Journal of Risk Model Validation

Risk.net

Smoothing algorithms by constrained maximum likelihood: methodologies and implementations for Comprehensive Capital Analysis and Review stress testing and International Financial Reporting Standard 9 expected credit loss estimation

Bill Huajian Yang

Smoothing algorithms for monotonic rating level PD and rating migration probability are proposed. The approaches can be characterized as follows:

  1. These approaches are based on constrained maximum likelihood, with a fair risk scale for the estimates determined fully by constrained maximum likelihood, leading to a fair and more robust credit loss estimation.
  2. Default correlation is considered by using the asset correlation and the Merton model.
  3. Quality of smoothed estimates is assessed by the likelihood ratio test, and the impacted credit loss due to the change of risk scale for the estimates.
  4. These approaches generally outperform the interpolation method and regression models, and are easy to implement by using, for example, SAS PROC NLMIXED.

In the process of loan pricing, stress testing, capital allocation, modeling of probability of default (PD) term structure and International Financial Reporting Standard 9 expected credit loss estimation, it is widely expected that higher risk grades carry higher default risks, and that an entity is more likely to migrate to a closer nondefault rating than a more distant nondefault rating. In practice, sample estimates for the rating-level  default rate or rating migration probability do not always respect this monotonicity rule, and hence the need for smoothing approaches arises. Regression and interpolation  techniques are widely  used for this purpose. A common issue with these, however, is that the risk scale for the estimates is not fully justified, leading to a possible bias in credit loss estimates. In this paper, we propose smoothing algorithms for rating-level PD and rating migration probability. The smoothed estimates obtained by these approaches are optimal  in the sense of constrained maximum likelihood, with a fair risk scale determined by constrained maximum likelihood, leading to more robust credit loss estimation. The proposed algorithms can be easily implemented by a modeler using, for example, the SAS procedure PROC NLMIXED.  The approaches proposed in this paper will provide an effective and useful smoothing tool for practitioners in the field of risk modeling.

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