# Definitions of VaR (Value at Risk)

Here is the definition of VaR (Value at Risk) taken from McNeil, Alexander J., Rüdiger Frey and Paul Embrechts (2015), Quantitative risk management: Concepts, techniques and tools: $$\textrm{VaR}_{\alpha}(L)=\inf\{l\in\mathbb{R}\colon P(L>l)\leq 1-\alpha\} =\inf\{l\in\mathbb{R}\colon F_{L}(l)\geq\alpha\}$$ where $$L$$ is the Loss distribution and $$F_L$$ its cumulative distribution function.

I would like to translate it in terms of Profit & Loss distribution $$X=-L$$ but I'm a bit unsure this is correct: $$\textrm{VaR}_{\alpha}(X)=-\inf\{x\in\mathbb{R}\colon P(X $$= -\inf\{x\in\mathbb{R}\colon F_{X}(x)\geq 1-\alpha\}$$

Obviously, the two values must be the same, that is:

$$\textrm{VaR}_{\alpha}(L)=\textrm{VaR}_{\alpha}(X)=\textrm{VaR}_{\alpha}$$

Could you please check my formula? In particular, should I use $$\sup$$ instead of $$\inf$$?

$$\textrm{VaR}_{\alpha}(X)=-\inf\{x\in\mathbb{R}\colon P(X
is wrong because I have to take the upper bound, since the set is made of negative numbers, so $$\textrm{VaR}_{\alpha}(X)=-\sup\{x\in\mathbb{R}\colon P(X is the correct formula.
$$\textrm{VaR}_{\alpha}(X)= -\inf\{x\in\mathbb{R}\colon F_{X}(x)\geq 1-\alpha\}$$ is correct, but to make it equivalent to the formula with the Loss distribution, even when there is no minimum, it should be: $$\textrm{VaR}_{\alpha}(X)= -\sup\{x\in\mathbb{R}\colon F_{X}(x)\leq 1-\alpha\}$$