Credit Risk: Simple Closed Form Approximate Maximum Likelihood Estimator

TL;DR

This paper introduces simple closed-form approximations to the maximum likelihood estimator for default probabilities in credit risk models, validated through empirical and simulated data, especially effective when default probabilities are small.

Contribution

It develops and rigorously analyzes a new closed-form approximation to the MLE for default probability models with Gaussian covariates, applicable under small probability regimes.

Findings

Approximate estimator performs similarly or slightly worse than MLE when correctly specified.

Both estimators are similarly affected by model misspecification.

Estimators become insensitive to data increases beyond a certain point.

Abstract

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 practically reasonable asymptotic regime where the default probabilities are small, say 1-3% annually, the number of firms and the time period of data available is reasonably large, we rigorously show that the proposed estimator behaves similarly or slightly worse than the MLE when the underlying model is correctly specified. For more realistic case of model misspecification, both estimators are seen to be equally good, or equally bad. Further, beyond a point, both are more-or-less insensitive to increase in data. These conclusions are validated on empirical and simulated data. The proposed approximations should also have applications outside finance, where logit-type models are used and probabilities of interest are small.