Credit-Risk Modelling

Starting from first principles, this chapter follows a chain of common sense logic to construct an initial model. Beginning from a sequence of independent Bernoulli trials, a binomial default-loss distribution emerges. A critical assumption of this introductory approach, however, is statistical independence between the default of various obligors in one’s portfolio. While of great mathematical convenience, this is an indefensible supposition. Not only is it economically questionable, imposition of modest amounts of default dependence has dramatic effects on risk estimates. It thus lacks conservatism. All realistic and reasonable credit-risk models, therefore, place the dependence structure at the centre of their methodology. That said, many lessons can still be drawn from this simple approach, which underpins many extant default-risk models. Maintaining the presumption of independence of default, the following discussion thus walks through its implementation—both numerically and analytically—in the context of our concrete portfolio example. It also explores its asymptotic properties and provides an alternative entry point, which will aid in expanding this foundation in the following chapters.

The independent-default model is deeply flawed. Not only is it fair to argue that dependence is the single most important aspect of credit-risk modelling, but the tails of the associated loss distribution are overly thin and its asymptotic behaviour is simply too well behaved. This chapter offers a family of approaches—generally referred to as mixture or actuarial models—to address each of these shortcomings. The principle idea behind this new methodology is the randomization of the default probability. Practically, a common state variable is introduced, which induces default dependence among all obligors. Conditionally, default events remain independent, but unconditionally they are related through the realization of the systematic state variable. The structure of the default-loss distribution thus depends on the statistical properties of one’s state-variable choice mixed with underlying binomial-default structure. A variety of possible choices are investigated, convergence properties are explored, and our portfolio example is examined from both analytic and numerical perspectives. Through the law of rare events, a separate class of Poisson-mixture models are explored ultimately leading to the celebrated CreditRisk+ model used widely in practice and first suggested by Wilde (1997, CreditRisk+: A credit risk management framework, Credit Suisse First Boston).

The binomial- and Poisson-mixture models offer a useful range of possible credit-risk implementations. Not only do we actively seek alternative approaches, but the mixture models are silent on the ultimate reason for default. In other words, the actuarial or mixture methodology is reduced form. A competing structural modelling family, referred to as the set of threshold models, is offered in this chapter. This approach is, in fact, a clever combination of a pragmatic, latent-variable approach and the classic Merton (1974, Journal of Finance, 29, 449–470) model. Default, therefore, occurs when a statistical proxy of the firm’s asset value falls below a predetermined threshold. The eponymous threshold is inferred from the obligor’s unconditional default probability. The basic structure and convergence properties of this technique are first examined in the Gaussian setting. This initial logic is then generalized—allowing for both thicker tails and tail dependence—through the introduction of the class of normal-variance mixture models. Parameter-calibration techniques are also reviewed and all models are exhaustively applied to our ongoing portfolio example.

The path-breaking work, Merton (1974, Journal of Finance, 29, 449–470), not only addressed a number of important asset-pricing and corporate-finance questions, but was also the genesis of the field of credit-risk modelling. This chapter focuses exclusively on this approach. Not only would it be an injustice to ignore this still-pertinent model, but it offers a range of useful insights into the class of threshold models. The Merton (1974, Journal of Finance, 29, 449–470) framework was conceived and developed in a continuous-time, mathematical-finance setting. To address this complicating factor, a significant amount of effort is allocated to the basic intuition, notation, and mathematical structure leading to a motivating discussion regarding the notion of geometric Brownian motion. Armed with this detail, the chapter proceeds to investigate two possible implementations of Merton (1974, Journal of Finance, 29, 449–470)’s model, which we term the indirect and direct approaches. The indirect approach will turn out to be quite familiar, whereas the direct method requires a significant amount of heavy lifting for its implementation. As in previous chapters, parameter calibration options are explored and both methods are applied to practical portfolio examples.

Quantitative analysts seek to construct useful models to assess the magnitude of credit risk in their portfolios and inform associated management decisions. Regulators, in a slightly alternative context, perform a similar task. They face, however, a rather different set of constraints and objectives. Regulators, in fact, use standardized models to promote fairness, a level playing field, monitor the solvency of individual entities, and enhance overall economic stability. Much can be learned from the regulatory perspective; indeed, actions and trends in the regulatory field are an important diagnostic for internal modellers. To underscore this point, this chapter focuses principally on the widely used internal-ratings-based (IRB) approach proposed by the Basel Committee on Banking Supervision. Closer inspection reveals that this approach is founded on a portfolio-invariant version of the Gaussian threshold model. Portfolio invariance implies that the risk of the portfolio is independent of the overall portfolio structure and depends only on the characteristics of the individual exposures. Expedient rather than realistic, this choice reduces the computational and system burden on regulated organizations, but ignores concentration risk. After applying the IRB approach to our portfolio example, we then carefully explore Gordy (2003, Journal of Financial Intermediation, 12, 199–232)’s granularity adjustment, which was proposed to address this shortcoming.

VaR and expected-shortfall are extremely useful portfolio risk measures. They are, however, reasonably difficult to compute and often challenging to interpret. Effective use and communication of one’s risk measures nonetheless requires significant insight into their underlying structure. In particular, it is inordinately useful to understand how individual obligors contribute to one’s risk estimates. This is referred to as risk attribution and is, it must be admitted, a non-trivial undertaking. To address this important area and do justice to its complexity, we allocate the entire chapter to this task. We begin with a surprising relationship between risk attribution and conditional expectation, which subsequently motivates development of a general-purpose numerical algorithm. To offer alternatives to this computationally intensive and often noisy approach, we examine two analytical techniques. The first, termed the normal approximation, provides insight into the underlying problem, but is unfortunately not a robust solution. The saddlepoint approximation, conversely, offers an accurate and fast alternative in the one-factor setting; it also enhances our technical understanding of the underlying risk measures. As always, analysis of all approaches are accompanied by detailed derivations and concrete examples.

Stochastic-simulation, or Monte-Carlo, methods are used extensively in the area of credit-risk modelling. This technique has, in fact, been employed inveterately in previous chapters. Care and caution are always advisable when employing a complex numerical technique. Prudence is particularly appropriate, in this context, because default is a rare event. Unlike the asset-pricing setting, where we typically simulate expectations in the central part of the distribution, credit risk operates in the tails. As a consequence, this chapter is dedicated to a closer examination of the intricacies of the Monte-Carlo method. Working from first principles, the importance of convergence analysis and confidence intervals is highlighted. The principal shortcoming of this method, its inherent slowness, is also explained and demonstrated. This naturally leads to a discussion of the set of variance-reduction techniques employed to enhance the speed of these estimators. The chapter concludes with the investigation and implementation of the Glasserman and Li (2005, Management Science, 51(11), 1643–1656.) importance-sampling method to the t-threshold model. This method, which employs the so-called Esscher transform, has close conceptual links to the saddlepoint technique introduced in the previous chapter.

If default dependence is the heart of credit-risk modelling, then an empirical estimate of its magnitude is of primordial importance. Unlike estimation of default probabilities, addressed in the previous chapter, the characterization of default dependence is model dependent rendering this task more difficult. Since different models incorporate the relationship between obligor defaults in alternative ways, dependence is governed by some subset of a model’s parameters. As usual, a variety of techniques are presented, examined, and concretely implemented. The first, based on the method of moments, applies quite generally and is conceptually similar to the calibration techniques employed in previous chapters. A second approach, using observed default outcomes, exploits conditional independence to build a likelihood function and applies in both mixture and threshold settings. This permits use of the maximum-likelihood framework for the production of both point and interval estimates. The final, somewhat complex and fragile, approach is only applicable to the family of threshold models. It enjoys the advantage of using all transition data, but simultaneously requires inference of the unobservable global state variable values. The robustness of the final two techniques are assessed within separate simulation studies.