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The transition matrices of rating agency data are unconditional on the state of the economy and reflect historical data. They provide the probabilities that an obligor in risk class k moves to risk class l during a certain period. Joint transition matrices provide the probability that a pair of obligors moves from one pair of credit states to another pair of credit states. Credit Metrics derives the probability of joint migrations from the correlated equity returns of firms. Joint migration matrices tabulate the final states of two obligors and assign probabilities to them. The matrix in Table 49.1 is the same one that served as an example when describing the modelling of correlations. We use this matrix to generate a loss distribution.

This is a simplified matrix. In general, there are more classes. For instance, with six risk classes plus the default state, there are seven transitions. The number of transitions for a two-loan portfolio is 7² = 49. When expanding the portfolio, the number of pairs of firms increases. It is equal to the number of combinations of two facilities among the N facilities in the portfolio, or N!/[2! x (N — 2)!], where the symbol ! represents the factorial operator. For N = 2, the number of pairs is one, for N = 3, it is three, for N = 4, it is six and for N = 5, it is 10. For large numbers, the number of pairs increases very quickly¹. Each pair corresponds to 49 transitions, or 490 for five obligors, and so on. This raises the issue of the number of calculations. That is why it might be simpler to use asset correlation to derive the structure of the portfolio by risk class at a given horizon.