Exact fit of simple finite mixture models

摘要

How to forecast next year's portfolio-wide credit default rate based on lastyear's default observations and the current score distribution? A classicalapproach to this problem consists of fitting a mixture of the conditional scoredistributions observed last year to the current score distribution. This is aspecial (simple) case of a finite mixture model where the mixture componentsare fixed and only the weights of the components are estimated. The optimumweights provide a forecast of next year's portfolio-wide default rate. We pointout that the maximum-likelihood (ML) approach to fitting the mixturedistribution not only gives an optimum but even an exact fit if we allow themixture components to vary but keep their density ratio fix. From thisobservation we can conclude that the standard default rate forecast based onlast year's conditional default rates will always be located between lastyear's portfolio-wide default rate and the ML forecast for next year. As anapplication example, then cost quantification is discussed. We also discuss howthe mixture model based estimation methods can be used to forecast total loss.This involves the reinterpretation of an individual classification problem as acollective quantification problem.