The value at risk of a random variable $X$ is defined as $$VaR_a(X)=\inf\{l\in\mathbb{R}|P(X>l)\leq 1-a\}$$ If $X\sim\mathcal{N}(\mu,\sigma^2)$, I want to show that $$VaR_a(X)=\mu+\Phi^{-1}(1-a)\sigma$$
If I start calculating $P(X>l)$, then $P(X>l)=1-\Phi(\frac{l-\mu}{\sigma})$ and thus, $P(X>l)\leq 1-a$ is equivalent to $\Phi(\frac{l-\mu}{\sigma})\geq a$, which is equivalent to $l\geq \Phi^{-1}(a)\sigma+\mu$ and $VaR_a(X)=\Phi^{-1}(a)\sigma+\mu$.
So why do i get $\Phi^{-1}(a)$ instead of $\Phi^{-1}(1-a)$?