  Expected Shortfall (ES, also known as CVaR) is the most important coherent
risk measure in finance, insurance, risk management, and engineering. Recently,
Wang and Zitikis (2021) put forward four economic axioms for portfolio risk
assessment and provide the first economic axiomatic foundation for the family
of ES. In particular, the axiom of no reward for concentration (NRC) is
arguably quite strong, which imposes an additive form of the risk measure on
portfolios with a certain dependence structure. We move away from the axiom of
NRC by introducing the notion of concentration aversion, which does not impose
any specific form of the risk measure. It turns out that risk measures with
concentration aversion are functions of ES and the expectation. Together with
the other three standard axioms of monotonicity, translation invariance and
lower semicontinuity, concentration aversion uniquely characterizes the family
of ES. In addition, we establish an axiomatic foundation for the problem of
mean-ES portfolio selection and new explicit formulas for convex and consistent
risk measures. Finally, we provide an economic justification for concentration
aversion via a few axioms on the attitude of a regulator towards dependence
structures.

Mathematical Finance