Conditional VaR (Expected Shortfall)
VaR says 'with 95% probability, the daily loss will not exceed X.' But if the remaining 5% hits, how much does it hurt? What answers that question is conditional VaR.
The question VaR cannot answer
VaR (Value at Risk) tells you 'up to a certain probability, the loss will not cross this line.' For example, if the '95% confidence 1-day VaR is about $740,' it means that on 19 of every 20 days, the loss is within about $740.
The problem is the remaining 1 day. VaR says nothing about whether, the moment you cross that line, the loss is about $748 or about $7,400. It tells you 'up to the threshold' and stays silent about 'beyond the threshold.'
During the 2008 financial crisis, many institutions trusted only VaR and then suffered far larger-than-expected losses in the tail (beyond the threshold).
Conditional VaR = the average loss beyond the threshold
Conditional VaR (CVaR) is also called Expected Shortfall (ES). Its definition is 'the average of those losses that exceeded the VaR line, once the loss crossed it.'
In other words, think of it as 'the loss you get by gathering only the worst 5% of scenarios and averaging them.' If VaR is 'the height of the threshold,' CVaR is 'on average, how deep the water is beyond the threshold.'
By definition, CVaR is always greater than or equal to the VaR at the same confidence level. Being the average beyond the threshold, it cannot be lower than the threshold. That's why it's regarded as a more conservative (safety-oriented) risk measure.
If returns have fat tails, the 97.5% CVaR can be much larger than the 99% VaR. CVaR better reveals how deep the extreme losses actually are.
Why regulation switched from VaR to ES
The 'Fundamental Review of the Trading Book (FRTB)' under the Basel banking regulations changed the basis for market-risk capital regulation from 99% VaR to 97.5% Expected Shortfall (ES).
There are two reasons. First, VaR ignores the 'severity (size of loss)' of the tail, whereas ES reflects it. Second, ES has a good mathematical property (sub-additivity) where risk does not inflate when several assets are combined. VaR can break this property.
Individual investors rarely calculate CVaR directly, but the mindset of 'don't view the worst case merely as a probability—look at its average depth as well' can be applied just as it is.
よくある質問
Q. Are CVaR and ES different measures?
In practice they are used with almost the same meaning. Conditional VaR (CVaR), Expected Shortfall (ES), and Average Value at Risk all refer to 'the average of the losses that exceed VaR.' If the distribution is continuous they are exactly the same, and only in strict academic definitions is there a subtle difference.
Q. Is it useful for individual investors too?
It's hard to compute the number directly, but the perspective is useful. Rather than viewing only 'the probability that a bad thing happens,' the habit of also imagining 'on average, how much it will hurt if that bad thing hits' is exactly the CVaR way of thinking. That's why it helps to look at maximum drawdowns and crisis cases in advance.
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