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I am very confused about the following:

Let us define $VaR_u(Y)$ as $\inf \{l:F_Y(l)\geq\ u \}$ where $F_Y$ is the CDF of the random variable Y.

Suppose $A $ and $X$ are independent random variables and suppose we have $VaR_u(AX)=5$. Furthermore, suppose we are interested in the following probability $P(AX>VaR_u(AX)|A=a)$. It seems as if there are two ways to deal with this probability, but surely there must be only one.

First we can take the stance that $VaR_u(AX)$ is a fixed quantity, namely 5. Thus we have $$P(aX>5)=P(aX>VaR_u(AX))$$

On the other hand one could argue that $VaR_u(AX)$ is dependent on the value of $A=a$ such that we have $$P(aX>VaR_u(aX))=1-u$$

I am inclined to believe that the first interpretation is the right one, but how does one 'choose'?

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The first interpretation is correct.

$VaR_u(Y)$ is the quantile function. It is a function of $u$ and of the true underlying distribution of $Y$. It does not depend on any particular value of $Y$ or any samples drawn from the true underlying distribution.

For example, if $u = \frac{1}{2}$, $VaR_u(Y)$ is the median of the underlying distribution of $Y$. If you draw samples from some distribution, it does not change the median of that distribution.

To avoid this confusion, you might consider using a different notation, such as $VaR(u, F) = \inf\{l:F(l)\geq u\}$.

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  • $\begingroup$ Thank you so much for clarifying the dependency $VaR_u(Y)$ has on $u$ and the distribution function of Y. This has been bothering me for two days. $\endgroup$
    – Joogs
    May 21, 2017 at 18:46

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