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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.

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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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  • $\begingroup$ Thank you! This is very insightful and exactly my problem. So calculating historical kurtosis won't affect the covariance matrix, or do I need to adjust that as well? The only other thing I can think of is to estimate some sort of multivariate non-normal distribution, and then just monte carlo sim from that. Maybe fit 5 years of data to some non-normal distribution, and the sim portfolio return paths. But yes, I can try to finagle the vol calc but unfort I'm limited to using first day of month and last day of month prices or changing time periods for estimation. Otherwise it seems weird. $\endgroup$ Jun 5, 2023 at 14:09
  • $\begingroup$ When you say factors, what do you mean? Like the Fama French factors or individual stocks themselves? $\endgroup$ Jun 5, 2023 at 14:11
  • $\begingroup$ You're v welcome. Running MC with distributions other than normal is hard. I was thinking of something like, some marke5 factor's historical standard deviation is some σ, its historical kurtosis is some k , so you wave your hands and use, say, volatility 2σ for the MC covatiance matrix, still normally distributer. Note that this won't give rise to 0 eigenvalue. As to what market factors to use, you could just have one for each stock, or you could have some factor model. If you have both single stocks and indices like S&P500, make sure their moves are consistent. $\endgroup$ Jun 5, 2023 at 18:48
  • $\begingroup$ Another possible approach is to use historical VaR, for whatever kurtosis really was there, rather than MC that assumes normal distribution. $\endgroup$ Jun 6, 2023 at 1:06

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