Questions about VaR and CVaR. Is there any relation between $VaR_{\alpha}(X)$ and $VaR_{\alpha}(-X)$, or $CVaR_{\alpha}(X)$ and $CVaR_{\alpha}(-X)$?

Question

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:

$$VaR_{\alpha}(X) := \inf \left\{x\in \mathbb{R}| Pr(X >x)\leq \alpha \right\}, \alpha \in [0, 1]$$

$$CVaR_{\alpha}(X) := \frac{1}{\alpha}\int_{0}^{\alpha}VaR_{s}(X)ds$$

Answer 1

We consider the case where the distribution function $F$ of $X$ is strictly increasing. Then \begin{align*} VaR_{\alpha}(X) &= \inf\{x: P(X >x) \le \alpha \}\\ &=\inf\{x: F(x)\ge 1-\alpha \}\\ &=F^{-1}(1-\alpha). \end{align*} Moreover, we note that the distribution function $G$ of $-X$ is defined by \begin{align*} G(x) &= P(-X \le x) \\ &=1-F(-x), \end{align*} Then, \begin{align*} VaR_{\alpha}(-X) &= G^{-1}(1-\alpha)\\ &=-F^{-1}(\alpha)\\ &=-VaR_{1-\alpha}(X). \end{align*} Furthermore, \begin{align*} CVaR_{\alpha}(-X) &=\frac{1}{\alpha}\int_0^{\alpha}VaR_s(-X)ds\\ &=-\frac{1}{\alpha}\int_0^{\alpha}VaR_{1-s}(X)ds\\ &={\color{red}{-}}\frac{1}{\alpha}\int_{1-\alpha}^1 VaR_s(X)ds\\ &=-\frac{1}{\alpha}\left(\int_0^1 VaR_s(X)ds - \int_0^{1-\alpha} VaR_s(X)ds\right)\\ &=-\frac{1}{\alpha}\int_0^1F^{-1}(1-s)ds +\frac{1-\alpha}{\alpha}CVaR_{1-\alpha}(X)\\ &=-\frac{1}{\alpha}\int_0^1F^{-1}(s)ds +\frac{1-\alpha}{\alpha}CVaR_{1-\alpha}(X)\\ &=-\frac{1}{\alpha}\int_{-\infty}^{\infty} x\, dF(x) +\frac{1-\alpha}{\alpha}CVaR_{1-\alpha}(X)\\ &=-\frac{1}{\alpha} E(X) +\frac{1-\alpha}{\alpha}CVaR_{1-\alpha}(X). \end{align*}