Asymptotic Behavior Of Extremal Events For Aggregate Dependent Risks

In the past decade there has been a rapidly growing interest in describing the dependent random variables. One driving force behind these increasing interests are the new regulatory frameworks of banking and insurance supervision. The financial industry has to come up with risk management tools that appropriately model the dependence structures between different risks and risk categories. Es-pecially, the recent2000-2001experience of high defaults frequencies in corporate bond markets have taught the lesson that joint defaults of risks may cause large losses and even jeopardize the solvency of financial institutions.To model a portfolio of risks, profound knowledge of the complete dependence structure of the random variables has to be known. Otherwise one does nor come to the right conclusions. In particular, it was understood in recent research that simple measures of dependence such as the correlation coefficient are insufficient to cover the full range of possible consequences of dependent events. Copulas are one approach that give a detailed picture of dependence structures. In this paper, we will use Archimedean copula. In most practical applications one usually is not able to justify one specific choice of a dependence structure. Reasons are that one often has not enough data to estimate a whole dependence structure. One way out of this dilemma is that one tries to get asymptotic behavior for certain subclasses of distributions. To study extremal events one then relies on this asymptotic behavior and the characteristics of its limiting distributions.Consider a portfolio of n identically distributed risks X1,..., Xn with depen-dence structure modelled by an Archimedean survival copula. It is known that if the underlying distribution function F belongs to the maximum domain of the Frechet distribution, the tail probability of the aggregate loss∑ni=1Xi scales like the probability of a large individual loss, times a proportionality factor. This fac- tor (limiting constant) depends on the dependence strength and the tail behavior of the individual risk, denoted by qfn. When F belongs to the maximum domains of the Weibull and Gumbel distributions, we have other two limiting constants qwn and qgn. The limiting properties of qfn has been studied carefully in the literature. The purposes of this thesis are to study the monotonicities and boundary values of qwn and qgn with respect to the dependence parameter and/or the tail behavior parameter, to investigate the diversification effect of the portfolio based on the Value-at-Risk risk measure, and to give the asymptotic behavior of conditional tail expectations of the aggregate risk∑ni=1Xi· Furthermore, we establish analogous results for an aggregate loss of the form g(X1,..., Xn) under the more general model, where g is a homogeneous function of order1. Properties of these factors are studied, and asymptotic VaR. behaviors of functions of dependent risks are also given.The main results in this thesis strengthen and complement some results in Alink et al.(2004,2005), Barbe et al.(2006), and Embrechts et al.(2009).