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J. Risk Financial Manag. 2014, 7(4), 150-164;

Exact Fit of Simple Finite Mixture Models

Prudential Regulation Authority, Bank of England, 20 Moorgate, London EC2R 6DA, UK
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
This paper is an extended version of our paper published in Fifth International Conference on Mathematics in Finance (MiF) 2014, organized by North-West University, University of Cape Town and University of Johannesburg, 24–29 September 2014, Skukuza, Kruger National Park, South Africa.
The opinions expressed in this note are those of the author and do not necessarily reflect views of the Bank of England.
Received: 6 September 2014 / Revised: 7 October 2014 / Accepted: 4 November 2014 / Published: 20 November 2014
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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. View Full-Text
Keywords: quantification; prior class probability; probability of default quantification; prior class probability; probability of default
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Tasche, D. Exact Fit of Simple Finite Mixture Models. J. Risk Financial Manag. 2014, 7, 150-164.

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