I found this definition of VaR (Value at Risk) in a paper:
VaR is defined as the “possible maximum loss over a given holding period within a fixed confidence level”. That is, mathematically, VaR at the $100(1 − α)%$ confidence level is defined as the upper 100α percentile of the loss distribution. Suppose X is a random variable denoting the loss of a given portfolio. Following Artzner et al. (1999), we define VaR at the $100(1 − α)%$ confidence level $(VaR_α(X))$ as
$VaR_α(X) = sup\{x|P[X \geq x] > α\}$
I am really struggling to understand what do they mean with upper percentile and how to understand this definition using a graph with the density.
For the following argument, it will be better if you can depict them by sketching the survival function $S(x) = \Pr\{X \geq x\}$ in various situtions.
If the CDF of the loss distribution is continuous, then by intermediate value theorem, there exist $x^* \in \mathbb{R}^+$ such that $\Pr\{X \geq x^*\} = \alpha$.
Further more, if the CDF is strictly increasing, i.e. the survival function is strictly decreasing, then such $x^*$ is unique for every given $\alpha$. Since only a single point satisfying the condition, the supremum is just $x^*$ itself.
If the survival function is only monotonic decreasing, and it happens to stay at the value $\alpha$ on an interval $I$, then there are infinite number of $x^* \in I$ satisfying $\Pr\{X \geq x^*\} = \alpha$. In such case, taking supremum means taking the right end point of the interval $I$, and therefore by taking supremum, this definition ensure the resulting value is unique.
The remaining case is when the survival function has jump discontinuity. So for some given $\alpha$, there may not exist $x$ satisfying $\Pr\{X \geq x\} = \alpha$, i.e. the survival function jumps from a value larger than $\alpha$, to a value less than $\alpha$. We cannot find the root of the above equation, so it will be an incomplete definition for VaR if we define this way.
To complete the definition, we simply take the largest $x$ that $\Pr\{X \geq x^*\} \geq \alpha$ which is the given definition. If you sketch the graph, it is simply the jump discontinutity point that jump over $\alpha$.