Limit of the Gaussian VaR-to-CVaR Ratio at High Confidence

Summary

This document poses a limiting question about the ratio of Value at Risk (VaR) to Conditional Value at Risk (CVaR) for a normally distributed variable. It states the familiar expressions for both measures in terms of the mean, standard deviation, the confidence level, and the standard normal density and quantile. The requested quantity is the ratio as the confidence level approaches one.

The text supplies the setup and formulas but gives no derivation or result. It therefore serves as a focused prompt for studying tail behavior of Gaussian risk measures, rather than a complete explanation. Any conclusion would depend on the stated convention for VaR and CVaR and on the limiting behavior of the normal quantile and density; those steps are not worked through here.

Key ideas

• The document asks for the limiting ratio of Gaussian VaR to CVaR as confidence approaches one.
• It gives formulas for both measures using the normal quantile and density.
• No solution or derivation is included, so the limiting result must be established separately.

Tags

# How to derive the limit of ratio between VaR and CVaR?


# How to derive the limit of ratio between VaR and CVaR?












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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.