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UBC Theses and Dissertations

Extreme value modelling with application to reverse stress testing Zhou, Menglin


Reverse stress testing of a financial portfolio aims to identify scenarios for risk factor changes that lead to a specified adverse portfolio outcome. The stress scenarios of interest naturally need to be extreme yet plausible. A statistical formulation of these requirements is to define a stress scenario at extreme threshold as the mode of the conditional density of the random vector of risk factor changes given that the loss on the portfolio exceeds the threshold. This thesis proposes and studies four different estimators of stress scenarios, varying in the assumptions made on the multivariate distribution of the risk factor changes and associated portfolio loss. The first one is built upon the work of Glasserman et al. [2015], estimating the conditional mode from conditional mean under the assumption of elliptical symmetry and regularly variation. We modify the estimator by replacing the empirical estimator of the conditional mean with an estimator based on extreme value theory. The estimator is shown to be consistent and asymptotically normally distributed. The second method relaxes the assumption of elliptical symmetry in the first method, deriving an asymptotic expression for the scaled conditional mode assuming that the random vector of risk factor changes asymptotically follows a skew-elliptical distribution. The third method aims to relax the model assumptions further. The idea is to estimate the conditional mode for a lower threshold using a non-parametric method and then extrapolate to the desired high threshold using a probabilistically justified extrapolation factor. Asymptotic theory is developed for this mode-based estimator. In addition to the above three methodologies based on extreme value techniques, we also present a pragmatic solution to the important task of reverse stress testing that banks and insurance companies rely on to ensure financial stability of their operations. The method utilizes a flexible multivariate modelling framework based on vine copulas, allowing a wide range of stochastic properties of the data. For each method, we investigate the finite-sample performance via simulation studies. Case studies with real-life financial portfolios suggest that the proposed methodologies can generate stress scenarios meeting the requirements of severity and plausibility.

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