Exact fit of simple finite mixture models

TL;DR

This paper demonstrates that a simple finite mixture model can exactly fit the current score distribution and provide optimal default rate forecasts, improving credit risk predictions.

Contribution

It shows that the maximum-likelihood approach yields an exact fit for simple finite mixture models with fixed component ratios, enhancing default rate forecasting methods.

Findings

ML approach provides an exact fit under certain conditions

Forecasts are bounded between last year's default rate and ML forecast

Mixture models can be used for total loss prediction

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 fix. 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, then cost quantification is 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.