Superquantile/CVaR risk measures: second-order theory

Superquantile risk, also known as conditional value-at-risk (CVaR), is widely used as a coherent measure of risk due to its improved properties over those of quantile risk (value-at-risk). In this paper, we consider second-order superquantile/CVaR measures of risk, which represent further “smoothing” by averaging the classical quantities. We also step further and examine the more general “mixed” superquantile/CVaR measures of risk with fundamental importance in dual utility theory. We establish representations of these mixed and second-order superquantile risk measures in terms of risk profiles, risk envelopes, and risk identifiers. The expressions facilitate the development of dual methods for mixed and second-order superquantile risk minimization as well as superquantile regression, a second-order version of quantile regression.