Decomposing Rating Migration Matrices From Market Prices Part C: Sources of Non-Linearity and Resolve

The study continues previous work on decomposing rating migration matrices from market prices. It further investigates the matter of the associated optimization problem, in plain form, yielding multiple possible local solutions. The sources of non-linearity and complexity of the optimization problem are outlined. This includes the rating category migration variance results of solutions, in terms of values and spacing, and within matrix rating category structures. Plain optimization problem decompositions struggle to surface, and correct both rating category migration variance values and spacing, and rating category matrix structures. Generally, full matrix decompositions require good initial solutions, to yield good results. Matrix averaging and matrix sampling are considered. Matrix averaging is based on limited coefficient counts or sets – not using the full coefficient count, but rather grouping coefficients. Matrix sampling forms an approximation of the matrix, in a sense through parsimony. Matrix averaging represents simple(r) optimization problems, and offers easy solutions, with good results and information, but offer poor initial solutions for full matrix decomposition. Matrix sampling can offer good indications, and good initial solutions for full matrix decomposition. Full matrix decomposition from initial solutions sourced through matrix sampling offers good results. Overall, revisiting and re-examining the way that the optimization algorithm searches optimal solutions based on the initial solution provided, can further improve decomposition results.