Abstract
The paper discusses methodological topics of bankruptcy prediction modelling—unbalanced sampling, sample bias, and unbiased predictions of bankruptcy. Bankruptcy models are typically estimated with the use of non-random samples, which creates sample choice biases. We consider two types of unbalanced samples: (a) when bankrupt and non-bankrupt companies enter the sample in unequal numbers; and (b) when sample composition allows for different ratios of bankrupt and non-bankrupt companies than those in the population. An imbalance of type (b), being more general, is examined in several sections of the paper. We offer an extended view of the relationship between the biased and unbiased estimated probabilities of bankruptcy—probability of default (PD). A common error in applications is neglecting the possibility of calibrating the PD obtained from a bankruptcy model to the unbiased PD that is population adjusted. We show that Skogsviks’ formula of 2013 coincides with prior correction known for the logit model. This, together with solutions for other binomial models, serves as practical advice for obtaining the calibration of unbiased PDs from popular bankruptcy models. In the final section, we explore sample bias effects on classification.
Highlights
Bankruptcy probability—or probability of default (PD)—is of interest to many participants in and observers of corporate financial markets
The purpose of this paper is to explore why bankruptcy models estimated with the use of unbalanced samples generate bankruptcy probabilities that are biased and to recommend techniques for their calibration to unbiased probabilities
The relevance of prior correction to calibrating the unbiased PD from estimated binomial models is shown in Section 7, which is preceded by Section 6 introducing the formula by Skogsvik and Skogsvik (2013)
Summary
Bankruptcy probability—or probability of default (PD)—is of interest to many participants in and observers of corporate financial markets. The purpose of this paper is to explore why bankruptcy models estimated with the use of unbalanced samples generate bankruptcy probabilities that are biased and to recommend techniques for their calibration to unbiased probabilities. The relevance of prior correction to calibrating the unbiased PD from estimated binomial models is shown, which is preceded by Section 6 introducing the formula by Skogsvik and Skogsvik (2013). This formula represents the relationship between the biased probability of bankruptcy obtained from the model and the population-adjusted unbiased probability of bankruptcy. The entire paper puts aside the classification problem and emphasizes questions of PD prediction that are the subject of growing demand from analysts and practitioners in accounting and corporate finance
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