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I have a question regarding Value-at-Risk and diversification? When one says that VaR "hides the risk in the tail", does one mean that if we for instance look at VaR at level p=0.05 say, we might get a value of 1000 say. But now if we look at level p=0.01, we might get a much higher value say 5000?

Also, how does VaR have a problem with diversifiation?

Thank you

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As you and @Malick noted, VaR only gives a certain threshold given a certain confidence but says nothing about what happens beyond that point (tail risk). For loss distributions with long tails, this would underestimate the risk.

Regarding VaR having a problem with diversification - VaR is technically not a coherent risk measure. In simple terms, we would expect a risk measure to show decreased risk as we increase diversification (to a point). However, VaR as a risk measure can sometimes increase with diversification. An easy and to-the-point example is found in wikipedia (https://en.wikipedia.org/wiki/Coherent_risk_measure#Value_at_risk)

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  1. Exactly, VaR is nothing more than a threshold loss value. But it does not tell you how big your loss can be (no information about the shape of the tail). To get more information about it you can use the Expected shortfall which is the expected loss given that a loss occurs in the tails.

  2. Diversification decreases the VaR, however extreme events may be, loosely speaking, correlated across stocks (and especially during a systemic crisis). Here again some tools have been developed to take into account this contagion effect. You can have a look to the CoVaR measure of Adrian and Brunnermeier.

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