I'm completely stuck at one of the introductory examples for the Value-at-Risk concept in the book I'm using…
The VaR is defined in a following way–>
For $\alpha \in \left< 0,1 \right>$, we define Value-at-Risk (VaR) of a random variable $X$, at confidence level $1-\alpha$, as $$\text{VaR}^{\alpha}(X)=-\inf \{ x : \alpha < F_X(x)\}.$$
In the example, the random variable is defined as
$$
X=
\begin{cases}
0,\ \text{with probability }p\\
1, \ \text{with probability }1-p\\
\end{cases},
$$
and then it is stated that for $p<\alpha$ we have
$$\text{VaR}^{\alpha}(X)=0.$$
Is this a typo? Shouldn't the actual $\text{VaR}^{\alpha}(X)$ be equal to -1?
I mean, is this –>
$$\{x : \alpha <F_X(x) \}=\left[ 1, +\infty \right>$$
correct?
Best Answer
Yes, you're right, this is an error and your last equation is correct.
By the way, Wikipedia's definition doesn't have that minus sign before the infimum, and it doesn't make much sense to me.