Multivariate Risk Measures and a Consistent Estimator for the Orthant Based Tail Value-at-Risk

Multivariate risk measures is a rapidly growing field of research. The advancement of dependence modelling has lent itself to this progress. Presently, a variety of parametric methods have spawned from these developments, extending univariate measures such as Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) to the multivariate context. With the inception of these measures comes the requirement to estimate them. In particular, the development of consistent estimators is crucial for applications in financial and actuarial industries alike. For adequate sample sizes, consistent estimation allows for accurate evaluation of the underlying risks without pre-imposition of a statistical model. In this thesis, several risk measures are presented in the univariate case and extended to the multivariate framework. Quantifying the dependence between risks is accomplished through the use of copulas. Several families of copulas, elliptical, Archimedean and extreme value, and examples of each are presented along with properties. With these dependence relations in place, multivariate extensions of VaR, TVaR and Conditional Tail Expectation (CTE) are all presented. Much of the focus is given to the bivariate lower and upper orthant TVaR. In particular, we are interested in developing consistent estimators for these two measures. In fact, it will be shown that the presented estimators are strongly consistent for the true parametric value. To accomplish this, the strong consistency of the orthant based VaR curve, which can be shown in two ways, is used in tandem with the dominated convergence theorem. With strong consistency established, some numerical examples are then presented demonstrating the strength of these estimators