Choosing Markovian Credit Migration Matrices by Nonlinear Optimization

Maximilian Hughes and Ralf Werner
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Maximilian Hughes: Fakultät für Informatik und Mathematik, Hochschule München, Lothstrasse 64, 80335 München, Germany
Ralf Werner: Institut für Mathematik, Universität Augsburg, Universitätsstrasse 2, 86159 Augsburg, Germany

Risks, 2016, vol. 4, issue 3, 1-18

Abstract: Transition matrices, containing credit risk information in the form of ratings based on discrete observations, are published annually by rating agencies. A substantial issue arises, as for higher rating classes practically no defaults are observed yielding default probabilities of zero. This does not always reflect reality. To circumvent this shortcoming, estimation techniques in continuous-time can be applied. However, raw default data may not be available at all or not in the desired granularity, leaving the practitioner to rely on given one-year transition matrices. Then, it becomes necessary to transform the one-year transition matrix to a generator matrix. This is known as the embedding problem and can be formulated as a nonlinear optimization problem, minimizing the distance between the exponential of a potential generator matrix and the annual transition matrix. So far, in credit risk-related literature, solving this problem directly has been avoided, but approximations have been preferred instead. In this paper, we show that this problem can be solved numerically with sufficient accuracy, thus rendering approximations unnecessary. Our direct approach via nonlinear optimization allows one to consider further credit risk-relevant constraints. We demonstrate that it is thus possible to choose a proper generator matrix with additional structural properties.

Keywords: credit risk; embedding problem; transition matrix; generator matrix; homogenization; best approximation of the annual transition matrix; quasi-optimization of the generator; constraints; optimization; regularization (search for similar items in EconPapers)
JEL-codes: C G0 G1 G2 G3 K2 M2 M4 (search for similar items in EconPapers)
Date: 2016
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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