<p>Quantifying changes in the credit rating of a bond is an important mathematical problem for the credit rating industry. To think of the credit rating as the state of a Markov chain is an interesting proposal leading to challenges in mathematical modeling. Since cumulative default rates are more readily available than credit migrations, a natural question is whether the credit transition matrix (CTM) can be determined from the knowledge of the cumulative default probabilities. Here we use a connection between the CTM and the cumulative default probabilities to set up an ill-posed, linear inverse problem with box constraints, which we solve by an entropy minimization procedure. This approach is interesting on several counts. To wit, we may have less data than unknowns, and even when we have as much data as unknowns, the matrix connecting them may not be invertible. Not only that, the procedure is robust, meaning that small changes in the data cause proportional changes in the solution. In addition to developing the necessary tools, we apply the method to several test cases to assess its performance.</p>