# MIGRATION MATRICES

Migration matrices provide discrete transition probabilities across rating classes, as illustrated in Chapter 39 on historical credit risk data. Migrations are relevant for modeling risk under full valuation mode. They are used for projecting the structure of a portfolio by rating class and they have a direct relationship with cumulative probabilities. Furthermore, they serve for valuing credit VaR. This calculation is expanded in Chapter 46 for calculating the standalone credit VaR of a single facility using full valuation mode, using the "matrix valuation" process, which consists of mapping credit spreads to the final credit ratings for revaluation at horizon. We address here the relationship between cumulative default probabilities and migration probabilities considering a standalone obligor.

## Migration Matrices and Cumulative Default Probabilities

A migration matrix is a squared matrix with initial credit state, or rating, in rows and final credit state in columns. There is usually an additional column, called "WR," for firms existing at the beginning of the period and whose rating was withdrawn within the period. We ignore the "WR" column in the example. The default state is a particular credit state among several. However, the default state is considered as an absorbing state. In plain words, it means that once the firm migrates to the default state, it does not get out of this special state.

Consider the simplified square transition matrix with annual transition probabilities. The basic assumption used here is that the matrix is stable through time. This assumption is restrictive. For multi-period models, one should use transition probabilities conditional on the state of the economy. This approach is used in one credit portfolio model, "Credit Portfolio View," described in the credit portfolio models section 12.

The example uses only three credit states, named A, B and D, inclusive of the default state (D), since this enough to highlight the essential properties of those matrices (Table 43.1).

TABLE 43.1

The matrix is squared and not symmetric. The highest transition probabilities are along the diagonal. A transition probability has two subscripts, the first one for the initial state and the second for the final credit state. The transition probability from initial credit state / to the final credit state *j* is *t .* The migration probabilities out of the default state are zero, because it is an absorbing state. The matrix has the Markov properties which imply time invariance and that *T* depends only on the credit states and does not change with time. Such transition probabilities are by definition unconditional.