Adjusting Normal VaR for Fat-Tailed Market Risk
Summary
The document considers why a portfolio VaR based on a covariance matrix and normally distributed market factors may exceed its stated confidence bounds more often than expected. It proposes decomposing portfolio risk into component VaR to identify which market factors contribute too little to the measured risk, then adjusting those factors’ volatilities under appropriate governance.
To estimate the size of an adjustment, the answer suggests checking historical kurtosis for the factors suspected of having fat tails. Another approach is to solve for volatility increases that bring component VaR to a chosen level. These are calibration ideas rather than a complete VaR model: the response gives no empirical example, and volatility changes require controls and validation. The document also does not specify how to model tail dependence or replace the normality assumption, so the proposed adjustments should be understood as targeted ways to increase measured VaR, not proof that the resulting estimate has accurate coverage.
Key ideas
- Component VaR can identify which market factors contribute too little to portfolio risk estimates.
- Volatility adjustments to selected factors can increase VaR when normality understates risk.
- Historical kurtosis can help estimate whether a factor has fatter tails than a normal model assumes.
- An inverse calculation can solve for volatility changes needed to reach a chosen component VaR.
- Any volatility adjustment needs appropriate governance and validation.
Tags
Full text
# Wider VaR for portfolio risk? # Wider VaR for portfolio risk? Is there a way to widen the 95% VaR by changing the distribution of a portfolio of stocks? When calculating 95% VaR of my portfolio using the holdings based approach (which requires the covariance matrix), more than 5% of the time it exceeds the upper/lower bounds. This is expected since in general the market has fatter tails. What are typical ways to expand the 95% VaR so that it's actually closer to 5%, and that can be easily implemented. The tools I have are simulation - I am open to simulate a distribution or a bunch of stock paths. Or just another quick computational method. ## Answer by Dimitri Vulis (score 1) https://quant.stackexchange.com/a/75755 If I understand the question correctly, you have a covariance matrix, you assume that your market factors are normally distributed, you calculate VaR, and the VaR comes out "too small". You're looking for a way to incrase the VaR being calculated, that would pass muster with others who might review / challenge / validate your methodology. I recently commented on some ways to debug VaR being "too large", and I will add another suggestion - just tweak the volatilities (with appropriate controls, of course). For analysis, divide and conquer - use "component VaR" to disaggregate the VaR into smallest pieces for which you can also attribute the P&L to market factors. This will tell you which of your market factors don't contribute "enough". Then increase their volatility in your covariance matrix. But make sure you have proper governance around this process. By how much should you increase the volatilities? If you suspect that the problem is that you assume normal distribution and in reality there should be fatter tails, then you can try to calculate the historical kurtosis of the problematic factors to verify this, and to guestimate by how much to increase each volatility to compensate. Or, an inverse problem, solve for the volatility increases that would sufficiently increase the (component) VaR.
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