# comparing modified VaR to ordinary VaR

What inferences can one draw when given a modified VaR at x% confidence and an ordinary VaR at x% confidence level. If the two are equal one inference can be that returns are gaussian but that also depends on the confidence level. At extremely high confidence levels mVaR can be equal to VaR even with skewness and kurtosis.

So how do risk managers look at these estimates? Is the uncertainty in the estimates a concern. What if VaR is lower than mVaR and also less uncertain than mVaR. What if VaR is higher than mVaR but less uncertain than mVaR.

Is mVaR always preferable? By eyeballing the loss distribution it is easy to see if there is skewness and excess kurtosis and so mVaR may be preferable but is there a case where using mVaR can be a bad idea compared to using VaR.

## 3 Answers

Using anything with "VaR" in the name, is basically a bad idea. But a modified VaR does not assume a normal distributed random variable. So maybe that makes people feel a little better.

mVaR might look equal to VaR at "high confidence levels" but it is well-known that both measures are inaccurate at high confidence levels. mVaR may even be worse, given the non-normality assumptions.

This is homework, right?

As described on this link, mVaR represents an empirical expression adjusted for skewness and kurtosis of the empirical distribution.

As we know, empirical returns are commonly skewed and peaked, such that assuming normal distribution is a bad fit to estimate VaR. Therefore, mVaR adjusts for skewness and kurtosis to better reflect the empirical VaR.

• The intuition is missing. I understand the use of cornish fisher expansion to the quantiles to include skewness and kurtosis but what inferences can be made but just looking at the numbers – Kumar Dec 19 '14 at 23:01
• @user2142 I don't think you can infer anything from VaR=mVaR, because you may get the same mVaR for different Skew and Kurtosis when they cancel out. You may calculate $(mVaR-q_\alpha)$ and $(VaR-q_\alpha)$ for different $\alpha\in(0,1)$ to get an overall error to the empirical distribution quantiles $q_\alpha$. The model with smaller errors is better. – emcor Dec 19 '14 at 23:04

In this presentation https://www.academia.edu/attachments/37039957/download_file?st=MTQyNjgyNTAxNCwyMDIuMTc0LjE3MC4xNjIsMTIyMTAxMg%3D%3D&s=work_strip I show the feasible bounds for the modified VaR calculation and provide a small test to show when it is outside those bounds.