Assessing the shape of the distribution of interest rates : lessons from French individual data

This paper considers the problem of fitting a finite Gaussian mixture with an unknown number of components to interest rates provided by a sample of credit institutions. Indeed, mixture models are able to represent arbitrarily complex probability density functions distributions, and hence outperform the usual non parametric kernel density estimates in several ways. Therefore, this tool enables us to improve our understanding of the credit market. The data are extracted from individual contracts and gathered on a quarterly basis. They provide a loan-by-loan disaggregation of average rates at bank level available in the monthly MIR reports. Loans to non-financial firms and to households are splitted up into eleven categories defined by instrument and maturity of loans. Each category is analyzed separately. Two major econometric issues are dealt with : how to estimate the number of components, and how to estimate the parameters defining the model. Several techniques are compared, which allows us to show how the results may be sensitive to the methodology used, especially when sample survey issues are properly taken into account : for instance, the model is enhanced in order to include discrete components which arise from clusters in the distribution. In a second step, this set-up allows us to estimate precisely the modes of the distribution. Then, these modes are interpreted as resulting from the segmentation of the credit market. We illustrate this topic through a kind of scoring of credit institutions based on ex-post probabilities for loans to be in the neighborhood of a particular mode. Special attention is also paid to the upper side of the distribution, where the usury law may affect the level of interest rates. The effect of the usury threshold on the distribution is first analyzed from a theoretical perspective, through simulations using loans which are not subject to the usury law. Then, for categories defined by the usury law, we provide a rough estimate of the proportion of loans which are not granted because of the threshold effect. Finally, introducing the time series dimension shows that the the shape of the distributions (number and amplitude of modes) is generally not constant over time. This instability raises another interesting issue, especially for short term analysis purposes. This is left for future research. ∗Monetary Studies and Statistics Directorate, Statistical Engineering and Coordination Division. I am grateful to Olivier Cousseran, Daniel Gabrielli, Emmanuel Gervais and Hervé Le Bihan for their helpful comments on previous drafts of this paper, and Frédérique Edmond and Bertille Mader for making all the data easy available to me. The views expressed in this paper are those of the author and do not necessarily reflect those of the Banque de France.