Expected shortfall (ES) is a risk measure, a concept used in finance (and more specifically in the field of financial risk measurement) to evaluate the market risk or credit risk of a portfolio. It is an alternative to value at risk that is more sensitive to the shape of the loss distribution in the tail of the distribution. The "expected shortfall at q% level" is the expected return on the portfolio in the worst % of the cases.
Expected shortfall is also called conditional value at risk (CVaR), average value at risk (AVaR), and expected tail loss (ETL).
ES evaluates the value (or risk) of an investment in a conservative way, focusing on the less profitable outcomes. For high values of it ignores the most profitable but unlikely possibilities, for small values of it focuses on the worst losses. On the other hand, unlike the discounted maximum loss even for lower values of expected shortfall does not consider only the single most catastrophic outcome. A value of 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 , and is defined to be the expected loss of portfolio value given that a loss is occurring at or below the -quantile.
If (an Lp space) is the payoff of a portfolio at some future time and then we define the expected shortfall as where is the Value at risk. This can be equivalently written as where is the lower -quantile and is the indicator function. The dual representation is
Informally, and non rigorously, this equation amounts to saying "in case of losses so severe that they occur only alpha percent of the time, what is our average loss".
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:
|of event||of the portfolio|
Now assume that we paid 100 at the beginning of the period for this portfolio. Then the profit in each case is (ending value−100) or:
From this table let us calculate the expected shortfall for a few values of :
To see how these values were calculated, consider the calculation of , the expectation in the worst 5% of 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 , 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
Similarly for any value of . We select as many rows starting from the top as are necessary to give a cumulative probability of and then calculate an expectation over those cases. In general the last row selected may not be fully used (for example in calculating we used only 10 of the 30 cases per 100 provided by row 2).
As a final example, calculate . This is the expectation over all cases, or
The Value at Risk (Var) is given below for comparison.
|0% ≤ < 10%||−100|
|10% ≤ < 40%||−20|
|40% ≤ < 80%||0|
|80% ≤ ≤ 100%||50|
The expected shortfall increases as increases.
The 100%-quantile expected shortfall equals the expected value of the portfolio.
For a given portfolio, the expected shortfall is greater than or equal to the Value at Risk at the same level.
Dynamic expected shortfall
The conditional version of the expected shortfall at the time t is defined by
This is not a time-consistent risk measure. The time-consistent version is given by
- Rockafellar, Uryasev: Optimization of conditional Value-at-Risk, 2000.
- C. Acerbi and D. Tasche: On the Coherence of Expected Shortfall, 2002.
- Rockafellar, Uryasev: Conditional Value-at-Risk for general loss distributions, 2002.
- Acerbi: Spectral measures of risk, 2005
- Carlo Acerbi; Dirk Tasche (2002). "Expected Shortfall: a natural coherent alternative to Value at Risk" (pdf). Economic Notes 31: 379–388. doi:10.1111/1468-0300.00091. Retrieved April 25, 2012.
- Föllmer, H.; Schied, A. (2008). Convex and coherent risk measures (pdf). Retrieved October 4, 2011.
- "Average Value at Risk" (pdf). Retrieved February 2, 2011.
- Julia L. Wirch; Mary R. Hardy. "Distortion Risk Measures: Coherence and Stochastic Dominance" (pdf). Retrieved March 10, 2012.
- Balbás, A.; Garrido, J.; Mayoral, S. (2008). "Properties of Distortion Risk Measures". Methodology and Computing in Applied Probability 11 (3): 385. doi:10.1007/s11009-008-9089-z.
- Detlefsen, Kai; Scandolo, Giacomo (2005). "Conditional and dynamic convex risk measures" (pdf). Finance Stoch. 9 (4): 539–561. doi:10.1007/s00780-005-0159-6. Retrieved October 11, 2011.[dead link]
- Acciaio, Beatrice; Penner, Irina (2011). Dynamic convex risk measures (pdf). Retrieved October 11, 2011.
- Cheridito, Patrick; Kupper, Michael (May 2010). "Composition of time-consistent dynamic monetary risk measures in discrete time" (pdf). International Journal of Theoretical and Applied Finance. Retrieved February 4, 2011.
- Embrechts P., Kluppelberg C. and Mikosch T., Modelling Extremal Events for Insurance and Finance. Springer (1997).
- Novak S.Y., Extreme value methods with applications to finance. Chapman & Hall/CRC Press (2011). ISBN 978-1-4398-3574-6.