CVaR formulation

Question

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 :

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

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

Answer 1

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.