# 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:

1. 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.

2. 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.

• I have trouble following your reasoning. But isn't $P[X \gt VaR_p(X)] \le 1-p$ formally equivalent to $P[X \le VaR_p(X)] \ge p$ ? In English saying "If I have 1000 in capital, there is a 10% chance I will go bankrupt tomorrow", is pretty much the same as saying "If I have 1000 in capital there is a 90% I will not go bankrupt tomorrow" ( i made these numbers up, of course). What substantive difference do do you see between interpretation 1 and 2? Dec 26 '20 at 13:18
• My problem is when we think of $VaR_p(X)$ as a maximum possible loss, not minimum capital. For instance if we take $VaR_p(X) + 1$, this is a greater number such that: $P[X \gt VaR_p(X)+1] \le 1-p$ and $P[X \le VaR_p(X)+1] \ge p$. So $VaR_p(X)$ would not be a maximum number that has this properties. Dec 26 '20 at 15:34