Multivariate Extreme Value Theory and Its Usefulness in Understanding Risk

This paper gathers recent results in the analysis of multivariate extreme values and illustrates their actuarial application. We review basic and essential background on univariate extreme value theory and stochastic dependence and then provide an introduction to multivariate extreme value theory. We present important concepts for the analysis of multivariate extreme values and collect research results in this area. We draw particular attention to issues related to extremal dependence and show the importance of model selection when fitting an upper tail copula to observed joint exceedances. These ideas are illustrated on four data sets: loss amount and allocated loss adjustment expense under insurance company indemnity claims, lifetimes of pairs of joint and lastsurvivor annuitants, hurricane losses in two states, and returns on two stocks. In each case the extremal dependence structure has an important financial impact.