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# Expected Shortfall

## What is Expected Shortfall (ES)?

Expected shortfall is a risk metric that includes expected losses above and beyond the VaR level. The predicted shortfall (also known as condition VaR or tail loss) is the loss that is expected if the loss is larger than the VaR level. The expected shortfall is calculated by averaging all of the returns in the distribution that are worse than the VAR of the portfolio at a given level of confidence.

For instance, the expected shortfall is calculated for a 95% confidence level by taking the average returns in the worst 5% cases.

## Example of Expected Shortfall (ES):

When losses are normally distributed with mean and standard deviation, the expected shortfall is:

$\mu\, +\, \sigma\, \frac{e^{\frac{-U^{2}}{2}}}{(1-X)\sqrt{2\pi }}$

where X is the confidence level and U is the point in the standard normal distribution that has a probability X% of being exceeded. Let’s take an example in which the loss was normally distributed with a mean —20 and a standard deviation of 30. The expected shortfall, given by this formula, is

$-20\, +\, 30\, \frac{e^{\frac{-2.326^{2}}{2}}}{0.01\sqrt{2\ }*3.1416} = 59.96$

## Why is the Expected Shortfall significant?

Expected shortfall is a statistic that provides more substantial incentives for traders than VAR. Conditional VAR, or tail loss, is another name for this phenomenon. Whereas VAR asks, “How bad may things become?” anticipated shortfall asks, “What is our expected loss if things do get bad?” Like VAR, the expected shortfall is a function of two parameters: N (time horizon in days) and X per cent (the confidence level). It’s the projected loss over an N-day period if it exceeds the Xth percentile of the loss distribution. With X = 99 and N = 10, for example, the predicted deficit is the average amount lost over a 10-days, assuming the loss is greater than 99%.

Owais Siddiqui
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