Interpreting Value at Risk as a Loss Quantile

Summary

The document examines two common explanations of Value at Risk for a loss variable: the capital level intended to keep the probability of insolvency below a chosen tail probability, and a loss threshold that is exceeded with only a specified probability. It gives the quantile definition using the cumulative distribution function and an equivalent expression based on the probability that loss exceeds a threshold. The author’s confusion centers on whether VaR is a maximum or minimum when distributions contain jumps or flat regions.

The useful lesson is that VaR is defined through a quantile, not as an absolute maximum possible loss. Its probability statements describe coverage at a selected confidence level, with boundary behavior depending on the loss distribution and the quantile convention. VaR therefore does not describe the size of losses beyond the threshold, and the document offers no worked distribution or resolution of the question. It is a conceptual prompt about interpreting the definition rather than a complete treatment of tail risk.

Key ideas

• Value at Risk for losses is defined as a quantile of the loss distribution.
• At the selected confidence level, the threshold bounds the probability of losses exceeding it.
• VaR is not the largest loss that can occur, since more severe losses may lie beyond the quantile.
• The equivalence of probability statements can depend on distributional boundary behavior and the quantile convention.
• VaR does not measure how large losses may be after the threshold is crossed.

Tags

# Interpretation of Value at Risk


# Interpretation of Value at Risk












Let $X$ be a Loss random variable (Positive values of X represents Losses) and let $p \in (0,1)$. I know that the Value at Risk at level $p$ of $X$ is defined as:

$$VaR_p(X) = inf{\{x \in \mathbb{R} : F(x) \ge p \}}= inf{\{x \in \mathbb{R} : P[X \gt x] \le 1- p \}}$$

(Also this infimum is equal to the minimum value because $F(VaR_p(X))\ge p$). My problem is the interepretation of this quantity:

- In some books (for example: Loss Models) $Var_p(X)$ is interpreted as the minimum capital required such that the probability of being insolvent is at most $1-p$: that is : $P[X \gt VaR_p(X)] \le 1-p$. This interpretation is fine with me.

- In some other references and books (for example Wikipedia) $VaR_p(X)$ is interpreted as the maximum possible loss (at the level $p$) such that the probability of loss being less than $VaR_p(X)$ is at least $p$: that is: $P[X \le VaR_p(X)] \ge p$

This second definition of maximum possible loss doesn't make sense to me because formally the definition of $VaR_p(X)$ is with an infimum (which coincides with the minimum)

I know that the value at risk is also equal to:

$$VaR_p(X) = sup{\{x \in \mathbb{R} : F(x) \lt p \}}= sup{\{x \in \mathbb{R} : P[X \gt x] \gt 1- p \}}$$

But if we try to intepret the $VaR_p(X)$ using this definition as a maximum possible loss it would be: The maximum possible Loss such that the probability of having a loss $X$ less than $VaR_p(X)$ is less than $p$ but again it still doesn't make sense to me.

I would really appreciate if someone can help me understanding this concept.

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