Expected shortfall (ES) is a risk measure—a concept used in the field of financial risk measurement to evaluate the market risk or credit risk of a portfolio. The "expected shortfall at q% level" is the expected return on the portfolio in the worst
Contents
Expected shortfall is also called Conditional Value at Risk (CVaR), Average Value at Risk (AVaR), and expected tail loss (ETL).
ES estimates the risk of an investment in a conservative way, focusing on the less profitable outcomes. For high values of
Expected shortfall is a coherent, and moreover a spectral, measure of financial portfolio risk. It requires a quantile-level
Formal definition
If
where
where
where
If the underlying distribution for
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".
Expected shortfall can also be written as a distortion risk measure given by the distortion function
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:
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
To see how these values were calculated, consider the calculation of
Now consider the calculation of
Similarly for any value of
As a final example, calculate
The Value at Risk (VaR) is given below for comparison.
Properties
The expected shortfall
The 100%-quantile expected shortfall
For a given portfolio, the expected shortfall
Dynamic expected shortfall
The conditional version of the expected shortfall at the time t is defined by
where
This is not a time-consistent risk measure. The time-consistent version is given by
such that