Asymptotic tail dependence of the normal copula

Copulas have lately attracted much attention as a tool in nance and insurance for dealing with multiple risks that cannot be considered independent. The normal copula, widely used in practice, is known to have the same tail dependence parameter as the product copula. The present paper brings into question the common interpretation of this fact as evidence that the normal copula lacks tail dependence, both by providing numerical examples and by mathematically determining the asymptotic behaviour of the tail dependence. Example 1. The function C(u;v) = uv is a copula and called the product copula. For a bivariate random variable (X;Y ), let FX and FY denote the marginal distribution functions and let FX;Y denote the joint distribution function: FX (x) = P (X ≤ x), FY (y) = P (Y ≤ y), and FX;Y (x;y) = P (X ≤ x; Y ≤ y) for x;y ∈ R. We say that (X;Y ) is continuous if FX and FY are both continuous. Theorem 1 (Sklar). If (X;Y ) is a continuous bivariate random variable, then there exists a unique copula CX;Y such that FX;Y (x;y) = CX;Y ( FX (x);FY (y) )