Distortion risk measures for sums of random variables

When we consider random couples (X1, Y1) and (X2, y2), both elements of R(Fx,Fy), relative riskiness of the sums Si= Xi+Yi results from dependency structure between the summands. In this paper we investigated the relation between a measure of risk for sums of random variable derived from distortion functions and traditional measures of dependencies like Pearson’s r, Spearman’s p and Kendall’s t. In the general case we proved that there is no relation between distortion risk measures and Pearson’s r. We also showed that for many classes of distortion risk measures (non-concave distortion risk measures, Tail Value-at-Risk, same holds true additionally for Spearman’s p and Kendall’s t. These findings aim to illustrate the problem of defining what the right measure of dependency is, and that risk measures widely used in practice are not always consistent with traditional measures of dependencies.