Value-at-Risk Tail Blindness and Diversification Limits
Summary
The document explains two limitations of Value-at-Risk. VaR reports a loss threshold at a chosen confidence level, but it does not describe the size or shape of losses beyond that threshold. For distributions with heavy or long tails, that omission can understate the severity of extreme losses. Expected shortfall is presented as a complementary measure that averages losses in the tail.
The discussion also addresses diversification. One answer notes that VaR is not a coherent risk measure and can, in some cases, rise when positions are combined. Another observes that diversification commonly lowers VaR, while cautioning that extreme losses may become more correlated across assets during systemic stress. A conditional measure such as CoVaR is mentioned as one way to examine spillover risk. These are conceptual explanations rather than a worked portfolio example; the document does not quantify tail behavior, dependence, or how alternative measures perform in practice.
Key ideas
- VaR gives a loss threshold at a selected confidence level, not the magnitude of losses beyond it.
- Long-tailed loss distributions can make VaR an incomplete guide to extreme risk.
- Expected shortfall supplements VaR by describing average losses in the tail.
- VaR can violate the expected diversification behavior of a coherent risk measure.
- Dependence between assets may strengthen in systemic stress, complicating diversification estimates.
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Full text
# Value-at-Risk "hiding risk in the tail" and diversification? # Value-at-Risk "hiding risk in the tail" and diversification? 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 ## Answer by compilation-error (score 2) https://quant.stackexchange.com/a/22517 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) ## Answer by Malick (score 1) https://quant.stackexchange.com/a/22515 - 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. - 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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