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It is well know that VaR is not subaddtive measure which means that condition

$$ \text{VaR}(X+Y) \leq \text{VaR}(X) + \text{VaR}(Y), $$

where $X$ and $Y$ are portfolios, is not satisfied. As a result, in some cases a simple division of portfolio to its subportfolios can lead to lower risk. Especially for this feature, VaR was criticised and subadditive measures like CVaR were introduced.

My question are:

  1. If VaR has so conterintuitive and unrealistic feature like not being subadditive, why is it still used?
  2. In which cases can we neglect that VaR is not subadditive?
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    $\begingroup$ VaR is theoretically not sub-additive however I am not sure how often it happens in practice. I was given once some examples but they were quite contrived. $\endgroup$ – Daneel Olivaw Apr 4 '20 at 15:44
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    $\begingroup$ If the underlying distributions of your portfolios are normal then VaR is always subadditive, so if you're calculating your VaR in the delts-normal way I think you would always have subadditivity. If you add a gamma term I guess you could find some examples like you say. Perhaps if you're selling some very ITM options. Another example would be owning a bond where the probability of default is lower than the p used for VaR so the risk gets hidden. Add a few more bonds and eventually the risk of one defaulting is above p. Like you said though I'm not sure this ever really comes up in practice. $\endgroup$ – Oscar Apr 4 '20 at 21:20
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    $\begingroup$ I saw a realistic situation once where VaR subadditivity was violated, I think it was around interest rates + credit spreads, in a historical simulation model. But I have seen that only once... $\endgroup$ – Kermittfrog Apr 5 '20 at 13:02
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    $\begingroup$ VaR is a lot more intuitive than CVaR. Backtesting VaR is fairly easy, backtesting CVaR is complicated and needs more data to be as significant. In short no measure is superior in every dimension. $\endgroup$ – PalimPalim Jun 1 '20 at 11:02

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