Let $x$ be a random variable which denotes the possible monetary loss on a portfolio over a fixed

time horizon $T($e$.$g$.$ one day, one week, or one year$).$ Here losses are counted as positive and

profits as negative, i$.$e$.x$ is the negative of the gain on the portfolio over the period $0,T.$ The

value at risk $($ VaR of the portfolio at confidence level $$ is the minimum value of the loss that

we are $(1-)\%$ certain will not be exceeded, i$.$e$.$

$Va{R}_{}(x)=$min$\{x$:$P(x>x)\}.$

Typical values of $$ are $5\%$ or $1\%.$

For example, if your portfolio has a $5\%$ one$-$day VaR of $1$ million, then we expect a loss of $1$

million no more than $5\%$ of days. On a given day, we are $95\%$ sure that the loss of the portfolio

will be $1$ million.

$($a$)$ Explain why $Va{R}_{}(x)=F_x^{-1}(1-).$