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Tuesday, September 27, 2011

VaR is not a maximum loss -- illustrating Condition VaR

Q: Everyone should know the theoretical maximum loss is 100% [1]. That's theoretical max. How about realistically? Can we say Value-at-risk is a realistic estimate of "maximum loss" in your portfolio, from a large number of extensive simulations and analysis? The original creators of VaR seems to say no. See https://frontoffice.riskmetrics.com/wiki/index.php/VaR_vs._Expected_shortfal.

Compared to ExpectedShortfall aka ConditionalVaR,  the original VaR measures the most optimistic level of loss i.e. the smallest loss within the fat tail.  Therefore, the magnitude of those big losses are not considered.

"Expected" is used in the statistical sense, like "average", or average-width [2] of normal bell curve.

Q: Does ES consider the magnitude of the loss in the worst, worst cases?
A: yes. Superior to VaR. Measures severity of fat tail losses.

For a given portfolio and a given period, the 5% expected shortfall is always worse (larger) than 5% VaR. This is Obvious on any probability density distribution curve, not just the Normal distribution bell curve. If PDF is hard to comprehend, try histogram.

-- Example --
"My 10-day 5% Expected Shortfall = \$5m" means in the worst 5% caseSSS, my AVERAGE-loss is that amount.
In contrast, "My 10-day 5% VaR == \$4m" means in the worst 5% caseSSS, my MINIMUM-loss is that amount. Most optimistic estimate.

VaR makes you feel confident "95% of the time, our loss is below \$4m" but remember, this level of loss is the SMALLEST-loss in the fat tail. How badly you lose if you are 5% unlucky and one of those 5% cases happen, you can't tell from VaR.

[1] In leveraged trading, you could lose more than 100% of the fund you bring in to the trading account, because the dealer/broker actually lends you money. If you lose all of the \$10,000 you brought in and lose \$2000 more, they could go to your house and ask you to compensate them for that loss.

[2] actually "average distance from the vertical-axis". Vertical-axis being the mean PnL = 0.