  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.
