Why Value at Risk Uses the Infimum of a CDF Threshold Set

Summary

The document asks why Value at Risk is defined as the infimum of losses whose cumulative distribution function reaches a chosen probability, rather than as the minimum of that set. It notes that a flat segment in the CDF can make VaR unchanged across a range of probability levels, and asks whether minimum would give the same result. No answer or worked example is supplied, so the text presents a mathematical question rather than a completed explanation.

The issue concerns whether the threshold set contains its lower endpoint. If it does, its infimum is also its minimum; otherwise, a minimum would not exist. The document does not establish which case applies to a loss distribution or discuss assumptions on the CDF, probability level, or support. Readers should treat it as a prompt to investigate generalized quantiles, not as guidance on VaR calculation or risk management.

Key ideas

• The document asks why VaR is defined using the infimum of a CDF threshold set.
• It observes that a flat region in the CDF can leave VaR unchanged across probability levels.
• It asks when the threshold set has a minimum and when it has only an infimum.
• No answer, example, or practical VaR calculation is provided.

Tags

# Why is infimum chosen to define value at risk as opposed to the minimum?


# Why is infimum chosen to define value at risk as opposed to the minimum?












I believe that the VaR is defined as the infimum of the generalized inverse of the CDF of the loss function (something like that, please correct accordingly):

$$\text{VaR}(\alpha)=\inf\{x: F_L(x)\geq \alpha\}=F_L^{\leftarrow}(\alpha)$$

I understand that whenever there is a sudden jump in the CDF and a plateau, the value-at-risk will not budge, but why is it infimum and not minimum on a case like this:

I see that the $$\text{VaR}(\alpha_1)=\text{VaR}(\alpha_2),$$ but $$\text{VaR}(\alpha_0)\neq \text{VaR}(\alpha_1).$$

However, the minimum would work just as well as the infimum.

> Why wouldn't the definition work out if the value at risk was defined identically, except for $\min$ instead of $\inf$? As in

$$\text{VaR}(\alpha)=\min\{x: F_L(x)\geq \alpha\}=F_L^{\leftarrow}(\alpha)$$

> I guess my question is in which scenario will the minimum not coincide with the infimum, and why?

Is the infimum chosen to work around situations like this:

?

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