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Why VaR at Complementary Confidence Levels Has No General Relationship

Article Quant Q&A · Author: Xinyuan
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Summary

The document asks whether Value at Risk at probability levels α and 1−α has a general relationship when the loss distribution is asymmetric. Using a definition based on a tail probability quantile, the answer says there is no such general relationship: VaR depends on the quantile at the chosen level, and asymmetry does not determine the value at the complementary level.

A pair of loss samples illustrates that the reported VaR can remain unchanged even when many observations before it differ. This also points to a limitation of VaR: it does not describe the size of losses beyond its threshold. The answer therefore mentions expected shortfall, which averages losses exceeding VaR, as an additional risk measure. The explanation is concise and illustrative rather than a formal treatment of quantile conventions, finite-sample estimation, or the precise behavior of expected shortfall when losses equal the VaR threshold.

Key ideas

  • VaR is a quantile-based measure, so complementary probability levels have no general relationship for an asymmetric distribution.
  • The examples show that observations away from the relevant quantile may not change the VaR value.
  • VaR does not measure how severe losses are beyond its threshold.
  • Expected shortfall supplements VaR by summarizing losses in the tail beyond the VaR level.

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Risk managementStatistics
Full text
# Answer by Klaus (score 1, accepted)


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

## Answer by Klaus (score 1, accepted)

https://quant.stackexchange.com/a/36435

No, because the VaR is defined as a quantil. For example, you have the loss-vector l=(-1,-2,3,4,5,6,7,8,9,10). The VaR(90%) is 9. And it is also VaR(90%)=9, if you have l=(8,8,8,8,8,8,8,8,9,10). The VaR is independent of the values before and after his value. This is also a disadvantage of the VaR and one reason to take also the expected shortfall (mean of the losses that are bigger than the VaR).

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