Rating Migration and Bond Valuation: Towards Ahistorical Rating Migration Matrices and Default Probability Term Structures

Abstract

The study examines rating migration, and default probability term structures obtained from rating migration matrices. It expands on the use of rating migration matrices with reduced form bond valuation models, by formally delineating the probability of default according to the likely rating paths of a bond, as implied by the rating migration matrix. Further, two alternatives are also considered. First, the cost of default is stipulated as the recovery of par according to the exit rating upon default. Also, in addition to stating the value of a bond in terms of expected cash flows, when considering the probability of default, the value of a bond is alternatively stated as the present value of all likely rating paths of the bond, discounted against the market risk-bearing bond forward rates of the different rating categories. The impact of term structure volatility and rating migration uncertainty on bond valuation is also considered.It is shown that the relationship between rating migration and default probability is complex, and the default probabilities of different rating categories are time-dependent and not isolated from each other. Also, rating migration resembles a delayed default process that influences default probabilities of subsequent intervals. The implications of a rating migration matrix may perhaps only be fully understood through simulation. This form one of the first points by which to evaluate rating migration matrices. The results of the valuation model show that historical rating migration matrices may not be optimal for pricing bonds ahistorically. A principal premise of the study is the dichotomy between historical values and ahistorical estimates, particularly with regards to rating migration. It is argued that historical estimates face two key shortcomings: they must be able to accurately forecast future rating migration and rating category intensities as a result, and they must specify a method to include rating migration uncertainty. An optimization model is delineated to extract ahistorical rating migration matrices from market prices. This too has implications that should be considered. In light of the above, reduced form models may have an advantage over structural models, in their ability to portray a far more sophisticated default process.

Rating Migration and Bond Valuation: Ahistorical Interest Rate and Default Probability Term Structures

“…In addition, even when accounting for the impact of the coupon size, significant differences exist between the default probabilities emanating from the plots, and the default probabilities contained in the rating migration matrix (table 2). With reference to equation 1, 2 and 3, and with reference to interval rating category intensities due to rating migration propagation (Barnard, 2017), only the primary rating category has non-zero intensity in the first interval, and because of this, only the primary rating category contributes to migration to default in the first interval. As a consequence, all default migration of the first interval is attributable to the primary rating category, and corresponds to the default probability of the primary rating category.…”

“…Taken from Barnard (2017), equation 1 states the reduced form model of Duffie and Singleton (1999), adapted for coupon paying bonds. Equation 1 has two components, a coupon paying component associated with non-default outcomes, and a recovery component associated with default outcomes.…”

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

“…Taken from Barnard (2017a), equation 1 states the reduced form model of Duffie and Singleton (1999), adapted for coupon paying bonds. Equation 1 has two components, a coupon paying component associated with non-default outcomes, and a recovery component associated with default outcomes.…”

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

“…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.…”

“…The x-to-x migration of rating category AAA recovered, but that of rating category BBB has dropped. Barnard, B., 2019 Table 10 shows the result when all rating categories' x-to-x migrations are constrained according to the following minimum values: 0.85, 0.5, 0.5, 0.7, 0.5, 0.65, 0.5. The x-to-x migration of rating category B is still relatively low, and is targeted next.…”