Rating Migration Matrices Research Articles

Structured dictionary learning of rating migration matrices for credit risk modeling

Structured dictionary learning of rating migration matrices for credit risk modeling

Statistical Surveillance of Structural Breaks in Credit Rating Dynamics.

The 2007–2008 financial crisis had severe consequences on the global economy and an intriguing question related to the crisis is whether structural breaks in the credit market can be detected. To address this issue, we chose firms’ credit rating transition dynamics as a proxy of the credit market and discuss how statistical process control tools can be used to surveil structural breaks in firms’ rating transition dynamics. After reviewing some commonly used Markovian models for firms’ rating transition dynamics, we present several surveillance rules for detecting changes in generators of firms’ rating migration matrices, including the likelihood ratio rule, the generalized likelihood ratio rule, the extended Shiryaev’s detection rule, and a Bayesian detection rule for piecewise homogeneous Markovian models. The effectiveness of these rules was analyzed on the basis of Monte Carlo simulations. We also provide a real example that used the surveillance rules to analyze and detect structural breaks in the monthly credit rating migration of U.S. firms from January 1986 to February 2017.

Financial liability stress tests: an approach based on the use of a rating migration matrix

Abstract The article addresses the issue of stress testing based on the probability of bankruptcy and a rating migration matrix. The analysis is conducted on a sample of listed companies in Poland in the years 1998–2016, and the forecasts are made for the years 2016–2018. Particular attention is paid to how the variable on which rating migration matrices are developed is defined. Stress tests are carried out on variables derived from rating migration matrices and economic indicators. The study provides information on the methodology for stress testing.

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.

Decomposing Rating Migration Matrices from Market Prices Revisited

The study revisits the method and premises of decomposing rating migration matrices from price data. Significantly easier, and more accurate method is presented. As an intermediary step, default and non-default probability term structures are first decomposed from market prices. This represents the composite default probability structure or process of a rating category, when considering rating migration and future ratings probabilities. A rating migration matrix is subsequently decomposed from the decomposed default probability term structures. Output residuals are low, yet the optimization problem finds many local solutions, rather than a solid global solution: different rating migration matrices can result in and produce the same decomposed default probability term structures. Furthermore, it is shown that this is not a matter of insufficient price data. Additional price data cannot solve this, and push the outcome to a global solution. The main implications are twofold: First, the underlying issue is that rating migration matrices state and reveal 2 components: a) default and non-default probability term structures, and b) non-default probability term sub-structures. The latter refers to a break-down of future non-default probabilities, per rating category – future rating probabilities in essence. Price data contain sufficient information to decompose the former, but not the latter. It is generally not possible to make non-default probability sub-structures (future rating probabilities) relevant in some way – to relate it to certain data – to allow decomposition, because there is little (market) data pertaining to this. The implication is that bond prices are not dependent on future ratings, over and above non-default probability term structures (“survival rates”), as a mirror of default probability term structures, as the latter already contains all price-relevant information. Secondly, rating migration matrices can be decomposed accurately enough, to provide some insight. At the same time, it may be unnecessary to decompose rating migration matrices, post default probability term structures. The latter can be done accurately and easily, and contain all the relevant price information.

Bond-Valuation and Default-Risk Re-examined: A Post Reduced-Form Model Based on Integer Migration and Portfolio Migration

The study re-examines bond valuation and default risk, by considering and relaxing assumptions regarding integer rating migration, and portfolio rating migration, and consequently the implied degree of diversification. A reduced form model, based on (a) rating migration (matrix), serve as basis and starting point. The study subsequently moves towards a post reduced form bond valuation model, by considering portfolio migration and integer migration. Portfolio migration refers to actual counts of issues migrating across rating categories, between the intervals of the portfolio – the number of issues with a particular rating at a particular point in time. Integer migration considers the fact that only integer counts of issues can migrate. Both portfolio and integer migration imply more sophisticated interpretation of rating migration matrices. Portfolio migration and integer migration result in interval rating category issue count probability distributions, and issue interval rating category intensity (probability) probability distributions, and consequently cause more sophisticated issue interval cash flow distributions to form. The resultant post reduced form model issue price results are encouraging and promising, and are briefly considered.

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

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: Towards Ahistorical Rating Migration Matrices and Default Probability Term Structures

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

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.

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

The study extends the theoretical framework proposed to decompose rating migration matrices from bond market price data. In this context, method to decompose default probability term structures for and from interest rate term structures for different rating categories, is also delineated and empirically evaluated. In principle, whenever it is possible to decompose an interest rate term structure for a rating category from ahistorical market data, it should also be possible to decompose a default probability term structure for the rating category, and to decompose a default probability term structure for the particular interest rate term structure of the rating category. Principal to the method and vantage point is that, when decomposing a default probability term structure, purely ahistorical market data is used, as is the case when decomposing interest rate term structures. In addition, greater emphasis is placed on surfacing actual market perceptions regarding default probability, which may differ from theoretical modelling of default probability. The method naturally allows a mapping and transitioning between interest rate term structures and default probability term structures. Consequentially, the study examines the corresponding interest rate term structures of the default probability term structures of a typical rating migration matrix, and the corresponding default probability term structures of a typical market interest rate term structure set. The sensitivity of the results to the coupon rate used is also examined, particularly in view of using an external (artificial) portfolio to facilitate the process. It is found that the default probability term structures decomposed from market interest rate term structures significantly differ from rating migration matrix based default probability term structures. This may point to differing views on default probability term structures. In addition, it is also shown that even external (artificial) portfolios can illuminate the default probability term structures of interest rate term structures with reasonable accuracy. Mapping to and fro interest rate term structures and default probability term structures introduces and additional level of triangulation and evaluation. It is expected that interest rate term structures should have valid and sensible default probability term structures, and vice versa. Another important implication of decomposing default probability term structures from ahistorical market data is that the default probability term structures would change at the frequency of the market and market data. This implies that market based default probability term structures may include a volatility component. Market based ahistorical default probability term structures offer a different and unique perspective on default probability term structures. If found to be an accurate representation of market views and perceptions, it may demand a reconsideration of the prominence of default probability as a bond valuation factor.

Connecting rating migration matrices and the business cycle by means of generalized regression models

The modeling of macroeconomic influence on rating migration matrices plays an important role in credit risk management, especially in stress testing. In contrast to approaches, which separately condition migration matrices by a qualitative assessment of the state of the business cycle, we promote the use of generalized regression models, which directly allow to consider macroeconomic covariates. We systemize, extend, and critically discuss different regression approaches and put an emphasis on violations of model assumptions as well as on sufficient treatment of such problems, an aspect, which has not been focused on to a satisfactory extent in the recent literature. Moreover, we introduce a framework for model evaluation and variable selection, which is based on the concept of out‐of‐sample forecasting, in order to avoid overfitting. Finally, we illustrate the concepts outlined by practical examples based on Standard & Poor's global corporate ratings data. Copyright © 2016 John Wiley & Sons, Ltd.

Reinforced urn processes for credit risk models

We propose a Bayesian nonparametric model to estimate rating migration matrices and default probabilities using the reinforced urn processes (RUP) introduced in Muliere et al. (2000). The estimated default probability becomes our prior information in a parametric model for the prediction of the number of bankruptcies, with the only assumption of exchangeability within rating classes. The Polya urn construction of the transition matrix justifies a Beta distributed de Finetti measure. Dependence among the processes is introduced through the dependence among the default probabilities, with the Bivariate Beta Distribution proposed in Olkin and Liu (2003) and its multivariate generalization.

Credit Migration Risk and Point in Time Credit Dynamics: A New Perspective for Credit Risk Management

This paper presents a credit migration model that aims to consistently capture the point-in-time dynamics of the credit worthiness of debt issuers and their obligations, and a calibration routine that permits the model to effectively fit historical ratings data. Our approach is to view the rating migration matrices as the operator semi-group associated to an approximated parametric infinitesimal generators. Our credit model accounts not only for default risk dynamics but also for the entire transition among states of the rating migration matrix. This modeling feature is fundamental for an efficient risk management of credit derivatives and credit risk portfolios conditionally on a state of the economy or specific macro factors. We fit our model to the historical average rating migration matrices published by Moody’s Investors Service, focusing on the banking sector over the period 1920-2005. Our results show that the model can identify the through-the-cycle transition across rating scales and that the point-in-time migration probabilities are only generated by stressed economic conditions and can only be justified by the influence of macro factors on the through-the-cycle unconditional probability values. The great level of modeling details and the accuracy of the produced results is an improvement over those of other available models.

신용등급전이행렬의 경험적 베이지안 추정과 비교

신용전이행렬을 추정함에 있어서 국내의 등급전이자료의 축적이 부족한 점을 극복하기 위하여 외국의 신용평가기관(무디스)의 전이행렬자료와 국내의 신용등급 부여자료를 이용하여 경험적 베이지안 추정방법에 의한 전이행렬을 도출하고, 이 전이행렬을 다른 전이행렬과 비교해보기 위하여 전이행렬의 동적인 요소를 평균전이확률의 개념으로 표시할 수 있는 특성척도를 개발하여 신용전이행렬의 시계열 특성과 통계적 특성을 비교한다. 시계열자료의 척도는 베이지안 추정행렬이 안정적임을 보여주는 반면 국내 행렬은 시간적으로 변화의 폭이 크고 무디스나 베이지안 행렬보다 상대적으로 인접전이의 비율이 높게 나타났다. 붓스트랩 검정을 통하여 세 가지 추정방법이 통계적으로 유의한 차이가 있음을 보이고 베이지안 행렬이 무디스 자료보다는 국내자료에 더 많은 영향을 받았음을 유추할 수 있다. 신용등급 전이에 따른 포트폴리오의 가치변화를 고려하는 몬테칼로 시뮬레이션을 통하여 신용 VaR를 구하여 비교하였다. 국내 전이행렬의 경우에 평균은 가장 크고 신용위험도 가장 큰 값을 보였다. 시뮬레이션에서도 베이지안 추정에 의한 결과가 국내자료에 의한 결과와 더 가깝다는 것을 알 수 있다. In order to overcome the lack of Korean credit rating migration data, we consider an empirical Bayes procedure to estimate credit rating migration matrices. We derive the posterior probabilities of Korean credit rating transitions by utilizing the Moody's rating migration data and the credit rating assignments from Korean rating agency as prior information and likelihood, respectively. Metrics based upon the average transition probability are developed to characterize the migration matrices and compare our Bayesian migration matrices with some given matrices. Time series data for the metrics show that our Bayesian matrices are stable, while the matrices based on Korean data have large variation in time. The bootstrap tests demonstrate that the results from the three estimation methods are significantly different and the Bayesian matrices are more affected by Korean data than the Moody's data. Finally, Monte Carlo simulations for computing the values of a portfolio and its credit VaRs are performed to compare these migration matrices.

Bayesian inference for issuer heterogeneity in credit ratings migration

Rating transition matrices for corporate bond issuers are often based on fitting a discrete time Markov chain model to homogeneous cohorts. Literature has documented that rating migration matrices can differ considerably depending on the characteristics of the issuers in the pool used for estimation. However, it is also well known in the literature that a continuous time Markov chain gives statistically superior estimates of the rating migration process. It remains to verify and quantify the issuer heterogeneity in rating migration behavior using a continuous time Markov chain. We fill this gap in the literature. We provide Bayesian estimates to mitigate the problem of data sparsity. Default data, especially when narrowing down to issuers with specific characteristics, can be highly sparse. Using classical estimation tools in such a situation can result in large estimation errors. Hence we adopt Bayesian estimation techniques. We apply them to the Moodys corporate bond default database. Our results indicate strong country and industry effects on the determination of rating migration behavior. Using the CreditRisk+ framework, and a sample credit portfolio, we show that ignoring issuer heterogeneity can give erroneous estimates of Value-at-Risk and a misleading picture of the risk capital. This insight is consistent with some recent findings in the literature. Therefore, given the upcoming Basel II implementation, understanding issuer heterogeneity has important policy implications.