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Reconciling CVaR Formulations for Profit and Loss

Article Quant Q&A · Author: Mohamed Chaaban
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Summary

The document explains why two common conditional value-at-risk (CVaR) formulations can appear to disagree: one maximizes a quantity expressed in terms of profit, while the other minimizes an expression for loss. It identifies the sign conversion as the link. Define loss as the negative of profit, set the tail probability parameter in one formulation equal to one minus the confidence parameter in the other, and change the optimization variable's sign. The resulting algebra shows that loss CVaR is the negative of profit CVaR under the conventions used.

The explanation uses the positive-part function and the relationship between supremum and infimum to transform one expression into the other. It cautions that the maximum-to-supremum step assumes the optimum is attained. The discussion is a conceptual clarification rather than a full treatment of CVaR conventions: signs and tail-probability parameters vary across sources, so they must be checked before comparing formulas or applying them to a risk-averse newsvendor objective.

Key ideas

  • A profit-based CVaR expression and a loss-based expression can differ by a sign because loss is defined as negative profit.
  • The conversion also requires matching the tail probability parameter to one minus the confidence level.
  • Changing the sign of the optimization variable transforms the profit formulation into the loss formulation.
  • The equivalence uses the relationship between supremum and infimum, with attainment needed if a maximum is written.

Tags

Risk managementStatistics
Full text
# CVaR formulation


# CVaR formulation












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 by Bjørn Kjos-Hanssen (score 3, accepted)

https://quant.stackexchange.com/a/45851

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.