Credit Risk: Simple Closed-Form Approximate Maximum Likelihood Estimator

We consider discrete default intensity-based and logit-type reduced-form models for conditional default probabilities for corporate loans where we develop simple closed-form approximations to the maximum likelihood estimator (MLE) when the underlying covariates follow a stationary Gaussian process. In a practical asymptotic regime where the default probabilities are small, say less than $3\mathrm{\%}$ annually, and the number of firms and the time period of data available are reasonably large, we rigorously show that the proposed estimator behaves similarly or slightly worse than the MLE when the underlying model is correctly specified. These approximations and conclusions are extended to the regularized MLE. For a more realistic case of model misspecification, both the estimators are seen to be equally good or equally bad. Further, in the presence of zero mean additive corruption in the data, the proposed estimator is somewhat better than the MLE. These conclusions are validated on empirical and simulated data.