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Suppose $X$ is a nonnegative random variable and $D$ is a constant, what is the relationship of $\text{VaR}_{\alpha}(min(X,D))$ and $VaR_{\alpha}(X)$? Here, $VaR$ stands for Value-at-Risk as, $$ VaR_{\alpha}(X) := \min_{x}\{x|P(X>x) \leq \alpha\}. $$

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It really depends on what level $D$ is at, as it caps $X$.

To put it in simple way, if $D$ is a really large value, then $min(X, D)$ is pretty much just $X$. On the other hand, if $D$ is really low (say 0), then your VaR can't be more than $D$.

let's use $\alpha = 95\%$, then while a large $D$ distorts the distribution of $X$ a little bit on the far right hand side, it shouldn't really matter for $P(X > x)$ when it's close to $\alpha = 95\%$. In fact, I'd say

for any $D > VaR_\alpha(X)$, $VaR_\alpha(min(X,D)) = VaR_\alpha(X)$

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