# Observed rating migration matrix to derive the generator matrix

I am doing some reading on the derivation of credit rating migration/transition matrices and probability of default term structures. I understand that a homogeneous Markov chain can be either discrete-time or continuous-time.

In the discrete-time case, the one year transition matrix can be derived using the cohort method which involves computing the proportion of observed migrations from the beginning of a year to the end of a year.

In the continuous-time case, we need to derive a generator matrix G. I am not sure I understood it correctly, but the generator matrix needs to be derived from an observed transition matrix, correct? If yes, what is the best way of obtaining the observed transition matrix? Simply the cohort method (and afterwards some smoothing etcetera to ensure that the generator matrix exists) or should I take into account all rating migrations during the one year time interval?

## 1 Answer

In case an answer is still useful for you after 11 months -> No, a generator matrix can be directly derived from observed rating transition data from which a transition probability matrix can be derived using the matrix exponential. For continuous-time Markov chains direct MLE is possible (and best) for discrete, you can look into applying an EM algortihm (also considered the best). Trying to find a generator matrix from a transition probability matrix makes you run into several mathematical problems (embeddability, identification etc.), thus you should try to coordinate the generator matrix unto rating transition data directly.

• Hi @user297530, your reply is still usefull (I did not know about the EM algorithm). Thank you! Jan 3, 2022 at 9:16
• The EM algortihm is not as easy to implement as other methods, but does allow for exact confidence intervals for the generator matrix and transition probability matrix. Useful article: Robust and consistent estimation of generators in credit risk - dos Reis
– user60544
Jan 3, 2022 at 13:18