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Can anyone give me some hints as to how to prove that

$$g(x) = \begin{cases} \frac{x}{1-\alpha}, &0 \leq x \leq 1-\alpha\\ 1 , &1-\alpha \leq x \leq 1 \end{cases}$$

Is the distortion function corresponding to $\text{CVaR}_\alpha(X)$?

Here I define $$\text{CVaR}_\alpha(X) = \frac{1}{\alpha} \int_{0}^{\alpha} F_X^{-1}(u) du$$ For more details, see Expected shortfall - Wikipedia .Of course the inverse is supposed to be understood as the generalized inverse.

My problem is that a direct calculation does not seem to work for me. Maybe I am missing some trick.

Any help would be appreciated.

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  • $\begingroup$ Some authors use $\alpha$ where some other authors use $1-\alpha$. Are you sure your expressions for g(x) and CVaR use compatible notation? $\endgroup$ – noob2 Nov 18 '16 at 9:28
  • $\begingroup$ @noob2 link According to this paper I should change $\alpha$ to $1-\alpha$ in the definition of CVaR and integrate from $\alpha$ to 1. But even then I cannot work it out. So far my approach has been to start with the distortion function and plug it into the Distortion Risk measure and then try to get some nice formula. $\endgroup$ – T-at-R Nov 18 '16 at 9:58
  • $\begingroup$ From the same paper, the distortion function is defined as $g$ such that $\text{CVaR}=\int_0^\infty g(1-F_X(x)) dx$ where F is the cumulative distribution of the P&L, with $g(0)=0$ and $g(1)=1$. $\endgroup$ – Alex C Nov 19 '16 at 17:54
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I have solved it myself. The key was to realize that for $X \geq 0$ and $S_X(t) = \mathbb{P}(X>t)$

$$ \int_0^\infty S(t) dt = \int_0^1 F_X^{-1}(u) du = \mathbb{E}\left[X \right].$$

This is elegantly explained in Characterization of $\mathbb{E}$.

Now this relationship can be extended for the whole real line, thus

$$ \int_0^1 F_X^{-1}(u) du = \int_0^\infty S_X(t) dt + \int_{-\infty}^0 S_X(t) -1 dt$$.

The rest of the proof is a matter of changing the variables in the indicator functions and considering the two cases of (a) $S_X(t) \geq 1-\alpha$ and (b) $S_X(t) \leq 1-\alpha$.

For a direct calculation @CaffeRistretto tipp was very helpful. So it is best to start with the definition of CVaR and work towards the distortion function.

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Maybe prove that

$$CVaR_\alpha (X) = \frac{1}{\alpha} \int_0^\alpha F^{-1}_X(u) du$$

has the distortion function

$$ g(u)= \begin{cases} \frac{u}{\alpha}, \quad \; u \leq \alpha \\ 1, \qquad u > \alpha\end{cases}$$

would be easier?

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