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:

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*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.
 A: In the end VaR is just a quantile in the loss distribution. Then, if the given interpretation agrees with the formal definition of quantile it should be fine.
Start from the definition of the p-quantile of the loss distribution:
$$VaR_p(X) = inf{\{x \in \mathbb{R} : F(x) \ge p \}}$$
Or in English, the minimum (assuming it coincides with the infimum) value such that the probability of X being lower or equal than it is greater or equal than p. This definition is consistent with the first interpretation you give and the following:
It is the minimum loss such that lower losses will happen at least with probability p.
It is the minimum loss such that higher losses will happen at most with probability 1-p.
Now, you can show that
$$VaR_p(X) = inf{\{x \in \mathbb{R} : F(x) \ge p \}} = sup{\{x \in \mathbb{R} : F(x) \lt p \}}$$
Or in English, the maximum (assuming it coincides with the supremum) value such that the probability of X being lower or equal than it is lower than p. This allows for interpretations in terms of maximum losses:
It is the maximum loss such that lower losses will happen at most with probability p.
It is the maximum loss such that higher losses will happen at least with probability 1-p.
Since we are rearranging the formal definitions and writting them in words, all interpretations are equivalent. However, the ones based on the infimum (minimum) are also more intuitive for me.
