A Comparative Anatomy of Credit Risk Models

Within the past two years, important advances have been made in modeling credit risk at the portfolio level. Practitioners and policy makers have invested in implementing and exploring a variety of new models individually. Less progress has been made, however, with comparative analyses. Direct comparison often is not straightforward, because the different models may be presented within rather different mathematical frameworks. This paper offers a comparative anatomy of two especially influential benchmarks for credit risk models, J.P. Morgan’s CreditMetrics and Credit Suisse Financial Product’s CreditRisk+. We show that, despite differences on the surface, the underlying mathematical structures are similar. The structural parallels provide intuition for the relationship between the two models and allow us to describe quite precisely where the models differ in functional form, distributional assumptions, and reliance on approximation formulae. We then design simulation exercises which evaluate the effect of each of these differences individually. JEL Codes: G31, C15, G11 ∗The views expressed herein are my own and do not necessarily reflect those of the Board of Governors or its staff. I would like to thank David Jones for drawing my attention to this issue, and for his helpful comments. I am also grateful to Mark Carey for data and advice useful in calibration of the models, and to Chris Finger and Tom Wilde for helpful comments. Please address correspondence to the author at Division of Research and Statistics, Mail Stop 153, Federal Reserve Board, Washington, DC 20551, USA. Phone: (202)452-3705. Fax: (202)452-5295. Email: 〈mgordy@frb.gov〉. Over the past decade, financial institutions have developed and implemented a variety of sophisticated models of value-at-risk for market risk in trading portfolios. These models have gained acceptance not only among senior bank managers, but also in amendments to the international bank regulatory framework. Much more recently, important advances have been made in modeling credit risk in lending portfolios. The new models are designed to quantify credit risk on a portfolio basis, and thus have application in control of risk concentration, evaluation of return on capital at the customer level, and more active management of credit portfolios. Future generations of today’s models may one day become the foundation for measurement of regulatory capital adequacy. Two of the models, J.P. Morgan’s CreditMetrics and Credit Suisse Financial Product’s CreditRisk+, have been released freely to the public since 1997 and have quickly become influential benchmarks. Practitioners and policy makers have invested in implementing and exploring each of the models individually, but have made less progress with comparative analyses. The two models are intended to measure the same risks, but impose different restrictions and distributional assumptions, and suggest different techniques for calibration and solution. Thus, given the same portfolio of credit exposures, the two models will, in general, yield differing evaluations of credit risk. Determining which features of the models account for differences in output would allow us a better understanding of the sensitivity of the models to the particular assumptions they employ. Unfortunately, direct comparison of the models is not straightforward, because the two models are presented within rather different mathematical frameworks. The CreditMetrics model is familiar to econometricians as an ordered probit model. Credit events are driven by movements in underlying unobserved latent variables. The latent variables are assumed to depend on external “risk factors.” Common dependence on the same risk factors gives rise to correlations in credit events across obligors. The CreditRisk+ model is based instead on insurance industry models of event risk. Instead of a latent variable, each obligor has a default probability. The default probabilities are not constant over time, but rather increase or decrease in response to background macroeconomic factors. To the extent that two obligors are sensitive to the same set of background factors, their default probabilities will move together. These co-movements in probability give rise to correlations in defaults. CreditMetrics and CreditRisk+ may serve essentially the same function, but they appear to be constructed quite differently. This paper offers a comparative anatomy of CreditMetrics and CreditRisk+. We show that, despite differences on the surface, the underlying mathematical structures are similar. The structural parallels provide intuition for the relationship between the two models and allow us to describe quite precisely where the models differ in functional form, distributional assumptions, and reliance on approximation formulae. We can then design simulation exercises which evaluate the effect of these differences individually. We proceed as follows. Section 1 presents a summary of the CreditRisk+ model, and introduces a restricted version of CreditMetrics. The restrictions are imposed to facilitate direct comparison of CreditMetrics and CreditRisk+. While some of the richness of the full CreditMetrics implementation is sacrificed, the essential mathematical characteristics of the model are preserved. Our