Expected Shortfall alternative definition

Question

Define:

$$q_\alpha(F_L)=F^{\leftarrow}(\alpha)=\inf\lbrace{x\in \mathbb{R}\mid F_L(x)\geq \alpha\rbrace}=VaR_\alpha(L)$$

I want to prove that:

$$ES_\alpha = \frac{1}{1-\alpha}\mathbb{E}[\mathbb{1}_{\lbrace{ L\geq q_\alpha(L)\rbrace}}\cdot L] \overset{!!!}{=}\mathbb{E}[L\mid L\geq q_\alpha(L)] $$

I get stuck as:

$$\mathbb{E}[\mathbb{1}_{\lbrace{ L\geq q_\alpha(L)\rbrace}}\cdot L]= \mathbb{E}[\mathbb{E}[\mathbb{1}_{\lbrace{ L\geq q_\alpha(F_L)\rbrace}}\cdot L\mid L\geq q_\alpha(F_L)]\ ] = \mathbb{E}[\mathbb{1}_{\lbrace{ L\geq q_\alpha(F_L)\rbrace}}\cdot\mathbb{E}[L\mid L\geq q_\alpha(F_L)]\ ]$$

Now I would like to use that $\Pr(L\geq q_\alpha(F_L) \ )=1-\alpha$, but I don't know how to proceed.

Answer 1

Do you know that

$$\mathbb{E}[X|A] = \frac{\mathbb{E}[X1_A]}{P(A)}?$$

Some take it as the definition of conditional expectation given an event. With $X=L$ and $A=(L\geq q_\alpha(L))$, you get your answer.