This study develops closed-form solutions for distortion risk measures (DRM) in extreme cases by utilizing the first two moments and the symmetry of underlying distributions. The resultant extreme-case distributions, encompassing the worst- and best-case distributions, are identified by the envelopes of the distortion functions. The findings of this study extend previous research on worst-case risk measures such as worst-case VaR, worst-case CVaR, worst-case RVaR, and worst-case spectral risk measure, by presenting a unified framework. Furthermore, the compact solutions enhance tractability in optimization problems involving these risk measures, particularly when the true underlying distribution is unknown, and the first two moments are uncertain. The application of the extreme-case DRMs is illustrated with real data sets through numerical examples.
