The theory of copulas provides a useful tool for modeling dependence in risk management. In insurance and finance, as well as in other applications, especially important is the dependence of extreme events, hence there is a need for the detailed study of the tail behaviour of the multivariate copulas.

In my presentation I am going to investigate the class of copulas having regular tails which allow the uniform expansion i.e. such that near the origin they can be approximated by a homogeneous function L(u) of degree 1. Having introduced the notion of the uniform tail expansion for the multivariate copulas I will describe its main properties and determine the set of all possible leading parts L(u). Between others I will show that L is concave. Next I will deal with the measure induced by L. I will show that it is a product of the Lebesque measure on the real half line and a measure on the unit simplex.

At the end I will present the example of an application of the uniform tail expansion to the problem of estimation of the extreme risk of the portfolio consisting of long positions in risky assets. The special attention will be given to the Value at Risk (VaR).

Key words: copulas, fat tails, tail expansions, dependence of extreme events, risk management, portfolio theory.

MSC2000: 62H05 91B28 91B30 62E20 62H20