How Capping a Random Variable Affects Its Value at Risk

Summary

The document considers how applying a fixed upper cap to a nonnegative random variable changes its Value-at-Risk (VaR). It defines VaR using an upper-tail exceedance probability and explains that the cap affects outcomes above the cap while leaving lower outcomes unchanged. Under the stated convention, when the cap is above the original VaR threshold, the capped variable has the same VaR; a lower cap can reduce the resulting VaR, which cannot exceed the cap.

The explanation is qualitative and gives no proof or worked distribution. Its conclusion depends on the stated VaR definition and on assumptions about the variable and probability level; it should not be read as a general treatment of alternative quantile conventions or other risk measures. The central practical point is that a cap matters to VaR only if it reaches into the tail region that determines the selected quantile.

Key ideas

• Capping a random variable changes only outcomes above the cap.
• If the cap is above the original VaR threshold, the document says VaR remains unchanged.
• A cap below the relevant tail threshold can reduce VaR, which cannot exceed the cap.
• The stated relationship uses an upper-tail exceedance definition of VaR.

Tags

# Answer by Will Gu (score 1, accepted)


# What is the relationship of Value-at-Risk of a random variable $X$ and a constant $D$ $VaR_{\alpha}(min(X,D))$and $VaR_{\alpha}(X)$?












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, $$ VaR_{\alpha}(X) := \min_{x}\{x|P(X>x) \leq \alpha\}. $$

## Answer by Will Gu (score 1, accepted)

https://quant.stackexchange.com/a/39910

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.