# Credit migration matrix – use and misuse

A rating is supposed to be a stable long term predictor of the creditworthiness of a borrower. Every once in a while, the Credit Rating Agencies (CRAs) would release their analysis upon rating migration and, more importantly, how their ratings compare with realised default frequencies. The information is often summarised in form of a credit migration/transition matrix.  It is a useful tool but sometimes be misused.

Moody’s, S&P and Fitch are the big three CRAs. All use similar rating scales. In the discussion below, I will ignore the rating modifiers and focus on S&P/ Fitch’s rating scale (investment grade as AAA, AA, A, BBB and speculative grade credits into BB, B, CCC and below, and of course, D/default as well).

Credit migration matrices are constructed either by cohort (count how many credits migrates from rating X to Y during the measurement period) or by duration (consider the actual time staying in each rating – e.g. a BBB credit downgraded first to BB at the end of Q1 and to B at the end of Q2 and remain B till the end of the year, duration method measures this as 25% BBB, 25% BB and 75% B). Constructing credit migration matrix by cohort is by far the more common approach. The credit migration matrix M is typically diagonal dominate (ie the most likely state for the next period is no change) and with D/default as the absorption boundary. Corporates emerge from bankruptcies are considered as new entities. For various reasons, such as retirement of all public debts or management refuses to grant enough access to CRAs anymore, some securities become non-rated (NR) after time . A sample credit migration matrix for global corporate credit from S&P is shown in Figure 1. It describes the one-year transition rate from the original rating (row) to the updated rating (column).

Usage

If we model credit migration as a Markov (“memoryless”) process, we can construct a credit migration matrix measured for any length of time via power of matrix calculation. A y-year credit migration matrix $M_y$ is related to the x-year migration matrix $M_x$ by $M_y = M_x ^ {y/x}$. Here we assume the original states are in rows and the updated states in columns. Starting from the one-year credit migration matrix $M_1$, the two-year matrix can be computed as $M_1*M_1$ and so on. Figure 2 shows the construction of a ten-year credit migration matrix from a one-year matrix.

The number of periods (y/x) can be a fraction. We can also obtain a credit migration matrix for a shorter time period from a longer term one (e.g. a three-year matrix from a seven-year one $M_3 = M_7^{3/7}$). Instead of simple matrix multiplication, such calculation would require more involved matrix manipulation*.
Once we obtain the desired credit migration matrix, we can evaluate the weight column vector x for the next period as $M' * x$ . The rating evolution for a typical static investment grade bond portfolio is shown in Figure 3. Due to the presence of default absorption border, all the credits will be modelled as defaulted given enough time.

Misuse

It is easy to overlook the fact that credit migration matrix is obligor specific. CRAs are not obliged to calibrate to their rating scales across different types of debt instruments. In order words, the default probability for a BBB rated corporate does not necessarily be (and usually is not) the same as a BBB non-agency mortgage tranche – rude awakening for many during the US subprime crisis.

The subtlety of the statistical assumptions is another issue. The performance of corporates are strongly influence by the multiyear business cycle. During the recovery upswing, corporates tend to be upgraded one after and another. The opposite is true once recession kicks started. Defaults tend to form clusters. While the Markov (“memoryless”) assumption may be reasonable in the long run, it is not so within the same business cycle. Also the use of historical credit migration matrix implicitly assumes that the market environment as time-invariant. Although market conditions tend to mean reverting, the process is unlikely to be strictly time-invariant either.

Lastly, a significant portion of credits could become non-rated (NR) esp for the lower rating entities. For example, S&P shows that 48.92% of originally B rated corporate credit becomes NR after 10 years. In general, analysts treat NR as uninformative and adjust the rest of the probabilities up proportionally. Given such a high proportion of NR, the assumption is worth further examination.

*Power of matrix with a fractional power $M_y = M_x ^ {y/x} = exp(y/x*log(M_x))$ can be calculated using matrix exponential which is in turn often approximated by Pade algorithm.