## Transition Probabilities

As we have seen, the measurement of long-term default rates can be problematic with small sample sizes. The computation of these default rates can be simplified by assuming a Markov process for the ratings migration, described by a transition matrix. Migration is a discrete process that consists of credit ratings changing from one period to the next.

The transition matrix gives the probability of moving to one rating conditional on the rating at the beginning of the period. The usual assumption is that these moves follow a Markov process, or that migrations across states are independent from one period to the next.4 This type of process exhibits no carry-over effect. More formally, a Markov chain describes a stochastic process in discrete time where the conditional distribution, given today's value, is constant over time. Only present values are relevant.

4 There is some empirical evidence, however, that credit downgrades are not independent but instead display a momentum effect.

Table 19-5 gives an example of a simplified transition matrix for 4 states, A, B, C, D, where the last represents default. Consider a company in year 0 in the B category. The company could default:

- In year 1, with probability D[t1 | B(f0)] = P(D1 \ B0) = 3%

- In year 2, after going from B to A in the first year, then A to D in the second, or from B to B, then to D, or from B to C, then to D. The total probability is

P(D2 | A1)P(A1) + P(D2 | B1)P(B1) + P(D2 | C1)P(C1)

 State Ending Total Starting A B C D Prob. A 0.97 0.03 0.00 0.00 i.00 B 0.02 0.93 0.02 0.03 i.00 C O.Oi 0.12 0.64 0.23 i.00 D O 0 0 i.00 i.00

The cumulative probability of default over the two years is then 3% + 3.25% = 6.25%. Figure 19-4 illustrates the various paths to default in years 1,2, and 3.

FIGURE 19-4 Paths to Default

Time 3 A

0 0