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.
Highlights
Numerous methods exist for embedding a given discrete-time transition matrix into continuous-time, i.e., for finding the generator matrix
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
Our research mainly follows the work of [1], where the proposed method is based on optimization. They suggest to avoid solving the true problem of interest, the best approximation of the annual transition matrix (BAM), but instead solving an approximation only, called quasi-optimization of the generator (QOG)
Summary
Numerous methods exist for embedding a given discrete-time (in most cases, one-year) transition matrix into continuous-time, i.e., for finding the (infinitesimal) generator matrix. Our research mainly follows the work of [1], where the proposed method is based on optimization They suggest to avoid solving the true problem of interest, the best approximation of the annual transition matrix (BAM), but instead solving an approximation only, called quasi-optimization of the generator (QOG). These checks are conducted on commonly-used one-year transition matrices in credit risk-relevant assessments, which are published on a regular basis by rating agencies.
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