3.1 Introduction

Tail dependence describes the amount of dependence in the tail of a bivariate distribution. In other words, tail dependence refers to the degree of dependence in the corner of the lower-left quadrant or upper-right quadrant of a bivariate distribution. Recently, the concept of tail dependence has been discussed in financial applications related to market or credit risk, Hauksson et al. (2001) and Embrechts et al. (2003). In particular, tail-dependent distributions are of interest in the context of Value at Risk (VaR) estimation for asset portfolios, since these distributions can model dependence of large loss events (default events) between different assets.

It is obvious that the portfolio's VaR is determined by the risk behavior of each single asset in the portfolio. On the other hand, the general dependence structure, and especially the dependence structure of extreme events, strongly influences the VaR calculation. However, it is not known to most people which are not familiar with extreme value theory, how to measure and model dependence, for example, of large loss events. In other words, the correlation coefficient, which is the most common dependence measure in financial applications, is often insufficient to describe and estimate the dependence structure of large loss events, and therefore frequently leads to inaccurate VaR estimations, Embrechts et al. (1999). The main aim of this chapter is to introduce and to discuss the so-called tail-dependence coefficient as a simple measure of dependence of large loss events.

Kiesel and Kleinow (2002) show empirically that a precise VaR estimation for asset portfolios depends heavily on the proper specification of the tail-dependence structure of the underlying asset-return vector. In their setting, different choices of the portfolio's dependence structure, which is modelled by a copula function, determine the degree of dependence of large loss events. Motivated by their empirical observations, this chapter defines and explores the concept of tail dependence in more detail. First, we define and calculate tail dependence for several classes of distributions and copulae. In our context, tail dependence is characterized by the so-called tail-dependence coefficient (TDC) and is embedded into the general framework of copulae. Second, a parametric and two nonparametric estimators for the TDC are discussed. Finally, we investigate some empirical properties of the implemented TDC estimators and examine an empirical study to show one application of the concept of tail dependence for VaR estimation.