Risk Management and Financial Institutions, + Web Site, 3rd Edition

APPENDIX J

Manipulation of Credit Transition Matrices

Suppose that A is an n × n matrix of credit rating changes in one year. This is a matrix such as the one shown in Table 18.1. Assuming that rating changes in successive time periods are independent, the matrix of credit rating changes in m years is Am. If m is an integer, this can be readily calculated using the normal rules for matrix multiplication.

Consider next the problem of calculating the transition matrix for 1/m years where m is an integer. (For example, when we are interested in one-month changes, m=12.) This is a more complicated problem because we need to calculate the mth root of a matrix. We first calculate eigenvectors x1, x2, ..., xn and the corresponding eigenvalues λ1, λ2,..., λn of the matrix A. These are explained in Appendix H. They have the property that

(J.1)

Define X as an n × n matrix whose ith column is xi and Λ as an n × n diagonal matrix (i.e., a matrix which has zero values everywhere except on the diagonal) where the ith diagonal element is λi. From equation (J.1), we have

so that

Define Λ* as a diagonal matrix where the ith diagonal element is λi1/m. Then

showing that the mth root of A, and therefore ...