Investigating Divergence Measures with Credit Risk Models

A divergence measure is a tool that applied to many fields. In particular, with the advancement of data-dependent modeling, it becomes one of the main tools in distributional analysis. In the financial context, it is applicable to see whether there is a shift in the estimated default probabilities over time and underlying explanatory features. Given the volatility of the financial data, it is almost inevitable to employ the divergence measure to rely on the model output for a specific period. Aside from how and where to apply divergence measures, it is of great importance to pick the correct divergence measure to apply. Out of the divergence measures, the Popula- tion Stability Index (PSI) has been the most widely used measure; however, other divergence measures, namely Jensen-Shannon divergence (JSD) and Kullback-Liebler Divergence (KLD), has also been employed in the literature. In this study, we want to evaluate these divergence measures and extend the work of (1) in terms of their statistical properties and applicability. Thus, the aim of this study is twofold. First, we derive the statistical properties of JSD and KLD, including distribution and bench- mark chi-square distribution values, and apply these measures to detect if there is any distributional change in the default probabilities obtained from Merton, Merton with the jump, and stochastic volatility with jump models. It is shown that JSD and KLD, with PSI, follow chi-square distribution and it also turns out that JSD produces more stable results, particularly in the presence of a low sample size