ES is an alternative to value at risk that is more sensitive to the shape of the tail of the loss distribution.
Expected shortfall is also called conditional value at risk (CVaR),[1] average value at risk (AVaR), expected tail loss (ETL), and superquantile.
[2] ES estimates the risk of an investment in a conservative way, focusing on the less profitable outcomes.
it ignores the most profitable but unlikely possibilities, while for small values of
On the other hand, unlike the discounted maximum loss, even for lower values of
the expected shortfall does not consider only the single most catastrophic outcome.
[citation needed] Expected shortfall is considered a more useful risk measure than VaR because it is a coherent spectral measure of financial portfolio risk.
[3] Note, that the second term vanishes for random variables with continuous distribution functions.
The domain can be extended for more general Orlicz Hearts.
[6] 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".
Then the profit in each case is (ending value−100) or: From this table let us calculate the expected shortfall
We select as many rows starting from the top as are necessary to give a cumulative probability of
This is the expectation over all cases, or The value at risk (VaR) is given below for comparison.
equals negative of the expected value of the portfolio.
Expected shortfall, in its standard form, is known to lead to a generally non-convex optimization problem.
However, it is possible to transform the problem into a linear program and find the global solution.
[9] This property makes expected shortfall a cornerstone of alternatives to mean-variance portfolio optimization, which account for the higher moments (e.g., skewness and kurtosis) of a return distribution.
The key contribution of Rockafellar and Uryasev in their 2000 paper is to introduce the auxiliary function
To numerically compute the expected shortfall for a set of portfolio returns, it is necessary to generate
turns the optimization problem into a linear program.
Using standard methods, it is then easy to find the portfolio that minimizes expected shortfall.
Closed-form formulas exist for calculating the expected shortfall when the payoff of a portfolio
For engineering or actuarial applications it is more common to consider the distribution of losses
follows generalized Student's t-distribution, the expected shortfall is equal to
follows the Laplace distribution, the expected shortfall is equal to[11] If the payoff of a portfolio
As the incomplete beta function is defined only for positive arguments, for a more generic case the expected shortfall can be expressed with the hypergeometric function:
, then the expected shortfall is equal to If the payoff of a portfolio
[14] The conditional version of the expected shortfall at the time t is defined by where
The time-consistent version is given by such that[17] Methods of statistical estimation of VaR and ES can be found in Embrechts et al.[18] and Novak.
[19] When forecasting VaR and ES, or optimizing portfolios to minimize tail risk, it is important to account for asymmetric dependence and non-normalities in the distribution of stock returns such as auto-regression, asymmetric volatility, skewness, and kurtosis.