Credit rating dynamics and Markov mixture models

Despite mounting evidence to the contrary, credit migration matrices, used in many credit risk and pricing applications, are typically assumed to be generated by a simple Markov process. Based on empirical evidence, we propose a parsimonious model that is a mixture of (two) Markov chains, where the mixing is on the speed of movement among credit ratings. We estimate this model using credit rating histories and show that the mixture model statistically dominates the simple Markov model and that the differences between two models can be economically meaningful. The non-Markov property of our model implies that the future distribution of a firm’s ratings depends not only on its current rating but also on its past rating history. Indeed we find that two firms with identical current credit ratings can have substantially different transition probability vectors. We also find that conditioning on the state of the business cycle or industry group does not remove the heterogeneity with respect to the rate of movement. We go on to compare the performance of mixture and Markov chain using out-of-sample predictions.     

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.     

Split bond ratings and rating migration

This paper examines the relationships between split ratings and ratings migration. We find that bonds with split ratings are more likely to have future rating changes. A one-notch (more-than-one-notch) split rating increases the probability of rating change within one year of initial issuance by about 3% (6%). Furthermore, we find that about 30% of split rated bonds have their two ratings converge after four years of initial issuance. The rating convergence tapers off after three years, and the rating agency with a higher (lower) initial rating generally maintains a higher (lower) rating in subsequent years if the two ratings do not converge. We also show that rating transition estimation can be improved by taking into consideration split ratings. We find that one-year rating transition matrices are significantly different between non-letter-split rated bonds and letter-split rated bonds, and we show that the difference has an economically significant impact on the pricing of credit spread options and VaR-based risk management models. Overall, our results suggest that split ratings contain important information about subsequent rating changes.     

Capturing model risk and rating momentum in the estimation of probabilities of default and credit rating migrations

We present two methodologies on the estimation of rating transition probabilities within Markov and non-Markov frameworks. We first estimate a continuous-time Markov chain using discrete (missing) data and derive a simpler expression for the Fisher information matrix, reducing the computational time needed for the Wald confidence interval by a factor of a half. We provide an efficient procedure for transferring such uncertainties from the generator matrix of the Markov chain to the corresponding rating migration probabilities and, crucially, default probabilities. For our second contribution, we assume access to the full (continuous) data set and propose a tractable and parsimonious self-exciting marked point processes model able to capture the non-Markovian effect of rating momentum. Compared to the Markov model, the non-Markov model yields higher probabilities of default in the investment grades, but also lower default probabilities in some speculative grades. Both findings agree with empirical observations and have clear practical implications. We use Moody's proprietary corporate credit rating data set. Parts of our implementation are available in the R package ctmcd.     

On a multivariate Markov chain model for credit risk measurement

In this paper, we use credibility theory to estimate credit transition matrices in a multivariate Markov chain model for credit rating. A transition matrix is estimated by a linear combination of the prior estimate of the transition matrix and the empirical transition matrix. These estimates can be easily computed by solving a set of linear programming (LP) problems. The estimation procedure can be implemented easily on Excel spreadsheets without requiring much computational effort and time. The number of parameters is O(s² m² ), where s is the dimension of the categorical time series for credit ratings and m is the number of possible credit ratings for a security. Numerical evaluations of credit risk measures based on our model are presented.     

An Estimation Model for the Term Structure of Yield Spread

An estimation model for term structure of yield spread has become an extremely important subject to evaluate securities with default risk. By Duffie and Singleton model, yield spread was explained by two factors, namely collection rate and default probability. An estimation of the collection rate is given from historical earnings data, but estimation of default probability is known to be a remaining problem.There are some approaches to express default probability. One of them is to describe it through hazard process, and the other is to represent it by risk neutral transition probability matrix of credit-rating class. Some models that use Gaussian type hazard process or Vasicek type hazard process have already constructed.An advantage of evaluation using a rating transition probability matrix is that it is easy to obtain an image of movement of the credit-rating class. We do not need to show the calculation basis of the threshold or an assumption for distribution of prospective yield spread. But the model that uses the risk neutral transition probability matrix has not established yet, because of the computational difficulty required to estimate large number of the parameters.At first, for the purposes of this article, we will estimate the term structure of credit spreads results from the possibility of future defaults. It is assumed that credit risk is specified as a discrete-state Markov chain. And we construct a model which can be used to estimate the baseline transition matrix of the credit-rating class, risk-adjusting factors, industrial drift factors, corporate drift factors and recovery ratio, from yield spreads for individual bond. This enables us to compute the implied term structure from market data. We are capable of computing the implied term structure from market date by this process. Next, we will provide a valuation model for the term structure of yield spread.     

Long memory versus structural breaks in modeling and forecasting realized volatility

We explore the possibility of structural breaks in the daily realized volatility of the Deutschemark/Dollar, Yen/Dollar and Yen/Deutschemark spot exchange rates with observed long memory behavior. We find that structural breaks in the mean can partly explain the persistence of realized volatility. We propose a VAR-RV-Break model that provides superior predictive ability when the timing of future breaks is known. With unknown break dates and sizes, we find that a VAR-RV-I(d) long memory model provides a robust forecasting method even when the true financial volatility series are generated by structural breaks.     

The impact of credit rating announcements on credit default swap spreads

We document the ability of the credit default swap (CDS) market to anticipate favorable as well as unfavorable credit rating change (RC) announcements based on more extensive samples of credit rating events and CDS spreads than previous studies. We obtain four new results. In contrast to prior published studies, we find that corporate RC upgrades do have a significant impact on CDS spreads even though they are still not as well anticipated as downgrades. Second, CreditWatch (CW) and Outlook (OL) announcements, after controlling for prior credit rating events, lead to significant CARs at the time positive CW and OL credit rating events are announced. Third, we extend prior results by showing that changes in CDS spreads for non-investment-grade credits contain information useful for estimating the probability of negative credit rating events. Fourth, we find that the CDS spread impact of upgrades but not downgrades is magnified during recessions and that upgrades and downgrades also differ as to the impact of simultaneous CW/OL announcements, investment-grade/speculative-grade crossovers, current credit rating, market volatility, and industry effects.     

Structural change and long-range dependence in volatility of exchange rates: either, neither or both?

In this paper, we test for the existence of long memory and structural breaks in the realized variance process for the DM/US$ and Yen/US$ exchange rates. While long memory is evident in the actual processes, a structural break analysis reveals that this feature is partially explained by unaccounted changes in regime. We then compare the forecasting performance of Markov switching models with that of an ARFIMA model. The results indicate that neglecting the break process is not important for very short term forecasting once it is allowed for a long memory component in the model, but that superior forecasts can be obtained at longer horizons by modelling both long memory and structural change.     

Analyzing the impact of credit migration in a portfolio setting

Credit migration is an essential component of credit portfolio modeling. In this paper, we outline a framework for gauging the effects of credit migration on portfolio risk measurements. For a typical loan portfolio, we find credit migration can explain as much as 51% of volatility and 35% of economic capital. We compare through-the-cycle migration effects, implied by agency rating transitions, with point-in-time migration, implied by EDF? (Expected Default Frequency) transitions, and find that migration of point-in-time credit quality accounts for a greater fraction of total portfolio risk when compared with through-the-cycle dynamics. In a stylized analytic setting, we show that, when controlling for PD term structure effects, higher likelihood of moving away from the current credit state does not necessarily imply greater risk. Finally, we review methods for generating high-frequency transition matrices, needed for analyzing instruments with cash flows or contingencies whose frequencies are asynchronous to an available transition matrix. We further demonstrate that the naïve application of such methods can result in material deviations to portfolio analytics.     

The ordered qualitative model for credit rating transitions

Information on the expected changes in credit quality of obligors is contained in credit migration matrices which trace out the movements of firms across ratings categories in a given period of time and in a given group of bond issuers. The rating matrices provided by Moody's, Standard & Poor's and Fitch became crucial inputs to many applications, including the assessment of risk on corporate credit portfolios (CreditVar) and credit derivatives pricing. We propose a factor probit model for modeling and prediction of credit rating matrices that are assumed to be stochastic and driven by a latent factor. The filtered latent factor path reveals the effect of the economic cycle on corporate credit ratings, and provides evidence in support of the PIT (point-in-time) rating philosophy. The factor probit model also yields the estimates of cross-sectional correlations in rating transitions that are documented empirically but not fully accounted for in the literature and in the regulatory rules established by the Basle Committee.     

Feedback effects of credit ratings

Rating agencies are often criticized for being biased in favor of borrowers, for being too slow to downgrade following credit quality deterioration, and for being oligopolists. Based on a model that takes into account the feedback effects of credit ratings, I show that: (i) rating agencies should focus not only on the accuracy of their ratings but also on the effects of their ratings on the probability of survival of the borrower; (ii) even when rating agencies pursue an accurate rating policy, multi-notch downgrades or immediate default may occur in response to small shocks to fundamentals; (iii) increased competition between rating agencies can lead to rating downgrades, increasing default frequency and reducing welfare.     

A numerical method to price exotic path-dependent options on an underlying described by the Heston stochastic volatility model

We consider the problem of pricing European exotic path-dependent derivatives on an underlying described by the Heston stochastic volatility model. Lipton has found a closed form integral representation of the joint transition probability density function of underlying price and variance in the Heston model. We give a convenient numerical approximation of this formula and we use the obtained approximated transition probability density function to price discrete path-dependent options as discounted expectations. The expected value of the payoff is calculated evaluating an integral with the Monte Carlo method using a variance reduction technique based on a suitable approximation of the transition probability density function of the Heston model. As a test case, we evaluate the price of a discrete arithmetic average Asian option, when the average over n = 12 prices is considered, that is when the integral to evaluate is a 2n = 24 dimensional integral. We show that the method proposed is computationally efficient and gives accurate results.     

An extended likelihood framework for modelling discretely observed credit rating transitions

The estimation of the parameters of a continuous-time Markov chain from discrete-time observations, also known as the embedding problem for Markov chains, plays in particular an important role for the modeling of credit rating transitions. This missing data problem boils down to a latent variable setting and thus, maximum likelihood estimation is usually conducted using the expectation-maximization (EM) algorithm. We illustrate that the EM algorithm is likely to get stuck in local maxima of the likelihood function in this specific problem setting and adapt a stochastic approximation simulated annealing scheme (SASEM) as well as a genetic algorithm (GA) to combat this issue. Above that, our main contribution is to extend our method GA by a rejection sampling scheme, which allows one to derive stochastic monotone maximum likelihood estimates in order to obtain proper (non-crossing) multi-year probabilities of default. We advocate the use of this procedure as direct constrained optimization (of the likelihood function) will not be numerically stable due to the large number of side conditions. Furthermore, the monotonicity constraint enables one to combine structural knowledge of the ordinality of credit ratings with real-life data into a statistical estimator, which has a stabilizing effect on far off-diagonal generator matrix elements. We illustrate our methods by Standard and Poor’s credit rating data as well as a simulation study and benchmark our novel procedure against an already existing smoothing algorithm.     

Variations in sovereign credit quality assessments across rating agencies

We investigate agency variation in credit quality assessment (Standard and Poor’s vs. Moody’s vs. Fitch) employing sovereign ratings data for 129 countries, spanning the period 1990–2006. While we find that the credit rating agencies often disagree about credit quality, it is usually confined to one or two notches on the finer scale. We find that several variables have varying importance in explaining ratings across agencies which leads us to conclude that material heterogeneity exists between them. Also, while watch and outlook procedures are generally strong predictors of rating changes relative to other public data, additional significant variables suggest that it might be possible to augment these agency data to provide better forecasts of future rating changes.     

Improving performance of corporate rating prediction models by reducing financial ratio heterogeneity

We introduce a new approach to improve the performance of rating prediction models for multinational corporations. In this segment, the low number of defaults poses a challenge, as it prevents rating models to be constructed for individual industry sectors or regions. We show that reducing group-level heterogeneity in financial ratios results in a rating prediction model with better performance than both unadjusted models and models adjusted by including industry dummies or other simpler procedures. Our approach fills a gap in cases where a limited dataset does not permit the construction of separate models for individual industries or regions.     

Oil price shocks and volatility spillovers in the Nigerian sovereign bond market

This paper investigates volatility spillover in the Nigerian sovereign bond market arising from oil price shocks, using Vector Autoregressive Moving Average ‐ Asymmetric Generalized Autoregressive Conditional Heteroscedasticity (VARMA‐AGARCH) model. The paper covers the period March 22, 2011 to April 14, 2016 and makes use of the daily data of the Nigerian Sovereign Bond, Brent oil and West Texas Intermediate (WTI), respectively. We endogenously and sequentially detect structural break points using the test of Bai and Perron (2003) framework. In order to accurately estimate the model, we modify it by incorporating the break points into the VARMA‐AGARCH model, a process which if ignored would lead to model misspecification. The results obtained demonstrate a significant cross‐market volatility transmission between oil and sovereign bond market with ample sensitivity to structural breaks. The study also computes optimum weight portfolio and hedge ratio both with and without structural breaks and results equally indicate sensitivity to structural breaks.     

Modeling of CPDOs – Identifying optimal and implied leverage

When the subprime crisis started to emerge, collateralized products based on credit default swap (CDS) exposures combined with security features seemed to be a more rational alternative to classic asset backed securities. Constant Proportion Collateralized Debt Obligations (CPDOs) are a mixture of Collateralized Debt Obligations (CDOs) and CPPIs with inverse mechanism. This new asset aims at meeting the investors’ demand for credit derivatives with security enhancements, but to our knowledge quantitative approaches for pricing other than simulation algorithms do not exist yet. CPDOs became famous notably by Standard & Poor’s rating model error which illustrated that closed-form analytical pricing is necessary in order to evaluate and understand complex derivatives. This article aims to shed a light on CPDOs’ specific structural enhancements and mechanisms. We quantify inherent risks and provide a dynamic closed-form pricing formula.     

Modelling the economic value of credit rating systems

In this paper we develop a model of the economic value of credit rating systems. Increasing international competition and changes in the regulatory framework driven by the Basel Committee on Banking Supervision (Basel II) called forth incentives for banks to improve their credit rating systems. An improvement of the statistical power of a rating system decreases the potential effects of adverse selection, and, combined with meeting several qualitative standards, decreases the amount of regulatory capital requirements. As a consequence, many banks have to make investment decisions where they have to consider the costs and the potential benefits of improving their rating systems. In our model the quality of a rating system depends on several parameters such as the accuracy of forecasting individual default probabilities and the rating class structure. We measure effects of adverse selection in a competitive one-period framework by parameterizing customer elasticity. Capital requirements are obtained by applying the current framework released by the Basel Committee on Banking Supervision. Results of a numerical analysis indicate that improving a rating system with low accuracy to medium accuracy can increase the annual rate of return on a portfolio by 30–40 bp. This effect is even stronger for banks operating in markets with high customer elasticity and high loss rates. Compared to the estimated implementation costs banks could have a strong incentive to invest in their rating systems. The potential of reduced capital requirements on the portfolio return is rather weak compared to the effect of adverse selection.     

Reject inference in consumer credit scoring with nonignorable missing data

We generalize an empirical likelihood approach to deal with missing data to a model of consumer credit scoring. An application to recent consumer credit data shows that our procedure yields parameter estimates which are significantly different (both statistically and economically) from the case where customers who were refused credit are ignored. This has obvious implications for commercial banks as it shows that refused customers should not be ignored when developing scorecards for the retail business. We also show that forecasts of defaults derived from the method proposed in this paper improve upon the standard ones when refused customers do not enter the estimation data set.