# What's the relationship between $VaR_{\alpha}(X)$ and $VaR_{1-\alpha}(X)$ if the probability distribution function is not symmetric?

If the probability distribution function $f(x)$ is not symmetric, is there any relationship between $VaR_{\alpha}(X)$ and $VaR_{1-\alpha}(X)$?

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

• If the distribution is not symmetric but fully general (discrete, continuous, mixed) then anything can happen ... – Richard Oct 13 '17 at 7:38
• In my answer to this question, I didn't assume any symmetric property. That is, the same relationship should be hold. What kind of symmetric property you are referring to? Can you revise your question to make it more specific? – Gordon Oct 13 '17 at 13:04

• The relationship $VaR_{\alpha}(-X) =-VaR_{1-\alpha}(X)$ should always hold. You can check with you examples. – Gordon Oct 13 '17 at 13:10