Models and Methods for Multivariate Risk Assessment

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Risk assessment lies at the core of risk management and risk mitigation in areas such as finance, insurance, hydrology, geoscience and engineering. The key ingredients in risk assessment are a probabilistic model describing the stochastic behaviour of the underlying system, an estimation procedure linking the data at hand to the model and a functional used as a measure of risk. The overarching goal of the proposal is to contribute to a more realistic and accurate modelling of risk in a variety of multivariate settings via development of flexible modelling frameworks and effective estimation methods that are specifically designed to deal with extreme values of the underlying processes and data sparsity in the risk regions of interest. Risk is intrinsically linked to extreme and hence rare events. The approaches we investigate involve a parametric modelling component in order to gain estimation efficiency in the face of data sparsity. However, reliance is made upon asymptotic approximations in the spirit of extreme value theory in order to develop methods that work in the tail regions of considered physical processes. As data in the real world exhibit different stochastic properties, a variety of settings need to be explored in order to accurately capture important data characteristics. The proposal covers three themes based on probabilistic settings considered for the data. The financial crisis of 2007-2009 highlighted the importance of systemic financial risk and its potential to destabilize the global economy. One popular measure of systemic risk is CoVaR. A methodology is proposed to estimate CoVaR semi-parametrically within the classical framework of multivariate extreme value theory. This framework covers heavy-tailed financial data, which exhibit what is known as tail dependence. While this setting is typical for many financial time series, several empirical studies indicated situations where asymptotic independence may be a more appropriate assumption. More revealing in this context are models with light tails, of which the multivariate Gaussian distribution is a standard example. Estimation of multivariate risk measures in the setting of light-tailed distributions with the property of asymptotic independence presents particular technical challenges both in terms of probabilistic approximations and inference, which will be explored as part of the research program. Finally, the third theme is devoted to the idea of bridging between the two paradigms of asymptotic dependence and independence by considering construction of models using a shape set, whose geometry is related to extremal properties of the underlying distribution and hence has the potential to capture the two situations in a unifying way.