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.     

On a Statistical Analysis of Implied Data

We propose a method of evaluating the accuracy of the implied default probabilities. We modify the model proposed by Duffie et al. (Rev Fin Stud 12:678–720, 1999) to allow the parametric statistical analysis. The pseudo maximum likelihood estimator is defined and to justify our method we shall prove the consistency and the asymptotic normality of the estimator. The key step is to define a pseudo score vector and apply the method of Wald (Ann Math Stat 20: 595–601, 1949) and a delta method. We also introduce the bootstrap for estimating the accuracies, which is similar to that for regression models. To implement our method to the real data, we shall recommend the bootstrap rather than asymptotic normality.     

Computation of Multivariate Barrier Crossing Probability and its Applications in Credit Risk Models

Abstract

In this paper we consider computational methods of finding exit probabilities for a class of multivariate diffusion processes. Although there is an abundance of results for one-dimensional diffusion processes, for multivariate processes one has to rely on approximations or simulation methods. We adopt a Large Deviations approach to approximate barrier crossing probabilities of a multivariate Brownian Bridge. We use this approach in conjunction with simulation methods to develop an efficient method of obtaining barrier crossing probabilities of a multivariate Brownian motion. Using numerical examples, we demonstrate that our method works better than other existing methods. We mainly focus on a three-dimensional process, but our framework can be extended to higher dimensions. We present two applications of the proposed method in credit risk modeling. First, we show that we can efficiently estimate the default probabilities of several correlated credit risky entities. Second, we use this method to efficiently price a credit default swap (CDS) with several correlated reference entities. In a conventional approach one normally adopts an arbitrary copula to capture dependency among counterparties. The method we propose allows us to incorporate the instantaneous variance-covariance structure of the underlying process into the CDS prices.     

Aggregating sectors in the infectious defaults model

Abstract

How to model the dependence between defaults in a portfolio subject to credit risk is a question of great importance. The infectious default model of Davis and Lo offers a way to model the dependence. Every company defaulting in this model may ‘infect’ another company causing it to default. An unsolved question, however, is how to aggregate independent sectors, since a naive straightforward computation quickly gets cumbersome, even when homogeneous assumptions are made. Here, two algorithms are derived that overcome the computational problem and further make it possible to use different exposures and probabilities of default for each sector. For an ‘outbreak’ of defaults to occur in a sector, at least one company has to default by itself. This fact is used in the derivations of the two algorithms. The first algorithm is derived from the probability generating function of the total credit loss in each sector and the fact that the outbreaks are independent Bernoulli random variables. The second algorithm is an approximation and is based on a Poisson number of outbreaks in each sector. This algorithm is less cumbersome and more numerically stable, but still seems to work well in a realistic setting.     

Parameter estimation risk in asset pricing and risk management: A Bayesian approach

Parameter estimation risk is non-trivial in both asset pricing and risk management. We adopt a Bayesian estimation paradigm supported by the Markov Chain Monte Carlo inferential techniques to incorporate parameter estimation risk in financial modelling. In option pricing activities, we find that the Merton's Jump-Diffusion (MJD) model outperforms the Black-Scholes (BS) model both in-sample and out-of-sample. In addition, the construction of Bayesian posterior option price distributions under the two well-known models offers a robust view to the influence of parameter estimation risk on option prices as well as other quantities of interest in finance such as probabilities of default. We derive a VaR-type parameter estimation risk measure for option pricing and we show that parameter estimation risk can bring significant impact to Greeks' hedging activities. Regarding the computation of default probabilities, we find that the impact of parameter estimation risk increases with gearing level, and could alter important risk management decisions.     

Moody's correlated binomial default distributions for inhomogeneous portfolios

This paper generalizes Moody's correlated binomial default distribution for homogeneous (exchangeable) credit portfolios, which was introduced by Witt, to the case of inhomogeneous portfolios. We consider two cases of inhomogeneous portfolios. In the first case, we treat a portfolio whose assets have uniform default correlation and non-uniform default probabilities. We obtain the default probability distribution and study the effect of inhomogeneity. The second case corresponds to a portfolio with inhomogeneous default correlation. Assets are categorized into several different sectors and the inter-sector and intra-sector correlations are not the same. We construct the joint default probabilities and obtain the default probability distribution. We show that as the number of assets in each sector decreases, inter-sector correlation becomes more important than intra-sector correlation. We study the maximum values of the inter-sector default correlation. Our generalization method can be applied to any correlated binomial default distribution model that has explicit relations to the conditional default probabilities or conditional default correlations, e.g. Credit Risk⁺, implied default distributions. We also compare some popular CDO pricing models from the viewpoint of the range of the implied tranche correlation.     

Credit risk drivers: Evaluating the contribution of firm level information and of macroeconomic dynamics

Understanding if credit risk is driven mostly by idiosyncratic firm characteristics or by systematic factors is an important issue for the assessment of financial stability. By exploring the links between credit risk and macroeconomic developments, we observe that in periods of economic growth there may be some tendency towards excessive risk-taking. Using an extensive dataset with detailed information for more than 30 000 firms, we show that default probabilities are influenced by several firm-specific characteristics. When time-effect controls or macroeconomic variables are also taken into account, the results improve substantially. Hence, though the firms’ financial situation has a central role in explaining default probabilities, macroeconomic conditions are also very important when assessing default probabilities over time.     

Modelling default contagion using multivariate phase-type distributions

We model dynamic credit portfolio dependence by using default contagion in an intensity-based framework. Two different portfolios (with ten obligors), one in the European auto sector, the other in the European financial sector, are calibrated against their market CDS spreads and the corresponding CDS-correlations. After the calibration, which are perfect for the banking portfolio, and good for the auto case, we study several quantities of importance in active credit portfolio management. For example, implied multivariate default and survival distributions, multivariate conditional survival distributions, implied default correlations, expected default times and expected ordered default times. The default contagion is modelled by letting individual intensities jump when other defaults occur, but be constant between defaults. This model is translated into a Markov jump process, a so called multivariate phase-type distribution, which represents the default status in the credit portfolio. Matrix-analytic methods are then used to derive expressions for the quantities studied in the calibrated portfolios.     

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.      

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.     

A Dynamic Analysis of Fixed- and Adjustable-Rate Mortgage Terminations

This paper provides a side-by-side comparison of loan-level statistical models for fixed- and adjustable-rate mortgages. Multinomial logit models for quarterly conditional probabilities of default and prepayment are estimated. We find that the estimated impacts of embedded option values for prepayment and default are generally quite similar across both FRM and ARM loans, providing additional empirical support for the basic predictions of the options theory. We also find that differences in estimates of conditional probabilities of prepayment and default associated with mortgage age, origination period, original LTV, and relative loan size, indicate the continued significance of these other economic and demographic factors for empirical models of mortgage terminations.     

Structural default model with mutual obligations

In this paper we consider mutual obligations in an interconnected bank system and analyze their impact on the joint and marginal survival probabilities for individual banks. We also calculate prices of the corresponding credit default swaps and first-to-default swaps. To make the role of mutual obligations more transparent, we develop a simple structural default model with banks’ assets driven by correlated multidimensional Brownian motion with drift. We calculate closed form expressions for many quantities of interest and use them for the efficient model calibration. We demonstrate that mutual obligations have noticeable impact on the system behavior.     

The credit rating process and estimation of transition probabilities: A Bayesian approach

The Basel II Accord requires banks to establish rigorous statistical procedures for the estimation and validation of default and ratings transition probabilities. This raises great technical challenges when sufficient default data are not available, as is the case for low default portfolios. We develop a new model that describes the typical internal credit rating process used by banks. The model captures patterns of obligor heterogeneity and ratings migration dependence through unobserved systematic macroeconomic shocks. We describe a Bayesian hierarchical framework for model calibration from historical rating transition data, and show how the predictive performance of the model can be assessed, even with sparse event data. Finally, we analyze a rating transition data set from Standard and Poor's during 1981–2007. Our results have implications for the current Basel II policy debate on the magnitude of default probabilities assigned to low risk assets.     

An Empirical Investigation of CDS Spreads Using a Regime-Switching Default Risk Model

Default risk in equity returns can be measured by structural models of default. In this article we propose a credit warning signal (CWS) based on the Merton Default Risk (MDR) model and a Regime-Switching Default Risk (RSDR) model. The RSDR model is a generalization of the MDR model, comprises regime-switching asset distribution dynamics, and thus produces more realistic default probability estimates in cases of deteriorating credit quality. Alternatively, it reduces to the MDR model. Using a dataset of U.S. credit default swap (CDS) contracts around the 2007-8 crisis we construct rating-based indices to investigate the MDR and RSDR implied probabilities of default in relation to the market-observed CDS spreads. The proposed CWS measure indicates an increase in implied default probabilities several months ahead of notable increases in CDS spreads.     

Measuring Credit Risk of Individual Corporate Bonds in US Energy Sector

In this paper, using the measures of the credit risk price spread (CRiPS) and the standardized credit risk price spread (S-CRiPS) proposed in Kariya’s (A CB (corporate bond) pricing model for deriving default probabilities and recovery rates. Eaton, IMS Collection Series: Festschrift for Professor Morris L., 2013) corporate bond model, we make a comprehensive empirical credit risk analysis on individual corporate bonds (CBs) in the US energy sector, where cross-sectional CB and government bond price data is used with bond attributes. Applying the principal component analysis method to the S-CRiPSs, we also categorize individual CBs into three different groups and study on their characteristics of S-CRiPS fluctuations of each group in association with bond attributes. Secondly, using the market credit rating scheme proposed by Kariya et al. (2014), we make credit-homogeneous groups of CBs and show that our rating scheme is empirically very timely and useful. Thirdly, we derive term structures of default probabilities for each homogeneous group, which reflect the investors’ views and perspectives on the future default probabilities or likelihoods implicitly implied by the CB prices for each credit-homogeneous group. Throughout this paper it is observed that our credit risk models and the associated measures for individual CBs work effectively and can timely provide the market credit information evaluated by investors.     

Modeling default probabilities: The case of Brazil

Using disaggregated data from the Brazilian stock market, we calculate default probabilities for 30 different economic sectors. Empirical results suggest that domestic macroeconomic factors can explain these default probabilities. In addition, we construct the Minimum Spanning Tree (MST) and the ultrametric hierarchical tree with the MST based on default probabilities to disclose common trends, which reveals that some sectors form clusters. The results of this paper imply that macroeconomic variables have distinct effects on default probabilities, which is important to take into account in credit risk modeling and the generation of stress test scenarios.     

A network model of systemic risk: stress testing the banking system1

Although there are many definitions of systemic risk, most agree that it manifests itself by an initial shock that results in the failure of one or more banks and then spreads out to the entire system by a contagion mechanism which can result in the failure of more banks in the system. Assuming that bank failures in the initial shock are randomly dependent on the failure probabilities of the individual banks and that the ensuing contagion process is deterministic, depending on interbank exposures, in this paper we propose a network model to analyse systemic risk in the banking system that, in contrast to other proposed models, seeks to obtain the probability distribution of losses for the financial system resulting from the shock/contagion process. Thus, calculating the probabilities of joint failures by simulation and assuming that the matrix of bilateral interbank exposures is known, we represent systemic risk in the financial system by means of a graph and use discrete modelling techniques to characterize the dynamics of contagion and corresponding losses within the network. The probability distribution of losses, risk profile for the Mexican banking system, is obtained through an efficient, complete enumeration procedure of all possible bank default events in the system. This, in turn, allows the use of the wide variety of well-established risk measures to describe the fragility of the financial system. Additionally, the model allows us to perform stress tests along both the bank default probabilities and the interbank exposures and is used to assess the risk of the Mexican banking system. Copyright © 2009 John Wiley & Sons, Ltd.     

Valuation of default-sensitive claims under imperfect information

We propose a valuation method for financial assets subject to default risk, where investors cannot observe the state variable triggering the default but observe a correlated price process. The model is sufficiently general to encompass a large class of structural models and can be seen as a generalization of the model of Duffie and Lando (Econometrica 69:633–664, [2001]). In this setting we prove that the default time is totally inaccessible in the market’s filtration and derive the conditional default probabilities and the intensity process. Finally, we provide pricing formulas for default-sensitive claims and illustrate in particular examples the shapes of the credit spreads.      

Default Risk in Corporate Yield Spreads

An important research question examined in the credit risk literature focuses on the proportion of corporate yield spreads attributed to default risk. This topic is reexamined in light of the different issues associated with the computation of default probabilities obtained from historical default data. We find that the estimated default risk proportion in corporate yield spreads is sensitive to the ex ante estimated term structure of default probabilities used as inputs. This proportion can become a large fraction of the spread when sensitivity analyses are made with respect to the period over which the probabilities are estimated and the recovery rates.     

Dynamic credit default swap curves in a network topology

Systemically important banks are connected and their default probabilities have dynamic dependencies. An extraction of default factors from cross-sectional credit default swap (CDS) curves allows us to analyze the shape and the dynamics of default probabilities. In extending the Dynamic Nelson Siegel (DNS) model to an across firm multivariate setting, and employing the generalized variance decomposition of Diebold and Yilmaz [On the network topology of variance decompositions: Measuring the connectedness of financial firms. J. Econom., 2014, 182(1), 119–134], we are able to establish a DNS network topology. Its geometry yields a platform to analyze the interconnectedness of long-, middle- and short-term default factors in a dynamic fashion and to forecast the CDS curves. Our analysis concentrates on 10 financial institutions with CDS curves comprising of a wide range of time-to-maturities. The extracted level factor representing long-term default risk shows a higher level of total connectedness than those derived for short-term and middle-term default risk, respectively. US banks contributed more to the long-term default spillover before 2012, whereas European banks were major default transmitters during and after the European debt crisis, both in the long-term and short-term. The comparison of the network DNS model with alternatives proposed in the literature indicates that our approach yields superior forecast properties of CDS curves.