Bayesian analysis of default and credit migration : latent factor models for event count and time-to-event data
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This thesis develops Bayesian models to explain credit default and migration risk. Credit risk models used in practice are based on an assumption of conditionally independent events given a realization of systematic risk factors. The systematic risk can be modelled with both observed and unobserved factors. On the one hand we consider generalised linear mixed models (GLMMs) for default count data where random e ects account for unobserved factor risk. On the other hand we consider survival models with shared frailties to model unobserved factors in time-to-default and time-to-rating-transition data. The latter models are developed in the Anderson-Gill counting process framework for the Cox proportional hazards model to allow multiple events and time-dependent covariates. Using Standard and Poor's data on default and rating transitions we control for observed macroeconomic factors in the xed e ect parts of the models. We allow the latent factors to have autoregressive time series structure. The results from both kinds of model show clear evidence of heterogeneity between industry sectors/countries and time period suggesting that di erent latent factor effects are present in di erent sectors. This is an important message that should be accounted for in risk analyses. We implement Bayesian inference for all our models and use the MCMC approach (Gibbs sampling). We show some tractable model formulations that capture the main sources and implement Bayesian model choice procedures to select the most explanatory models. There are couple of contributions in this thesis: First, this is an analysis of industry e ects on default and migration rates using vector-valued random e ects in default count models and vector-valued dynamic frailties in time-to-event/survival models. While this has been done before in models for default counts (McNeil-Wendin) it is quite novel for time-to-event models. Koopman, Lucas and Schwaab (2012) which has some similarities but the estimation is by Monte Carlo maximum likelihood, not by Bayesian methods. Second, estimation of rating transition model with shared dynamic frailties for di erent industry sectors and macroeconomic covariates using Bayesian techniques (MCMC). This is a new model which is based on a simpler model used in medical statistics (Manda & Mayer(2005)) that has been adapted and extended for the credit risk application. We show how to estimate the new model using a Bayesian approach. Finally, we use the model to compute point-in-time dynamic estimates of rating transition probabilities for di erent industry sectors and forecast these into the future, while taking into account macroeconomic factors. This can be very useful for risk management applications and economic scenario generation.