# Robust and Consistent Estimation of Generators in Credit Risk

## TL;DR

This paper improves the estimation of generator matrices in credit risk models by providing closed-form formulas for the EM algorithm, ensuring convergence, and demonstrating superior performance over existing methods in benchmark tests.

## Contribution

It introduces directly computable formulas for the EM algorithm in CTMCs, proves convergence under weak conditions, and benchmarks the method showing improved accuracy and efficiency.

## Key findings

- Faster computation of generator matrices with closed-form formulas.
- Proven convergence of the EM algorithm under weak conditions.
- Outperforms existing algorithms in credit risk benchmarks.

## Abstract

Bond rating Transition Probability Matrices (TPMs) are built over a one-year time-frame and for many practical purposes, like the assessment of risk in portfolios or the computation of banking Capital Requirements (e.g. the new IFRS 9 regulation), one needs to compute the TPM and probabilities of default over a smaller time interval. In the context of continuous time Markov chains (CTMC) several deterministic and statistical algorithms have been proposed to estimate the generator matrix. We focus on the Expectation-Maximization (EM) algorithm by Bladt and Sorensen (2005) for a CTMC with an absorbing state for such estimation. This work's contribution is threefold. Firstly, we provide directly computable closed-form expressions for quantities appearing in the EM algorithm and associated information matrix, allowing to easily approximate confidence intervals. Previously, these quantities had to be estimated numerically and considerable computational speedups have been gained. Secondly, we prove convergence to a single set of parameters under very weak conditions (for the TPM problem). Finally, we provide a numerical benchmark of our results against other known algorithms, in particular, on several problems related to credit risk. The EM algorithm we propose, padded with the new formulas (and error criteria), outperforms other known algorithms in several metrics, in particular, with much less overestimation of probabilities of default in higher ratings than other statistical algorithms.