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I am a research intern and I am working on a topic about a profit maximization of a risk-averse newsvendor by using Conditional Value-at-Risk.The problem is that I found different expressions of CVaR. In a risk-averse newsboy problem papers, I have found the following formula :

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But, in risk management papers (finance etc), I have found the following one with its proof : enter image description here

The first formula is a maximization problem and the second one, it is a minimizarion.

The problem is that I coudn'd find the link between the two formulas.

π(μ,D) : π is a profit function which depends on some factors that we can control μ (decision variables vector) and D represents randomness and this case it is random demand. Y : is a random variable that represents loss function. α is variable. It does not have a special signification. But we can prove that Value-at-Risk is a solution of the second optimization problem. I thing there is something missing but I dont know what because first we talk about profit and then we talk aboout loss. Maybe there is something missing related to this.

Could you please help me?

Thanks

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    $\begingroup$ I would suggest you edit the question by adding more context and definitions for all notations. $\endgroup$ – Gordon May 29 at 13:17
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    $\begingroup$ Keep in mind that maximization of $x$ is equivalent to minimization of $-x$. So it is important to understand what $a$ and $\nu$ represent. For example are we trying to maximize a gain or minimize a loss. And do we lose when $\{\cdots\}$ is too high or do we lose when it is too low. $\endgroup$ – Alex C May 29 at 14:27
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    $\begingroup$ Thanks for the edit, but still not clear. Please provide definitions to all notations. For example, what is $\mu$, what is $D$, and what is $\pi(\mu, D)$? Similarly, what are $Y$ and $\alpha$? $\endgroup$ – Gordon May 29 at 14:53
  • $\begingroup$ π(μ,D) : π is a profit function which depends on some factors that we can control μ (decision variables vector) and D represents randomness and this case it is random demand. Y : is a random variable that represents loss function. α is variable. It does not have a special signification. But we can prove that Value-at-Risk is a solution of the second optimization problem. I thing there is something missing but I dont know what because first we talk about profit and then we talk aboout loss. Maybe there is something missing related to this. I hope it's clearer this time. Thanks $\endgroup$ – Mohamed Chaaban May 29 at 19:08
  • $\begingroup$ $Y$ and $\pi(\mu, D)$ must have some relationships to make the comparison meaningful- you can not just comparing two random variables. Please edit directly in your question. $\endgroup$ – Gordon May 29 at 19:17
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If $Y=-\pi(\mu,D)$ then the first formula is $$\mathrm{CVaR}_\eta(-Y)=\max_{\nu\in R}\left\{\nu+\frac1\eta E((-Y-\nu)^-)\right\}$$ where $X^-=\min (X,0)$ and $X^+=\max(X,0)$. Note that $(-X)^-=-(X^+)$.

If we let $1-\alpha=\eta$ and $\nu=-a$ this becomes (assuming $\max=\sup$, i.e. the sup is attained, and using $\sup(\mathcal A)=-\inf(-\mathcal A)$): $$\begin{eqnarray*}\max_{\nu\in R}\left\{\nu+\frac{1}{1-\alpha} E(-((Y+\nu)^+))\right\}&=&\sup_{\nu\in R}\left\{\nu+\frac{-1}{1-\alpha} E((Y+\nu)^+)\right\}\\ =\sup_{a\in R}\left\{-a+\frac{-1}{1-\alpha} E((Y-a)^+)\right\} &=&-\inf_{a\in R}\left\{-\left(-a+\frac{-1}{1-\alpha} E((Y-a)^+)\right)\right\}\\ &=&-\inf_{a\in R}\left\{a+\frac{1}{1-\alpha} E((Y-a)^+)\right\}\end{eqnarray*}$$

Now let's imagine $\pi(\mu,D)$ is profit and $Y=-\pi(\mu,D)$ is a corresponding loss.

So the CVaR of $Y$, the loss, according to the second formula, is the negative of the CVaR of the profit, $-Y$, according to the first formula.

So I guess when dealing with a loss we take the CVaR to be negative, see e.g. an answer by Kozarevic.

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  • $\begingroup$ Thanks. What you did sound correct but personaly, I stil missing something. In the case of a profit function, let call it Y, CVaR is the expected profit if Y exceeds or doesnt exceed VaR(Y). If Y is a loss function, CVaR is the expected loss if Y exceeds or doesnt exceed VaR. I hope my question is clear. Thanks I am really confused between loss and profit functions and how VaR and CVaR expressions vary with respect to if Y is a loss or a profit function. $\endgroup$ – Mohamed Chaaban May 29 at 20:52
  • $\begingroup$ @MohamedChaaban OK I added something, maybe it helps $\endgroup$ – Bjørn Kjos-Hanssen May 30 at 0:31
  • $\begingroup$ @kjos-hanssen thanks a lot man. $\endgroup$ – Mohamed Chaaban May 30 at 2:36

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