Next Article in Journal
The Wasserstein Metric and Robustness in Risk Management
Next Article in Special Issue
Nested MC-Based Risk Measurement of Complex Portfolios: Acceleration and Energy Efficiency
Previous Article in Journal
On the Capital Allocation Problem for a New Coherent Risk Measure in Collective Risk Theory
Previous Article in Special Issue
Lead–Lag Relationship Using a Stop-and-Reverse-MinMax Process
Article Menu

Export Article

Open AccessFeature PaperArticle
Risks 2016, 4(3), 31; doi:10.3390/risks4030031

Choosing Markovian Credit Migration Matrices by Nonlinear Optimization

1
Fakultät für Informatik und Mathematik, Hochschule München, Lothstrasse 64, 80335 München, Germany
2
Institut für Mathematik, Universität Augsburg, Universitätsstrasse 2, 86159 Augsburg, Germany
*
Author to whom correspondence should be addressed.
Academic Editor: Alexander Szimayer
Received: 4 July 2016 / Revised: 12 August 2016 / Accepted: 22 August 2016 / Published: 30 August 2016
(This article belongs to the Special Issue Applying Stochastic Models in Practice: Empirics and Numerics)
View Full-Text   |   Download PDF [929 KB, uploaded 30 August 2016]   |  

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. View Full-Text
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 credit risk; embedding problem; transition matrix; generator matrix; homogenization; best approximation of the annual transition matrix; quasi-optimization of the generator; constraints; optimization; regularization
Figures

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).

Scifeed alert for new publications

Never miss any articles matching your research from any publisher
  • Get alerts for new papers matching your research
  • Find out the new papers from selected authors
  • Updated daily for 49'000+ journals and 6000+ publishers
  • Define your Scifeed now

SciFeed Share & Cite This Article

MDPI and ACS Style

Hughes, M.; Werner, R. Choosing Markovian Credit Migration Matrices by Nonlinear Optimization. Risks 2016, 4, 31.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Risks EISSN 2227-9091 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top