  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.
