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VaR and heavy tails

Written by Mukul Pareek

Created on Friday, 23 October 2009 00:40

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Value at risk is affected by tails and there is so much stuff in the PRMIA handbook about dealing with heavy tails.  This can be confusing as the handbook sort of presumes an understanding of how tails affect VaR - so here is a short tutorial to explain how heavy tails affect Value at Risk.

A heavy tailed distribution has more weight in the tails.  However, since the total "weight" has to be 1 (ie the area under the curve will equal the total porbability which would be 100%), the extra weight in the tails comes from the middle of the distribution.  So this leads to interesting effects on VaR calculations.

Generally, one can quite easily remember that at high levels of confidence, heavy tailed distributions will produce a higher & more conservative VaR number.  However, there would be cases where at lower levels of confidence they may actually indicate a lower VaR than say, a normal distribution.

Look at the graphic below to understand this effect.



At a high level of confidence, say 95% (ie 5% level of significance), VaR is greater for the distribution with heavy tails.  At lower levels of confidence, say 85%, VaR is lower for the distribution with the heavier tails.  Remember that if the actual distribution is heavy tailed but a normal distribution is assumed, then:
- VaR is being underestimated at higher confidence levels, and
- VaR is being overestimated at lower confidence levels.