Rating Migration and Bond Valuation: Decomposing Rating Migration Matrices from Market Data Via Default Probability Term Structures

Abstract

The study builds on previous research that decomposes rating category default probability term structures from rating category interest rate term structures, and proposes a method to decompose rating migration matrices from market data, via decomposed default probability term structures. To investigate the power and accuracy of the proposed method, it was examined to what extent an existing, known rating migration matrix could again be surfaced by the method. Overall, the results are more than satisfactory, and the method promises to be accurate. Although not considered here, the main objective is the application of the method to market data. The outcome should be insightful in itself, and can be used to evaluate historical rating migration matrices commonly devised by rating agencies, and to form a better understanding of the default probability term structures embedded in market data.

Sovereign Credit Rating, Rating Migration, and the Risk-Free Rate: A Joint Markov Process and Random Walk Modelling of the Risk-Free Rate

“…The objective is to see how accurately the original migration matrix can again be surfaced. To decompose a rating migration matrix, the method of Barnard (2017) is followed. In all cases, a simple barrier method is used to implement all optimizations.…”

“…After an initial solution migration matrix is constructed, it is further optimized by simultaneously or collectively optimizing all rating category migration matrix entries. Barnard (2017) iterates between optimizing rating category migration matrix entries individually, and collectively. But in this case, the bonds comprise a single sovereign and thus a single (initial) rating category.…”

“…From the probabilities of the risk-free rate instances (values) of the rating categories and the universal risk-free range, and of each discrete time Markov step, the value of a risk-free bond can be determined. The method is further expanded on by Barnard (2019) and Barnard (2017). Equation set 1 stipulates the optimization problem to decompose such a risk-free rate rating migration matrix from market prices.…”

“…Equation 1.b stipulates the rating migration matrix equality constraint -at all times, the sum of the migration probabilities of each rating category should equal 1. Barnard (2017) details how this constraint can be met during optimization. Equation 1.c and 1.d note additional constraints to guide the solution.…”