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Choosing Markovian Credit Migration Matrices by Nonlinear Optimization

Author

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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)

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.

Suggested Citation

  • Maximilian Hughes & Ralf Werner, 2016. "Choosing Markovian Credit Migration Matrices by Nonlinear Optimization," Risks, MDPI, vol. 4(3), pages 1-18, August.
  • Handle: RePEc:gam:jrisks:v:4:y:2016:i:3:p:31-:d:76960
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    References listed on IDEAS

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    1. Robert A. Jarrow & David Lando & Stuart M. Turnbull, 2008. "A Markov Model for the Term Structure of Credit Risk Spreads," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 18, pages 411-453, World Scientific Publishing Co. Pte. Ltd..
    2. Robert B. Israel & Jeffrey S. Rosenthal & Jason Z. Wei, 2001. "Finding Generators for Markov Chains via Empirical Transition Matrices, with Applications to Credit Ratings," Mathematical Finance, Wiley Blackwell, vol. 11(2), pages 245-265, April.
    3. Yasunari Inamura, 2006. "Estimating Continuous Time Transition Matrices From Discretely Observed Data," Bank of Japan Working Paper Series 06-E-7, Bank of Japan.
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    Cited by:

    1. Linda Möstel & Marius Pfeuffer & Matthias Fischer, 2020. "Statistical inference for Markov chains with applications to credit risk," Computational Statistics, Springer, vol. 35(4), pages 1659-1684, December.
    2. Tamás Kristóf, 2021. "Sovereign Default Forecasting in the Era of the COVID-19 Crisis," JRFM, MDPI, vol. 14(10), pages 1-24, October.

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