This issue consists of three papers, two of which, reflecting some econometric zeitgeist, have a great deal in common. The first paper, "Estimating long-run probability of default, asset correlation and portfolio-level probability of default using Vasicek models" by Bill Huajian Yang, proposes a family of Vasicek-type models for estimating portfolio level probability of default (PD). With these models, both asset correlation and long-run PD for a risk-homogenous portfolio have analytical solutions. In addition, longer external time series for market and macroeconomic variables can be included, and the traditional asymptotic maximum likelihood approach can be shown to be equivalent to least squares regression, which greatly simplifies parameter estimation. The author explicitly quantifies the contribution of uncertainty to an increase in long-run PD. To validate the proposed models, the author estimates asset correlations for thirteen industry sectors using corporate annual default rates from Standard & Poor's for 1981-2011 and long-run PD and asset correlation for a US commercial portfolio, using US delinquent rates for commercial and industry loans from the US Federal Reserve. This is a valuable exercise but represents the outlier in this issue.
Our second paper, "A mixture vector autoregressive framework to capture extreme events in macro-prudential stress tests" by Paolo Guarda, Abdelaziz Rouabah and John Theal, examines the impact of high-impact and low-probability events that, the authors claim, are fundamental aspects of systemic financial stress. These unlikely adverse events arise from the extreme tail of a probability distribution and are therefore very poorly captured by traditional econometric models that rely on the assumption of normality. In order to address the problem of extreme tail events in a stress testing framework, the authors adopt a mixture vector autoregressive (MVAR) framework that allows for a multimodal distribution of the residuals. They use permutation tests to facilitate a comparison between the respective results from a vector autoregressive (VAR) model and an MVAR model, and the results suggest that the mixture of distributions allows for a better assessment of the effect that adverse shocks have on counterparty credit risk, on the real economy and on banks' capital requirements.
In the issue's third paper, "The daily returns of the Portuguese Stock Index: a distributional characterization" by Sameer R. Rege, João C. A. Teixeira and António G. de Menezes, the authors compare the fitting of the normal, generalized hyperbolic, normal inverse Gaussian and Student t distributions to the daily returns of the Portuguese Stock Index PSI-20 over the period 1992-2013. They find that the distribution of the actual returns of the PSI-20 exhibits much higher kurtosis and more extreme values compared with the normal distribution. In general, and in particular in the tails, the best fit is provided by the Student t and generalized hyperbolic distributions. It is worth commenting that validating distributional assumptions is an important, but rather neglected, aspect of risk model validation. All three papers, perhaps implicitly in the case of the first paper and explicitly in the case of the second and third, focus on the important but neglected role of distributional assumptions for returns and spreads in the validation process.
Finally, we are pleased to include two detailed comment pieces. The first comment, from Lawrence R. Forest Jr., Gaurav Chawla and Scott D. Aguais, is in response to the paper "A methodology for point-in-time-through-the-cycle probability of default decomposition in risk classification systems" by Magnus Carlehed and Alexander Petrov thatwas published last year in this journal; the second is a reply from the authors of the original paper. This kind of lively, informed and robust debate is something we encourage at The Journal of Risk Model Validation: we aim to inspire vigorous discussion in the risk model validation field.
Trinity College, University of Cambridge
A mixture vector autoregressive framework to capture extreme events in macro-prudential stress tests
Estimating long-run probability of default, asset correlation and portfolio-level probability of default using Vasicek models
Comment in response to “A methodology for point-in-time–through-the-cycle probability of default decomposition in risk classification systems” by M. Carlehed and A. Petrov