  We present two methodologies on the estimation of rating transition
probabilities within Markov and non-Markov frameworks. We first estimate a
continuous-time Markov chain using discrete (missing) data and derive a simpler
expression for the Fisher information matrix, reducing the computational time
needed for the Wald confidence interval by a factor of a half. We provide an
efficient procedure for transferring such uncertainties from the generator
matrix of the Markov chain to the corresponding rating migration probabilities
and, crucially, default probabilities.
  For our second contribution, we assume access to the full (continuous) data
set and propose a tractable and parsimonious self-exciting marked point
processes model able to capture the non-Markovian effect of rating momentum.
Compared to the Markov model, the non-Markov model yields higher probabilities
of default in the investment grades, but also lower default probabilities in
some speculative grades. Both findings agree with empirical observations and
have clear practical implications.
  We illustrate all methods using data from Moody's proprietary corporate
credit ratings data set. Implementations are available in the R package ctmcd.
