  Risk measures such as Expected Shortfall (ES) and Value-at-Risk (VaR) have
been prominent in banking regulation and financial risk management. Motivated
by practical considerations in the assessment and management of risks,
including tractability, scenario relevance and robustness, we consider
theoretical properties of scenario-based risk evaluation. We propose several
novel scenario-based risk measures, including various versions of Max-ES and
Max-VaR, and study their properties. We establish axiomatic characterizations
of scenario-based risk measures that are comonotonic-additive or coherent and
an ES-based representation result is obtained. These results provide a
theoretical foundation for the recent Basel III & IV market risk calculation
formulas. We illustrate the theory with financial data examples.
