In a multivariate setting, the dependence between random variables has to be accounted for
modeling purposes. Various of multivariate risk measures have been developed, including
bivariate lower and upper orthant Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR). The
robustness of their estimators has to be discussed with the help of sensitivity functions, since
risk measures are estimated from data.
In this thesis, several univariate risk measures and their multivariate extensions are presented.
In particular, we are interested in developing the bivariate version of a robust risk
measure called Range Value-at-Risk (RVaR). Examples with different copulas, such as the
Archimedean copula, are provided. Also, properties such as translation invariance, positive
homogeneity and monotonicity are examined. Consistent empirical estimators are also presented
along with the simulation. Moreover, the sensitivity functions of the bivariate VaR,
TVaR and RVaR are obtained, which confirms the robustness of bivariate VaR and RVaR
as expected.