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Estimating VaR and CVaR Sensitivity with Bump and Revaluation

Article Quant Q&A · Author: Xinyuan
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Summary

The document asks how to differentiate Value-at-Risk and Conditional Value-at-Risk for a capped random variable, formed by taking the minimum of a random variable and a threshold. It does not derive the sensitivity or specify assumptions about the distribution, quantile convention, or treatment of points where the risk measure is not differentiable.

The answer gives a general numerical fallback: evaluate the risk measure at the current threshold and at a slightly increased threshold, then divide the change by the size of that bump. It advises using a closed-form derivative when one is available, since that can be faster. The finite-difference estimate depends on the bump size and may be inaccurate near discontinuities or where the measure is nonsmooth; the brief response provides no error analysis or worked example.

Key ideas

  • A closed-form derivative is preferable when it can be obtained for the chosen risk measure.
  • Bump and revaluation estimates sensitivity by changing the threshold slightly and measuring the resulting change in VaR or CVaR.
  • Finite-difference accuracy depends on bump size and local smoothness.
  • The discussion does not derive a formula or state distributional assumptions.

Tags

StatisticsRisk managementDerivatives pricing
Full text
# Answer by MaPy (score 1)


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












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 by MaPy (score 1)

https://quant.stackexchange.com/a/40782

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. $$\frac{\partial V}{\partial d} = \frac{V(d+h) - V(d)}{h}$$

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