Extreme Value Theory in Finance and Insurance

When modelling high-dimensional data, dimension reduction techniques such as principal component analysis are often used. In the first part of this thesis we will focus on two drawbacks of classical PCA. First, interpretation of classical PCA is often challenging because most of the loadings are neither very small nor very large in absolute value.  Second, classical PCA can be heavily distorted by outliers since it is based on the classical covariance matrix. In order to resolve both problems, we present a new PCA algorithm that is robust against outliers and yields sparse PCs, i.e.  PCs with many zero loadings. The approach is based on the ROBPCA algorithm that generates robust but non-sparse loadings. The construction of the new ROSPCA method is detailed, as well as a selection criterion for the sparsity parameter. An extensive simulation study and a real data example are performed, showing that it is capable of accurately finding the sparse structure of datasets, even when challenging outliers are present. 

Stock market crashes such as Black Monday in 1987 and catastrophes such as earthquakes are examples of extreme events in finance and insurance, respectively. They are large events with a considerable impact that occur seldom. Extreme value theory (EVT) provides a theoretical framework to model extreme values such that e.g. risk measures can be estimated based on available data. In the second part of this PhD thesis we focus on applications of EVT that are of interest to finance and insurance.

A Black Swan is an improbable event with massive consequences. We propose a way to investigate if the 2007-2008 financial crisis was a Black Swan event for a given bank based on weekly log-returns. This is done by comparing the tail behaviour of the negative log-returns before and after the crisis using techniques from extreme value methodology. We illustrate this approach with Barclays and Credit Suisse data, and then link the differences in tail risk behaviour between these banks with economic indicators.

The earthquake engineering community, disaster management agencies and the insurance industry need models for earthquake magnitudes to predict possible damage by earthquakes. A crucial element in these models is the area-characteristic, maximum possible earthquake magnitude. The Gutenberg-Richter distribution, which is a (doubly) truncated exponential distribution, is widely used to model earthquake magnitudes. Recently, Aban et al. (2006) and Beirlant et al. (2016) discussed tail fitting for truncated Pareto-type distributions. However, as is the case for the Gutenberg-Richter distribution, in some applications the underlying distribution appears to have a lighter tail than the Pareto distribution. We generalise the classical peaks over threshold (POT) approach to allow for truncation effects. This enables a unified treatment of extreme value analysis for truncated heavy and light tails. We use a pseudo maximum likelihood approach to estimate the model parameters and consider extreme quantile estimation. The new approach is illustrated on examples from hydrology and geophysics. Moreover, we perform simulations to illustrate the potential of the method on truncated heavy and light tails.

The new approach can then be used to estimate the maximum possible earthquake magnitude. We also look at two other EVT-based endpoint estimators and endpoint estimators that are used in the geophysical literature. To quantify uncertainty of the point estimates for the endpoint, upper confidence bounds are also considered.  We apply the techniques to provide estimates, and upper confidence bounds, for the maximum possible earthquake magnitude in Groningen where earthquakes are induced by gas extraction. Furthermore, we compare the methods from extreme value theory and the geophysical literature through simulations.

In risk analysis, a global fit that appropriately captures the body and the tail of the distribution of losses is essential. Modelling the whole range of the losses using a standard distribution is usually very hard and often impossible due to the specific characteristics of the body and the tail of the loss distribution. A possible solution is to combine two distributions in a splicing model: a light-tailed distribution for the body which covers light and moderate losses, and a heavy-tailed distribution for the tail to capture large losses. We propose a splicing model with the flexible mixed Erlang distribution for the body and a Pareto distribution for the tail. Motivated by examples in financial risk analysis, we extend our splicing approach to censored and/or truncated data. We illustrate the flexibility of this splicing model using practical examples from reinsurance.