Formulación alternativa esperada de déficit

Question

Definir:

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

Quiero probar que:

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

Me quedo atascado como:

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

Ahora me gustaría usar ese %-%-%, pero no sé cómo proceder.

Answer 1

Tenga en cuenta que \begin{align*} \mathbb{E}\big(L \mid L\geq q\alpha(L)\big) &= \frac{\mathbb{E}\big(\pmb{1}{{L\geq q\alpha(L)}} L\big)}{\mathbb{P}\big(L\geq q\alpha(L) \big)}. \end-------------------------------------------------------------------------------- La fórmula sigue inmediatamente.