Value at risk (VaR) conditioning argument

Question

Let $X_1$ be a random variable and let $\text{VaR}_p(X_1)$ denote the quantile of $X_1$ such that $$ P(X_1>\text{VaR}_p(X_1))=1-p. $$ Now, let $X_2$ be another random variable with $\text{VaR}_p(X_2)$ such that $$ P(X_2>\text{VaR}_p(X_2))=1-p. $$

Let $Y$ be yet another random variable. Now let $\widetilde{\text{VaR}}_p(X_1)$ be $\text{VaR}_p(X_1)$ conditional on $Y$. That is, $$ P(X_1>\widetilde{\text{VaR}}_p(X_1)|Y=y)=1-p. $$ Clearly, it holds that $$ \mathbb{E}(P(X_1>\widetilde{\text{VaR}}_p|Y))=1-p=P(X_1>\text{VaR}_p). $$

I am wondering if it also holds that $$ \mathbb{E}(P(X_1>\widetilde{\text{VaR}}_p(X_1), X_2>\widetilde{\text{VaR}}_p(X_2) |Y))=P(X_1>\text{VaR}_p(X_1), X_2>\text{VaR}_p(X_2))? $$

Answer 1

It is not 100% clear what you are asking, since, as has been remarked by ssdecontrol, you confuse the conditioning somewhat. But I'll try to have a stab at it anyway and I'll interpret the conditioning in the density sense, i.e. conditioning on $Y=y$ means you evaluate the densities of $X_1$ and $X_2$ under the condition that $Y=y$.

Then the answer is NO. Your last equation does not hold in general. The problem is that you are fixing the probability of the margins by introducing new $VaR$ thresholds but you do not control the dependency. And of course conditioning will not only change the margins but also the dependency.

For an example, assume $X_1$, $X_2$ are independent standard normal variables, $Y = X_1 - X_2$ and condition on $Y=0$ or $X_1=X_2$. This will produce two new joint normal variable $(Z_1, Z_2)$ which are 100% correlated. Hence the left side of your last equation will be equal to $1-p$ because if one $Z$ is larger than the $VaR$ the other will be as well. But the right side is $(1 - p)^2$ due to independence.