How to calculate $\frac{\partial\ \text{CVaR}_{\alpha}(\min(X,d))}{\partial d}$ and $\frac{\partial\ \text{VaR}_{\alpha}(\min(X,d))}{\partial d}$?

Question

How to calculate $\frac{\partial\ \text{CVaR}_{\alpha}(\min(X,d))}{\partial d}$ and $\frac{\partial\ \text{VaR}_{\alpha}(\min(X,d))}{\partial d}$?

Here, $\text{CVaR}$ is short for Conditional Value-at-Risk and $\text{VaR}$ is short for Value-at-Risk. And $X$ is a random variable.

Answer 1

It really depends on the way you calculate your Var and CvaR. If your are able to get a closed form solutions for the derivative then you must use those, for faster results. Otherwise, you can use bump and reval. $$\frac{\partial V}{\partial d} = \frac{V(d+h) - V(d)}{h}$$