Statistical inference for Markov chains with applications to credit risk
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Statistical inference for Markov chains with applications to credit risk Linda Möstel1 · Marius Pfeuffer1 · Matthias Fischer1 Received: 23 April 2018 / Accepted: 10 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The focus of this paper is on the derivation of confidence and credibility intervals for Markov chains when discrete-time, continuous-time or discretely observed continuous-time data are available. Thereby, our contribution is threefold: First, we discuss and compare multinomial confidence regions for the rows of discrete-time Markov transition matrices in the light of empirical characteristics of credit rating migrations. Second, we derive an analytical expression for the expected Fisher information matrix of a continuous-time Markov chain which is used to construct credibility intervals using a non-informative Jeffreys prior distribution and a Metropolis-Hastings Algorithm. Third, we concretize profile and estimated/pseudo likelihood based confidence intervals in the continuous-time data settings, which in contrast to asymptotic normality based intervals explicitly consider non-negativity constraints for the parameters. Furthermore, we illustrate the described methods by Moody’s corporate ratings data with exact continuous-time transitions. Keywords Transition matrix · Generator matrix · Bootstrap · Jeffreys prior · Metropolis-Hastings algorithm
1 Motivation Forecasting a counterparty’s creditworthiness is a crucial and fundamental issue when banks are required to quantify the credit risk related to their business activities. Typically, counterparties are assigned to different (homogenous) rating classes, where each rating (grade) represents the probability that a counterparty will default (i.e., not be able or willing to meet the agreed payments) within the next year. In the course of time, the counterparty’s creditworthiness may change and, hence, the original rating grade is also expected to change (termed as rating migration) if the counterparty would
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Marius Pfeuffer [email protected] Institute for Statistics and Econometrics, Lange Gasse 20, 90403 Nuremberg, Germany
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be re-rated. Hence, the modeling of (credit) rating migrations constitutes an important issue in quantitative risk management. In this context, discrete- and continuous-time Markov chains are very popular statistical models for which transition probability or transition intensity parameters can be estimated from historical rating migration data. Credit rating migrations may then be further applied as an input factor for credit portfolio models, financial stress tests as well as the pricing of debt or derivative financial instruments, see e.g., Trück and Rachev (2009). Moreover, they can be employed in the estimation of expected credit losses as required by the FASB accounting regulation FAS 5, see e.g., Betancourt (1999) or lifetime expected losses as required by the novel accounting standard IFRS 9 which took effect on January 1, 2018, see e.g., Pfeuffer et al. (2019)
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