I know if $X \sim N(\mu,\sigma^2)$, then $VaR_{\alpha}(X) =\mu + \sigma\Phi^{-1}(\alpha)$ and $CVaR_{\alpha}(X) = \mu + \sigma \frac{\phi(\Phi^{-1}(\alpha))}{1-\alpha}$

But how to evaluate $\lim_{\alpha \to 1}\frac{VaR_{\alpha}(X)}{CVaR_{\alpha}(X)}$?

  • $\begingroup$ The Mills Ratio may be useful here en.wikipedia.org/wiki/Mills_ratio $\endgroup$ – Alex C Oct 8 '17 at 15:19
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    $\begingroup$ This is basically a calculus question: \begin{align*} \lim_{\alpha\rightarrow 1} \frac{VaR_{\alpha}}{cVaR_{\alpha}} &= \lim_{\alpha\rightarrow 1}\frac{\mu+\sigma\Phi^{-1}(\alpha)}{\mu + \sigma \frac{\phi(\Phi^{-1}(\alpha))}{1-\alpha}}\\ &=\lim_{x\rightarrow \infty}\frac{\mu+\sigma x}{\mu + \sigma \frac{\phi(x)}{1-\Phi(x)}}. \end{align*} $\endgroup$ – Gordon Oct 8 '17 at 15:44

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