Smoothing Algorithms by Constrained Maximum Likelihood

In the process of loan pricing, stress testing, capital allocation, modeling of PD term structure, and IFRS9 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 non-default rating than a farther away non-default rating. In practice, sample estimates for rating level default rate or rating migration probability do not always respect this monotonicity rule, and hence the need for smoothing approaches. Regression and interpolation techniques are widely used for this purpose. A common issue with these approaches 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 modeller 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.