A new characterization of second-order stochastic dominance

We provide a new characterization of second-order stochastic dominance, also known as increasing concave order. The result has an intuitive interpretation that adding a risk with negative expected value in adverse scenarios makes the resulting position generally less desirable for risk-averse agents. A similar characterization is also found for convex order and increasing convex order. The proofs techniques for the main result are based on properties of Expected Shortfall, a family of risk measures that is popular in financial regulation.