Abstract

We present two methodologies on the estimation of rating transition probabilities within Markov and non-Markov frameworks. We first estimate a continuous-time Markov chain using discrete (missing) data and derive a simpler expression for the Fisher information matrix, reducing the computational time needed for the Wald confidence interval by a factor of a half. We provide an efficient procedure for transferring such uncertainties from the generator matrix of the Markov chain to the corresponding rating migration probabilities and, crucially, default probabilities. For our second contribution, we assume access to the full (continuous) data set and propose a tractable and parsimonious self-exciting marked point processes model able to capture the non-Markovian effect of rating momentum. Compared to the Markov model, the non-Markov model yields higher probabilities of default in the investment grades, but also lower default probabilities in some speculative grades. Both findings agree with empirical observations and have clear practical implications. We use Moody's proprietary corporate credit rating data set. Parts of our implementation are available in the R package ctmcd.

Highlights

  • Credit risk modelling and financial regulations have received added attention from Mathematics and Economics disciplines since the 2008 financial crash

  • The question we look to answer is, are we capturing the data better or just overfitting? To do this we calculate the Bayesian Information Criterion (BIC), it is a common test used in statistics for model selection and is known to penalize model complexity more than other statistical tests, such as the Akaike information criterion (see (Claeskens and Hjort 2008, Chapter 3))

  • In the first part of this paper we have shown how one can evaluate errors in the transition matrices of continuous-time Markov chains at the level of discretely observed data using new closed-form expressions

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Summary

Introduction

Credit risk modelling and financial regulations have received added attention from Mathematics and Economics disciplines since the 2008 financial crash. When one can access the full data set (continuous-time observations), it is possible to construct tractable models that capture non-Markov effects and this is one of our contributions. If the company transitions from B to A, it is instead rated A+ which has a smaller probability of downgrade than A This construction allows us to maintain the Markov property; we must calibrate many more parameters and, in real-world data, we do not observe a company belonging to the excited or non-excited state. D’Amico et al (2016) apply a semi-Markov model to capture the observed effect that companies move through states not following an exponential distribution They still rely on the Markov transition structure and they need to expand the state space in order to include momentum.

Data description
Direct Differentiation for Gradient and Hessian of the Likelihood
The Delta method - Confidence Intervals for probabilities
Extending Markov Processes to Capture Rating Momentum
Testing for non-Markovian phenomena
Our new Model to capture Rating Momentum
An MCMC calibration algorithm for the model
Bayesian Information Criterion
Examples and testing
Summary
Acknowledgements and Funding
Full Text
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