<p>In this paper, the mathematical properties of monotonicity and distribution discriminatory, which have been shown to be requisite properties when comparing transition probability matrices in credit risk modelling, are proved to hold for the risk-adjusted difference indices. This is a key result, and it implies that the risk-adjusted difference indices are suitable for comparing transition probability matrices; as they will be able to capture key properties that are peculiar to transition probability matrices in credit risk modelling. To the best of the researchers' knowledge, for the first time an index that has been proved to satisfy the requisite properties is used to compare the performance of the embeddability problem solving algorithms. The results of the non-parametric Kruskal-Wallis test and the Wilcoxon rank sum test show that the differences in the performances of the algorithms are statistically significant.</p>