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

New class of distortion risk measures and their tail asymptotics with emphasis on VaR