Business — Banking — Management — Marketing & Sales

Migration matrices group historical frequencies of transitions across risk classes over a specified horizon, for example 1 year. A migration frequency is the count of firms migrating from one risk class to another risk class, or the ratio of these firms to the original firms in the initial risk class. Starting from one risk class, there are as many migrations as there are final states. The final states include all risk classes plus the default state. The default option model applies to migrations since it models the default probabilities that map with risk classes.

The option framework helps to model migrations in two ways. The first application moves from modelled asset values to migration probabilities. The process implies modelling the distribution of asset values at the horizon. This provides three different results: the expected value, from which the expected default probability results; the distribution of probabilities of migrating to any of the values in the spectrum of final asset values, which are the migration probabilities; the corresponding distance to default and the related default probabilities. Since default probabilities map to ratings, it is possible to convert any band of final default probabilities into a final rating.

The second application moves in the opposite direction, from observed default and migration probabilities across ratings classes to the asset value bands and standardized distances to default at the horizon corresponding to final default probabilities. The process is the reverse of the preceding one. It moves from given migration probabilities to probabilities of reaching bands of asset values at the horizon. The purpose is the same as the simplified model of the section above. It allows us to model correlations between migrations of pairs of firms from correlations of their unobservable asset values once we know their initial default probabilities.

From Modelled Asset Values to Migration Probabilities

In the KMV universe of firms, modelled asset returns result in a distribution of final values at the horizon. Each corresponds to a distance to default and a default probability. Hence, the same process that generates the expected value at the horizon to obtain the expected default probability also provides the migration probabilities for all asset levels and the corresponding default probabilities. In Figure 38.5, the distance to default decreases when moving downward from one time path of asset returns to a lower one. Accordingly, the default probability increases. The distribution of asset values at the horizon is identical to the distribution of migrations, and provides the migration probabilities. Notice that the default probability expected at the horizon results from one single asset value, which is the expected asset value. When the time path hits a level other than this expected value, it does not change the default probability between the current date and the horizon, but it does change it from the horizon to the next period. For each new asset value along the unique distribution at the horizon, there is a subsequent distribution which is higher or lower than the one at the horizon, depending on where the asset value ends up at the horizon. Figure 38.5 shows the unique expected default probability at the horizon, not the different future default probabilities corresponding to each final state.