Robust and consistent estimation of generators in credit risk

Bond rating Transition Probability Matrices (TPMs) are built over a one-year time-frame and for many practical purposes, like the assessment of risk in portfolios or the computation of banking Capital Requirements (e.g. the new IFRS 9 regulation), one needs to compute the TPM and probabilities of default over a smaller time interval. In the context of continuous time Markov chains (CTMC) several deterministic and statistical algorithms have been proposed to estimate the generator matrix. We focus on the Expectation-Maximization (EM) algorithm by Bladt and Sorensen. [J. R. Stat. Soc. Ser. B (Stat. Method.), 2005, 67, 395–410] for a CTMC with an absorbing state for such estimation. This work’s contribution is threefold. Firstly, we provide directly computable closed form expressions for quantities appearing in the EM algorithm and associated information matrix, allowing to easy approximation of confidence intervals. Previously, these quantities had to be estimated numerically and considerable computational speedups have been gained. Secondly, we prove convergence to a single set of parameters under very weak conditions (for the TPM problem). Finally, we provide a numerical benchmark of our results against other known algorithms, in particular, on several problems related to credit risk. The EM algorithm we propose, padded with the new formulas (and error criteria), outperforms other known algorithms in several metrics, in particular, with much less overestimation of probabilities of default in higher ratings than other statistical algorithms.     

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.     

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.     

Revolving versus Convenience Use of Credit Cards: Evidence from U.S. Credit Bureau Data

Credit card payments and revolving debt are important for consumer theory but a key data source—credit bureau records—does not distinguish between current charges and revolving debt. We develop a theory-based econometric methodology using a hidden Markov model to estimate the likelihood a consumer is revolving debt each quarter. We validate our approach using a new survey linked to credit bureau data. We estimate that for likely revolvers: (i) 100% of an increase in credit becomes an increase in debt eventually; (ii) credit limit changes are half as salient as debt changes; and (iii) revolving status is persistent.     

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.     

A sparsity algorithm for finding optimal counterfactual explanations: Application to corporate credit rating

Machine learning methods used in finance for corporate credit rating lack transparency as to which accounting features are important for the respective rating. A counterfactual explanation is a methodology that attempts to find the smallest modification of the input values which changes the prediction of a learned algorithm to a new output, other than the original one. We propose a “sparsity algorithm” which finds a counterfactual explanation to find the most important features for obtaining a higher credit score. We validate the novel algorithm with synthetically generated data and we apply it to quarterly financial statements from companies in the US market. We provide evidence that the counterfactual explanation can capture the majority of features that change between two quarters when corporate ratings improve. The results obtained show that the higher the rating of a company, the greater the “effort” required to further improve credit rating.     

A Hidden Markov Approach to Disability Insurance

Point and interval estimation of future disability inception and recovery rates is predominantly carried out by combining generalized linear models with time series forecasting techniques into a two-step method involving parameter estimation from historical data and subsequent calibration of a time series model. This approach may lead to both conceptual and numerical problems since any time trend components of the model are incoherently treated as both model parameters and realizations of a stochastic process. We suggest that this general two-step approach can be improved in the following way: First, we assume a stochastic process form for the time trend component. The corresponding transition densities are then incorporated into the likelihood, and the model parameters are estimated using the Expectation-Maximization algorithm. We illustrate the modeling procedure by fitting the model to Swedish disability claims data.     

Credit Ratings and Earnings Management around IPOs

This study examines the impact of having a credit rating on earnings management (EM) through accruals and real activities manipulation by initial public offering (IPO) firms. We find that firms going public with a credit rating are less likely to engage in income‐enhancing accrual‐based and real EM in the offering year. The monitoring by a credit rating agency (CRA) and the reduced information asymmetry due to the provision of a credit rating disincentivise rated issuers from managing earnings. We also suggest that the participation of a reputable auditing firm is crucial for CRAs to effectively restrain EM. Moreover, we document that for unrated issuers, at‐issue income‐increasing EM is not linked to future earnings and is negatively related to post‐issue long‐run stock performance. However, for rated issuers, at‐issue income‐increasing EM is positively associated with subsequent accounting performance and is unrelated to long‐run stock performance following the offering. The evidence indicates that managers in unrated firms generally manipulate earnings to mislead investors, while managers in rated firms tend to exercise their accounting and operating discretion for informative purposes.     

Modeling of Contagious Credit Events and Risk Analysis of Credit Portfolios

We present a new model of the occurence of credit events such as rating changes and defaults for risk analyses of some portfolio credit derivatives. The framework of our model is based on a so-called top-down approach. Specifically, we first consider modeling the point process of each type of credit event in the whole economy using a self-exciting intensity process. Next, we characterize the point processes of credit events in the underlying sub-portfolio using random thinning processes specified by the distribution of credit ratings in the sub-portfolio. One of the main features of our model is that the model can capture credit risk contagion simultaneously among several credit portfolios. We present a credit event simulation algorithm based on our model and illustrate an application of the model to risk analyses of loan portfolios.     

Analyzing rating transitions and rating drift with continuous observations

We consider the estimation of credit rating transitions based on continuous-time observations. Through simple examples and using a large data set from Standard and Poor's, we illustrate the difference between estimators based on discrete-time cohort methods and estimators based on continuous observations. We apply semi-parametric regression techniques to test for two types of non-Markov effects in rating transitions: Duration dependence and dependence on previous rating. We find significant non-Markov effects, especially for the downgrade movements.     

Interacting particle systems for the computation of rare credit portfolio losses

In this paper, we introduce the use of interacting particle systems in the computation of probabilities of simultaneous defaults in large credit portfolios. The method can be applied to compute small historical as well as risk-neutral probabilities. It only requires that the model be based on a background Markov chain for which a simulation algorithm is available. We use the strategy developed by Del Moral and Garnier in (Ann. Appl. Probab. 15:2496–2534, 2005) for the estimation of random walk rare events probabilities. For the purpose of illustration, we consider a discrete-time version of a first passage model for default. We use a structural model with stochastic volatility, and we demonstrate the efficiency of our method in situations where importance sampling is not possible or numerically unstable.      

Numerical Approach to Asset Pricing Models with Stochastic Differential Utility

In this paper employing two heuristic numerical schemes, we study the asset pricing models with stochastic differential utility (SDU), which is formulated by either of backward stochastic differential equations (BSDEs) or forward-backward stochastic differential equations (FBSDEs).The first scheme is based upon a traditional lattice algorithm of option pricing theories, involving the discretization scheme of coupled FBSDEs, which is combined with a technique of solving numerically a certain type of nonlinear equations with respect to the backward state variables. The second one is based upon the four step scheme of Ma et al. (1994) which solves quasi-linear partial differential equations associated with the FBSDEs. We demonstrate that our practical implementation algorithms can successfully solve the asset pricing models with generalized SDU and the large investor problem with market impact which are typical examples such that the usual four step scheme is difficult to implement. As other numerical applications we study the optimal consumption and investment policies of a representative agent with SDU, and the recoverability of preferences and beliefs from observed consumption data.     

Credit rating dynamics in the presence of unknown structural breaks

In many credit risk and pricing applications, credit transition matrix is modeled by a constant transition probability or generator matrix for Markov processes. Based on empirical evidence, we model rating transition processes as piecewise homogeneous Markov chains with unobserved structural breaks. The proposed model provides explicit formulas for the posterior distribution of the time-varying rating transition generator matrices, the probability of structural break at each period and prediction of transition matrices in the presence of possible structural breaks. Estimating the model by credit rating history, we show that the structural break in rating transitions can be captured by the proposed model. We also show that structural breaks in rating dynamics are different for different industries. We then compare the prediction performance of the proposed and time-homogeneous Markov chain models.     

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.     

Gradient-based simulated maximum likelihood estimation for stochastic volatility models using characteristic functions

Parameter estimation and statistical inference are challenging problems for stochastic volatility (SV) models, especially those driven by pure jump Lévy processes. Maximum likelihood estimation (MLE) is usually preferred when a parametric statistical model is correctly specified, but traditional MLE implementation for SV models is computationally infeasible due to high dimensionality of the integral involved. To overcome this difficulty, we propose a gradient-based simulated MLE method under the hidden Markov structure for SV models, which covers those driven by pure jump Lévy processes. Gradient estimation using characteristic functions and sequential Monte Carlo in the simulation of the hidden states are implemented. Numerical experiments illustrate the efficiency of the proposed method.     

Gradient-based simulated maximum likelihood estimation for Lévy-driven Ornstein–Uhlenbeck stochastic volatility models

This paper studies the parameter estimation problem for Ornstein–Uhlenbeck stochastic volatility models driven by Lévy processes. Estimation is regarded as the principal challenge in applying these models since they were proposed by Barndorff-Nielsen and Shephard [J. R. Stat. Soc. Ser. B, 2001, 63(2), 167–241]. Most previous work has used a Bayesian paradigm, whereas we treat the problem in the framework of maximum likelihood estimation, applying gradient-based simulation optimization. A hidden Markov model is introduced to formulate the likelihood of observations; sequential Monte Carlo is applied to sample the hidden states from the posterior distribution; smooth perturbation analysis is used to deal with the discontinuities introduced by jumps in estimating the gradient. Numerical experiments indicate that the proposed gradient-based simulated maximum likelihood estimation approach provides an efficient alternative to current estimation methods.     

Portfolio selection with qualitative input

We formulate a mean-variance portfolio selection problem that accommodates qualitative input about expected returns and provide an algorithm that solves the problem. This model and algorithm can be used, for example, when a portfolio manager determines that one industry will benefit more from a regulatory change than another but is unable to quantify the degree of difference. Qualitative views are expressed in terms of linear inequalities among expected returns. Our formulation builds on the Black-Litterman model for portfolio selection. The algorithm makes use of an adaptation of the hit-and-run method for Markov chain Monte Carlo simulation. We also present computational results that illustrate advantages of our approach over alternative heuristic methods for incorporating qualitative input.     

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.     

Regulation of credit rating agencies

Financial regulators recognize certain credit rating agencies for regulatory purposes. However, it is often argued that credit rating agencies have an incentive to assign inflated ratings. This paper studies a repeated principal-agent problem in which a regulator approves credit rating agencies. Credit rating agencies may collude to assign inflated ratings. Yet we show that there exists an approval scheme which induces credit rating agencies to assign correct ratings.     

信用评级变动能反映企业真实财务信息吗?——基于财务重述的视角

本文以2007-2018年拥有主体信用评级的A股上市公司为研究对象,利用企业财务错报在未来被重述这一场景,检验主体信用评级变动能否反映企业真实财务信息。研究发现,评级机构在发债企业财务错报年显著下调了主体信用评级,而在重述公告发布年没有上述现象,这表明主体信用评级下调反映了企业的真实财务信息。在控制内生性影响后,结论仍然成立。进一步研究发现,发债企业当期财务错报涉及盈余时,主体信用评级被下调的幅度更大,说明评级机构更加关注与盈余相关的财务信息。机制分析表明,评级机构维护自身声誉是主体信用评级变动能够反映企业真实财务信息的主要机制。此外,主体信用评级被下调还导致了资本市场投资者的负面反应。本文的研究结果为主体信用评级变动反映企业真实财务信息提供了直接的证据支持,揭示了主体信用评级的信息含量,也对理解中国情境下评级机构调整主体信用评级的行为动机提供参考。