# LO 19.3: Describe a rating migration matrix and calculate the probability o f

LO 19.3: Describe a rating migration matrix and calculate the probability o f default, cumulative probability o f default, marginal probability o f default, and annualized default rate.
A migration frequency represents how often ratings change from one class to another. A migration matrix shows relative frequencies of counterparties that move from one rating class (shown in each row) to another class (shown in each column). Figure 1 shows a one- year Moodys migration matrix across a 30-year period (19702007), with WR representing withdrawn ratings.
Figure 1: One-Year Moodys Migration Matrix
Aaa 89.1 1.0 0.1 0.0 0.0 0.0 0.0 0.0
Aaa Aa A Baa Ba B Caa Ca-C
Final Rating Class (%)
Aa 7.1 87.4 2.7 0.2 0.1 0.0 0.0 0.0
A 0.6 6.8 87.5 4.8 0.4 0.2 0.0 0.0
Baa 0.0 0.3 4.9 84.3 5.7 0.4 0.2 0.0
Ba 0.0 0.1 0.5 4.3 75.7 5.5 0.7 0.4
B 0.0 0.0 0.1 0.8 7.7 73.6 9.9 2.6
Initial Rating Class
Caa 0.0 0.0 0.0 0.2 0.5 4.9 58.1 8.5
Ca-C Default WR 3.2 0.0 0.0 4.5 4.1 0.0 0.0 5.1 8.8 0.0 0.6 10.4 12.8 3.6 19.8 38.7
0.0 0.0 0.0 0.2 1.1 4.5 14.7 30.0
It is worth noting that migrations are correlated and dependent transitions that occur over time (as opposed to being random walks). Observations over time have shown that when initial ratings are low (high), they become better (worse) than expected. However, default frequencies do have inherent limitations tied to the different applied methodologies of rating agencies. These limitations include differences in definitions, observed populations, amounts rated, and initial ratings.
Several key measures are used to assess the risk of default. The first is the probability of default (PD), which is shown in the following equation:
defaulted!^ PDk = ———-5
namest
where: PD = probability of default defaulted = number of issuer names that have defaulted in the applicable time horizon names = number of issuers k = time horizon
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2018 Kaplan, Inc.
Topic 19 Cross Reference to GARP Assigned Reading – De Laurentis et al., Chapter 3
A cumulative probability of default, given a sequence of default rates, can be calculated as follows:
p yj cumulative
i=t+k
defaulted j i= t____________
names t
Comparing the two previous equations, a marginal probability of default can be calculated as follows:
marginal k
p Q cumulative
t+k
cumulative t
Finally, the annualized default rate (ADR) can be computed for both discrete and continuous time intervals as follows:
discrete: ADRt = 1 j/(l PD cumulative i In 1 -P D fcumulative