Exercise: Proving ES is coherent from the dual representation

Expected Shortfall at level $\alpha$ has the dual (Kusuoka) representation

Tasks

1. Verify that the set $\mathcal{Q}_\alpha$ is convex: if $\mathbb{Q}_1, \mathbb{Q}_2 \in \mathcal{Q}_\alpha$ and $\lambda \in [0,1]$, then $\lambda \mathbb{Q}_1 + (1-\lambda)\mathbb{Q}_2 \in \mathcal{Q}_\alpha$.

2. Monotonicity (A1). Show that if $X \le Y$ a.s., then $\text{ES}_\alpha(X) \ge \text{ES}_\alpha(Y)$. (Hint: work directly from the dual.)
3. Translation invariance (A2). Show that $\text{ES}_\alpha(X + c) = \text{ES}_\alpha(X) - c$ for any constant $c$.
4. Positive homogeneity (A3). Show that $\text{ES}_\alpha(\lambda X) = \lambda\,\text{ES}_\alpha(X)$ for $\lambda \ge 0$.
5. Subadditivity (A4). Show that $\text{ES}_\alpha(X + Y) \le \text{ES}_\alpha(X) + \text{ES}_\alpha(Y)$. (Hint: the supremum of a sum is at most the sum of suprema.)

6. Find a pair $X, Y$ for which $\text{ES}_\alpha(X+Y) < \text{ES}_\alpha(X) + \text{ES}_\alpha(Y)$ strictly. What does this say about using ES-based risk capital allocation?

Hint

(5) is the key property and follows from a one-line argument using the supremum representation. (6) is any pair where the optimising $\mathbb{Q}$ differs for the individual and joint problems.