Expected shortfall

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Expected shortfall is a concept used in Finance (and more specifically in the field of financial risk measurement) to evaluate the market risk of a portfolio. It is an alternative to Value at risk. The "expected shortfall at q% level" is the expected return on the portfolio in the worst $q$ % of the cases.

Expected shortfall is also called Conditional Value at Risk or CVaR.

ES evaluates the value (or risk) of an investment in a conservative way, focusing on the less profitable outcomes. For high values of $q$ it ignores the most profitable but unlikely possibilities, for small values of $q$ it focuses on the worst losses. On the other hand, unlike the discounted maximum loss even for lower values of $q$ expected shortfall does not consider only the single most catastrophic outcome. A value of $q$ often used in practice is 5%.

Expected shortfall is a coherent, and moreover a spectral measure of financial portfolio risk. It requires a quantile-level $q$ , and is defined to be the expected loss of portfolio value given that a loss is occurring at or below the $q$ -quantile.

1 Formal definition
2 Examples
3 Properties
4 References

[edit] Formal definition

$E S q = E (x | x < μ)$ where $μ$ is determined by $P r o b (x < μ) = q$ and $q$ is the given threshold.

(Some authors define it as the negative of the above, so as to result in a positive number for a loss).

[edit] Examples

Example 1. If we believe our average loss on the worst 5% of the possible outcomes for our portfolio is EUR 1000, then we could say our expected shortfall is EUR 1000 for the 5% tail.

Example 2. Consider a portfolio that will have the following possible values at the end of the period:

Notice that, for convenience, the outcomes have been ordered from worst (first row) to best (last row). Also, the probabilities add up to 100% by construction.

probability	ending value
of event	of the portfolio
10%	0
30%	80
40%	100
20%	150

Now assume that we paid 100 at the beginning of the period for this portfolio. Then the profit in each case is (ending value-initial investment) or:

probability
of event	profit
10%	-100
30%	-20
40%	0
20%	50

From this table let us calculate the expected shortfall $E S q$ for a few (arbitrarily chosen) values of $q$ :

$q$	expected shortfall $E S q$
5%	-100
10%	-100
20%	-60
40%	-40
100%	-6

To see how these values were calculated, consider the calculation of $E S 0.05$ , the expectation in the worst 5 out of 100 cases. These cases belong to (are a subset of) row 1 in the profit table, which have a profit of -100 (total loss of the 100 invested). The expected profit for these cases is -100.

Now consider the calculation of $E S 0.20$ , the expectation in the worst 20 out of 100 cases. These cases are as follows: 10 cases from row one, and 10 cases from row two (note that 10+10 equals the desired 20 cases). For row 1 there is a profit of -100, while for row 2 a profit of -20. Using the expected value formula we get $[(10 / 100)( - 100) + (10 / 100)( - 20)] / (20 / 100) = - 60$ .

Similarly for any value of $q$ . We select as many rows starting from the top as are necessary to give a cumulative probability of $q$ and then calculate an expectation over those cases. In general the last row selected may not be fully used (for example in calculating $E S 0.20$ we used only 10 of the 30 cases per 100 provided by row 2).

[edit] Properties

The expected shortfall $E S q$ increases as $q$ increases.

The 100%-quantile expected shortfall $E S 1.0$ equals the Expected value of the portfolio. (Note that this is a special case; expected shortfall and expected value are not equal in general).

For a given portfolio the expected shortfall $E S q$ is worse than (or equal) to the Value at Risk $V a R q$ at the same $q$ level.