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Wikipedia / Expected shortfall

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 cases. 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), average value at risk (AVaR), tail value at risk (TVaR), conditional tail expectation (CTE), expected tail loss (ETL), and superquantile. These names are often used interchangeably, although several definitions exist in the literature. These definitions coincide in many cases, but may differ for certain types of loss distributions.

Background

Risk measures are used both in mathematical finance and in actuarial science, and the value-at-risk and expected shortfall measures are often expressed using different sign conventions and tail conventions in these disciplines. The discussion that follows takes the mathematical finance point of view.

. This can take positive or negative values, and downside risk is defined to be the absolute value.

Expected shortfall is often considered preferable to VaR because it accounts for the severity of the failure, not only the chance of failure. Further, it is a coherent spectral measure is always more conservative than VaR at the same level, i.e., ES is always at least as big as VaR at the same level.

Formal definition

is

is the value at risk.

Several other definitions appear in the literature under the names ES, TVaR, AVaR, CTE, and CVaR. The formulation above as an integral of VaR values is coherent and well-defined in the general case. Other definitions typically coincide under common assumptions such as continuity of the loss distribution, but may differ for distributions with atoms.

The above definition is equivalent to

-quantile is the indicator function.

Some authors define expected shortfall, tail conditional expectation, or related quantities directly as a conditional expectation beyond the relevant quantile,

, but may differ for distributions having atoms at the quantile. Indeed, the second term in the formula just preceding this one vanishes for random variables with continuous distribution functions, and this conditional expectation formula follows.

.

There are a number of related, but subtly different, formulations for TVaR in the literature. A common case in literature is to define TVaR and average value at risk as the same measure. Under some formulations, it is only equivalent to expected shortfall when the underlying distribution function is continuous.

The canonical tail value at risk is the left-tail (large negative values) in some disciplines and the right-tail (large positive values) in other, such as actuarial science. This is usually due to the differing conventions of treating losses as large negative or positive values. Using the negative value convention, Artzner and others define the tail value at risk as:

Given a random variable then the tail value at risk is defined by-quantile is in some Lp-space are 5% and 1%.

The dual representation is

is the set of probability measures which are absolutely continuous almost surely. is the Radon–Nikodym derivative.

spaces (Lp space dual space. The domain can be extended for more general Orlicz Hearts.

is a continuous distribution then the expected shortfall is equivalent to the tail conditional expectation.

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::, 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.

, 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

we used only 10 of the 30 cases per 100 provided by row 2).

. This is the expectation over all cases, or

The value at risk (VaR) is given below for comparison.

Properties

decreases.

equals negative of the expected value of the portfolio.

level.

Optimization of expected shortfall

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. 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.

Suppose that we want to minimize the expected shortfall of a portfolio. The key contribution of Rockafellar and Uryasev in their 2000 paper is to introduce the auxiliary function is a loss function is convex simulations of the portfolio constituents; this is often done using copulas turns the optimization problem into a linear program. Using standard methods, it is then easy to find the portfolio that minimizes expected shortfall.

Formulas for continuous probability distributions

:

in this case are 5% and 1%.

are 95% and 99%:

Since some formulas below were derived for the left-tail case and some for the right-tail case, the following reconciliations can be useful:

Normal distribution

follows the normal (Gaussian) distribution is the standard normal quantile.

.

Generalized Student's t-distribution

follows the generalized Student's t-distribution is the standard t-distribution quantile.

.

Laplace distribution

follows the Laplace distribution with the p.d.f.

and the c.d.f.

.

follows the Laplace distribution, the expected shortfall is equal to

Logistic distribution

follows the logistic distribution.

follows the logistic distribution.

Exponential distribution

follows the exponential distribution.

Pareto distribution

follows the Pareto distribution.

Generalized Pareto distribution (GPD)

follows the GPD with p.d.f.

and the c.d.f.

then the expected shortfall is equal to

and the VaR is equal to

Weibull distribution

follows the Weibull distribution is the upper incomplete gamma function.

Generalized extreme value distribution (GEV)

follows the GEV is the upper incomplete gamma function is the logarithmic integral function.

follows the GEV is the lower incomplete gamma function is the Euler-Mascheroni constant.

Generalized hyperbolic secant (GHS) distribution

follows the GHS distribution is the dilogarithm is the imaginary unit.

Johnson's SU-distribution

follows Johnson's SU-distribution is the c.d.f. of the standard normal distribution.

Burr type XII distribution

follows the Burr type XII distribution is the hypergeometric function.

Dagum distribution

follows the Dagum distribution is the hypergeometric function.

Lognormal distribution

follows lognormal distribution is the standard normal quantile.

Log-logistic distribution

follows log-logistic distribution is the regularized incomplete beta function.

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.

is the incomplete beta function.

Log-Laplace distribution

follows log-Laplace distribution, then the expected shortfall is equal to

Log-generalized hyperbolic secant (log-GHS) distribution

follows the GHS distribution, then the expected shortfall is equal to

is the hypergeometric function.

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

such that

See also

Methods of statistical estimation of VaR and ES can be found in Embrechts et al. and Novak. 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.

References