Bond rating Transition Probability Matrices (TPMs) are built over a one-year time-frame and for many practical purposes, like the assessment of risk in portfolios  , one needs to compute the TPM for a smaller time interval. In the context of continuous time Markov chains (CTMC) several deterministic and statistical algorithms have been proposed to estimate the generator matrix. We focus on the Expectation-Maximization (EM) algorithm by \mbox{%DIFAUXCMD BladtSorensen2005 for a CTMC with an absorbing state for such estimation. This work's contribution is fourfold . Firstly, we provide directly computable closed form expressions for quantities appearing in the EM algorithm  . Previously, these quantities had to be estimated numerically and considerable computational speedups have been gained. Secondly, we prove convergence to a single set of parameters under reasonable conditions . Thirdly, we derive a closed-form expression for the error estimate in the EM algorithm allowing to approximate confidence intervals for the estimation . Finally, we provide a numerical benchmark of our results against other known algorithms, in particular, on several problems related to credit risk. The EM algorithm we propose, padded with the new formulas (and error criteria), is very competitive and outperforms other known algorithms in several metrics  .
Bond rating Transition Probability Matrices (TPMs) are built over a one-year time-frame and for many practical purposes, like the assessment of risk in portfolios or the computation of banking Capital Requirements (e.g. the new IFRS 9 regulation) , one needs to compute the TPM and probabilities of default over a smaller time interval. In the context of continuous time Markov chains (CTMC) several deterministic and statistical algorithms have been proposed to estimate the generator matrix. We focus on the Expectation-Maximization (EM) algorithm by Bladt and Sorensen (2005) for a CTMC with an absorbing state for such estimation. This work's contribution is threefold . Firstly, we provide directly computable closed-form expressions for quantities appearing in the EM algorithm and associated information matrix, allowing to easily approximate confidence intervals . Previously, these quantities had to be estimated numerically and considerable computational speedups have been gained. Secondly, we prove convergence to a single set of parameters under very weak conditions (for the TPM problem) . Finally, we provide a numerical benchmark of our results against other known algorithms, in particular, on several problems related to credit risk. The EM algorithm we propose, padded with the new formulas (and error criteria), outperforms other known algorithms in several metrics , in particular, with much less overestimation of probabilities of default in higher ratings than other statistical algorithms .
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or the computation of banking Capital Requirements (e.g. the new IFRS 9 regulation)
R
for
and probabilities of default over
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\mbox{%DIFAUXCMD BladtSorensen2005
Bladt and Sorensen (2005)
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fourfold
threefold
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closed form
closed-form
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and associated information matrix, allowing to easily approximate confidence intervals
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reasonable conditions . Thirdly, we derive a closed-form expression for the error estimate in the EM algorithm allowing to approximate confidence intervals for the estimation
very weak conditions (for the TPM problem)
D
is very competitive and
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, in particular, with much less overestimation of probabilities of default in higher ratings than other statistical algorithms