  Distortion risk measures are extensively used in finance and insurance
applications because of their appealing properties. We present three methods to
construct new class of distortion functions and measures. The approach involves
the composting methods, the mixing methods and the approach that based on the
theory of copula. Subadditivity is an important property when aggregating risks
in order to preserve the benefits of diversification. However, Value at risk
(VaR), as the most well-known example of distortion risk measure is not always
globally subadditive, except of elliptically distributed risks. In this paper,
instead of study subadditivity we investigate the tail subadditivity for VaR
and other distortion risk measures. In particular, we demonstrate that VaR is
tail subadditive for the case where the support of risk is bounded. Various
examples are also presented to illustrate the results.
