Suppose we know the observed transition matrix for annual migrations between credit ratings, $T_{ij,t}$, for $N$ years. How is the through-the-cycle (TTC) transition matrix defined?

Sometimes the following definition is used, just like other TTC quantities which are long-run averages (through a full economic cycle): $$ \overline{T}_{ij} = \mathbb{E}(T_{ij,t}) $$

However, since the dynamics of credit rating migrations comes about by multiplying transition matrices, the above definition in terms of an arithmetic mean looks problematic in my opinion.

Shouldn't a more accurate definition be given in terms of the geometric mean? $$ \overline{T} = \left(\prod_{t=1}^{N}(T_{t})\right)^{1/N} $$ or, alternatively $$ \overline{\log T} = \mathbb{E}(\log T_{t}) $$ Now, it is well known that taking logs and roots of transition matrices is fraught with non-uniqueness and non-positivity issues but there are some ways to tackle this -- there is some discussion here.

My question is what should be the correct definition of the through-the-cycle transition matrix? (For those who can relate, this question is similar to calculating quenched vs. annealed averages for disordered systems in physics.)



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