Abstract
How to forecast next year’s portfolio-wide credit default rate based on last year’s default observations and the current score distribution? A classical approach to this problem consists of fitting a mixture of the conditional score distributions observed last year to the current score distribution. This is a special (simple) case of a finite mixture model where the mixture components are fixed and only the weights of the components are estimated. The optimum weights provide a forecast of next year’s portfolio-wide default rate. We point out that the maximum-likelihood (ML) approach to fitting the mixture distribution not only gives an optimum but even an exact fit if we allow the mixture components to vary but keep their density ratio fixed. From this observation we can conclude that the standard default rate forecast based on last year’s conditional default rates will always be located between last year’s portfolio-wide default rate and the ML forecast for next year. As an application example, cost quantification is then discussed. We also discuss how the mixture model based estimation methods can be used to forecast total loss. This involves the reinterpretation of an individual classification problem as a collective quantification problem.
Highlights
The study of finite mixture models was initiated in the 1890s by Karl Pearson when he wanted to model multimodal densities
We explore a specific property of simple finite mixture models, namely that their maximum likelihood (ML) estimates provide an exact fit of the observed densities if the estimates exist
We have suggested a Gauss–Seidel-type approach to the calculation of the Kullback–Leibler distance estimator that triggers an alarm if there is no solution with all components positive
Summary
The study of finite mixture models was initiated in the 1890s by Karl Pearson when he wanted to model multimodal densities. More recently the natural connection between finite mixture models and classification methods with their applications in fields like machine learning or credit scoring began to be investigated in more detail. In these applications, often it can be assumed that the mixture models are simple in the sense that the component densities are known (i.e., there is no dependence on unknown parameters) but their weights are unknown. A practical consequence of exact fit is an inequality for the finite mixture model estimate of the class probabilities in a binary classification problem and the so-called covariate shift estimate (see Corollary 7 below for details).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
