Risk Management and Financial Institutions, 4th 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 A^m. 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, …, x_n and the corresponding eigenvalues λ₁, λ₂, …, λ_n 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 x_i 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 λ_i. From equation (J.1), we have

so that

Define Λ* as a diagonal matrix where the ith diagonal element is λ^1/m_i. Then

showing that the mth root of A, ...