Proof that conditional value at risk is coherant risk measure

Question

Conditional value at risk, sometimes called expected shortfall, is defined as:
$$CVaR_{\alpha}(X) = E[X|X \ge VaR_{\alpha}(X)]$$
For a risk measure $\left( \rho(x)\right)$, to be coherent, we require following properties:
$(1)\ \rho(0) = 0$
$(2)\ \rho(X + c) = \rho(X) + c$
$(3)\ $ For $X \ge Y$, $\rho(X) \ge \rho(Y)$
$(4)\ \rho(\lambda X + (1 - \lambda) Y) \le \lambda \rho(X) + (1 - \lambda)\rho(Y)$
$(5)\ \rho(\lambda X) = \lambda\rho(X)$
Note that (4), (5) directly imply subadditivity $\rho(X + Y) \le \rho(X) + \rho(Y)$. Also, note that I think of an X as a loss, so positive values of X are losses and negative values of X are gains.
First two properties are easy to prove, but I am struggling with last 3 properties.

Answer 1

Your definition of CVaR is consisten with the usual definition in the case where $X$ is a continuous random variable.

I will use the following definition for CVaR: \begin{align} CVaR_{\alpha}(X) := \frac{1}{1-\alpha} \int_{\alpha}^{1} Q_{\beta}(X) dx, \end{align} where $Q_{\beta}(X) := \inf \{x \ | \ F(x) \geq \beta \}$.

Axiom $(3)$ follows directly: Assume $X \geq Y$. Then, \begin{align} Q_{\beta}(X) \geq Q_{\beta}(Y) \end{align} for every $\beta \in (0, 1)$. Thus \begin{align} \frac{1}{1-\alpha} \int_{\alpha}^{1} Q_{\beta}(X) dx \geq \frac{1}{1-\alpha} \int_{\alpha}^{1} Q_{\beta}(Y) dx, \end{align} so axiom $(3)$ clearly holds.

Axiom $(5)$ also follows directly by linearity of integration.

Axiom $(4)$ is harder. You will find the details in Theorem 2 of.

Rockafellar, R. Tyrrell, and Stanislav Uryasev. "Optimization of conditional value-at-risk." Journal of risk 2 (2000): 21-42.

Very roughly, they use that \begin{align} CVaR_{\alpha}(X) = \inf_{c} \left[c + \frac{1}{1-\alpha} E\left[ \left[ X-c\right]^+ \right]\right], \end{align} where $[t]^+ = \min(t, 0)$. By appealing to the facts that $[t]^+$ is convex, and that minimizing a convex function of two variables with respect to one variable yields a convex function of the other variable, the result follows.