I have some questions when dealing with Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR).
Is there any relationship between $VaR_{\alpha}(X)$ and $VaR_{\alpha}(- X)$, or $CVaR_{\alpha}(X)$ and $CVaR_{\alpha}(-X)$ ?
Here, $VaR$ and $CVaR$ are defined as:
$$VaR_{\alpha}(X) := \inf \left\{x\in \mathbb{R}| Pr(X >x)\leq \alpha \right\}, \alpha \in [0, 1]$$
$$CVaR_{\alpha}(X) := \frac{1}{\alpha}\int_{0}^{\alpha}VaR_{s}(X)ds$$