on a small number of indicators, such as accounting ratios, have been successful. There are a number of classical studies explaining scoring systems and providing numerical results.